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EDUCAT MONOGRAPH NUMBER
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ISSN0883-9530
FOR JOURNAL
IN RESEARC
MATHEMATIC
EDUCAT MONOGRAPH NUMBER
SI
L*A-
*I
National Council of Teachersof Mathematics
2
P
Copyright
)
1987 by
THE NATIONALCOUNCILOF TEACHERSOF MATHEMATICS,INC. 1906 Association All
Drive, rights
Reston,
VA 22091
reserved
Council of Teachers of the National The publications The views of viewpoints. Mathematics present a variety unless otherwise noted, or implied in this publication, of the Council. as official not be interpreted positions
Printed
in the United
States
of America
of expressed should
ACKNOWLEDGEMENTS
The set
of studies
out without
carried we thank
Second,
helpful.
who were willing
They were patient
observed.
Next,
Education
at the University and help
hospitality In Wisconsin
the information
particularly
indepted their
Kay Schultz. for
editing
this
of the Center
when the study
it
had been collected.
work.
Egener,
Teri
for
organize,
and Martha drafts
of
Donna Mlsna,
and
must thank both Doug Grouws and Deborah
volume with
care.
iii
and
We are
and retyping
Frailey,
for
College
was conducted.
clean,
Jacob Evanson, For typing
way
and
John Fischer
members helped
to Anne Buchanan,
and
and in every
enthusiasm
staff
after
First,
persons.
of Sandy Bay Infant
their
and St.
1979-80
during
conscientious
We also
for
to the staffs
we thank Dorothy
the manuscripts,
and staffs
School
not have been
interviewed
us and our questions
of Tasmania
many project
summarize
Romberg for
Primary
we are indebted
support.
their
with
of several
to be tested,
we thank the teachers
and Waimea Heights
School
monograph could
and cooperation
the help
the children
in this
reported
Stewart
Table
of Contents
Page Abstract
. . . . . . . . . . . . . . . . . . . . . . . . .
List
of Tables
List
of Figures
........................xiii
1.
Introduction
Chapter
2.
of Groups of Children Who Differ Identification ......... Capabilities Cognitive-Processing
Chapter
Chapter
Chapter References
3.
4.
5.
6.
vii ix
.........................
Chapter
Chapter
.
1
.................... in
PerforCognitive-Processing Capacity and Children's mance on Verbal Addition and Subtraction Problems . Cognitive-Processing mance on Standard
PerforCapacity and Children's Addition and Subtraction Problems.
Cognitive-Processing Capacity Instruction .................... Summary, Conclusions,
.......
v
56
85
and Classroom 117
and Implications
.............
21
....... ...
146 168
of Tables
List Table
Page of Word Problems
Classification
5
........
1
Semantic
2
Characteristics
3
Frequency
4
Correlations
5
of Classifications That are the Number and Percentage Same, Higher for the First Test, and Lower for the First Test for all Test Comparisons .............
32
6
Factor
32
7
Estimated Vectors for the Six Groups Derived from a Where the Distance Between Score Cluster Analysis Vectors Is Less than 1.50 ................
35
Tests Included in Each Set, Sequence of Administration, and Rules for Selecting ............ Subjects
42
of the Ten Cognitive Intercorrelations Development Tests . . . . . . . . . . . . . . . . . . . . . . . . .
45
8
9
of Sample
of Scores
on the M-Space Tests
of Scores
Analysis
for
for
28
........
the Four Memory Tests
the Four Memory Tests
. . . .
.......
10
Factor
Cognitive
Development
Tests
11
of the Eight Cognitive Correlations and the Four M-Space Tests .......
Development .......
Tests
for Eight Cognitive Factor Analysis and the Four M-Space Tests .......
Development .......
Tests
12
Analysis
for
24
...............
Eight
30
. . 46
49
50
Percent Correct for the Six M-Space Groups on the Ten Cognitive Development Tests ...............
52
14
Children
58
15
Problem Types
16
Frequency and Percent Correct All SN Tasks .....................
by Cognitive
Frequency and Percent Correct All LN Tasks ................
by Cognitive
13
17
in Each Cluster
Group in Each Class
......
......................
59
ix
Group for 65 Group for .
67
Table 18
19
20
21
22
Page Frequency and Percent Correct All NR, R Tasks . . .
by Cognitive
Frequency and Percent Each SN Task ....
Correct
by Cognitive
Frequency and Percent Each LN Task ....
Correct
Group for 67 Group for 69
by Cognitive
Group for 70
Frequency and Percent Correct Each NR, R Task ..................
by Cognitive
f or Group for ..
Frequency of Use of Strategies by Cognitive Category for All SN Tasks . ...
72
Group and 74 Group
23
24
25
26
27
28
by Cognitive Frequency of Use of Strategies Category for All LN Tasks . ... by Cognitive Frequency of Use of Strategies Category for All NR Tasks . ...
and 76 and
Frequency of Use of Strategies Category for Each SN Task ....
Group and
by Cognitive
Frequency of Use of Strategies Category for Each NR, R Task
and
and
by Cognitive Frequency of Use of Strategies Category for Each LN Task . ...
in Addition Assessed Objectives Achievement Monitoring Battery
76
78
Group * . .
80
Group and by Cognitive . . . . . . . . . . . . .
82
Group in each Grade
at each Cognitive
30
33
Group Group
Children
32
74
Group
by Cognitive Frequency of Use of Strategies . ... Category for All R Tasks
29
31
and
and Subtraction . ...
88
. ...
..
89
Percent Correct by Administration
and Composite Objectives for Objectives Time for Grade 1, Form K ......
94
Percent Correct by Administration
and Composite Objectives for Objectives Time for Grade 2, Form S ......
96
Percent Correct by Administration
and Composite Objectives for Objectives Time for Grade 3, Forms S, V . . ..
99
x
..
Table 34
35
36
37
38
39
40
41
42
43
44
45
46
47
Page and Subtraction Percent Correct for Addition Algorithms Timed Tests by Problem Type for Grade 3, Form V . . ..
101
and Composite Percent Correct for Common Objectives Growth Across Grades 1, for Cross-sectional Objectives 2, and 3 . . . . . . . . . . . . . . . . . . . . . ..
103
Frequency and Percent Correct for Composite Objectives Times for Group for All Administration by Cognitive Grade 1, Form K ...................
.
104
Frequency and Percent Correct for Composite Objectives Times for Group for All Administration by Cognitive Grades 2, Form S ...................
106
Frequency and Percent Correct for Composite Objectives Times for Group for All Administration by Cognitive ... Grade 3 .....................
107
Frequency and Percent Correct for Cognitive Group 1 for All Across Grades .....................
for Composite Administration
109
Frequency and Percent Correct for Cognitive Group 2 for All Across Grades .....................
for Composite Administration
Frequency and Percent Correct for Cognitive Group 3 for All Across Grades .....................
for Composite Administration
Children in Each Cognitive in the Observation Study
Objectives Times 110 Objectives Times 112
Group, Class ...............
of Time Spent on Mathematical Percentage .............. Log Data by Grade--Teacher
and Grade Used 118 Content
Observed Minutes and Percent by Grade ......................
of Time of Pupil
Minutes and Percent .......................
of Time of Pupil
Observed Minutes and Percent by Cognitive Group ................
of Time of Pupil
Observed by Class
Objectives Times
Area 123
Actions . Actions 127 Actions
Observed Minutes and Percent of Time of Pupil Actions Group Within Class 1, Grade 1 ...... by Cognitive xi
125
128
130
Table 48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
Page Observed Minutes and Percent of Time of Pupil Actions by Cognitive Group Within Class 2, Grade 2 .......
131
Observed Minutes and Percent of Time of Pupil Actions by Cognitive Group Within Class 3, Group 3 .......
132
Observed Minutes and Percent of Time of Pupil Actions by Cognitive Group Within Class 4, Grade 3 .......
133
Observed Minutes and Percent of Time of Pupil Action by Cognitive Group Within Class 5, Grade 3 .......
134
Observed Minutes and Percent of Time of Teacher Behaviors by Grade ...................
136
Observed Minutes and Percent Behaviors by Class .................
of Time of Teacher 137
Observed Minutes and Percent of Time of Teacher Behaviors by Cognitive Group ..............
139
Observed Minutes and Percent of Time of Teacher and Pupil Engagement by Grade ........ Behaviors
141
Observed Minutes and Percent of Time of Teacher and Pupil Engagement by Class Behaviors ........
142
Observed Minutes ment for Various
143
Frequency Different
and Percent Interactions
of Time of Pupil Engageby Cognitive Group . . . .
and Percent Correct for Each Task on Items for All Students ............
Performance and Common Use of Strategies Group 1 ........................
for
Performance and Common Use of Strategies . .. .. . ... .. Group 2 . . ..
for . ..
Cognitive ... ...
Performance and Common Use of Strategies Group 3 ........................
for
Cognitive
Performance and Common Use of Strategies . ............ Group 4 .........
for
and Common Use of Strategies ....................
for
Performance Group 5,6
xii
152
Cognitive 153
155
156 Cognitive 158 Cognitive ...
159
List
of Figures
Page
Figure 1
2
for the six M-space Pattern of scores correct) (percent tests grouped by process groups on ten cognitive . .. . . ... . .. .. .. . . . . .. factors . ..
53
and crossmean growth (unshaded planes) Longitudinal in sectional growth (shaded plane) for students grades 1, 2, and 3 ..................
86
xiii
DIFFERENT WAYSCHILDRENLEARNTO ADD AND SUBTRACT
Thomas A. Romberg Wisconsin Center for Education Research of Wisconsin-Madison University
Kevin F. Collis of Tasmania University
Edited by Douglas A. Grouws of Missouri-Columbia University
In 1979 the Research Committee of the Graduate School at the University the Wisconsin Center for Education Research, of Wisconsin-Madison, and of Tasmania jointly to the University funded the principal investigators of studies children's carry out the series relating cognitive capacity and to the strategies to their performance they used when working addition The Wisconsin Center for Education and subtraction problems. of Institute in part by a grant from the National Research was supported The opinions in this Education (Grant No. NIE-G-81-0009). expressed or endorsement of reflect the position, paper do not necessarily policy, the National of Education. Institute
ABSTRACT This monograph summarizes carried
out by the authors The overall
1979-80. children
in grades
and subtract
children cluster
whether
learned
capacity
to add
two studies
in cognitive
for
was
of
to form groups
Six groups the primary
being
study
the same population
were used
capacity.
memory capacity
The second
on a variety
tasks
to determine
designed
of children.
developmental
with
analysis,
in
Australia,
survey
differences
performance
who differed
studies
ways.
Data from these
children.
related
was to examine
in cognitive
of a population
related
mathematically
studies
was a cross-sectional
study
to portray
designed
of the
1-3 who differed
the memory capacity
from five
in Sandy Bay, Tasmania,
purpose
in different
The first
the findings
of of
were formed via distinguishing
characteristic. The third, from the six both
cluster
the performance
structured
set
study
involved
items
measuring
last
study
classroom were taught cognitive
groups
studies
across
examined
children
used
to solve
word problems.
of the children's
related
to addition
children
and their
teachers
instruction
in mathematics
and whether
or not
and subtraction. were observed
was related
a
The fourth
performance
to see how addition
instruction
of students
study
objectives
these
a sample
The third
these
and subtraction
assessment
repeated
each used
grades.
and the strategies
of addition
on In the
during and subtraction
to the children's
capacity.
The results reflected
and fifth
fourth,
in their
in the strategies
show that
children's
performance they
used
differences
on both verbal
to solve vii
problems.
in capacity and standard However,
were
problems instruction
and did
not vary is
for
these
one of children
strategies, to see
those
the value
a variety
were limited
by their
to solve
that
of the
Finally,
the capacity
procedures
students
invented
instruction
a variety
of children
to solve
was carried
each other.
viii
procedures
out in schools
emerges
concepts
to process
capacity of problems
by using
They dismissed
in solving
to process
a variety
that
of important
had not been taught. taught
problems.
in which
The picture
classrooms.
to learn
Most were able
information.
failed
within
struggling
Some children
and skills.
invented
children
these
information,
of problems,
or
the
and the way
did not seem related
to
1
Chapter
INTRODUCTION
For several centuries being able to find "one's sums and one mark of a schooled has been considered differences" Although today we may have expanded our expecperson. we still tations about what constitutes literacy, expect on all children to efficiently carry out operations whole numbers. Yet, in spite of these expectations and substraction, there about the skills of addition has been little consensus about how such skills develop. (Romberg, 1982, p. 1) The basic
under
question
in cognitive-processing this
children's
performance
related
five
question
was assumed
instruction
tasks
they
who differ differently?
the evolution
of
as addition
(such
To examine
receive.
in Sandy Bay,
This monograph summarizes
and
cognitive
developing
were conducted
studies
in 1979-80.
Australia,
that
to their
both
Do children
to add and subtract
on mathematical
and to related
abilities
those
it
question,
must be related
subtraction)
learn
capacity
In raising
was,
investigation
this
Tasmania,
the findings
from
studies. for
The rationale (Romberg, how, for
Carpenter,
mathematical
for
Education between
and the acquisition learning.
of a complete Processes
Center
1978).
(1968-1976),
on the relationship
and materials
studies
& Moser,
a decade
nearly
at the Wisconsin efforts
these
is
In that
(DMP) (Romberg,
in a conceptual paper
the Studies
instructional
program,
Moser,
Harvey, 1
project
paper describe
in Mathematics
processes,
of mathematical
mathematics
the authors
had concentrated
Research
The work in that
elementary
detailed
skills led
its methods,
associated
with
to the development
Developing
& Montgomery,
project
Mathematical 1974,
1975,
2
Introduction
1976).
DMPwas based
Although
learning,
development,
questions
were raised
In particular,
and instruction
it
perform
mathematical
material,
lessons.
content,
this
capacity,
and classroom
Addition
and Subtraction
We chose
to teach
what might
mean learning
relationship
operate
elementary
Research for school
this
such as
the
interactions
the three
areas
involved
in
cognitive
children's
investigation
the
represents
first
as formal
represent
on the symbols,
on logically
level
between
to be carefully
for
that
schools
attempt
mathematics.
a problem
By this
situation
and interpret
analyzing
mathematical (e.g.,
skills
Carpenter
the
early
reasons
the
work had been done at the Wisconsin
these
of
interactions
There were several
be recognized
considerable
Education
for
to symbolically
word problems),
instruction
instruction.
area
this
First,
material,
and subtraction),
(addition
a
mathematical
needed
describe
and subtraction.
choice.
was a
presentations
that
and instruction
as the vehicle
work in addition
Second,
work indicated
sections
of
picture
and teacher-pupil
past
content
investigation:
with
teacher
engagement,
capacity,
developed.
of classroom
tasks,
of
a number of
to be learned,
capacity
pupil
The following
examined.
1977),
content
cognitive
on learning
Thus,
cognitive
Romberg,
What was needed
of the features
and an identification how children
and theories
a complete
that
was lacking.
of children's
description
during
became clear
of the mathematical
characterization
(see
evidence
as the program was being
instruction
mathematics
on empirical
this make we
(often
via
result. Center
for
semantic-syntactic
as they & Moser,
apply 1983;
at the early Moser,
1979).
3
Introduction
Research subtraction. only
successful
young children
strategies subtraction a clinical
to solve
use
(see
problems
for
and subtraction
and
1982).
performance
had been developed
tasks
several
& Romberg,
assessing
were
those
using
addition
elementary
and
materials
problems
had identified
Moser,
Carpenter,
schedule
observation
to solve
researchers
those
using
Education
addition
to teach
in classrooms
in learning
various
Fourth,
operations.
addition
children
However,
moderately
materials
instructional
had developed
for
Center
at the Wisconsin
the staff
in the 1970s
Third,
Finally,
on some & Moser,
(Carpenter
1979). To solve
Word problems. problem,
one first
must understand
the element
Quantifying
how many).
be expressed
in the
must be able
to carry
Then,
out the procedural
Most children
procedures,
some knowledge
opportunity
Thus,
to,
solve
word problems. Semantics.
subtraction work uses
Not all
have the four
of objects context
variations
and after
prior
semantic
comes next
formal
same semantic
of the problem
of adding
steps
must be
operations well
developed
and some understanding such as "joining" researchers
involving
structure.
counting of
and
have a unique process
when they
addition In fact,
and subtraction
must
Next the child
(algorithmic) of these
a unit
choosing
and subtraction.
instruction
of addition
meaning.
(e.g.,
semantics
in how children
word problems
broad classes
implied
word
and subtraction
to such problems
of numbers,
from this
to examine during,
bring
on sets
operations
"separating."
the results
Finally,
addition
the implied
of addition
syntax
expressed.
physical
its
of the problem
and counting
and subtracting.
a typical
information attempt
to
and most current problems:
4
Introduction
Combine,
Change,
are two basic
that
change-separate
causes
and separate
there
upon which
depending
and compare problems
set
particular
the two subsets
static
involve
to find
asked
the comparison
and the other
of problems,
unknown--the set
there
exist
can be either six
compare and change in the change
compared;
possible
The third
sets. then
set
equalize
There is
but it
is
of their is
involve
one set
is
the referent
entity
in these
larger
set
set is
problems
exceeds
entities
could
the other. be the The
compared set.
or the compared set.
is
based
posed,
problems,
the same sort
Thus,
are a hybrid of action
on the comparison
As in the compare problems, the question
exist:
types
and the solver
one set
no
among a
the size
Because
or the
is
of compare problems.
types
of problems,
problems,
the
the referent
there
Compare problems
to label
set,
problems.
Both combine
1).
are given
sets.
referent
different class
to find
any one of the three the
difference,
The final
disjoint
is
Table
Two problem
subset.
or the amount by which
class
larger
asked
the join
of problems
types
subsets.
disjoint
the compared set.
the difference, In this
it
both
existing
and the union
of two distinct,
compared to the other,
distinct
Within
the relationship
of the other
the size
time.
In both
set.
for which
and one is
are given
over
or
For
quantity.
relationships
two disjoint
or one of the subsets
union,
occurs
action.
and a direct
removed from a given
is unknown (see
involve
and its
is
There
1983).
involve
quantity
in that
are three
quantity
Combine problems
action.
an initial
a subset
the change
classes,
both of which
an increase
problems,
of problems,
classes
is
& Moser,
(Carpenter
problems,
there
problems,
action
implied
of change
types
In change-join
and Equalize
Compare,
two disjoint
What could
of
as found
of two sets
are
be done to one of the
Semantic
Table 1 Classification of Word Problems & Moser, 1983) (Carpenter
Join
1.
Introduction
Separate
Change Connie had 5 marbles. Jim gave 2. her 8 more marbles. How many marbles does Connie have altogther?
Connie had 13 marbles. She How gave 5 marbles to Jim. many marbles does she have left?
3.
Connie has 5 marbles. How many more marbles does she need to have 13 marbles altogether?
4.
Connie had 13 marbles. She Now she gave some to Jim. has 8 marbles left. How many marbles did Connie give to Jim?
5.
Connie had gave her 5 she has 13 marbles did start with?
6.
Connie had some marbles. Now she has gave 5 to Jim. 8 marbles left. How many marbles did Connie have to start with?
some marbles. Jim more marbles. Now marbles. How many Connie have to
Combine 7.
Connie has 5 red marbles blue marbles. How many marbles does- she have?
9.
Connie has 13 marbles. Jim has 5 marbles. How many more marbles does Connie have than Jim?
10.
Connie has 13 marbles. Jim has 5 marbles. How many fewer marbles does Jim have than Connie?
11.
Jim has 5 marbles. Connie has 8 more than Jim. How many marbles does Connie have?
12.
Jim has five marbles. He has 8 fewer marbles than Connie. How many marbles does Connie have?
13.
Connie has 13 marbles. She has 5 more marbles than Jim. How many marbles does Jim have?
14.
Connie has 13 marbles. Jim has 5 fewer marbles than Connie. How many marbles does Jim have?
15.
Connie has 13 marbles. Jim has 5 marbles. How many marbles does Jim have to win to have as many marbles as Connie?
16.
Connie has 13 marbles. Jim has 5 marbles. How many marbles does Connie have to lose to have as many marbles as Jim?
17.
Jim has 5 marbles. If he wins 8 marbles, he will have the same number of marbles as Connie. How many marbles does Connie have?
18.
Jim has five marbles. If Connie loses 8 marbles, she will have the same number of marbles as Jim. How many marbles does Connie have?
19.
Connie has 13 marbles. If Jim wins 5 marbles, he will have the same number of marbles as Connie. How many marbles does Jim have?
20.
Connie has 13 marbles. If she loses 5 marbles she will have the same number of marbles as Jim. How many marbles does Jim have?
and 8
8.
Connie has 13 marbles. Five are red and the rest are blue. How many blue marbles does Connie have?
Compare
Equalize
5
6
Introduction
to make it
sets
the smaller
to the other?
equal
of the two sets,
On the other
if
hand,
then
the action
If it
the action
becomes
an equalize-join
to be performed results.
then an equalize-separate
problem
the unknown can be varied
to produce
to be performed
As with
three
set,
larger
compare problems,
distinct
equalize
of
problems
each type. To build
sentence the
could
apply
forms are ever It is
1984).
procedures
they
Development
are taught
for
(e.g.,
grades
Center
symbolic
procedures
that
of problems Vergnaud,
to use one semantic
to the symbolism
and then
a few of the
of connecting
context to relate
In
semantic & Joillet,
Janssens, have found
little
and the symbolic
During
Education
had been recognized.
so that
1982).
materials. for
The symbolic
of the problems.
Verschaffel, students
the
No serious
structure
K-6 (Romberg et al.,
program the problem
this
meaning
types
of instructional
of the Wisconsin
was decided
DeCorte,
same
Traditionally,
were assigned
procedures.
only
the
is how to
problem
of word problems.
semantic
then,
different
creating
it
(see
the
for which
problems.
in many texts
that
no surprise,
between
DMP curriculum
symbolic
to the
included
connection
the staff
their
is now clear
it
the semantic
and some word problems
was given
consideration
forms
the pedagogical
independently
taught
forms and relevant
as a model to introduce
used
appropriate,
were taught,
students
semantic
are many semantic
to all
symbolism
procedures
fact,
is
has been
symbolism
there
Because
symbolism. symbolic
between
one form is usually
symbolism,
relate
the connection
on
problem.
on the
is
is
the early
Research 1974,
1975,
produced
the
1976).
In
word problems
For addition
1970s,
and
and subtraction,
to introduce the symbolism
and to give to other
Introduction
context
This
used.
for
context
initial
of addition
variety
(see
strategies addition identified:
objects,
physical
basic
(counting
strategy
or fingers two sets
addition
sequence
distinct
with
begins
similar
children
do not use physical
addends.
However,
strategies
require steps
1983).
For
have been or
fingers
use physical
and then
and
sequences, in the most
objects
the union
of the
represent
action
the second use
but a substantial accompanying
to play
a very
their different
their
the answer with
track
is
models
to represent
counting.
except the
counting
of the number of
to keep
fingers
reached.
strategy
to know when to
addend in order
number give
role
the counting
strategy,
or fingers
objects
sequences
counting
and the two following
strategy
Most children
number of counts,
all
some method of keeping
that
counting.
appear
this
until
to the counting
that
they
& Moser,
involving
one and continues
is
physical
use numerous
children
In the most elementary
problems.
strategy
stop
the
children
In addition,
models),
1980).
to solve
on the use of counting
strategies
This
counting
1979,
counted.
There are three for
with
of
was used as the
with
modeling
number facts.
all
set
levels
strategy
each of the addends,
to represent is
Carpenter
on direct based
on recalled
based
strategies
1982;
basic
three
strategies
A revised
In order
word problems
et al.,
based
strategies
and
(Kouba & Moser,
and subtraction
was
teachers
both
forms.
word problems.
solving
and subtraction
for
semantic
instruction
Carpenter
of DMP, equalizing
version
in which part-part-whole
developed
for
Strategies
other
examined
was later
materials
initial
to be difficult
proved
when they
students
basic
In the
situations.
semantic
7
track
no evidence
When fingers than in the direct
of the of any
are used, modeling
8
Introduction
strategy.
In this
per se,
but are used
the counting to have
mechanical
application
counting
begins
forward
protracted
forward
counting
strategies
case,
When concrete
separates,
one at a time, Counting
answer.
There
also
counting
down from.
a child
that
The
the child
to occur
appears
solve
ultimately
over
a
addition
simple rather
of number combinations
classes
at the direct
than by
in the
quantity
larger
the given
is
a parallel A child
is
removed from it.
called
strategy
is
set
and then
strategy
equal
a backward
separating
from.
away or
takes
to the number given
objects
based
In
action.
initially
subsequently
of remaining
initiates
and counting
a subtractive
subtraction
a number of objects the set
of subtraction
modeling
involves
strategies
are used, the
the entire
strategies.
smaller
constructs
is
number facts
quantity
objects
strategies,
strategy,
except
a
of the two addends.
a number of distinct
larger
in the problem.
identical
larger
by recall
these
addend in the problem.
most children
or modeling
and the
represented
The child
the
have been observed
the
is
of basic
One of the basic
levels.
have
and imply
to reconstruct
on from first
the first
with
problems
For subtraction
this
with
span of time,
and subtraction
when they
tell
in
do not appear
often
In applying
is not necessary
strategy
learning
Although
of counting.
it
on from larger
counting
incremented
are more efficient
strategies
In the counting
sequence. counting
using
that
recognizes
begins
addend
number of fingers. two counting
counting
children
fingers,
but can immediately
fingers,
The other
a child
the second
of the number of steps
When using
their
to count
do not represent
fingers
to keep track
sequence.
put up a certain
less
the
case,
yields
on counting counting
the called
sequence
Introduction
beginning contains last
with
the given
as many counting in the
The separating
to strategy
that
except
number of objects problem. Similarly,
words
the backward
constructs
The child
the new collection
added on gives
(counting
up from given),
strategy
beginning
with
with
larger
the
counting
given
feasible sets
basic
A fifth
standing
strategy
Counting (choice)
up from given,
In an and
the child
on),
number (an
one at a time until Counting
In the parallel
number.
determines
the unmatched
the answer. is
Matching
one of the given
ends
of the number of
track
The child
puts
cubes
on which
only
out two
numbers.
a combination
depending
counting
The sequence
matching.
the
counting
a forward
initiates
involves
down to
quantity
number.
the child
called
for
set
given
are available.
objects
one-to-one.
down from and counting
is
strategy
the answer.
action.
given
by keeping
Again,
in the sequence,
each set
are then matched
to that
given
in the
and the number of
(adding
to the smaller
smaller
number.
when concrete
of cubes,
answer.
the
words uttered
The fourth
objects
a child
the
of the problem.
the answer.
strategy
reached
the smaller
to the larger
equal
until
number given
an additive
with
then adds objects is
number of objects
involves
equal
set
in the counting
the solution
With concrete
larger.
from
removed provides
number is
starts
out a number of objects
addend).
is
sequence
The
the answer.
to the separating
sequence
sequence
number.
smaller
to the smaller
equal
smaller
the
the child
solution, the
is
of strategies
pair
counting
are removed from the larger
counting
until
in the counting
additive
similar
the number of objects
Counting
The third
is
elements
is
sequence
counting
remaining
continues
strategy
sets
number words as the given
number uttered
strategy
The backward
number.
larger
9
gives
The sets the
of counting is
the most
10
Introduction
In this
efficient.
number of counts
fewest
As with
a child
case,
addition,
and solves
of their
are calling
they
the logical
of the
children
strategies
at the outset by addition
of this
and subtraction
have developed subtraction
word problems
least
to formal
prior
solution.
different
elicit
of the operations initial
of semantic
for
the variety
However,
little
be affected
addition
arise.
and
or at
experiences "efficient"
methods
a logical
seems to imply
analysis these
that
as the number or the
increases. goal
of mathematics algorithmic
and subtraction
several
How will
by the number,
aspects learning type,
of
form of word problems
generalizable
of addition
is known about
number of questions procedures
solution
more formal,
to teach
solving
for
children
or the numbers become larger,
seems to be a reasonable
instruction
can be solved
Second,
Finally,
word problems
increased,
necessary
It therefore
were noted
become more and more inefficient
should
forms is
number of steps
from children.
and related
strategies
learning
in the semantic
strategies
that
to solve
on consolidated
instruction
and the
form.
strategies
to school
prior
differences
Third,
word problems
or "child"
"primitive"
between
any,
and subtraction
in semantic
differ
if
link,
such problems.
addition
First,
project.
Of significant
forms of problems
about
give
Children's
facts.
combinations.
use to solve
points
eventually
the number combinations
must be the
semantic
actually
Several
Summary.
addition
and teachers
accordingly.
strategies
that
the
requires
strategy
or derived
suggest
upon are often
analysis
the problem
number facts
solutions
to researchers
interest
which
and counting
modeling
way to the use of recalled explanations
decides
procedures
word problems.
of this
process
of the mathematical and success
of the
and a
Introduction
who are successful
How do children
with
strategies
existing should
teachers
formal
of functioning
in this
area?
of the relationship by children
possesses?
combine
their
modes of presentation? account
to performance
functioning
mathematical to take
to a consideration
leads
child
solvers
problem
instruction
adapt
level
demonstrated
an individual
strategies
problem-solving
preexisting
11
How
of a child's these
Raising
of general
on addition
questions
cognitive word
and subtraction
problems.
Capacity
Cognitive
for
Concern
education.
mathematics on claims
abilities
is well
entrenched
The approach
adopted
in this
cognitive
from two sources:
differential
in research
in
was based
project
and cognitive
abilities
development. Differential educational
solve is
in the Thurstone
psychologists we decided
abilities,
addition
to use
to attempt
and subtraction
test
abilities,
scores
traits,
characteristics
of students
These
traits
biological individuals describe
have been ordered
are assumed in origin,
to be fixed,
which
describe
in the same way as height, physical
characteristics.
of learning, ability
Although
approach
differential
For example,
such
field
to low on those
characteristics,
intellectual weight,
in this
to
have been identified,
from high stable
mental
of students
to identify
and so forth.
rate
and spatial
the ability
analyses
styles,
as intelligence,
of distinct
The procedure
problems.
aptitudes,
work of a number of
tradition
to measure
and psychometric
independence/dependence, samples
Based on the extensive
abilities.
differences stature,
and traits.
largely between
and so forth
we did not utilize
tests
12
Introduction
from this
developed
to each student
administering the
relating
scores,
Our initial
mathematical that
the psychometric
a number of tests,
and classifying
students
was to find
and administer
task
that
functioning
we used
perspective,
appeared
material.
related
logically
seemed to be related
could
Only instruments
that
contain
related
mathematical
tasks
conservation
to early
and counting
Cognitive
the notion
We chose
Development.
evolve
gradually Rather function usually
of growth.
Children
are at a concrete
(are
egocentric),
they
should
to
such as number
learning
in the primary
not be expected
with
to reason
on
based
the environment
individuals
as a
example,
in terms near
and
stages.
are viewed
for
grades,
referents
in the
is
discontinuous
think
stage,
of concrete
to be used
perspective
through between
operations
and think
This interact
processes
differences
fixed,
of tests
be shown prima facie
the measures
adaptively
intellectual
than being
of
learning
of cognitive
level
development.
individuals
that
of cognitive
measures
were selected.
from work in cognitive
study
scores.
to use a battery
to the children's
development.
on their
to the
we decided
However,
the tests,
scoring based
of
strategy
of themselves
at hand.
Hence,
external
about hypothetical,
situations. The choice children's following expectations. investigate
of tests
from this
the
failure
the clinical
interview
interested and learning
developmental material
research
of the "new math" programs
Psychologists
mathematical
This
of mathematics.
understanding
grew out of work on
perspective
(e.g.,
Collis,
as a technique
gained
to live
in mathematics
up to early learning
phenomena by using 1975). for
These studying
impetus
began to
elementary
investigators the mathematical
used
Introduction
that
concepts
Much of the work was stimulated
had formed.
children
& Piaget,
the notions
of Jean Piaget
was related
to the work on memory capacity
Case
This
(1972).
mathematical
of cognitive
work with
in their evolution
This
1974b, Biggs
to describe
1975, & Collis,
(Inhelder
& Piaget,
tentative
explanations
terms
of Case's
recent
papers
skills
model which,
1980b,
it
although
on the increasing
after
papers,
1982) for
allows
& Biggs,
stage
theory
about
1976,
(Case,
items
provide in
The most
1975).
an intellectual
describe the stage
phenomenon,
of responses
complexity
(1971,
phenomena found earlier
theory
& Collis,
kept
use mathematical
Piaget's
of the developmental
Biggs
of
Collis
1982;
papers
to modify
information-processing (e.g.,
the emphasis
1980a,
The later
1958).
of development
the work of Collis
through
to some extent,
and,
psychologists
children.
The earlier
1982).
and
(1976)
of the phenomena that
individual
1978,
1976,
enabled
by
interest
by Pascual-Leone
of stages
can be traced
Later
1958).
functioning
to an explanation
thinking
appearing
1979;
(Inhelder
from the mere description
to turn
1974a,
view
13
places
a given
within
stage. At the various
time
both
structural
then most theorists Case,
1985;
1984;
Seigler,
that
selected
time
and process
1981;
1980;
Halford,
Sternberg,
their
to be the most applicable
were in
were of
of developmental
Klahr,
to the content
tasks.
positions 1984;
The investigators
two reasons.
that
systems
and that
theoretical
1980;
1984).
the Case model for
theoretical
components
to a broad range
have published
Fischer,
a number of theorists
began,
and generalizing
in relation
significance
project
project
of refining
stages
included
this
First, area
Since (e.g.,
Pascual-Leone, in this it
seemed at
and the
14
Introduction
that
methodology that
could
Central of the
the concept
attention
this
processing
mental
space
retention
and retrieval
available
processing
study
the
to solve
by Collis
(1973)
elicited
this
A simple 1980b)
(Collis,
example
may help
of
that
1985)
and is
shared
mental
operation
individual the given
In
the
between
and the If
exhibits
the symptoms
problem.
An
phenomenon in relation by Collis
quoted
to explain
two
to
in a later overload.
cognitive
6 or 7 stage (circa statement 3+2+4; a
What number does 3+2+4 equal? 3+2=5 (pause) what was the other number? I said, "What number does 3+2+4 equal?" Ah yes. Now, 3 plus (pause) what is the sum again? (p. 87)
Tester: Child: Tester: Child:
to be happening
Let us suppose
that,
available
for
space
in
to the number of
(Case,
A child at the early concrete operational is asked to find the value of the years) interview goes as follows: typical
What appears
we used
of such an operation.
unable
exercises.
of tests
the
to at any one time.
of an ongoing
exceeded,
basic
conception
refers
constant
and is
mathematical publication
is
it
overload
of cognitive early
is
of the product space
as Baldwin's
proposal
available
This to process
set
is
project
has a long history
can attend
Case's
the execution
activities,
construct
Basically
an individual
that
we have adopted
project
overall
1895).
to this
ability
the first
Thus,
back as far
going
(Baldwin,
elements
available
(M-space).
to a child's
The M-space
M-space.
span
memory capacity
presented.
theory,
psychological
had tests
and crucial
theory
was central
material
was to measure
mental
to Case's
short-term
we believed
mathematical
it
Second,
using.
in the project.
be utilized
M-Space.
variable
we envisaged
may be explained
in the diagram processing
data.
below,
the
by using
rectangle
At the early
a diagram.
represents
concrete
operations
the
Introduction
it
stage
and one operation
elements
(Collis,
meaningfully
element,
the
1975).
the data
necessary
for
of data,
another
another
exceeded
outcome
is
the
forced
in these
circumstances
needed
to solve
the problem
in the working
order
to obtain
a satisfactory
solution.
level.
to our investigation, only measure data
we should
in the research
work.
Moreover,
decade
now that
apply
it
to M-space,
specific
areas also
appropriate
obvious
make because
it
that
literature
has been very
we felt
of relevance
be an influence. to mathematical
results;
forced
was clear
other
clear
and so on. the information
at the
this
same time
could
not be the
from the correlational
influences
must also
to mathematics
that
(see
a child's
to the content
Bauersfeld,
be at for
educators theory
rarely In
1979).
developmental
level
in
area under consideration
Thus we incorporated learning
in
basic
appeared
drawn from pure psychological learning
of
one
and retrieves
construct
that
part
out of
has all
space
was also
to mathematics
directly
never
the M-space
deductions
addition
could
the child
fully and
operation
situation
Although it
however,
out of the space
Hence,
Development
is
in two
calculation
is,
and overflow
realizes
to take
the necessary
to introduce
a successful
piece
sufficient
space
As the subject
consideration.
is
space
The processing
is
available
space
the
and to perform
If one now attempts
occupied.
piece
that
can be demonstrated
15
and used both
Piagetian
tests
the M-space
and
a
16
Introduction
data
developmental mathematical
to define
cognitive
with
mathematics
for
capacity
which would be more useful
material, concerned
primarily
a construct,
in a study
instruction
rather
that
was
than cognitive
theory. In summary, to identify of tests.
The first
The second
child's
level
included
battery
of cognitive
development
such as conservation
model,
mathematical
of children
tests
with
specific
to measure
related
to
factor
procedures,
the data
from both batteries
we assumed
that
well-defined could
characteristics
cognitive
the
from the Piagetian
and presumably
approach,
the short-
mathematical
processing
on dimensions
to interpret
From this
to measure
constructed
and transitivity,
analyses
and to group children. groups
for
We then used psychometric
ability. and cluster
analysis,
was designed
of the child
(M-space)
we gave two batteries
capacity,
of tests
battery
term memory capacity material.
cognitive
be
identified.
Classroom
Instruction
Throughout
this
to add and subtract
of students
at grades
and subtraction was selected
skills to reflect
in five
1,
in the study
To identify
in school.
we observed
instruction,
the children
project
some aspects
classrooms
2, and 3.
It
is
to gather at these
are taught.
The sample
differences
in cognitive
Data on the performance achievement
monitoring
This battery
provides
subtracting,
and in several
of the students
battery information
developed
grades
of students
taught
of classroom
data
on a sample that
addition
we observed
capacity.
were collected
using
an
by Buchanan and Romberg (1983).
on a variety
administrations
were being
of aspects
profiles
of adding
of growth
and
can be
Introduction
then
The profiles
obtained.
to observe
we decided
Third,
interactions
teacher-pupil in cognitive
for
had been central
mathematics
education
For example,
developed,
behaviors
were specified.
that
the
Berliner
(1975)
pointed
to examine
the
particularly
conceptualization between
teacher
logical
analyses
of teacher tasks
and pupil
are as far
out on mathematics
the logical
application
development
and learning
Perhaps
what was needed
the program
evidence
available
is
for
teacher
framework
this
to
subsequent
programs
to mathematics look
relationship that
possible
realities
programs
theories
as the analyses and 1960s of
of a decade
at the problem.
the
rather
in the 1950s
psychologists'
in
the
to Berliner's
from classroom
curriculum
is
and pupil
to progress
and the assumed direct It
who
behaviors
for
of
lack
researchers
facing
performance.
was a fresh
descriptions
in teaching
as a major impediment
of general
(IGE)
are all
reasons
between
tasks
Research.
actions.
inadequate
of the problem
overview
little
of problems
the relationships
work done on
as DMPwas being
to the probable list
make a
Education
1977)
were to use
of teacher
who differed
Guided Education
In addition,
He saw methodology
performance.
carried
a long
for
Center
efforts,
level
"teachers
(Klausmeier,
teachers
importance
and identified
that
and
actions,
pupil
at each grade
Individually
these
Despite
substantiate
pessimistic
in the
are to take.
teachers
actions,
to much of the previous
programming model
of actions
area,
children
at the Wisconsin
the steps
instructional
this
teacher
The proposition
capacity.
difference"
attempt
of the
of instruction.
effectiveness
data
as indicators
can be used
17
ago.
or
18
Introduction
In this
as they
actions
teachers'
teachers
approach on that
The approach
actions.
conceptualizing that
used used
investigation
by Carroll
on pupils'
of data
(1963,
the notion The
performance.
Study
based
approach
(BTES), which
in
and
the observational
instructional
time with
instrument
for
developed
even
to code in this
advances
of recent
Carnahan, of both
the behavior
account
of the
Moreover,
was not usual
it
Small,
at that
understanding
or surveys.
advantage
DMP (Romberg,
into
a clear
were employed, to take
We decided
was the lack,
tests
by objective
techniques
takes
toward
Bloom (1974),
1973),
gave researchers
that
gathered
instrument
Evaluation
made by Berliner
area by using
This
and testing
teachers'
(1975).
major criticism
actions.
tasks
Teacher
to the
make some progress
should
was a "time-on-task"
when observational pupil
related
reactions
in this
of instruments
meaning
characteristics
effect
and Wiley
Another
on teachers'
of known cognitive
instructional
in the Beginning
Harnischfeger
attention
have some discernible
turn was influenced
time,
to children
related
on the same children's
and, moreover, initiating
to concentrate
we decided
study
of
the study
& Cookson, teachers
1979). and
children. is
The instrument as target
identified based
observational
each minute content students
form for
are then
a limited
pupil
and classroom
variables.
This methodology
information
about how time
spent
mean class
reliable
fills
out a time
activities,
Data from target
characteristics.
provides
who are
At the end of
teacher
activities,
to estimate
is
observer
each day of instruction.
codes
aggregated
of pupils
sample
Then a trained
students.
the observer
categories,
used with
time on the
and generalizable
in classrooms.
19
Introduction
Conclusion studies
The five
data
draw together
whether
in combination
students
different
cognitive
strategies
children
problems.
The third
use
to solve
verbal
achievement
procedures.
The final
approach,
the direct
perspective,
was to use a time-on-task how features
from the
interview
data
of classroom
the
perspective,
test
using
monitoring
instruction
observation
teaching to
procedure relate
instruction
about
and subtraction
addition
in student
determine
with
students
approach,
changes
was to assess
approach,
was to use
from the quasi-experimental
approach,
how
portray The first
to identify
was to gather
perspective,
cognitive-processing
better
perspective,
The second
capacities.
was to see
our intent
skills.
differences
in two studies
techniques
psychometric
and subtraction
individual
from the classical
could
to
Each
perspectives.
However,
the perspectives
addition
develop
different
own right.
in its
an attempt
monograph represent
from four
gathered
is viable
perspective
in this
reported
to student
engagement. were designed
The studies the four
not only
described
perspectives
the four
between
interactions
would be of considerable
interests
and to examine
on the interaction and the other We first specific M-space
some new hypotheses children's
but in view we decided
cognitive
on
the
a number of
Obviously, interest,
data
of our
to concentrate
processing
capability
variables. identified
cognitive (study
between
factors.
and analyze to examine
but also
above,
interactions
to gather
a sample
characteristics.
1) and measuring
of children
aged 4-8 years
Sample selection cognitive
development
required (study
with measuring 2) of a
20
Introduction
of 4- to 8-year-olds.
population
In clinical
3-month period.
Achievement
3).
(study
subtraction
tasks
children's
strategy
questions
processing
children
5 we attempt
studies
problems.
to relate
change
and
to relate
achieved
and
and to specific In this
in performance
the data
used, four
chapters.
way, we
and
in studies
In chapter
Chapters
individual
and strategies
level
is
further
on
presented.
In
to teacher-pupil
a summary of the findings
for
clinical
used by
4 achievement
and subtraction
3 and 4 on
performance
3 the
the understandings
some direction
2 is
the cognitive
1 and 2.
In chapter
and their
gathered, Chapter
to characterize
cognitive
6 provides
draw together
and suggests
3).
of addition
Chapter
level
had used.
teacher
of each group to their
(study
tests
that
and
provided
we would be able
capability
coded for both performance
paper-and-pencil
conclusions
studies
about
techniques
level
are presented
interactions.
child's
examined
and subtraction data
addition
in classroom
terms of performance
the means we used
the cognitive
interview
five
in the next
capabilities
addition
problems
causes.
possible research
with
written
of the instruction
cognitive
various
are described
concerned
chapter
to a child's
and their
analysis
standard
obtained research.
a
and
performance
and subtraction
were determined
(in
the
The various
relate
time
activities
consider
strategy
from these
at a given
instructional
with
over
5).
(study
adopted)
the sample
provided
addition
The nature
4).
the mathematics
the children's
verbal
and engagement
We assumed that performance
with
was measured
(study
actions
observations
could
interviews
were determined
strategies
and instruction
used,
strategies
performance,
we studied
Next,
and some through
the
2
Chapter
IDENTIFICATION OF GROUPSOF CHILDRENWHODIFFER IN COGNITIVE-PROCESSINGCAPABILITIES
In this according
to their
materials
is
label
as determined
were the basis the developmental that
are applicable
mathematical is
of measures
of children
a derived
of working
of cognitive The M-space
model.
into
measures and
categories,
of developmental
gave an indication
tests
with
of the level
by the Piagetian
of the classification
groups
capability
on a combination and measures
(M-space)
memory capacity development
Cognitive-processing based
into
capabilities
cognitive-processing
presented.
categorization
of children
the classification
chapter
criteria
each category.
within
Study 1--M-space
can be characterized
functions stored,
in terms
and operated
accessed,
terms of an intake
register
environment
the system,
in which
enters
the actual
in which knowledge The working appears
on the idea
that
of the way information
Mental
structures
which
information
a working
occurs
is
are discussed
in
from the
or short-term
processing
mental
memory (M-space)
and a long-term
memory
stored. growing
memory's
as a fundamental
capacity
characteristic
number of theories
(Bruner,
children
limited
are quite
on.
through
information is
are based
theories
Information-processing
1966; in their 21
to process
information
of cognitive
Case,
1978a;
ability
development
Flavell,
to deal
with
1971). all
in a Young
the
22
Cognitive-Processing
Capabilities
demands of complex
information be a critical
developmental
instructional
situations
Pascual-Leone the development
from factors
theory,
or patterns)
available
chunks
new scheme is
limited.
also
the processes
with
the functional
Learning
to process
capacity
To generate both
tasks,
the
all
about
addresses
the problem
M-space
demands of the of assessing
a
learned
is
is
concerned of
processing
on the child's information.
performance
capacity
limited,
by the
theory
depends
children's
is
to produce
of schemes
incoming
information-processing
and the information-processing study
Since
a
produces behavioral
represented
and the mental
instruction
resulting
Learning
are constrained
of the essential
hypotheses
in behavior
complexity
of development
through
or M-space.
Pascual-Leone's
system.
operationalizes
can be coordinated
of learning
aspects
that
system.
learner.
the
Therefore,
psychological
information.
a theory
a change
that
in
1978b).
(internally
to the
seems to
capacity learning
capacity
is
of schemes
the number of information
developing
1978a,
proposed
learning
limited
constrains
to the psychological
extrinsic
limited;
1975,
1976)
in the repertoire
change units
(Case,
that
of information-processing
to this
According
factor
(1970,
Their
tasks.
(M-space)
on specific of the child
must be known.
task
This
information-processing
capacity. The rationale construct
is
for
based
on the results
(1979),
in which
a measure
predict
learning
of mathematical
(1978)
in which
Cucui,
and digit
three
different
administering
placement)
of two recent
of M-space
different
tests
(backward
skills measures
were given.
studies, digit
and another
this
one by Hiebert span)
did not
by Case and Kurland
of M-space Although
to measure
(counting
span,
Mr.
in Case and Kurland's
study
correlations
positive
between
the consistency
tests,
by Case and associates that
suggests
difficult
that
Their
indicate
that
supposed
in determining
previously use the
three
tests
backward digit
also
variables
three
Recent
work
M-space
demands. study
estimate
of a child's
in terms
with
to see whether
to
the together The tests
M-space.
of the task
than
we decided
Thus, along
measure
of tasks.
range
may be more important
study
a reliable
the
one general
on a wide
from Hiebert's
seemed appropriate
in learning
to construct
performance
task
was not high.
from Case and Kurland's
span test
they would yield chosen
predict
were found between
Daneman, & Emmanuel, 1979)
Kurland,
of M-space data
will
.60)
the measures
(Case,
may be very
it
to
(.50
23
Capabilities
Cognitive-Processing
variables
involved
to add and subtract.
Method Sample All School
of the
in Hobart,
located
River
of Tasmania.
2 gives
details
K-2 at the Sandy Bay Infant for
this
The school
is
in Sandy Bay, a suburb of Hobart near
the
The community about
the number of children
in grades were tested
Tasmania,
on the Derwent
University Table
139 children
the age,
is middle
grade
study.
to upper-middle
and gender
of the
class.
sample
and
involved.
Tests Counting Conceptually, counting. counting
span. it
is
The items operations.
This
test
was developed
straightforward. that
The operation
must be stored
Children
by Case and Kurland required
are the products
are presented
with
(1978).
is
of a series
a sequence
of arrays
of
24
Cognitive-Processing
Capabilities
2
Table
of Sample
Characteristics
Class
and Grade
1
2
3
4
5
6
K-AM
K-PM
Prep
Gr. 1
Gr. 1/2
Gr. 2
Boys
16
11
8
8
15
15
73
Girls
9
9
13
14
9
12
66
Total
25
20
21
22
24
27
139
Characteristic
Total
Gender
Age
Youngest
4.9**
5.0
5.4
6.2
6.5
7.3
Oldest
5.1
5.7
6.1
7.3
7.10
8.2
Average
4.11
5.4
5.10
6.7
7.3
7.8
*Gr. 1/2 was a mixed class **4.9
means 4 years
with
both Grade 1 and Grade 2 students.
9 months as of October
1, 1979.
of geometric
to count
shapes
in the arrays
objects finished
in the set is
M-space
can count
assumed while
The test
is
incremented
maintaining
at any one of five
trials
a modified
1977).
Children
until
it
M-space basal"
were presented
was determined
the level
of complete
and children's that
they
items
were
recall. However,
at most,
levels.
To reduce
with
sets
at what level
success
have
The number
card.
to trial
method was used
They were then presented
failed.
as soon as they
stimulus
from trial
perfect
"ceiling
the number of
to the maximum number of arrays
33 items.
scored
trial
on the current
to be equal
includes
to recall
the current
preceding
the shapes
counting
of arrays
and are asked
25
Capabilities
Cognitive-Processing
the
total
and the level
& Denny, levels
M-space
and at what level
passed
a larger
number of
(Bachelder
from different
they
with
five
only
number of trials
of complete
they
until
failure
had been
determined. Mr. Cucui.
This measure
for use with
by DeAvila, (DeAvila
& Havassy,
suitable
for use with
Cucui.
After
what parts blank
are colored. levels;
it
for
test
or use numbers.
is is
the only Instead,
as older
children.
the outline
five
they
are told
five
to point
different
items
recall
does not require
of spatial
location
with
to the parts
a that
at each of five
as the number of body parts one that
of Mr.
to remember
They are then presented
of Mr. Cucui and told
defined
and is
quickly
with
seconds,
laboratory
command of English
are presented
body are colored.
drawing
an imperfect
as well
four-year-olds
There are 25 items,
a level This
viewing
with
in Pascual-Leone's
It can be administered
children
of his
outline
children
1974).
On each trial,
was designed
that students is
are colored. to count
required
to
26
Cognitive-Processing
respond
This
placement. by Case.
standardized
of M-space factor
general The basic
It
(cf.
out of order from view,
is
the children
for which
items
for
the
on this
measured
7).
are asked
the task test,
9, 12,
series.
for
with
the
(Case & Globerson,
with
1974).
of numbers.
a set
The
of magnitude
After
the numbers have disappeared where
to indicate M-space
number
to the maximum set There are 15
successfully.
each of three
and the nth is
the final
corresponds
levels
levels;
All
above are not tested.
in the two tests
highly
order
can be executed
five
the same norms as other
M-tests
subjects
and
developed
and to correlate
1972)
are in ascending 2, 5,
(e.g.,
of M-space
known to yield
to present
in the original
belongs
is
a measure
by more lengthy
defined
procedure
is
Case,
n - 1 of these
first
method was followed
Span Test.
Digit
size
The ceiling-basal
correctly.
Counting
tests
Capabilities
items
1 and 5 as were given
to
each subject. Backward digit Hiebert
The subject
corresponds test,
On each trial,
(1979).
digits.
there
The form used
span.
is
the experimenter
to repeat
to the maximum series
two tests
study
was developed
reads
a series
them in reverse size
correctly
(10 at each of four
are 40 items
in the first
in this
is not tested
and all
order.
of
M-space
repeated.
levels;
level
items
are given
In this 1 as measured to each
student).
Test
Administration A research
administer
the
One interviewer
and two experienced
assistant tests.
All
administered
were trained the counting
before
teachers
were hired
the testing
span test;
by
to
proceeded.
a second
the Mr.
Cognitive-Processing
and the third
Cucui test;
Children
span tests. the
All
took
and the
total
correct
were at least
followed.
different
Most
two a day or two later.
aim to measure
administered
five
at a level
scored
incorrect. details
and will
Second,
two of the
to each child;
than where the child
items
responded
those
rules
and for
basal"
were not actually
but at a level correct
responded
were devised
did not
for
lower
and all
correctly
were
each test.
in Romberg and Collis
The (1980a)
and Discussion of scores
frequency the
means and standard for
some items
tests
here.
Results 3 shows the
the "ceiling
were scored
are available
to
would need to be
not administered
correctly
Four scoring
regarding
in each class
tests,
were designed
as two of the since
level,
was not
procedure
scores
than where the child
higher
not be reported
Table
item
or incorrect
M-space
in each test
levels--especially levels.
correct
each child's
why this
of items
of M-space,
those
all
items
to estimate
sets
levels
was used with
procedure
be scored
two sound reasons
to reflect
weighted
scores
to come to
10 days.
obviously
counted
because
First,
measure
class
teacher
to an interviewer.
assigned
within
item could
each
Although
full
by their
the Tests
Scoring
there
selected
on one day and the other
two tests
was completed
testing
and the backward digit
placement
were randomly
room and randomly
interview
children
the digit
27
Capabilities
total
population
deviations
the four memory tests
(M-space for
each test.
are presented. provide
level)
for
children
In addition,
The distributions
two interesting
results.
of
28
Cognitive-Processing
Capabilities Table of Scores
Frequency
3
on the M-Space Tests
Score
Class 0
1
2 Counting
(1) (2) (3) (4) (5) (6)
K-AM K-PM Prep Gr. 1 Gr. 1/2 Gr. 2
Totals
1
1
3
4
M
SD
1.12 1.50 1.52 1.96 2.29 2.44
.33 .69 .60 .57 .55 .70
0
1.83
.75
.58 .64 .67 .84 .93 1.02
5
Span Test
22 9 11 4 1 2
3 9 9 15 15 12
1 1 3 8 12
1
49
63
25
1
Mr. Cucui Test (1) (2) (3) (4) (5) (6)
K-AM K-PM Prep Gr. 1 Gr. 1/2 Gr. 2
12 5 4 1
0
Totals
1
12 11 12 9 6 6
1 2 5 8 9 7
4 7 11
2 2
1.56 1.75 2.05 2.68 3.21 3.26
25
56
32
22
4
2.45
1.05
1.08 1.00 1.10 1.23 2.00 3.30
.40 .00 .30 .53 1.25 1.20
1.68
1.17
1.48 1.90 1.95 2.18 2.33 2.85
.51 .31 .22 .40 .48 .66
2.14
.64
Digit (1) (2) (3) (4) (5) (6)
K-AM K-PM Prep Gr. 1 Gr. 1/2 Gr. 2
Totals
0
Placement
24 20 19 18 12 5
2 3 6 1
98
12
1
Backward Digit (1) (2) (3) (4) (5) (6) Totals
K-AM K-PM Prep Gr. 1 Gr. 1/2 Gr. 2
13 2 1
0
16
Test
1 2
6 19
4
25
0
Span Test
12 18 20 18 16 8
4 8 15
4
92
27
4
0
First,
among children
of scores
are clearly
Scores
children
older
although
at different
by age.
Second,
individual
children
across
across
could
tests)
a cue that scores
are allowed
level,
it
that
for
of Scores
Each of the
material. covered
it
the
variation with
investigate similar
population;
(2)
of the
level
partial
at a higher
from the protocols, is
gradual.
(1)
the data
for
all
math-related
student
Three
it
was important
pairs
matrix
tests
statistical
matrix
from the four of tests
population
demonstrated
or not the different
were the
on the correlation
the
Thus,
a correlation
the scores
the amount of early
and the children
of children.
see how many classifications was performed
processing
were different,
some care whether
between
to another
students
may give
on a test
would reflect
in performance.
on the data:
correlations
for
children
range,
classifications
performed
items
frequencies
if
on the evidence
was hoped,
tasks
a wide age/grade
considerable
for
on the Tests
to the
However,
striking.
level
of the text
answering
of M-space
quite
to be specifically
In addition,
questions.
deduction,
tests,
available
M-space
is
in within-class
(variation
children
a reasonable
levels
the overlap
scores,
of M-space
the context
the move from one level
Relationship
to determine
were
for
up to show the the
were cross
same; and (3)
yielded
procedures
was set
tests
to
total tabulated
a factor
analysis
the dimensionality
scores.
Correlations Table
that
them answer
helps
is
the variation tests
imply
grade
but do not appear
age-related
determined
have higher
generally
29
Capabilities
Cognitive-Processing
of test
4) are positive
scores.
and statistically
Although
all
significant,
the correlations they
are not
(see
to
30
Cognitive-Processing
Capabilities
Table Correlations
of Scores
Test Counting Mr. Cucui Digit
CS
Span (CS) (MC)
Placement
Backward Digit
for
(DP) Span (BDS)
4 the Four Memory Tests
MC
DP
BDS
1.00 .49
1.00
.61
.50
1.00
.52
.40
.64
1.00
different M-space
The highest
high.
particularly
of scores
between
of students
different
This in different
there
If
tests
various
cutoff
are found
that
the same
multifactor
solution
extractions
with
in Table
6.
total model.
iterative
the Mr. Cucui test
The
test.
shown in Table
and in
The
4.
classified
in the
factor
the only
the
tests
this
on this
children
classify
a single
are along would mean that
each
one dimension.
However,
more than one dimension,
of the cross-tabulation more critical.
question matrix
presented
then each
All
extractions
A factor in Table
from this
was extracted.
did not load
analysis
5 for
were principle
of commonalities,
The data
one of the
made
the
The model used was a
population.
amount of the variance is
is
that
points
estimates
was used.
A single
considerable
we
tests,
in the same categories
problem;
The results
the
across
procedure
each other
four
the
different.
on the correlation
rotation
To examine
classifications
to measure
something
analysis.
was performed
Cucui test
these
the dimensionality
examining
that
into
on the
based
with
demonstrates
different
is measuring Factor
test
is not a serious
identifies tests
each
tests.
who were differently
tabulation
ways.
dimension,
four
clear
from 68% to 46%.
ranges cross
schemes
who were classified
of individuals
comparisons
these
for
the four
for
in each comparison
categories
percentage
test
children
classify
classification
the data
tabulated
proportion
if
It seems
.64.
levels.
similiarity
test
only
do not necessarily
tests
Cross-tabulation
cross
is
31
Capabilities
Cognitive-Processing
heavily is
still
four
that
and the varimax
factor
However, on this
factor
analysis it
be noted
should
factor,
unaccounted
appear
for.
and a The Mr.
does not ask children
to
32
Cognitive-Processing
Capabilities
Table
5
of Classifications Number and Percentage the First Test, and Lower for the First Test
That are the Same, Higher for Test for all Test Comparisons
Comparisons
(A/B)
Classification
CS/DP N(%)
CS/MC N(%)
CS/BDS N(%)
DP/MC N(%)
DP/BDS N(%)
MC/BDS N(%)
Same S(Am) (A=B)
58(42)
47(34)
75(54)
49(35)
44(32)
57(41)
Higher (A>B)
36(26)
16(12)
13(9)
19(14)
31(22)
55(40)
Lower (A
45(32)
76(55)
51(37)
71(51)
64(46)
27(19)
Note:
CS = Counting Span DP = Digit Placement MC = Mr.
Cucui
BDS= Backward Digit
Span
Capabilities
Cognitive-Processing
Table Factor
Analysis
for
6
the Four Memory Tests
Factor
2.59
Eigenvalue % variance
64.8 factor
Raw (rotated) Counting Digit
matrix .44(.56)
Span
Placement
.54(.72)
Mr. Cucui Backward Digit
.30(.37) Span
.44(.51)
33
34
Cognitive-Processing
count,
and it
factor
is
counting
Capabilities
also
is
less
a quantitative
English
memory of spatial
tests
to classify
Thus,
best
to administer
then
to classify
on the other
measure
hand,
requires
appears,
level.
The next
section
basis
of the results
regard
can reliably indicates
of three
quantitative it
would
as was done in this
to that
underlying
classify
children
should
be fairly
seem and
study
structure.
No
an M-space
into
a classification
that
tests
factor,
levels,
M-space
of tests
a combination with
one primary
into
children
children
it
test,
the
orientation.
In summary, the four
single
that
suggests
memory of number or
involving
The Mr. Cucui test,
sequences.
M-space.
factor
M-space
This
dependent.
made on the reliable
for most
children.
Cluster
Analysis
Since
this
This
1The usual
A cluster points,
analysis
indicated
that
the
last
Euclidian
four
for
groups
distance
seemed desirable
analysis
between
group vectors
one dimension
the children
distances
the estimated analysis,
it
to classify
dimension.
single
showed that
analysis
of the variance,
of the project
Euclidian
used,
factor
two-thirds
nearly stage
the
the six (3,
4,
between
for
procedure, for
were six groups
the classification. groups.
identified.
in four
Table
- y)2
+ (x2 - y2)2
+ (x3 - y3)2
7 gives
In the together
dimensions
i.e., dA- (x1
along
which uses
5, and 6) were closer
points
for
the next
in the population
was used there
accounted
+ (x4 - y)2
was
Capabilities
Cognitive-Processing
Table
35
7
Estimated Vectors for the Six Groups Derived from a Cluster Analysis Between Score Vectors is Less than 1.50 Where the Distance
Test Amalgamated distance
Number of children
1
1.05
59
2
1.44
3
Group
CS
MC
Overall M-space classification
DPT
BDS
1.32
1.07
1.73
1.61
1
38
1.90
1.66
2.13
2.76
2
1.43
16
2.25
2.10
2.25
3.69
2S+
4
1.03
11
2.91
4.00
2.91
2.46
3S-
5
1.06
4
2.17
3.83
2.67
4.50
3S+
6
1.23
6
2.50
4.00
3.75
3.75
4S-
36
Cognitive-Processing
Capabilities
than Groups 1 and 2. that
This
that
suggests
5, and 6, although
Groups 3, 4,
Groups 1 and 2 are distinct
different
from each other,
and
are less
distinct. Group 1 is for
levels
MC, Level
this 1.
children
group is
at Level
1; BDS, Level
basic
M-space
reach
Level
2.
Level
They are below
scores
on the Mr. Cucui test.
perception
and nearly
spatial
perception
on that
test,
have labeled
16 members,
reached is
are above
level
Group 4 has 11 members. at Level
3; on DPT they
are only
at Level
spatial
perception
2.
involved
the lowest
BDS could
that
children
some
or they
for
other
on the spatial On two tests,
basic
and
tests,
2 on three Either
their
to chunk information
M-space
level
the fact
of 2.
that
We
these
test. are
CS and DBS, the children
4, but on the Mr. Cucui test,
M-space
to
contributed
above Level
2S+ to highlight
from the
found the
information
are able
a basic
a
exhibit
seemed,
the children
slightly
2;
on the DPT and nearly
level
4 on the Mr. Cucui test.
are at Level
Their
These
to check
exhibit
group Level that
1; and
group are CS, Level
than instructions
scored
high
still
but they this
Level
quite
1,
These differences
or ability
spatial
Level
this
2.
factors:
for DPT more complex
Group 3, with
for
2; and MC, Level
either
children
The levels
3 on the Mr. Cucui test.
instructions
1; BDS, Level
Only for
measured.
to be due to contextual
protocols,
tests
at M-space
the
separately,
2.
Group 2 has 38 members. DPT, Level
1; DPT, Level
clearly
in the domain being
be placed
For the tests
59 members.
group are CS, Level This
level
M-space
with
largest,
level
in Mr. Cucui appears
is
probably
3.
they
Their
not to be as highly
developed
as their
four members who have a similar
Group 5 has only
in Group 4 except
to children
Their
Cucui test. and therefore
basic
on this
this
It
case.
test,
needs
but overall
we have labeled
strategies
versa.
be simply
group is
is
not clear
Level
3,
4 on
what the
did not assist
a sampling
in
or testing are so small.
than we were able
lower
to perform
in
than Group 5 on the Mr. Cucui
skills
quantitative
are at Level
in children's to instruction
occasion.
even
This
though
suggests
4.
Therefore,
they
possible
prior
ability
significant or their
same basic
hypothesize
and develop
number skill
on the child's
have the
One could
mechanism.
cognitive
strategies
and number development
to be interwoven achieve
effect
use of problem-solving
potential.
(qualitative) appear
an underlying
suggest
has a significant
on any given
but some children vice
at M-space
and the protocols
examination
results
setting
cognitive-processing development
Level
them 4S-.
The contextual
reception
their
these
Overall,
differences
this
on the Mr.
them at M-space
the numbers in the category
closer
However,
It
3 on CS.
implies,
since
very well
They are basically
of course,
especially
study.
to respond
test
of levels
pattern
them 3S+.
below Level
could,
This variation
Group 5 scores seems to place
pattern
members.
but score
tests
discrepancy
this
that
we have classified
Group 6 has six
variation
we have
Therefore,
them 3S-.
classified
three
abilities.
quantitative
37
Capabilities
Cognitive-Processing
that
spatial
(quantitative) close
to spatial
together skill
in time, and others
38
Cognitive-Processing
Capabilities
Study The reason
for wanting
the development
Piaget,
a battery
on the theory
is based
development
2--Cognitive
process,
is
is
position
that
measure
is
Development
to
According
from the growth
of
a broad-based
of situations. in the following
summarized
cognitive
(1974).
inseparable
faculties.
to a wide variety
generalizing
Piaget's
of tests
of Jean Piaget
of cognition
and psychological
biological
Development
statement:
and this option I think that development explains learning, is a to the widely held opinion that development is contrary sum of discrete (1974, p. 176) learning experiences. The phrase learning
"development is
experience
That is,
capabilities. extent
a large
For this the early
in part
the entire
of 10 tests
(or explained)
with
to
measuring
elementary
premathematical
to relate
already
all
to work with
ability
in order
to characteristics
characteristics
defined
was devised,
concerned
concepts
population
of a
level.
of the child's
and logical
quantitative
is
potential
a battery
project
the outcome
for by developmental
accounted
learning
that
implies
learning"
by developmental
development
We tested
explains
derived
skills.
developmental from the M-space
tests.
Development
Cognitive
As stated intent
to examine
performance were selected (1968);
earlier,
Tests the choice
of cognitive
the relationship
on addition
and subtraction
from a large
two from tests
battery
devised
tests
of specific
tasks.
of tests
was based
capability
to children's
Of the 10 tests,
constructed
by Romberg, Carpenter,
on our
seven
by Fullerton and Moser (1978);
by the authors
and one was constructed
in Romberg and Collis
can be found
test
Extension
of dots
extend
portion
of the number scale.
correct
responses
that
the
is
child
The test
to set
by Fullerton
boxes
has the
refers
of each
answer
to the fact
up a one-to-one
that
the
to a higher
12 items. is
(1968).
same number
of subitemization
contains
A correct
scored.
able
choice
level
the usual
Details
study.
(1980b).
The term extension
beyond
is
this
was developed
of three
which
box.
as a sample
number sets
group test
This
(E).
are to decide
Children
for
39
Capabilities
Cognitive-Processing
The number of to mean
interpreted
correspondence
between
also
developed
sets. Ordinal
the format
Fullerton
(1968),
Extension
Test.
This
test
responses
is
scored.
correct
the child
that
is
In this
(OC).
Correspondence
the items
for
also
contains A correct
to establish
able
group test, is
similar
12 items. answer
an ordinal
is
by
in the
to that
The number of as meaning
interpreted
between
correspondence
sets. Conservation
by Fullerton
developed Wohlwill is
items
A correct
to preserve
also
work (1960),
developed
is
(i.e.,
able
(Wohlwill, by Fullerton
are interspersed
because
of the similarity
in that
a single
object
with
between is
based
either
group test,
on an earlier
test
to mean that
interpreted
between
to overcome AS-W). (1968) those
sets
and based
the two tests.
responses
after
This
added to or subtracted
is
one set
distractions). for
this
group
on Wohlwill's
of the previous
by
the child
perceptual
The items
also
developed
The number of correct
correspondence
Addition-Subtraction test,
is
response
This
CN-W).
are given.
one-to-one
has been rearranged
is
(1968),
Six
(1960).
scored.
able
of Number (Wohwill,
test test
earlier (CN-W)
differs
from the
only
40
Cognitive-Processing
of objects
collection response
is
increase
or decrease
in front
are no longer
sets
the Coordination
because
to attend
transitivity.
A correct
to preserve
both equivalence
score
is
recorded
Coordination group test those
for
as the child
Class items
Inclusion
was developed
response
is
set
distinct
into
Additive administered that
A correct
response
Test
group test
(CRE, described and a linear
was designed
to assess
as the child
interpreted
only able
being
A total
relationships.
Test
correct
the fixed
is
set
able
also
to preserve
six-item
are similar
to
transformed
A correct
together).
This
(CRE).
The items
(1968).
here
response
is
equivalence
The scoring
rearrangement.
This
(CI).
by Romberg,
is
procedure
administered
individually
and Moser (1978).
Carpenter,
as a child
being
able
to subdivide
of two
test
A correct a
logically
subsets.
Composition
ask children
are
T.
interpreted
test,
that
being
even after for
items
transitivity
of Equivalence
or heaped
interpreted
to that
is
by Fullerton
except
shortened,
identical
test
and order
of Relations
(lengthened,
relationships
Six
each child.
was developed
in the T test
correspondence
six-item
Equivalence
to both
response
this
developed
The present
of sets.
rearrangement
an
that
recognizes
in one-to-one
a correct
scored.
of Relations
a child
requires
case
in such correspondence.
The authors
(T).
Transitivity
In this
the child
in one of two sets
and the number correct
given
of the children.
to mean that
interpreted
means these
next)
Capabilities
of Number (ACN).
developed to respond implies
by Fullerton to three the child
quite
This (1968),
individually includes
different
can establish
three
composition an equivalence
items tasks.
by the common practice
relationship
when distracting
correspondence
On (CO).
Counting developed three onto 20,
by Romberg, for
items
a number less and a large
that
is
they
answer
four
the
two of three
Back (CB).
typical
used
is
the same as in CO.
Test
Administration Because
important,
the order and because
two decisions the tests
separate children.
times.
2 were given
four
is
less
like
formal
than 15?
tests
were made to gather into
not all
Second,
are given
to all
children.
2 (CN-W and AS-W) was assumed
a level is
if
then
in this
CO; however, back starting
at 15
The scoring
procedure
were administered
was
sets
tests
of the tests
test
the number
to children
of varying
the data more efficiently.
four
of the
10 and
The typical
score
Could you count
these
number
2, or 3).
1,
they would be administered
were separated
small
at 13 to find
(0,
includes
a number between
A total
passed test
in which
The organization
was to take which set
is
on:
are marked as passing
asked was,
question
was
The test
10 and 20.
counting
correctly.
This
test
of counting
a number between
items
the number that
First,
number onto
Children
to find
ages,
small
Could you start
more than 13?
Counting
and Moser (1978). levels
such a
presented.
administered
three
of the number of levels
recorded
case
number onto
and preserve
is
individually
Carpenter,
than 10,
asked was,
question
information
This
each of the
of sharing
41
Capabilities
Cognitive-Processing
in Table
to be administered were given
and the rules 8.
A child
to have passed
to all for
The interview
passing set
at
who
selecting tests
the two tests 1 and was given
and in set set
3.
42
Cognitive-Processing
Capabilities
Table Tests
Order
8
Included in Each Set, Sequence of Administration, and Rules for Selecting Subjects
Set
Rule
(tests)
1
Interview (ACN, CI, CO, CB)
All
children
2
Set 2 (CN-W, AS-W)
All
children
3
Set 1 (E, OC)
Children failing test in set 2
either
4
Set 3 (T, CRE)
Children passing in set 2 tests
both
Cognitive-Processing
if
However,
a child
failed
and the child
administered,
On the interview
tests
and the other
Tests, children
room (the
Children
Children
two tests
took
Shortly
after
given.
Set
time
the
were working
by their
teachers
following observed
on the correct
page,
copying
from others.
Set
testing
was completed
within
was in a corner
two a day to two after.
the group batteries
of six
to eight
assistant
presented
a script
and using
the children
at a
the stimulus a large
to make sure
followed
next,
were
children
in the right
responding
1 was given
Again
to an interviewer.
assigned
to groups
assistants
3.
the CO and CB
Each interviewer
were completed,
The research
each test
set
1 was
to come to the
on one day and the other
first
2, set
the CI and the ACN Tests.
lounge).
interviews
The other
board.
administered
were randomly
2 was given
for
one assistant
selected
from each class.
information
in set
to have failed
teachers'
of the room.
tests
was assumed
administered
were randomly
interview
of the
either
43
Capabilities
magnet they and not
place,
by set
3.
All
four weeks.
Results Intercorrelations Full are given between procedure
summary tables
for
the tests
Development
the raw score
by Romberg and Collis
(1980b).
and the structure
Tests
data
for
each of the tests
To examine
of the battery
the relation
itself,
a two-step
to organize
the battery
was followed.
Fullerton tests
Among Cognitive
(1968)
he developed.
established Unfortunately,
an order that
used
scalogram
He found for
the
tests tests
method fails
analysis that
grouped
together,
based
on test
difficulty.
to establish
the underlying
and he
of
44
Cognitive-Processing
Capabilities
of the data matrix
dimensionality
A more satisfactory
assumed hierarchy.
a hierarchy
then
unidimensional,
The intercorrelations
but fairly
positive
fall
correlations
low,
they were baseline
of the tests.
across
the whole
appear
in Table
for
for
the is
the 10
The correlations
9.
17 of the
We decided
matrix
tests
first
If the matrix
population
to .79;
from .24
.40 and .58.
correlation
to determine
can be established.
ranging
between
from the
OC tests that
tests
processing
of the
structure
method is
of the intercorrelations
dimensionality
cognitive
or the possible
further
are all 28 the E and
to exclude
on the grounds
analysis
on which most children
at the
scored
ceiling.
Factor
of Cognitive
Analysis
the dimensionality
To determine
was performed
analysis
solution
were principal
factor
commonalities;
the varimax
of this
analysis
factor
examination
is
extractions
considerably
of this
in Battery
of counting
rotated
quantitative
factor
All
iteration
The other
At best
influenced
of
was employed.
than that
This
four
extractions
estimates
although
matrix
factor
E and
tests
The results
10.
on the rotated
not to the same degree.
model was used.
procedure
larger
4, T and CRE.
skill.
8 with
was derived,
solution
(CB, CO) load heaviest tests
rotation
a factor
shown in Table
with
are shown in Table
A two factor first
of the intercorrelations,
on the matrix
A multifactor
OC excluded.
Tests
Development
factor
tests
for
the second
shows that
first
also
may reflect load
by the ability
it
is
to count.
An tests
by the
a mature
on this
the
factor.
the counting followed
factor,
we can say that
for
the eigenvalue
level but
factor,
probably
a
The second
Capabilities
Cognitive-Processing
Table
of the Ten Cognitive
Intercorrelations
E
E
OC
9
ACN
CN-W
Development
CI
AS-W
CB
CO
T
CRE
1.00
OC
.45
ACN
.22
.25
CN-W
.30
.32
.35
AS-W
.35
.37
.48
.51
CI
.13
.13
.32
.24
.28
CO
.22
.28
.55
.43
.42
.44
CB
.15
.21
.49
.40
.39
.45
.79
T
.13
.16
.43
.42
.36
.47
.52
.61
CRE
.17
.21
.51
.55
.48
.39
.58
.62
1.00 1.00 1.00 1.00
Maximum
12
12
3
6
6
Mean
10.88
10.63
1.97
4.86
5.03
1.56
1.34
Std. deviation
Tests
1.90
2.28
.94
1.00 1.00 1.00 1.00 .68
1.00
3
3
6
6
.51
1.35
1.06
3.93
1.75
.81
1.29
1.21
4.68
1.49
2
45
46
Cognitive-Processing
Capabilities
Table Factor
Analysis
for
Eight
10
Cognitive
Development
Tests
Factors 1 Eigenvalue
4.34
% variance
54.30
Raw (rotated)
factor
2 .92 11.50
matrix
ACN
.64(.46)
.07(.45)
CN-W
.61(.25)
.35(.65)
AS-W
.60(.24)
.37(.66)
CI
.52(.49)
-.13(.23)
CO
.80(.76)
-.22(.33)
CB
.83(.85)
-.33(.26)
T
.73(.60)
-.06(.41)
CRE
.81(.56)
.11(.59)
seems more qualitative,
factor
Class
One test, Since
factor.
Class
Inclusion
of the first
interpretations The factor
dimensions
interpretable
factor
the first
since
variance tests
using
simplex
(1954)
the correlation
examines
at this
functioning
from all
the evidence,
ordering
of these
on the
factor
quantitative for
over
ability
half
In this information
dimension.
can be explained influenced
Between section
in terms
and see
of the analysis, tests
in
are not the
tests
five of
aspects
It
seems
a hierarchical tests
development two-thirds
to count, factor
which that
without
a
accounts involves
counting.
and M-Space Tests
we attempted and the cognitive
to combine
do
of the
of two dimensions,
transformations
Development
for
about
Rather,
and a qualitative
Cognitive
from the M-space
cognitive
of the
as one
the criteria.
is no basis
by the ability
of the variance,
to make comparisons
Relationships
there
to test
satisfy
In summary, the
a single
tests
do not
that
the tests
be considered
might
clear
of the matrix,
a subset
of the
ordering
It is
procedures. that
However,
proportion
hierarchical
as a whole
level
then,
tests.
not seem to measure variance
simplex
were two
there
on the tests.
such a large
the possible
matrix
(ACN, CN-W, CI, T, CRE) that logical
for
Even when we take
order.
to our
seemed to show that performance
underlying accounted
Guttman's
the
and is
reasoning credence
gives
on either
two factors.
we examined
(54.30%),
finding
the factor
rotated
heavily
logical
of the data
analysis
not load
involves
this
test,
nonquantitative
only
on this
heaviest
does
Inclusion,
to make comparisons In particular,
to count.
having
(AS-W and CN-W) load
tests
Wohlwill load.
without
transformations
and see
the ability
involving
47
Capabilities
Cognitive-Processing
the
development
tests
an
48
Capabilities
Cognitive-Processing
describable
distinct
To begin four M-space
tests
and OC being
omitted
from .29
tests
tests
with
Factor
is not
To check
this
the eight
in Table
appear
analysis
of the cognitive The two factors
appeared. that
appeared
the first
At this a pattern
six
groups
proportion set
memory the
that
suggests
dimension.
and M-Space Tests a factor
relationship,
the four M-space
Again,
were added to
tests
for
that
factor
as was the case
with
the factor
The data
tests
development
(see
have the same structure
Table
two factors
9),
as the two factors
The memory tests
analysis.
we decided
formed by the cluster
out in Table
of the different This
tests.
tests.
in the achievement
correct
However,
on
load heavily
but not the second.
point,
for
tests.
unidimensional
in the earlier
factor
than some of the other
Development
12.
the counting
both
undoubtedly
out in which
analysis
with
tests
a single
development
cognitive
The
The counting
processing
suggested
was carried
occur
tests
range
.40 and .59.
between
32 falling
the E
(tests
The correlations
in correlations
along
tests
development
surprising.
variation
is
have
11) was drawn up for
(Table
earlier).
given
the M-space
of Cognitive
Analysis
analysis
20 of the
the cognitive
correlation
positive
reasons
memory capacity
no apparent
is
cognitive
This
a larger
there
and the eight
with
(CO, CB).
rquire
matrix
with
to .79,
correlations
higher
a correlation
for
that
categories
characteristics.
cognitive
with,
into
the children/pupils
to grouping
a view
with
that
on all analysis
in each cognitive 13; a graphical
to look
we had enough information
test
cognitive
tests
of the M-space for
representation
for
each of the
tests.
each M-space
The
category
is
of the same information
Table Correlations
11
of the Eight Cognitive Development and the Four M-Space Tests
Cognitive M-Space Tests
49
Capabilities
Cognitive-Processing
Processing
Tests
Tests
ACN
CN-W
AS-W
.54
.32
.39
.43
.63
.61
.47
.53
.54
.44
.41
.45
.77
.79
.69
.63
Mr. Cucui
.46
.32
.37
.48
.53
.55
.46
.47
Backward Digit Span
.48
.48
.50
.38
.61
.58
.55
.54
Counting Digit
Span
Placement
CI
CO
CB
T
CRE
50
Cognitive-Processing
Capabilities
Table Factor
12
for Eight Cognitive Analysis Development and the Four M-Space Tests
Tests
Factors 1
2
Eigenvalue
6.52
1.02
% variance
54.40
8.50
Raw (rotated)
factor
analysis
ACN
.65(.56)
.08(.36)
CN-W
.58(.40)
.41(.50)
AS-W
.59(.37)
.41(.60)
CI
.55(.56)
-.12(.12)
CO
.83(.78)
-.18(.28)
CB
.84(.85)
-.25(.15)
T
.73(.73)
-.01(.16)
CRE
.78(.70)
CS
.71(.68)
-.13(.25)
DP
.86(.74)
-.19(.10)
MC
.63(.68)
-.09(.25)
BDS
.73(.62)
.16(.31)
.13(.43)
is
shown in Figure
1.
can be seen
It
there
that
are clear
between
Groups 1 and 2 and the other
four
groups,
Groups 3 and 4 differ
from each other
Groups 5 and 6, which
are also
Group 1 children in all
four
tasks.
with
They could
with
skills,
Group 1 on all
tests.
level
on the
although
with
on and counting
back,
they
better
qualitative
scored
specific than
correspondence
somewhat lower
than the
of number test. level
2S+ were high
the specific
test.
reasoning
Their
performed
with
and all
reasoning
counting in their logical
on qualitative
skills
of counting
use of those reasoning
better
considerably
M-space
test.
In fact, composition
Groups 5 and 6 with
M-space
on these
correspondence
tests.
tests,
level
the quantitative
on the additive
profiles
only
skills
was also
than Groups 1 or
test.
correspondence
3 only
quantitative
without
considerably
handle
but were inadequate
Group 4 children
logical
groups
and transformations
2 were also
performed
they
M-space
had developed
although
level
They could
correspondence,
on the transitive
they
conservation
Group 3 children
2 on that
the other
of handling
incapable comparisons
M-space
although
the
at an acceptable
deficient,
from
level.
quantitative
groups
but do differ
1 were below
level
make qualitative
Group 2 children
other
M-space
the latter
similar.
very
and were in general
areas
at a moderate
little
differences
Within
groups.
51
Capabilities
Cognitive-Processing
they test levels
They reached scoring
a little
3S- were high
on qualitative
but inadequate
tests, differed
significantly
and the
transitivity
3S+ and 4S- presented the ceiling higher
on the from Group test. similar
on the qualitative
than Groups 2, 3, and 4.
52
Cognitive-Processing
Capabilities
Table
13
Percent Correct for the Six M-Space Groups on the Ten Cognitive Development Tests
Test Factor
Group (M-space level)
1 (Quantitative)
Factor 2 (Qualitative)
Baseline
Logical T
CI
6
2
6
35
43
14
20
82
78
73
53
50
87
93
87
80
90
50
100
100
91
75
100
100
88
100
93
100
87
100
80
90
E
OC
AS-W
CN-W
ACN
CO
CB
CRE
1(1)
90
84
46
48
45
9
3
2(2)
100
100
91
71
80
52
3(2S+)
100
100
87
93
71
4(3S-)
100
100
100
90
5(3S+)
100
100
100
6(4S-)
100
100
100
Cognitive-Processing
Capabilities
*---e A--A
53
Group6 Level 4SGroup5 Level3S+
0CI I o ...o *--I
w
\
Group 4 Level3SGroup 3 Level 2S+
0
0
/
\I
I
?r. D----0 Group 2 Level 2
1o-
*-*
OC , ASW
E
I
z
.J
CN-W ACN , CO
Ty
0HZ c
Z
W
Z
CB
CRE
T
Group 1 Level 1
Cl
T- UJ
-I
d> Z
0o_
Z
< a0 oC 0 Figure
1.
Pattern of scores for the six M-space correct) (percent on ten cognitive tests grouped by factors. process
groups
54
Cognitive-Processing
Capabilities
Like Group 4 children,
in these
Children
tests.
From these
had high
they
series
on all
the quantitative
on the
class
of students
a sample
groups,
3, 4, and 5 in this
Studies
were high
groups
cluster
scores
in the
inclusion
test.
was drawn for school
following
year.
Summary and Conclusion Based on data
well-defined
but different
with
students
of a factor
similar
analysis,
of responses.
we found
that
counting
how the six
examining
on the cognitive scores
This
last
formed five
step
mathematical
was used
interaction
with
In conclusion, that
researchers
to study
analysis
differed for
the data
the following and practitioners:
several
gathered
propositions
tests mature Third,
by
performed
the cognitive-processing
systematically. of the project.
the remainder (cluster
groups
related
capabilities
instruction
mathematics
involves
by the M-space
In the following
materials.
that
of
groups
development
factor.
of students
groups
known cognitive
information
suggest
groups
six
correspondence
that
using
from the results
Second,
factor
we demonstrated
was the basis
distinct with
combined)
tests,
of the six
of five
defined
groups
First,
steps.
the cognitive
and a qualitative
strategies
who had This
we identified
scores,
a quantitative
of children
capabilities.
in the following
patterns
on two factors:
loaded
cognitive
groups
cognitive-processing
on the memory test
analysis
and eight
to identify
was accomplished
identification cluster
we were able
tests,
development
from four memory tests
aspects
learning
we describe
of
how this
of the children's
in early
school.
elementary
and analyzed deserve
5 and 6 were
to the
chapters,
We
in this
close
chapter
attention
by both
1.
A global children's
2.
mathematical level
M-space
The suggested elementary be:
4.
in the early
thinking
seems to be related
in
apparent
school
year;
to the development
of other
related
in mathematically
years
--
in the preschool
sequence
developmental
comparison
logical
is
skills;
cognitive 3.
distinction
qualitative/quantitative
55
Capabilities
Cognitive-Processing
qualitative
to early
reasoning --
correspondence
to
appears
--
quantitative
operations;
An M-space
level
of 1 is
enough for handling
level
of 2 is
enough for understanding
simple
comparison
tasks; 5.
An M-space
correspondence
and is
a prerequisite
for
qualitative
the development
of number
skills; 6.
An M-space cated
level
counting
In all,
these
of 3 seems necessary
data
suggest
and order)
is
by a qualitative
understanding changed
transitivity develops.
that
correspondence between
circumstances.
on and counting tasks.
simple
on sophisti-
back develop,
Finally,
correspondence
to be the first
appears
how correspondence
under varying
counting
success
tasks.
equivalence followed
for
two sets
Next,
is
that
for
by their logical
This
involves or
preserved
the quantitative
followed
the capacity
to develop.
ability
capacity
(both
skills use in
reasoning
of
3
Chapter
COGNITIVE-PROCESSINGCAPACITYAND CHILDREN'S PERFORMANCE ON VERBALADDITION AND SUBTRACTIONPROBLEMS
In this purpose
was to study
capacity,
and their
and subtraction the concepts
is
subtraction
problems,
independent
Ginsburg,
1977;
and selected on three April,
should
master
and
addition
that
demonstrated
such mathematical
solving
Carpenter
learn
be
must first
& Moser,
1983;
many of the strategies
they
than the
more insight
over May).
problems
in the previous
tested
different
to reflect
and 26-28
each student's
(cf.
and demonstrate
of the children
occasions
subtraction
addition
are taught.
that
A sample
verbal
for
In fact,
1978).
on verbal
children
has been clearly
of instruction
use are more sophisticated procedures
that
of strategies
Resnick,
cognitive
of knowing how children
to solve
it
However,
a variety
develop
assumed
Its
reported.
and subtraction
of addition
frequently
problems.
is
among children's
The importance
and then begin
skills
computational
set
and use of strategies
performance
and procedures It
in this
study
the relationships
problems.
self-evident.
children
the third
chapter,
cognitive
a 3-month period In each interview,
was given
performance
in 1980 a set
56
(27-29
2)
(Chapter
was interviewed
capabilities
to each child.
and strategies.
studies
February,
of verbal
9-1
addition
The interviewer
coded
and
Performance
on Verbal
Problems
57
Method Sample The children since
school
from the earlier
previous
in different Heights
Primary
Our intent
students
in each grade.
was made.
imbalance
across
44 students in Table
the grade
who and
grade
were enrolled
however,
in
2 students
now were in third
However,
at Waimea
after
but should
rosters
school
from each
of students
to another.
from
selection
some third
began,
This
not have affected
are shown by cognitive
study
students
Then an initial
level.
were switched
classes
in this
of two to four
We began with
cognitive
in one class
originally
Most,
a grade
School.
and their
each grade
in October
was to have a sample
level
cognitive
School
schools.
primary
had advanced
Furthermore,
testing.
were in Sandy Bay Infant
studies
of
graders
created
some
the results.
and grade
class,
group,
The
14.
Procedure Types of problems. (tasks)
given
under four
problems
solvable
solvable
by subtraction
terms six
of semantic
problem
(1979,
An interview
types
of six
by addition
conditions.
The six
of the two given
of the two given The semantic
is
in Moser (1979)
problem
types
included
types
numbers and four
numbers.
structure. detailed
of six
consisted
The types
characterization
problems
differed for
and in Carpenter
two
in
these and Moser
1983). Table
problems problem
15 presents
representative
were administered type
differed
to the children.
but the semantic
in the order
problems
The actual
structure
in which
wording
remained
for
constant.
the each
58
Performance
on Verbal
Problems
Table Children
14
in Each Cluster
Sandy Bay Infant School
Group in Each Class
Waimea Heights
Class
Primary
School
Class
1
2
3
4
5
Grade 1
Grade 2
Grade 3
Grade 3
Grade 3
1
3
2
0
0
0
5
2
3
6
0
4
0
13
3
1
2
2
3
3
11
4
0
0
2
3
3
8
5,6
0
0
3
1
3
7
Totals
7
10
7
11
9
44
Cognitive Group
Total
Performance
Table
on Verbal
Problems
15
Problem Types
Task
Sample Problem
1.
Change/Join
2.
Change/Separate (Subtraction)
3.
Combine/Part (Subtraction)
4.
Combine/Whole (Addition)
5.
Compare (Subtraction)
Her brother Todd Angie has 4 lady bugs. has 7 lady bugs. How many more lady bugs does Todd have than Angie?
6.
Change/Join, Change set Unknown (Subtraction)
Gene has 5 marshmallows. How many more marshmallows does he have to put with them so he has 8 marshmallows altogther?
(Addition)
Pam had 3 shells. Her brother 6 more shells. How many shells have altogether?
gave her did Pam
She gave 5 erasers Jenny had 7 erasers. to Ben. How many erasers did Jenny have left? Unknown
Unknown
There are 5 fish in a bowl. 3 are and the rest are spotted. How striped fish are in the bowl? many spotted Matt has 2 baseball cards. He also has 4 football How many cards does cards. Matt have altogether?
59
60
on Verbal
Performance
Within
two of three
each problem, by x + y = z,
defined
numbers from a number triple
x < y < z, were given.
x and y were presented,
problems,
In the four
first.
Problems
subtraction
The order
presented.
with
number x always
z and the larger
of y and z varied
of presentation
y,
z)
In the two addition
the smaller
problems,
(x,
given
addend y were
among problem
types. The six
semantic
conditions,
problem not all
although
conditions and absence
In the larger
number problems
the sum was 11 < z < 15.
In the smaller
LN problems),
(called
under six conditions. zs with
same sets
with
given
presence
number problems
on
the restriction
SNp and LNp were given
sets
Four
of 5 < z < 9 was imposed.
guideline
Problem
the
present;
manipulatives
zs and larger
smaller
the addition
to all
responded
materials.
of manipulative
SN problems),
(called
children
from crossing
resulted
used were presented
types
with absent
manipulative
were
SNa and LNa.
called
For the two-digit
In the two-digit
In the NR probems,
algorithm
computational
For the two-digit
was required.
restricted
to numbers in the
20s and 30s.
took the LN, NR, and R problems. used are reported Interview interviews
in Romberg,
method.
(see
interviewer-training
In the
Three
(borrowing
subdomain
problems,
the sum z was
the third-grade
details
(1981).
trained
interviewers
administered
for
details
and reliability).
children
of the procedures
and Buchanan
1980,
or
R problems,
Collis,
Cookson & Moser, procedures
Complete
second
All
two subdomains
or sum when a
a difference
was used.
regrouping
domain,
no regrouping
to determine
was required
the domain of
children,
third-grade
numbers was included.
were identified. carrying)
with
interviews
of One interviewer
the
on Verbal
Performance
worked at Sandy Bay Infant
a day,
interviews
from distracting
separate
to the
numbers or relationships required
(or one for
problems.
four
No child
each
in detail
were coded for
and,
responses
to the tasks
chapter
was compiled for
all
and are reported
the given interview
for LN tasks
and the other
The sessions
lasted
the same sequence
other
of
in one day. codings
of student
in Cookson and Moser (1980).
error.
rooms
were read and
remembering
of the possible
each child:
incorrect,
are the basis
twice
All
responses.
are presented
if
child
was interviewed
strategy,
in this
the SN tasks
task
At the schools,
An individual
receiving
elements
profiles
no difficulty.
with
each,
and on the
tasks
so that
for NR and R).
Coding student responses
The verbal
as necessary
caused
from 8 to 12
which were quiet
areas,
the LN and the other
minutes
15-25
schedules
interview
one for
two sessions,
schools'
activities.
as often
child
to conduct
than the SN tasks.)
longer
were assigned
the interviewers
reread
took
61
two at Waimea Heights
was able
on the
depending
(The LN tasks
level.
and the other
Each interviewer
School.
Primary
School
Problems
model used,
A record
Three or
correctness,
of each subject's
from the coding statistical
sheets.
These
information
in Romberg, Collis,
appearing
and Buchanan
(1981).
Data Aggregation The interview general children differing children
strategy.
and Analysis data
The data
are summarized cognitive
are summarized for
percent
by examining
processing
in terms
of percent
correct
and
answered
correctly
by
of items
the differences
capabilities.
in Group 5-6 would answer more items
for
children
It was anticipated correctly
than those
with that in
62
Performance
on Verbal
Problems
Group 4, who in turn would answer more items and so forth.
children, Pupil any),
or process
strategy for
Direct
objects
for
a number other a given
than 1.
a second
requires
direct 4.
mental
the
fact
string
the entry
of counting point
or until
a desired
or some sort
is
words,
in the sequence in either
may proceed
in the problem)
operations--use
mental
direc-
number (usually This
reached.
of a tracking
of memorized
mechanism,
number facts
for
of some other
6 + 8 can be derived
in the problem,
of a nonmemorized
operations--derivation
manipulation
Inappropriate
at all.
where Counting
counting
more than the easily 5.
or relationship
by
recall.
Nonroutine through
on the
performed
by the use of fingers.
aided
Routine
Actions
of the
number of counts,
one of the numbers given
3.
general
or fingers,
provided,
to the action
sequences--use or backward,
forward
often
Five
any).
(if
are the following:
entities.
correspond
Use of counting
tion
(if
of model used
in the problem.
described
is
and errors
of the manipulatives
the problem
generally
either
used,
to type
according
SN and LN problems
the
modeling--use
to stand
2.
was categorized
strategy
categories 1.
than the Group 3
correctly
recalled
remembered doubles
Behaviors--guessing, adding
instead
using
As an example,
fact.
by determining fact
it
to be two
of 6 + 6.
one of the given
of subtracting,
fact
or giving
numbers no answer
on Verbal
Performance
For the NR and R data, tasks
were used
write
a sentence,
6.
Correct
other
This
place
value.
paper
and pencil.
ones
1982)
that
behavior
Algorithmic
Correct
sentence/nonalgorithmic.
work is
done mentally
After
was required
Inappropriate
sentence.
This
seen
(e.g.,
of
by use of
a sentence,
in problems
(Moser,
behavior
includes
considerations
writing
as was frequently
the wrong sentence
did
as any "invented"
must be exhibited
(NR tasks)
with
as well
involve
no regrouping
working
students
of behavior
category
in school
taught
If
were used.
categories
algorithms & Moser,
a sentence.
the in which
1980).
involves
writing
and
addition
instead
of
subtraction). of what specific
Details
form these
model,
are presented
categories
and error
strategy,
data were used
in Romberg, Collis,
to
and Buchanan
(1981). The plan primary
(three with
in this
involves
interviews)
of six
six
tasks
classes
dimension
in children's
problem
is
sets
children
within
incomplete
in each grade
in the level cognitive
crossed
level,
on the
of problem The
capacity.
repeated
two
assessment
(SNp, SNa, LNp, LNa, NR, and R),
(combine/join,
includes
and in turn,
data was based
study--differences
a completely
in each set
The data matrix represented
of the aggregated
and differences
dimension
The student within
analyses
dimensions
administered problem
for
and so on).
combine/separate, nested
in cognitive
levels
grades. since grade
not all
cognitive
1 and grade
63
the SN and LN
used with
categories
did not write
sentence/algorithmic.
(Carpenter
8.
students three
standard
the
7.
if
the five
Problems
levels
2 children
are
did not
64
Performance
take
on Verbal
Problems
the NR and R problems, The small
problems. extensive
number of subjects,
and testing
3 children
did not take
the unequal
of the matrix
incompleteness
frequencies
and the grade
limited
cell
with
SN
and the
sizes,
us to describing
a few of the differences
the
the
chi-square
statistics. For purposes
of this
report,
frequency
of use of strategy
cognitive
processing
within
each set. level
grade
Romberg,
are presented
for
The data
capabilities.
(SN, LN, and NR and R combined)
sets
problem
and percent
frequency
Other analyses
are not reported
Collis,
here.
and Buchanan
children
with
are presented and for
for
performed
correct
each
Those analyses
and
different
for
three
each semantic interview
task
and by
can be found in
(1981).
Results Performance All cognitive responses, 16,
by Cognitive
SN and LN tasks.
separate
percent
for
2 children correct
To examine
are reflected
capacity
the data
grade
Groups
tables
in different
are presented
the SN problems clearly
whether
that
show that
(x2= 47.19,
percentages each problem
were given
there
(56% to 75% to 88%) for
2, and 3, respectively
for
or not differences
is
only
a significant
children
in
of correct In Table
set.
to grade
1 and
increase
in cognitive
in
Groups 1,
p< .01).
Because of the large number of trials and lack of a systematic of .01 was arbitrarily an alpha level chosen plan to test differences, In addition, to test significance. tests that yielded probability values between an alpha of .05 and .01 (.05> j> .01) were considered All X2 values were calculated via 2 x 2 significant. marginally tables where frequency of correct answers or strategy was contingency dichotomized.
Performance
Table Frequency
and Percent
on Verbal
16
Correct by Cognitive All SN Tasks
Group for
Correct Cognitive
Group
N
Total
Problems
Responses
Frequency
Responses Percent
1
5
180
100
56
2
9
312a
235
75
3
3
108
95
88
4
-
-
5,6
-
-
Total
17
600
-
430
were present aWhen all children for all three interviews, equals N times 12 problems (6 SNp and 6 SNa) times three
72
number of trials occasions,
65
66
Performance
on Verbal
For the LN problems in different
children differences while
Problems
cognitive
are striking.
the percent
children,
groups
is
shown in Table
The Group 1 children
in Group 5-6
children
to all
given
17.
Group 2 to Group 3 (65% to 81%, X2= 26.74, 4 to Group 5-6 correct
(83% to 96%,
between since
surprising,
these
differed
groups
little
very
from Group
of difference
Group 3 and Group 4 children
Cognitive
from
p<.01),
and again
jp<.01),
The lack
<.01).
The
a significant
from Group 1 to Group 2 (22% to 65%, X2 = 94.38,
increase
for
22% correct,
got
There is
96% correct.
got
only
correct
is
in percent not
on the cognitive
tests. All
NR and R tasks. the pattern
children,
children 11.76,
students,
responses
on these
the difference
in Cognitive p <.01),
between
in performance
are combined
18.
for (x2=
the differences
Again,
sets
of problems
significant.
Overall, correctly
our predictions
about
were found to be accurate,
Groups 3 and 4 differed
very
little
of the
percentage except in terms
that
items
children
of their
answered in Cognitive
general
performance.
Performance Within (change/join;
by Task each problem
set,
change/separate;
one item representing combine/part
each of six
tasks
unknown; combine/whole
3
for
Groups 4 and
Groups 3 and 4 on both
Cognitive
in Table
significant
Cognitive
E <.01).
Thus,
correct
percentage
between
to grade
only
similar.
Groups 2 and 3 (49% and 67%) is
as are the differences
between
given
were very
problems
(62% and 83%, X2 = 30.05,
5-6 children
are not
of correct
the data
summary purposes, For these
For the D and E problems
Performance Table Frequency
and Percent
on Verbal
17
Correct by Cognitive All LN Tasks
Group for
Correct Group
Cognitive
N
Total
Problems
Responsesa
Response Percent
Frequency
1
5
180
40
22
2
13
456
206
65
3
11
396
320
81
4
8
264
220
83
5,6
7
252
241
96
44
1548
1117
72
Total
for all three interviews, aWhen all children were present equals N times 12 problems (6 LNp and 6 LNa) times three
Table Frequency
and Percent
18
Correct by Cognitive All NR, R Tasks
Group for
Correct Cognitive 1
Group
N -
Total
number of trials occasions.
Responsesa -
Frequency -
Response Percent -
2
4
144
71
49
3
8
264
176
67
4
8
252
155
62
5,6
7
252
210
83
27
912
612
67
Total
aWhen all children were present for all three interviews, number of trials equals N times 12 problems (6 NR and 6 R) times three occasions.
67
68
Performance
on Verbal
Problems
unknown; compare and change/join; Because
the different
cognitive
demands,
tasks.
Following
a harder
problem)
to follow
each task
in the
SN set
of differences
3 children
between better
performing
for
each task thirds
of the for
criterion
the children correct
is
(combine/part
the LN set items
in the higher apparent.
Group 3 children
on those
those
the SN level
data
for
then data.
this
Group
to this
20.
If
two
pattern
of
as many or more items
pattern
was on Task 3
did not do as well
Group 1 children
group on
as a rough
a consistent
group getting
as easy.
children.
in Table
was used
Again,
successfully.
all
each cognitive
are presented
the Group 4 children tasks.
were just
4 and 6, however,
correct
cognitive
to work any of the LN problems
with
1 and 2
The one exception
unknown),
consistent
The
19.
Tasks
of problems
for
in Table
for
were correct,
success
group
better
The percent
Each LN task.
of memory.
perform
and Task 5 was hard for
Task 3 was more difficult,
because
each cognitive
is
groups
addends
of the missing
more units for
it
than Group 2 who, in turn,
Tasks
children.
all
for
missing
location
are presented
cognitive
As expected
than the Group 1 children. for
of problems
data
given
in difficulty,
to be hardest
schema which requires correct
tasks
a change/cause
6 and 3 (the
of the
because
the
and
demand only
they
Tasks
schema;
The percent
Each SN task.
were easy
1 and 2 (change/join for
different
of the six
unknown) to be next
in difficulty
elicit
would vary with
performance
categorization
Tasks
combination
a comparison
involves
pattern
type
and Task 5 (comparison/subtraction)
information;
for
(1980)
Task 4 (combine/whole
involves
it
that
to be the easiest,
change/separate) schema;
of each problem
we anticipated
15) we expected
Table
(see
semantics
Greeno's
unknown) was given.
change/set
were generally
The majority
as the unable
of Group 2
Table Frequency
Cognitive
N
Group
Total
Task 1 1 2 3
15 26 9
4 5,6
Total
Change/Join
Correct
Correct
Response
Frequency
Percent
1 2 3
Cognitive
N
Group Task 4
23 44 18
-
-
-
50
100
77 85 100
2
85
Change/Separate
85
Total
Combin 15 26 9
1 2 3
50 Task 3 15 26 9
21 42 16
70 81 89
100
79
79
30 52 18
-
Total
50 Task
30 52 18
Combine/Part
5,6
(-)
--
5,6
Unknown (-) 12 40 16
1 2 3 5,6 5,6 Total
Task 6 40 77 89
1 2 3
---
50
-
100
68
68
5
15 26 9 .
-
.
50
Change/Join 15 26 9
4
4
Total
Each SN
Group for
(+)
30 52 18
15 26 9
Total
5,6
by Cognitive
4
Task
1 2 3
Responses
and Percent
19
5,6
-
Total
50
Table Frequency
and Percent
Correct Cognitive
Group
N
Total
Task 1 1 2 3 4 5,6 Total
15 38 33 22 21 129
15 38 33 22 21 129 Task 3
1 2 3 4 5,6 Total
15 38 33 22 21 129
Joining
Frequency
Separating
30 76 66 44 42 258
Each LN T
Response Percent
Cognitive
Group
N
30 72 86 91 95 78
15 38 33 22 21 129
1 2 3 4 5,6 Total
(-)
Task 5 7 52 51 34 40 184
addend
Total Task 4
9 55 57 40 40 201
30 76 66 44 42 258 PPW, missing
Group for
by Cognitive
(+)
30 76 66 44 42 258
Task 2 1 2 3 4 5,6 Total
Responses
Correct
20
23 68 77 77 95 71
1 2 3 4 5,6 Total Task 6
(-) 6 54 56 33 41 190
15 38 33 22 21 129
20 71 85 75 98 74
1 2 3 4 5,6 Total
Joining, 15 38 33 22 21 129
Co
Performance
children
worked all were able
groups difficult
problems
to work all
comparison
Task 5.
except
The children
However,
problems.
Problems
71
in the higher for
except
the tasks
(Task 5),
problems
on Verbal
the
were of equal
difficulty. Each NR and R task.
are shown in Table
problems
for
except
The same data
the Cognitive lower
marginally
21.
than Group 3 children on Task 5.
successful
Tasks
all
4, and 6.
1,
tasks.
on Task 1.
However,
computational
Task 6 is notions,
are easier
problems
a subtraction making it
problem,
easier
In summary, although due to problem is
clear
that
set
children
cognitive-processing addition factors.
(size
use
were successful
were successful
that
algorithms,
was often
on
on all 3 and 5 for
when problems
have large
the
because
were
implied
than the semantics. problems.
solved
using
Although
additive
than Task 2. there
were important
of number) to specific who had been identified
capabilities
and subtraction
Group 2 children
than subtraction it
evident
1 and 2 and was about
Overall,
become more important
procedures
addition
Thus,
should
suggest
of
was
Task 2 was as hard as Tasks
These data
children
enough numbers,
on Tasks
Group 3 and 4 children
unexpectedly,
is
whose performance
And, Group 5-6 children
of the children.
the NR and R sets
the same pattern
Again,
Group 4 children
the same as Group 2 children only
for
tasks
performed regardless
variations task,
and to grade,
as having differently
of the other
in performance
different on these important
it
Table Frequency
and Percent
Correct Cognitive
N
Group
Total
Task 1
Responses Change/Join
Frequency
Correct
by Cognitive
Group for
Each NR, R
Response Percent
Cognitive
Group
N
Task 4
(+)
1
2 3 4 5,6 Total
21
Total
Combine/W
1
12 22 21 21 76 Task
16 39 33 37 125
24 44 42 42 152 2
Change/Separate
67 89 78 88 82
2 3 4 5,6 Total
12 22 21 21 76 Task
(-)
5
1
2 3 4 5,6 Total
12 22 21 21 76 Task 3
24 44 42 42 152 Combine/Part
10 27 21 31 89
42 61 50 74 58
Task 6
Unknown (-)
1
2 3 4 5,6 Total
2 3 4 5,6 Total
12 22 21 21 76 Change/Join,
1
12 22 21 21 76
24 44 42 42 152
9 22 22 35 88
38 50 52 83 58
2 3 4 5,6 Total
12 22 21 21 76
on Verbal
Performance
used
children
strategies
the SN and LN problem mental
and eight
no-sentence
five
(direct
modeling,
mental for
categories
as for
categories correct
the data
chapter,
in terms
are summarized
sets
sentence-algorithmic,
of this
part
nonroutine
operations,
strategy)
first
in the
As outlined
73
Used
Children
Strategies
Problems
of five
counting
on for
categories
routine
sequences,
and inappropriate
operations,
the NR and R problem
sets
SN and LN tasks
correct
plus
(the
same
and incorrect
sentence-nonalgorithmic,
sentence). We expected use
that
operations. children
SN tasks.
each problem children
(Table
use of routine
34.80,
categories All
as expected,
mental
operations
? <.01).
23.
in Cognitive
for
given
by
only
children here
Group 1 either
cognitive
to grades
1 and 2 increase
strategy
over given
cognitive to all
cognitive
is more dramatic. directly
with
modeled
(39% to 18% to
of use of the other
the frequency
in different
in
and a corresponding
p<.01)
constant
for
are presented
(8% to 27% to 35%) by children
However, remained
different
was a significant
there
For the LN problems
The picture
with
mental
frequency
tables
in use of an inappropriate
unexpectedly
data
strategy
with
separate
(x2= 36.97,
capacity
LN tasks.
children
SN problems
22),
decrease
significant
and routine
in increasing
strategies,
For the
cognitive
7%; x2=
To show whether
set.
Children
of competency.
different
used
capacities
higher
levels
would either
capacity
model problems.
sequences
would be used
Algorithms at higher
All
Table
would use counting
capacities
higher
low cognitive
or directly
strategies
inappropriate
with
children
levels.
children,
groups
are shown in
As anticipated, the problems
the
children
(28% of the
Table Frequency
Direct Cognitive Group
Responses
of
Use
of
Modeling
Counting
Percent
Frequency
Strategies
Frequency
22 Group and Category
by Cognitive
Routine Mental Operation
Sequences Percent
Frequency
Percent
180
69
38
20
11
15
8
2
312
120
38
43
14
85
27
3
108
39
36
17
16
39
36
600
228
38
80
13
139
23
Table Frequency
Direct Cognitive Group
Responses
Frequency
of
Use of
Modeling Percent
Strategies
Counting Frequency
1
180
50
28
1
2
456
166
36
82
3
396
71
18
4
264
30
5,6
252
Total
1548
by Cognitive
Sequences Percent 0+
All
Non
1
Total
for
Frequ
23 Group and Category Routine Mental Operation Frequency
for
All
Nonr
Percent
Frequ
2
1
18
59
13
2
130
33
104
26
3
11
79
30
92
35
3
32
13
101
40
105
42
1
349
22
393
25
362
23
11
Performance
or used
trials)
an inappropriate
an inappropriate cognitive
strategy
Direct
children).
down consistently
goes
is
modeling counting
mental
by Groups 4 and 5-6
operations
sequences
NR tasks.
the third-grade
Data for
use of counting
and routine used
counting
p <.01).
used other
strategies R tasks.
at about The data
and R tasks,
algorithms
As expected,
a significant
at all
the R problems,
for
in Table
there
increase
in
and a
R<.01)
strategies
p <.01)
As for
25.
from 29% to 2% levels
cognitive
given
increased
strategies
only
to third
from Group
significantly
and the use of inappropriate
was no appreciable in higher
also
the NR problems,
from 44% to 5% (X2= 46.52,
by children
24.
to
only
the same frequency.
the use of counting
decreased
is
children
Unexpectedly,
from 4% to 32% (X2= 20.50, strategies
in Table
in use of inappropriate
are summarized
2 to Group 5-6,
there
which were given
from 12% to 33% (X2= 10.40,
strategies
(X2= 30.86,
tasks
in
used by Cognitive
who also
the NR problems,
Groups 2 and 5-6
decrease
corresponding
graders,
most often
children
are summarized
children,
Cognitive
All
increase
by Group 3 children;
sequences
Use of
frequently.
All
between
with
75
to 0% for Group 5-6
the strategy
Group 2 children;
Problems
(70% of the trials).
strategy
(70% for Group 1 children
capacity
on Verbal
cognitive
j <.01).
increase groups
For both NR
in the use of (NR, 21% to 25%; R,
26% to 25%).
Strategies Within (change/join;
by Task Type each problem
set,
change/separate;
unknown; compare,
and change
one item
representing
combine/part join/change/set
each of six
tasks
unknown; combine/whole unknown) was given.
From
Table of Use of Strategiem
Frequency
Direct Coglnitive Group
Responses
%
Group and Category
Nonrout ne ental Operation
Operstion
Fre unc
equnc
by Cognitive
Routine ental
Counting Sequences
Modeling
24
Inappropriate
% %
Frequency requency
AlINR
for
A %
Frequency
2
72
14
1r
9
12
8
11
3
4
21
29
3
132
29
22
27
20
24
18
5
4
21
16
4
127
25
20
28
22
23
18
9
7
15
12
5,6
126
20
16
42
33
27
21
1
1
3
2
Table of Use of Strategies
Frequency
Direct Modeling
Cognitive
Group
Responses
Frequency
%
Frequency
%
Frequency
%
2
72
5
7
3
132
28
21
4
127
29
23
17
13
17
13
5,6
126
28
22
40
32
16
13
3
4
7
10
30
23
17
13
Fre Fr
25
by Cognitive
Routine MentalMental Operation
Counting Sequences
Tasks
Group and Category
for
All
R Tasks
Nonroutine Operation Frequency
Alg
Inappropriate %
Frequency
%
2
3
32
44
O
0
30
23
1
1
32
25
1
1
6
5
Frequ
on Verbal
Performance
past
research
different
(e.g.,
would be used
strategies
structures
The strategy
for
of problems
inverse
between
relationship level
cognitive
is
used with
consistent
across
direct
modeling
(Tasks
5 and 6),
used
counting
task
for
use of direct
most frequently
Cognitive
as cognitive
for
operations
by task.
Again,
all
direct
modeling
operations
Again,
strategies
or algorithms
and algorithms often
28.
The data the
used direct
modeling
for
were used
on Task 2 (the
capacity
increases
with
or used
is
on Tasks
subtraction
as does
inappropriate varies
strategies
with
tasks.
1 and 4 (the
tasks
sequences
capacity.
evident.
subtraction
the
Again,
the use of counting
subtraction
simplest
on all
27.
the NR and R sets
same pattern
only
group on each
in Table
not used often for
students
Task 6.
The use of other is
used
In particular,
increases
modeled
directly
modeling
Each NR and R task. shown in Table
generally
of tasks.
types direct
while
strategies,
seem to be
addend tasks
missing
each cognitive
decreases
Group 1 children
strategies
for
are presented
modeling
mental
with
data
of various
few students
group.
of problems
the use of inappropriate and routine
of cognitive
The strategy
the LN set
For example,
groups.
and
the patterns
differ,
each
A consistent
26.
strategies
the percentages
tasks
group for
in Table
the compare and change/join
sequences
3 and 6,
Tasks
each cognitive
are presented
each of the
regardless
Each LN task.
for
data
Although
cognitive for
addend problems,
that semantic
differing
use of inappropriate
apparent.
strategies
with
77
Task 5).
Each SN task. the SN set
we anticipated
1982),
on tasks
on the missing
(particularly
and on the compare problem,
task
& Moser,
Carpenter
Problems
task).
Tasks
5 and 6.
of problems Students Routine
used mental
addition Students
but rarely
are
for
tasks)
Table Frequency
Direct Cognitive
Group
N
Frequency
Modeling
of Use of
Counting
Percent
Frequency
Strategies
by
Sequences Percent Task
1
15
16
53
2
26
23
44
5
10
3
9
6
33
3
17
50
45
45
11
11
Total
3
26
1
10
Task 2 1
15
17
57
1
3
2
26
27
52
3
6
3
9
10
56
1
6
50
54
54
5
5
Total
Task 3 1
15
12
40
1
3
2
26
26
50
1
2
9
9
50
2
11
50
47
47
4
4
3 Total
Task 4 1
15
17
57
3
10
2
26
25
48
8
15
3
9
10
56
2
11
50
52
52
13
13
Total
Task 5 1
15
2
7
1
3 13
2
26
10
19
7
3
9
3
17
1
6
50
15
15
9
9
Total
Task 6 1
15
5
17
11
37
2
26
9
17
19
36
9
1
6
8
44
50
15
15
38
38
3 Total
Group and Category
Cognitive
Routine Mental Operation Frequency Change/Join
for
Each SN Task
Nonroutine Mental Operation
Percent
Frequency
Percent
Inappropriate Frequency
Percent
Trials
(+)
3
10
4
13
4
13
30
17
33
3
6
4
8
52
7
39
2
11
0
0
18
27
27
9
9
8
8
100
Change/Separate
(-)
2
7
0
0
10
33
30
16
31
2
4
4
8
52
6
33
0
0
1
6
18
24
24
2
2
15
15
100
Combine/Part
Unknown (-)
2
7
0
0
15
50
30
14
27
1
2
10
19
52
5
28
2
11
0
0
18
21
21
3
3
25
25
100
Combine/Whole
Unknown (+)
3
10
1
3
6
20
30
14
27
0
0
5
10
52
6
33
0
0
0
0
18
23
23
1
1
11
11
100
2
7
0
0
25
83
30
4
8
1
2
30
58
52
8
44
0
0
6
33
18
14
14
1
1
61
61
100
0
11
37
30
6
52
(-)
Compare
Change/Join, 3
Change
Set
Unknown (-)
10
0
20
38
1
2
3
7
39
1
6
1
6
18
30
30
2
2
15
15
100
Table 27 Frequency of Use of Strategies
Direct Cognitive
Group
N
Frequency
Modeling Percent
Counting Frequency
by
Sequences Percent Task 1
1 2 3 4 5,6 Total
15 38 33 22 21 129
11 32 12 1 1 57
37 42 18 2 2 22
0 13 23 15 14 65
0 17 35 34 33 25 Task 2
1 2 3 4 5,6 Total
15 38 33 22 21 129
12 33 15 5 8 73
40 43 23 11 19 28
0 5 17 17 20 59
0 6 26 39 48 23 Task 3
1 2 3 4 5,6 Total
15 38 33 22 21 129
11 32 19 4 7 73
37 42 27 9 17 28
0 14 17 14 13 58
0 18 26 32 31 22 Task 4
1 2 3 4 5,6 Total
15 38 33 22 21 129
11 34 9 7 5 66
37 45 14 16 12 26
0 18 28 11 16 73
0 24 42 25 38 28
Task 5 1 2 3 4 5,6 Total
15 38 33 22 21 129
1 14 9 9 7 40
3 18 14 20 17 16
0 12 24 12 20 68
0 16 36 27 48 26 Task 6
1 2 3 4 5,6 Total
15 38 33 22 21 129
4 21 7 4 4 40
13 28 11 9 10 16
1 20 21 10 18 70
3 26 32 23 43 27
Cognitive
Group and Category
Routine Mental Operation Percent
Frequency Change/Join
1 8 17 11 20 57
Frequency
Percent
Inappropriate Frequency
Percent
Trials
0 6 7 7 6 26
0 8 11 16 14 10
19 10 5 3 0 37
63 13 8 7 0 14
30 76 66 44 42 258
0 5 12 5 4 26
0 6 18 11 9 10
18 21 10 5 0 54
60 28 15 11 0 21
30 76 66 44 42 258
0 5 8 8 2 23
0 6 12 18 5 9
18 17 5 7 0 47
60 22 8 16 0 18
30 76 66 44 42 258
0 13 26 39 48 25
0 0 4 7 1 12
0 0 6 16 2 5
19 14 8 2 0 43
63 18 12 4 0 17
30 76 66 44 42 258
0 5 24 32 36 19
0 6 2 5 0 13
0 8 3 11 0 5
29 40 15 4 0 88
97 53 23 9 0 34
30 76 66 44 42 258
0 6 8 14 2 6
24 20 10 4 0 58
80 26 15 9 0 22
30 76 66 44 42 258
0 20 29 41 50 28
Change/Separate
Combine/Part
Nonrputine Mental Operation
(+)
0 15 19 18 21 73
0 12 12 12 10 46
for Each LN Task
(-) 0 16 18 27 24 18
Unknown (-) 3 10 26 25 48 22
Combine/Whole Unknown (+) 0 10 17 17 20 64
Compare (-) 0 4 16 14 15 49
Change/Join, 1 10 23 20 19 73
Change Set Unknown (-) 3 13 35 45 45 28
0 5 5 6 1 17
Table Frequency
of
Use of
Strategies
28 by
No Sentence
Direct Cognitive
Group
N
Frequency
Modeling
Counting
Percent
Frequency
Sequences Percent
Routine Mental Operation Frequency
Percent Task
2 3 4 5,6 Total
12 22 21 21 76
1 7 2 4 14
4 16 5 10 9
2 6 2 3 13
8 14 5 7 9
2 15 16 16 49
1
8 34 38 38 32
Task 2
2 3 4 5,6 Total
12 22 21 21 76
4 15 13 13 45
17 34 31 31 30
0 7 6 5 18
0 16 14 12 12
2 2 6 2 12
8 4 14 5 8
Task
2 3 4 5,6 Total
12 22 21 21 76
4 13 13 14 44
17 29 31 33 29
2 10 7 17 36
8 23 17 40 24
3 4 3 5 15
3
12 9 7 12 10 Task 4
2 3 4 5,6 Total
12 22 21 21 76
1 7 5 4 17
4 16 12 9 11
1 2 4 6 13
4 4 9 14 9
4 12 8 11 35
17 27 19 26 23
Task
2 3 4 5,6 Total
12 22 21 21 76
4 8 9 6 27
17 18 21 14 18
4 16 13 25 58
17 36 31 60 38
2 3 3 3 11
5
8 7 7 7 7
Task 6
2 3 4 5,6 Total
12 22 21 21 76
5 7 12 7 31
21 16 29 17 20
3 16 13 26 58
12 36 31 62 38
2 5 4 6 17
8 11 9 14 11
Group and Category
Cognitive
for
Each NR, R Task
Correct Nonroutine Mental Operation Frequency Change/Join
0 2 2 0 1
Change/Separate
(-)
0 1 0 0 1
0 4 0 0 0
Combine/Whole
0 0 0 0 0
Frequency
Percent
Strategies
Frequency
Percent
Trials
25 4 12 0 9
12 12 16 19 59
50 27 38 45 39
1 1 0 0 2
4 2 0 0 1
0 0 0 0 0
0 0 0 0 0
24 44 42 42 152
8 6 4 2 20
33 13 10 5 13
8 13 11 16 48
33 29 26 38 31
1 0 1 2 4
4 0 2 5 3
1 0 1 2 4
4 0 2 5 3
24 44 42 42 152
10 11 13 2 36
41 25 31 5 23
4 5 4 3 16
17 11 9 7 10
0 0 0 1 1
0 0 0 2 1
0 1 1 0 2
0 2 2 0 1
24 44 42 42 152
0 0 0 0 0
10 10 6 0 26
42 23 14 0 17
8 13 19 21 61
33 29 45 50 40
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
24 44 42 42 152
0 2 12 2 5
12 11 10 4 37
50 25 24 9 24
2 3 0 2 7
8 7 0 5 5
0 0 0 0 0
0 0 0 0 0
0 2 2 1 5
0 4 5 2 3
24 44 62 42 152
29 25 21 2 18
0 3 1 1 5
0 7 2 2 3
1 0 0 0 1
4 0 0 0 1
2 0 0 0 2
8 0 0 0 1
24 44 42 42 152
Unknown (-)
Unknown (+)
(-)
0 1 5 1 7
Change/Join,
4 2 3 1 10
Frequency
All
Non-Algorithm
Percent
Sentence
6 2 5 0 13
4 0 2 0 1
1 0 1 0 2
Compare
Frequency
Percent
Incorrect
(+)
0 1 1 0 2
Combine/Part
Algorithm
Inappropriate
Percent
Sentence
Change
17 4 7 2 7
Set
Unknown (-)
7 11 9 1 28
84
on Verbal
Performance
and often
addition, problems
Cognitive
increase
the missing
with
sequences
in use of counting
Group 2 to Cognitive
a significant cognitive
counting
3 and 6) and the compare problem
(Tasks
considerable
used
Problems
Group 5-6 on these
is
from
apparent
tasks.
use of inappropriate
between
relationship
A
(Task 5).
sequences
addend
There remains and
strategies
group.
Summary and Conclusion In summary, there with
problem
set
to be important
are important
between
interactions as having
cognitive-processing
capabilities
different
subtraction be understood with
capacities tasks, that
other
tasks.
the strategies
associated
There also
appear
used by children
who
different and problem
use different
regardless
cognitive-processing
in strategies
of number) and specific
(size
have been identified
with
variations
capacity.
and task. on these
strategies
of the other
factors,
set
important
such as grade,
factors.
Children addition It
and
should
have been confounded
4
Chapter
COGNITIVE-PROCESSINGCAPACITYAND CHILDREN'S PERFORMANCE ON STANDARDADDITION AND SUBTRACTIONPROBLEMS
In this
the fourth
chapter,
purpose
was to relate
to their
performance
the
children's
The procedure
monitoring
(Romberg & Braswell,
various
aspects
of addition
design
involved
Figure
2 shows the design
combining
cross-sectional
groups
for
by the relative
represented
of students
perpendicular
design sets
both
longitudinal
axis
&
of items
on
designs.
the longitudinal
The shaded
time-of-testing
(Campbell
The quasi-experimental
growth
of the unshaded
heights
and
repeatedly
and cross-sectional
describing
in each grade.
to the
to addition
were objective-referenced
longitudinal
level
was achievement
involves
and subtraction.
Its
reported. and grade
related
study which
1973),
is
capacity
of items
The within-grade
data.
series
in a quasi-experimental
The measures
1963).
Stanley,
set in this
used
of students
groups
in this
cognitive
on a standard
subtraction.
measuring
study
plane
and is
planes
for
across
grades
the
cross-sectional
represents
growth. The data
common scales made for
across
students
grades.
for
scales
Finally,
discussed
they
each grade
Third, to the
we relate used
are summarized
to solve
in Chapter
3. 85
first
to portray
in terms
summarizations
groups
these
third-grade
the verbal
for
on the
of performance
same cognitive data
of
longitudinal
are presented
growth profiles
grades.
belonging
to the strategies problems)
study
cross-sectional
Second,
growth.
across
on the
correct
percent
in this
gathered
problems
by grade
are and
children
(NR and R
86
Performance
on Standard
Problems
LEVEL OF PERFORMANCE
TIME
cross-sectional and planes) (unshaded growth mean 2. Longitudinal Figure Figure2.
mean Longitudinal (shaded plane) growth (shaded plane) growth
for
students
grades in grades
2, and 3. 1,, 2,
on Standard
Performance
Problems
87
Method Sample
studies
in the previous
a set
administered
in 1980
period
on all
performance from those
test
A battery
of paper-and-pencil
skills
The battery
contained
In each
29.
to each student. This
report
Each the
presents
three
addition
on certain (Romberg,
objectives
included
objectives
were assessed
administered
test
forms
at all
is
grade
at each grade
as well
1983).
The items
were topics
as to measure
A summary of all
1978).
in Table however.
levels,
one of the
(Form K at grade
and
and noninstructional
provided
to be tested,
on addition
of 10 experimental
objectives & Moser,
had
& Romberg,
each grade.
objectives
in the battery
of students
for
and subtraction
Carpenter,
achievement
2, and 3 (Buchanan
prerequisite
objectives
sample
student
the instructional
to teach
performance
1,
tests
objective-referenced
to assess
at grades
to assess
at grade
was given
The
administrations.
subtraction
small
a 3- to 4-month
shown in Table
is
was scored.
items
been developed
previously
designed
over
and 28 May or 6 July).
level
items
2 and 3) were
Chapters
occasions
examined
of the Tests
Description
written
of test
a set
(see
11 April,
at each grade
from the population
1-3
series
on three
(29 February,
administration,
data
in this
of items
number of children
child's
in grades
of 44 children
A sample
three
30.
Not all of the
Because
forms was
1, Form S at grade
2, Form V
3).
Form K was a 30-minute multiple-choice
subtest
test
containing
and two separate
three 9-item
subtests: subtests
a 15-item assessing
88
Performance
on Standard
Problems
Table Children
29
at each Cognitive
Group in each Grade
Sandy Bay Infant School
Waimea Heights Primary School
Grade 1
Grade 2
Grade 3
1
3
2
0
5
2
3
6
4
13
3
1
2
8
11
4
0
0
8
8
5,6
0
0
7
7
Total
7
10
27
44
Cognitive
Group
Total
Performance
Table Objectives
Assessed Achievement
Prerequisite
Instructional
Monitoring
Numerousness 0-10 11-20 0-99, writes 0-99, represents Place Value Ordering, one-to-one sets, correspondence numbers 0-20 numbers 0-99, orders numbers 0-99, notation Instructional Objectives A Topic Series
for
the S and
Open Sentences add 0-20 subt 0-20 0-20 Sentence-Writing add-simple joining subt-simple separating subt-part part whole-addend add- part part whole subt-comparison subt-join-addend 0-99 Sentence-Writing add-simple joining subt-simple separating subt-part part whole-addend add-part part whole subt-comparison subt-join-addend Algorithms add 0-99 subt 0-99
Problems
89
30
in Addition
Objectives
on Standard
and Subtraction Battery
Non-instructional
Objectives
0-20 Problem-Solving add-simple joining subt-simple separating subt-part part whole-addend add-part part whole subt-comparison subt-join-addend 0-99 Problem-Solving add-simple joining subt-simple separating subt-part part whole-addend add-part part whole subt-comparison subt-join-addend Counting on back
9-31
Basic Facts--Speeded add 0-20 subt 0-20 Algorithms--Timed add 0-99 add 0-99
Test
Test
90
Performance
on Standard
of addition
recall
a 35-minute
test
Problems
and subtraction four
containing
to the Form K subtests
similar
form a 19-item
subtest
Form V for
third
this
the two recall
case,
Form S multiple-choice were 24-item
subtest,
test
Five
six
containing
items
measure. subtests.
In
subtest
were
from the
were dropped
14 items.
leaving
were
tests.
sentence-writing
of performance
timed measures
recall
and the sentence-writing
subtests
the Form S subtests.
for
identical
and two 12-item
grade was a 40-minute
subtests
and some added to
dropped
free-response
Form S was
limits.
of the
three
some items
subtest
was a 4-item
under time
subtests;
with
multiple-choice
The fourth
facts
The two new subtests
on addition
and subtraction
algorithms. subtests.
Multiple-choice numerousness,
ordering,
For the two objectives
were assessed.
back for numbers to 18,
counting
an additional because
problem
Four individual
objectives
for
for
and 0-99.
Since
students
problem
on the
test
(see
and then
situation.
page.
This
Chapter
3).
were represented
1, these
For grade
relative
items
by
contained
2 and 3 the number domains were 11-15
grades
a sentence,
read aloud
the verbal
numbers
in each test
interest
was no way in a multiple-choice
write
actually
represented printed
there
larger
sentence-writing
in each form.
on and
counting
numbers was of potential using
numbers 5-9 or 11-15;
counting,
they
was one item per form; however,
there
situations
item
a multiple-choice
form on which
for numbers to 31 was included
on these
information
to interview
item
counting
for
were
and algorithms
in each test
item
of
in the areas
objectives
open sentences,
value,
place
by one multiple-choice
represented
verbal
Individual
the
items
choosing
required the
The problem prevented
sentence situation
reading
format listening that itself
difficulties
to have to a correctly was not and also
with
was in keeping
were presented
problems
for
while
for
the problem
Forms S and V, four
case
of the verbal
for
sentence-writing
story
and the "puzzled
not learned
this
yet."
not read or explained
the
four
and problem-solving
The response
The children the solution,
that
an option
face,"
and pictures
symbols,
(with
the exception
two
"I have
indicated
choices,
to the children
in the
were repeated;
choices:
response
booklets.
of the tests
was read twice.
situation
then marked an X on one of the distractors,
section
the key phrases
and
objectives,
in the student
and then
problems
the entire
objectives,
the sentence-writing
in the multiple-choice
were read to the children
The
were included.
objectives
were not printed
situations
of the questions
All
the
area were
the problem-solving
number domains were the same as for again,
in which
the interviews,
orally.
For Form K, two objectives assessed
for
the procedures
91
Problems
on Standard
Performance
were
of the puzzled
face). The puzzled
to use
this
face
they
choice
allowed
subtests.
not yet
useful
covered
to 18. recall
the
The test subtests;
facts
we expected
Although
because
testing
and/or
at the baseline
period.
mastered, Marking the
children
to give
a positive
response
learned
to find
the answer
to a question.
There were 9 addition
from 4 to 9; the
administered then
unnecessary
introduced
Form K and 12 on each of Forms S and V. case
to avoid
the achievement
throughout
be objectives
had not yet
Speeded
was provided
random guessing.
was particularly
option
puzzled that
this
would always
there
option
and to reduce
frustration students
face
specific
introduced directions
and 9 subtraction
The first last
indicating
six
three
the addition on a tape
problems
(or six)
facts
in each
involved
and subtraction recording
on
preceded
10
92
on Standard
Performance
the items
and 2 seconds designated short
with
presented,
leaving
spaces,
were assessed
employed
in which
three-digit
that
problem next
allowed
Test
if
for
(the
problems
in
There was a
the numbers 0-20 format
and not
required
regrouping,
preceded
preceded
were 6 did
regrouping problems
were alphabetized)
the
three-digit problems
in which both
were instructed
The students
to solve
For example,
of difficulty.
and three-digit
who
These
subtests.
The items
regrouping
was
per test.
24 items.
18 items in order
12 individual
to the students,
items
timed
to do a particular
unable
for
types)
the situation
algorithms
the ones were regrouped
in order
Form K
answers
empty.
A free-response
and two 0-99
regrouping,
were regrouped.
problem
for
not requiring
required
and tens
problem
each contained
were arranged
problems
in which only ones
a sentence
or three-digit;
The items
their
Four of the
problem was read twice
and subtraction
two-digit
problems
a verbal
in Form V only,
subtests,
unknown facts
in Forms S and V.
to write
Addition
time for
working wrote
subtests.
(verbal
There were two 0-20
sentence.
for
free-response
and 0-99
were directed
The children
spaces
objectives
sentence-writing
of 4 seconds
the two subtests.
Sentence-writing
not.
intervals
Forms S and V.
for
break between
either
Problems
to try
each
and to go on to the Six minutes
example.
were
each subtest.
Administration The three
carried
assistants
out that
achievement
tests
task
who gathered in this
were provided
study.
data
in Study
Guidelines
to each assistant.
for
3 (Chapter administering The guidelines
3) also the
on Standard
Performance
indicated
which
were to be given,
tests
dates
for
Problems
93
and so
administration,
forth. The first
Professor
admininstrations
administrations
Romberg had returned
to the United
except
in the third
Second,
that
failed children
break,
to administer not reflect between Center for
to administer returned
All
data were then shipped
summary information
responses
Results Growth Within
Longitudinal Grade 1. individual
The percent
objectives
administrations
is
and in May many items
than May. but the
term, After
term.
the data
of this
a short The
and was advised
administration
there
would
had been a break
to Madison and scored
were recorded in this
appearing
rather
time.
the next
since
instruction,
Each subject's
staff.
The results
Form
were not given.
the end of the
gather
2, one
First,
since
problem,
at that
still
classes
in July
to start
she should
Form V in July.
terms.
all
the tests
to school
much additional
near
At grade
mixups.
tests
Form V was given
test
not read that
three
algorithms
was scheduled
asked whether
assistant
the timed
class,
The May administration assistant
to all
in April
This was not a serious
to two of the classes. are the same,
could
out
These
States.
were two administrative
than Form V was given
S rather
Romberg and
were carried
as scheduled.
so students
copy well,
3 there
At grade
question.
1 went smoothly
at grade
on Form S did not
item
by Professor
and third
The second
went smoothly. after
was supervised
administration
by
and are the basis
paper.
and Discussion
Grades correct
and composite shown in Table
for
students
objectives 31.
Overall,
at grade for
1 on the
each of the three
the data
show that,
at
Administration
Results of Objectives
Description
Time for
for
Grade 1, Form K
Results
Objectives
Number of Items
Feb. N=7
April N=7
May N=7
Number of Items
1 1
100 71
100 43
100 86
2
1 1
86 100
71 100
86 86
2
Open sentences Add 0-20 Subt 0-20
1 1
43 14
57 14
86 14
2
0-20 Sentence-writing (11-15) separating Subt-simple (5-9) Subt-comparison (11-15) Add-simple joining (11-15) Subt-part part whole-addend
1 1 1 1
14 29 29 14
0 14 14 14
0 0 57 29
4
0-20 Problem solving Add-part part whole (5-9) (11-15) Subt-comparison
1 1
100 29
100 14
100 71
on 9-31 back 9-31
2 1
29 0
43 14
57 14
Prerequisite
instructional
objectives
Numerousness 0-10 11-20 Ordering Sets, one-to-one Numbers 0-20 Instructional
Noninstructional
Counting Counting
Recall of basic Add 0-20 Subt 0-20
correspondence
objectives
for
S topics
objectives
facts--speeded
3
test 9 9
the
of the
start
school
the prerequisite
objectives
problems
(but probably
not by addition),
to an open addition write
problems,
By the end of the autumn term (May), had improved
skills
recall
they
to 76%.
However, a verbal
solving
the
comparison
in grade
addition
a correct
for
33%
facts,
Only for
subtraction.
(29% to 71%) and for
marked improvement.
for
increased
subtraction instruction
Obviously,
1 had some effect. The picture
Grade 2. (see
Table
nine
students
three
items
is
generally
numerousness
for
grade
of the school
year,
this
had a low percent
did more than half
The students sets
open sentences
(30% to 63%) and writing instruction
on several
were comfortable
to any level
had an effect,
for verbal but increases
problems
(0-99)
answer. in
composite
on basic
of mastery),
(17% to 88%), and had improved sentences
on only
with
(56% to 75%), had improved
(29% to 51% and 23% to 53%, but not yet simple
numbers
of
sample
the correct
get
large
By May, improvement
was apparent. of larger
Form S used
2 students
In fact,
correct.
of the students
was that
of the questions.
objectives
somewhat different
At the beginning
32).
of the difficulty
several
be said
problem
(29% to 56%) was there
facts
correct
29% to 57%; and addition
on,
same cannot
addition
students'
these
43% to 86%; writing
29% to 57%; counting
sentence,
Again,
the
find
nor could
back,
addition
subtraction
not solve
The percent
dramatically.
an open sentence,
solving
solve
the verbal
had
facts.
basic
Part
solve
and some (43%) could
in or count
count
of students
They could
problem.
sentences,
sample
and could
acquired
answer
this
(February),
year
95
Problems
on Standard
Performance
facts could
in counting
(28% to 59%).
in performance
were not
Table 32 Percent
Correct
Administration
Results Description
Prerequisite
of Objectives
instructional
objectives
0-20, 0-99 Sentence-writing (multiple choice) (11-15) separating Subt-simple (0-99)
(0-99) Add-simple joining Subt-part part whole-addend (11-15) 0-20, 0-99 Sentence-writing (free response) (0-99) separating Subt-simple Subt-part part whole-addend (0-99) Add-part part whole (11-15) (11-15) Subt-join-addend Algorithms Addition algorithm Subtraction algorithm
by
Results
for Objectives
for Composit
Number of Items
Feb. N=9
1
56
25 13
2
6
78 0
100 75
2
17
33
33
25
0
0
0
1 1
11 22
11 11
25 13
4
17
1
56
44
75
1
0
0
0
1 1
56 0
89 78
100 63
4
28
1 1
11 11
33 0
13 38
2
11
Number of Items
Feb. N=9
April N=9
May N=8
1 1
56
67
75
1 1
11 0
0 0
1 1
22 11
1 1
a
for S and
Open sentences Add 0-20 Subt 0-20
Subt-comparison
Objectives
Time for Grade 2, Form S
objectives
Numerousness Writes 0-99 Represents 0-99 Ordering, place value Ordering 0-99 Place value 0-99 Instructional A topics
and Composite
for Objectives
Table
Results Description
of Objectives
32 (continued)
for
Results
Objectives
Number of Items
Feb. N=9
April N=9
1 1 1 1
0 22 44 22
22 56 67 11
25 50 13 13
2 1
33 22
28 44
81 25
May N=8
Number of Items
Noninstructional
objectives Problem-solving 0-20, 0-99 Add-part part whole (0-99) Subt-comparison (11-15) Subt-part part whole-addend (0-99) Subt-join-addend
Counting Counting
on 9-31 back 9-31
Recall of basic Add 0-20 Subt 0-20
Students
(11-15)
facts--speeded
were unable
3
test 12 12
to complete
item because
tests
duplicated
poorly.
98
for
apparent
ordering
Table
except
on two items,
on all
composite
was the item on place
When all
34.
regrouping. that
because
last
items
22 children
were tested
of the
timed
conditions
on the addition
May, and only
no children
12 in July.
considerably
better.
They still
three-addend
addition
problems
but the
improvement
In summary, for grade
level,
addition across
in every this
objectives.
is
this
In addition,
did not attempt
test
the items
again
students'
of children
sample
clear.
was the
did
in April
performance with
or
was
the problems,
regrouping
striking.
each grade is
without
of the difficulty
had some difficulty
case
and
regrouping.
and the subtraction
small
growth within
and subtraction
those
By then,
shown
but had considerable
requiring
were given
on
they
subtraction Part
items
items
the subtraction
is
problems
who did reach
regrouping
test
algorithms
most students
Those children
test.
high
addend).
in February,
did poorly.
they
Unfortunately,
timed
two-digit
testing
others,
with
difficulty
were not yet
addition-without-regrouping
On all
in the
well
fairly
on the
items
performance
and part-part-whole
performance
on the three
acceptably
80%)
The exception
80%.
but scores
(comparison
on the six
well
performed
July)
(above
Sentence
for numbers 0-99.
situations
Grade 3 students'
one approached
had improved,
3 students
was not high
performance
except
value
skills
some subtraction
in Table
their
grade
but by the end of May (or early
objectives
writing-selecting
for
was more encouraging
In February,
33).
written
selecting
solving,
problem
and algorithms.
problems,
The picture
Grade 3. (see
numbers,
large
for verbal
sentences
Problems
on Standard
Performance
assessed
on some aspects
Growth, overall
however,
level
at each
associated is
with
not uniform
of performance
on many
Table Percent
Correct
for
Administration
Objectives Time for
33 and Composite
Objectives
by
Grade 3, Forms S, V
Res of Objectives
Description
Num of It
Number of Items
Feb. N=22
Numerousness Writes 0-99 0-99 Represents
1 1
45 91
32 91
64/92 100/91
2
Ordering, place value Ordering 0-99 Place value 0-99
1 1
36 23
91 50
64/75 0/42
2
0-20, 0-99 Sentence-writing choice) (multiple (11-15) separating Subt-simple (0-99) Subt-comparison (0-99) Add-simple joining (11-15) part whole-addend Subt-part
1 1 1 1
60 18 77 10
91 14 91 50
73/100 18/58 64/100 9/75
4
0-20, 0-99 Sentence-writing (free response) (0-99) separating Subt-simple (0-99) Subt-part part whole-addend Add-part part whole (11-15) (11-15) Sabt-join-addend
1 1 1 1
36 5 68 45
77 23 95 60
Prerequisite
Instructional A topics
instructional
objectives
April N=22
May/Julya N=11/12
objectives
for
S and
82/92 18/67 100/92 55/75
4
Table
33 (continued)
for
Results Description
Noninstructional
of Objectives
Feb. N=22
1 1 1 1
55 91 77 45
April N=22
May/Julya N=11/12
Num of I
objectives
0-20, 0-99 Problem-solving Add-part part whole (0-99) (11-15) Subt-comparison Subt-part part whole-addend (0-99) Subt-join-addend Recall of basic Add 0-20 Subt 0-20
Number of Items
Re
Objectives
facts--speeded
(11-15)
68 77 95 73
4
test
12 12
test Algorithms--timed Addition algorithm Subtraction algorithm
24 24
aForm S was used in April bForm S did not assess
64/92 100/100 91/83 64/75
this
and May; Form V was used objective.
in February
and July.
Performance
Table Percent
Correct
Timed Tests
for Addition
on Standard
34 and Subtraction
Algorithms
by Problem Type for Grade 3, Form V
Item Type
Number of Items
Percent
Correct
Feb. N=22
July N=12
Addition 2-digit
(without
regrouping)
3
86
100
3-digit
(without
regrouping)
3
93
94
2-digit
(with
regrouping)a
6
49
89
3-digit
(with
regrouping)
9
16
78
3-digit
addends
3
0
44
Subtraction 2-digit
(without
regrouping)
3
68
94
3-digit
(without
regrouping)
3
33
89
2-digit
(with
regrouping)a
6
8
75
3-digit
(with
regrouping)
12
0
47
aThree
items
Problems
are 2-digit
+ 1-digit.
101
102
is
objectives
not high.
To portray were assessed
whole
three
in all
addend
missing recall
(11-15);
and 2, and the composite
subtraction-part-part-
scale
subtraction-comparison of basic
and recall
were also
scales
value
facts-
to grades
administered
place
ordering,
objectives
1
was administered
at
2 and 3.
grades
data
The cross-sectional 35.
to master
subtraction-
writing:
solving
problem
five
2),
Figure
sentence
facts-addition;
Two composite
subtraction.
(see
growth
sentence-writing:
of basic
had yet
Grades
grades:
(11-15);
students
grade,
or subtraction.
cross-sectional
separating
(11-15);
addition
Growth Across
Cross-sectional
simple
By mid-third
of either
many aspects
Problems
on Standard
Performance
data,
of Children
Performance
in different
scores
into
a total
Small
sample
size
must be regarded
sample
is
across
score
that
dictates
as tentative
all
is
evident.
we summarize
conclusions
data
for
each group's of the
administrations
three
as with
Grades
by aggregating
and should
But,
or smooth.
Groups Within
of students
in Table
are presented
not uniform
groups
cognitive
also
growth
in Cognitive
of the small
Because children
the growth
scales
these
considerable
On each objective,
the longitudinal
for
tests. data
drawn from these of further
be the subject
study. Grade 1. in grade
The relative
1 in different
were three in Cognitive
children
cognitive
groups
in both Cognitive
Group 3.
on the
performance
The differences
is
test
items
shown in Table
for 36.
Groups 1 and 2, but only in performance
for
children There one child
the eight
Table Percent Objectives
Correct
for
for
35
Common Objectives
Cross-sectional
Growth Across
and Composite Grades
Percent Description
Feb. Grade 1 N=7
of Objective
1,
Correct
April Grade 2 N=9
Sentence writing (11-15) separating Subt-simple (11-15) Subt-part part whole-addend
14 14
33 11
Problem solving Subt-comparison
29
56
33 29
35 30
Recall of basic Add 0-20 Subt 0-20
facts--speeded
Feb. Grade 2 N=9
aData gathered
6
value on these
G
29 19
Open sentences Counting on and back
place
Ma G
test
Feb. Grade 1 N=7
Ordering,
2, and 3
dates
have been
combined.
Ma G
Table 36 Frequency
and Percent
Number of Items instructional
Prerequisite Numerousness
Instructional S topics
Open sentences
Noninstructional
Frequency
Group 1 (N=3) Percent
Trials
2
14
78
18
2
16
89
18
for the
objectives
0-20
Sentence-writing
Cognitive
by
objectives
0-20
0-20
2
7
39
18
4
4
11
36
objectives
Problem solving
0-20
Counting Addition
for Composite Objectives
of Objectives
Description
Ordering
Correct
facts
Subtraction test
test
recall--speeded
facts
recall--speeded
2
12
67
18
3
2
7
27
9
24
30
81
9
24
30
81
Table 37 Frequency
and Percent
Correct
of Objectives
Description
Cognitive Number of Items
instructional
Prerequisite Numerousness Ordering,
value
Objectives
0-99
Group 1 (N-2)
Frequency
Percent
1a
4
67
6
2
3
25
12
Trials
for the S
objectives
Open sentences
2
4
33
12
Sentence-writing 0-20, choice) (multiple
0-99
4
1
4
24
0-20,
0-99
4
9
38
24
2
1
8
12
4
11
46
24
3
6
33
18
Sentence-writing (free response) Algorithms Noninstructional
objectives
Problem solving
0-20,
0-99
Counting Addition Subtraction test
by
objectives
0-99
place
Instructional and A topics
for Composite
facts
test recall--speeded facts recall--speeded
12
17
24
72
12
16
22
72
for the numerousness aTwo items were administered so data for this item were discarded.
objective;
students
had
Cognitive
Times for Grade 1, Form K
Group for All Administration
Group 2 (N-3)
Cognitive Frequency
Percent
Trials
Cognitive Frequency
Total
Group 3 (N=1) Percent
Trials
Frequency
Percent
Trials
17
94
18
4
67
6
35
83
42
15
83
18
6
100
6
37
88
42
7
39
18
2
33
6
16
38
42
12
15
18
84
9
25
36
2
17
13
72
18
4
67
6
29
69
42
16
59
27
2
22
9
20
32
63
65
80
81
11
41
27
100
53
189
49
60
81
8
30
27
81
43
189
Cognitive
Times for Grade 2, Form S
Group for All Administration
Cognitive
Group 2 (N-5)
Frequency
Percent
Trials
Cognitive
Total
Group 3 (N-2)
Frequency
Percent
Trials
Frequency
Percent
Trials
8
58
14
5
84
6
17
67
26
0
0
28
1
8
12
4
8
52
13
46
28
7
58
12
24
46
52
11
20
56
4
17
24
16
15
104
25
45
56
14
58
24
48
46
104
4
14
28
4
33
12
9
17
52
104
11
20
56
8
33
24
30
29
14
33
42
12
67
18
32
41
78
58
35
168
43
60
72
118
38
312
50
30
168
43
60
72
109
35
312
difficulty
reading
one of the items
due to poor quality
of the test
duplication
106
on Standard
Performance
six
with
composites,
the Group 2 children
some of the differences
the Group 2 students
addition, May over
all
The single
performance
shown in Table
is
cognitive
groups
Cognitive
Groups 1 and 3 and five
children
than Group 1 children.
better example,
open sentences
However,
there
Group 1 children to 33%).
and counting verbal
skills
arithmetic
is
further
children skills
than either
(but not
on those
four
close
as the year
Group 3
are striking; facts
they
items,
the
(46% to 20% algorithms, to the
found answers with
better
may have attempted This
to use
explanation
of Group 1
in performance
progressed
for
(60%-35%-24%).
groups
but made errors.
by the decrease
items
each in
who in turn performed
of the other
to mastery)
in
in different
problem-solving
that
to
any pattern.
In general,
The children
strategies.
only
There were two children
suggest
the problems,
substantiated
improved
were low on facts,
children
these
other
to solve
skills
For the
In
from February
2 children
of grade
in Group 2.
the results
using
problems
those
since
skills,
37.
in Group 1 on large.
to fit
fails
(58%-46%-33%) and addition
did better
However,
quite
Some of the differences
one anomaly.
is
those
in performance
than Group 2 children,
better
performed
being
Group 3 child
The relative
Grade 2.
increased
over
but the Group 1 students
the objectives,
of facts.
recall
favor
objectives
composite
Problems
and as their
arithmetic
in different
cognitive
improved. Grade 3.
groups
Results
are striking
5-6 children
differing
the grade
3 students
but somewhat ambiguous
performed
and Group 2 children However,
for
better
on all
were lower
than other
of these
Table
groups
consistently.
two groups
The Group
38).
than any other
objectives
Groups 3 and 4 did not differ characteristics
(see
on all
group,
the objectives. the
Obviously,
are not related
to
Table 38 Frequency and Percent Correct for Composite Objectives for All Administration
of Objectives
Description
instructlonal rrerequlslie Numerousness 0-99
Number of Items
Instructional and A topics
objectives
Group 2 (N-4) Percent
Cognitive
Trials
Frequency
Group
Group 3 (N-8) Percent
Trials
Group 4 (N-8)
Cognitive Frequency
Percent
Trials
2
15
63
24
29
73
40
24
67
36
2
8
33
24
20
50
40
14
39
36
0-99
4
21
44
48
42
53
80
39
54
72
0-20,
0-99
4
25
52
48
49
61
80
37
51
72
tfree
response)
Freq
for the S
0-20, Sentence-writing (multiple choice) Sentence-writing
Cog
objectives
place value 0-99
Ordering,
Cognitive Frequency
by Cognitive
Times for Grade 3
Noninstructional
objectives Problem solving 0-20, 0-99 testa Addition algorithms--timed Subtraction Addition
algorithms--timed facts recall--speeded
Subtraction test
facts
testa
recall--speeded
aThis objective was assessed assessed in April.
test
4
34
71
48
58
73
80
55
76
72
24
30
31
96
134
51
264
111
51
216
1
12
12
13
96
78
30
264
55
25
216
1
12
65
45
144
152
63
240
133
62
216
1
12
61
42
144
135
56
240
126
58
216
1
all
cognitive
in all
groups excep
in February for 22 students
representing
groups and in May for 12 students
108
Performance
differences
on Standard
Problems
in performance.
Most of the differences
5-6 and the Group 2 children
are large
68% to 33%, subtraction
ordering
Performance
of Children
Performance
data
from Tables
capacity
(Children
in Groups 4 and 5-6 are only
2 is
The performance
shown in Table
at both
is
grades
small
numbers is
little
difference
difference is very
their
in performance
grades.
Children
capacity
gains
For example,
in this
of these Only for
items,
1 and
children ordering there
There is
is
a marked
although
for
by grade
these
group are at all children
are apparent,
performance
1 to grade
(0-99)
solving
decrease
was there
in performance facts.
the increased
number of facts
and decreased
not due to having
suggests committed
that
the facts
in Table
1 to grade
time
from
sentences
there
Also,
2 on recall
is undoubtedly for
response
performance
to memory.
very
Only for
3.
(20% to 71%).
The decrease
the high
grade
goes
on writing
2 to grade
from grade
and subtraction
clearly
open sentences
gain
both addition
This
are presented
2 and performance
a large
three
but in most cases
on solving
from 45% to 52% from grade
response)
a large
data
performance
39% to 46% from grade
forms.
group at grades
More striking,
adequate.
on the counting
Performance
modest.
is
in this
capacity.
between
section.
3.)
the performance their
performance
The comparative
problem
with
in
low.
levels.
(free
of children
In general,
consistent
in grade
(7% to 33%) only
Group 2.
40.
39.
children
2, and 3 are compared in this
Groups 1,
cognitive
Group 1.
Grades
and 37 for
36,
65% to 44%,
51% to 13%, and so forth).
Groups Across 35,
the Group
sentences
(selecting
algorithms
in Cognitive
between
Also,
due to
over
at grade it
of
should
1 was be
39
Table and Percent
Frequency for
Cognitive
for
Correct
Group 1 for All
Objectives
Composite
Times Across
Administration
Grades
Grad
Grade 1 (N=3) Description
of Objectives Frequency
Prerequisite
instructional
Numerousness 0-20 Numerousness 0-99 Ordering 0-20 Ordering, place value Instructional and A topics
Trials
Frequency
objectives
0-99
objectives
Open sentences 0-20 Sentence-writing 0-20, Sentence-writing choice) (multiple 0-20, Sentence-writing (free response) Algorithms Noninstructional
Percent
for
14 16 -
78 -89 --
18 18 --
7 4 -
39 11 --
18 36 --
4
--
9
-
1
4 3
the S
0-99 0-99
-
-
12 2 24 24
67 7 30 30
1
objectives
0-20 Problem solving Problem solving 0-20, 0-99 Counting facts recall--speeded Addition facts Subtraction recall--speeded
test test
18 27 81 81
11 6 17 16
P
Table Frequency for
Cognitive
i
rti?
anr
of
instructional
0-20 Numerousness 0-99 Numerousness Ordering 0-20 place value Ordering, Instructional and A topics
Across
Grade
Grades
2 (N=6)
Percent
Trials
Frequency
Percent
Trials
objectives 17 --
15 -
0-99
objectives
Open sentences 0-20 Sentence-writing 0-20, Sentence-writing choice) (multiple 0-20, Sentence-writing (free response) Algorithms Noninstructional
Times
Administration
bi ectivefi Frec luency
Prerequisite
All
Objectives
Composite
1 (N=3)
Grade n
for
Correct
and Percent Group 2 for
40
for
the
94
18
--
--
83 --
18
--
-
8
58 --
14 --
0
0
28
13 11
46 -20
28 _56
25
45
56
4
14
28
11 14 58 50 ---
20 33 35 30 ---
56 42 168 168 ---
--
--
S 7 9
18 36 --
39 25
0-99 0-99
objectives
0-20 Problem solving 0-99 Problem solving 0-20, Counting facts Addition recall--speeded facts Subtraction recall--speeded Addition algorithms Subtraction algorithms
13
18
72 --
test test
16 65 49
59 80 60
--
27 81 81 ---
---
noted
3, these
at grade
that,
and subtraction
addition overall
performance
with
algorithms
of these
to use
had not learned
children
any facility.
reflects
students
Problems
on Standard
Performance
111
the the
Again,
more than grade
capacity
level. in this
children
2 and 3 are compared in Table
grades
at this
in performance
differences
sentence-writing
for
However,
solving.
Overall
grades.
other
all
the addition
of Performance
Relationship
and from 33% to 73% on problem
fair
is
performance
and students
but not with
algorithm,
performance
from 17% to 53% on
value;
place
scales,
is
performance
group at
capacity
For example,
grade.
choice);
(multiple
cognitive
There are some important
41.
from 8% to 50% on ordering,
increases
with
for
The data
Group 3.
on Algorithms
similar
across
show some facility
the subtraction
algorithm.
Used to Solve
to Strategies
Problems One general have students
problems
algorithm
in the
to the
to use
when solving
those
such verbal
62 items
is
We now examine
third-grade
children
they
used
whether
and subtraction
The strategy in Chapter
timed
problems
data were 3.
We were
or not students algorithms
the
on the
verbal
to solve
to
in Chapter
presented
algorithm.
discussed
in examining
and subtraction
as those
algorithms.
study
the addition
For addition
(such
of the
strategies
interview
interested
particularly
attempted
problems
on addition
or subtraction
be done using
could
collected
verbal
of the performance
relationship
learned
solve
of instruction
an addition
3) by using
that
goal
chose
who had to use
them
problems.
problems and got
requiring 57 correct
no regrouping, (92%); in July,
at Time 1, students students
answered
41
Table and Percent
Frequency for Cognitive
Correct
Group 3 for All
for
Composite
Administration
Objectives
Times Across
Grades
Grade 2 (N=2) Frequency Prerequisite
instructional
Numerousness Ordering, place Instructional and S topics
Percent
Trials
Frequency
objectives 5 7
84 8
6 12
29 20
7 4
58 17
12 24
-
0-99 0-99
14
58
24
49
4
33
12
8 12 43 43 --
33 67 60 60 ---
24 18 72 72 --
in this
comparison.
value
0-99
objectives
for
Open sentences 0-20, Sentence-writing choice) (multiple 0-20, Sentence-writing (free response) Algorithms Noninstructional
Gra
of Objectives
Description
the A
42
objectives
Problem solving 0-20 Counting Addition facts recall--speeded Subtraction facts recall--speeded Addition algorithms Subtraction algorithms NOTE: The one group 3 child
test test
at grade
1 was not included
58 152 135 134 78
on Standard
Performance
all
36 items
they
these
students
517),
of use
increased
The data students
for
74 items
and got
66 correct
could
the students
who made six
student
The interview performance, addition
data
at the start (with
in spite
At Time 1, about
an algorithm
(46% correctly).
using
an algorithm
(48% correctly),
students
although
of the year,
by July,
the exception
of one
of this
high
did not use the algorithms
using
with
Thus,
(89%).
By Time 2, they
(69%).
problems).
show that,
many students
problems.
at Time 1 were similar;
regrouping
in six
errors
used
At Times 2 and 3, the
66 correct
regrouping
add with
at Time 1, students
regrouping
and got
with
(student
to 79% and 72%, respectively.
95 items
had some difficulty all
with
113
numbers without
(54% correctly).
but only
addition
attempted
attempted
interviews
on the
59% of the time
only
algorithms percent
knew how to add two-digit
However,
regrouping.
With one exception
correctly.
attempted
Problems
half
of
level
to solve
verbal
(54%) of the children
tried
had changed
At Time 2, this and by Time 3,
to 60%
78% used an algorithm
no errors. For subtraction
performance verbal
correct
on three
subtraction and got
55 items
without
achievement problems
these
can conclude
that
regrouping.
However,
of the strategies of the year.
only
(82%).
used
between
on the four
At Time 1, students
attempted
By Time 2, 34 of 36 attempts made any errors
were able
four verbal
used were algorithmic
At Time 2, this
of a comparison
and strategies
one student
students on the
items
were similar.
45 correct In fact,
(94%).
results
regrouping,
to subtract
subtraction (only
had increased
were
in July.
One
without problems
9% correct)
only
at the
to 25% and finally
14%
start
to 34% by
114
on Standard
Performance
Time 3.
over half
Furthermore,
2 (simple
items
no-regrouping their
in this It is
capability.
must assume that problems
had they
students
attempted
only
there
66 items
most were capable
On the first
problems. items
As with
correct). on the
simple
On this
last
they
most errors
set
got no items (only
Group 5-6 students
attempted
used
in
difficulty
this it
interview,
for
procedures
algorithms
interview,
verbal
on only
13%
had increased
was 35% (26% were
most of the attempts
problems,
the cognitive
to use algorithms
correct
10% correct)
Thus,
(44%).
of verbal
attempts
time,
them to solve
to use
no-regrouping,
tasks
separating
made the most total though
subtraction
(July),
problems.
the algorithmic
second
and by the third
to 23% (11% correct),
One
algorithm.
students
On the
(5% correct).
of
by the end of the autumn term
of knowing
interview,
measure
At this
(82%).
had considerable
did not attempt
most children
subtraction,
six
administration
on the six
students
a subtraction
in spite
however,
Again,
of the
of using
the first
only
12
to do the regrouping
55 correct
in February,
regrouping
and got only
why they were so slow.
made more than one error that
subtraction
was no real
By the second
and got
for
38 items
so there
test,
them.
was evidence
with
subtracting
occurred
attempted
hard to imagine
attempted
problem.
managed to complete timed
(59%) were on Task
attempts
they would have been unable
two students
although
but more pronounced,
Many children
(32%).
total
subtraction
At Time 1, students
regrouping.
correct
of the
the most obvious
separate),
The same pattern, with
Problems
(35% of the time),
on the achievement on verbal
problems.
to use algorithms
Group 2 students
only
test
even
and made the
In contrast,
the
22% of the time.
between
The relationship subtraction is
Most of the
interesting.
these
to solve
strategies
until
in their
they were not proficient
even though most third
graders
algorithms
but chose
structures
(verbal
to use other semantics)
In fact,
the problems.
that
the realization
that
recognized
the
to use
these
could
other algorithms
suggests
that
be solved
using
The problem
strategies. how the
influenced
clearly
This
problems
familiar
students
worked
seemed to be more important
semantics could
the problems
in using
the taught
use.
problems
strategies
who had not acquired
tended
problems
other
used
115
and
verbal
became confident
they
Group 2 children
However,
algorithms.
students
third-grade
and so forth)
fingers,
(counting,
to solve
the algorithms
and using
algorithms
addition
in performing
proficiency
Problems
on Standard
Performance
than
be done algorithmically.
Summary and Conclusions The overall
the complex
to learn
solved basic
three-digit
with
place
solved
with
should interview
one important
groups
consistent
verbal
within
differences
associated to solve
value
with
verbal
arithmetic
between
addition
they
improved competence,
and The
correctly even though students
problems.
exception,
each grade.
struggling
problems.
even though
as being
who were identified
Children
in other
With little
some simple
of children
Work on algorithms
problems.
were weak.
facts
correctly
group,
skills
is
presents
skills
those
had difficulty
children
data
arithmetic
and to use
subtraction
these
picture
in a particular
performed The
was the lack
Groups 3 and 4 at grade
be noted
that
Group 3 at grade
tasks
(see
Chapter
3 also
3) and only
than children
differently
exception
3.
did not differ
differed
cognitive
of
Again,
it
on the
on transitivity
on the
116
tasks
cognitive children
Problems
on Standard
Performance
(see
Overall,
2).
Chapter
in cognitive-processing
who differed
2, Groups 3 and 4, and Group 5-6) specific
it
Although for
cannot
some differences
and subtraction
tasks,
be denied
children's
it
striking
performance
appears
to be consistent
performance
between
groups
differences
one would expect,
quantitative
skills,
differently
and within based
memory capacity).
teaching
with
that
apparent
(Group 1, Group of
regardless
or experience
performance that
is
or grade.
time,
that
in the is
capacity
performed over
instruction
objectives,
it
however,
on standard
the actual capacity.
groups
across
on the nature
accounts
level
of
Differences grades
addition
in
are
of the groups
(e.g.,
5
Chapter
COGNITIVE-PROCESSINGCAPACITY AND CLASSROOM INSTRUCTION
and last
The fifth Its
chapter. differ
is
in this
reported
the question, different
receive
capacity
series
in this
was to examine
purpose
in cognitive
study
Do children
who
instruction?
Method Sample A sample in this
series
three-month
Table
(see
in 1980
influence
3 and 4 over
a
The number of
May 28).
and grade
class,
group,
in studies
instruction
during
27 through
(February
in each cognitive
was to determine
certain
actions
children
teacher
affect
in learning
is
shown in
segmentation
and sequencing"
to influence
teacher
organization,
the allocation
within
classroom,
Small,
& Carnahan, this
in actual
planning,
model,
and, 1979,
instruction
and "content which
of instructional
eventually, for
pupil
a complete
data were gathered
classroom
settings
for
117
was examined. in which
time,
engaged explication
periods
to which
To do this
"content were hypothesized
influences
on various
several
the extent
structuring"
in turn
of content
and in turn how
In particular,
mathematics
a model of classroom
aspects
instruction
during
outcomes.
pupil
are engaged
the way in which
behaviors
we developed
test
used
42. Our attempt
these
2) was observed
chapter
period
observed
children
from the population
of 35 children
classroom
verbal
time
(see
interactions Romberg,
of the model).
components of time
To
of the model (see
Romberg,
118
Classroom
Instruction
Table Children
42
in Each Cognitive
Group, Class
Grade Used in the Observation
Sandy Bay Infant School
Study
Waimea Heights
Class
Cognitive Group
and
Primary
School
Class
1
2
Grade 1
Grade 2
3
1
2
2
2
3
4
3
1
2
Grade 3
4
5
Grade
3
Grade 3
3 2
2
2
4
2
2
2
5-6
3
1
2
7
8
6
Totals
6
8
Instruction
Classroom
& Cookson,
Carnahan,
Small,
the relationship
data
a description
of coding
of coding
categories).
of the model to classroom
instruction
as detailed
used as well
for
1979,
explanations
119
procedures With such can be
examined.
and Aggregation
Summary of the Coding Procedure Data were collected involved
pupil
behaviors
using
two procedures
Content.
First,
to estimate
the teachers
instruction
in nine
"other
areas
dealt
arithmetic"
division
activities,
activities
content with area
where one observed
observe
instruction
observation
period.
mathematics
each target
Seven of
child.
The
on both multiplication
spent
encompassed
all
and
other
or geometry.
fractions, trained
observers data
School
gathered
in Studies
and observed
Each observer
approximately
both
who did not distract
sat
either
Primary to
the
in a class teacher
One
the grade
was able
24 days during
the observers
the data.
3 and 4.
two worked at Waimea Heights
At the schools,
of
to add and subtract.
mathematics"
two classes.
in a class
time became fixtures
for
who gathered
The other
School,
children.
time
Three
observation.
2 classes.
in Romberg, Collis,
the number of minutes
of learning
aspects
and "other
to log
spent
worked at Sandy Bay Infant
and grade
over
areas
included
These were the same persons observer
and
of mathematics
on various
time spent
were asked
such as measurement,
Classroom
appear
teacher
1982).
objectives,
the nine
details
and on certain
and learning
in the teaching
(complete
& Romberg,
Buchanan,
covered
on content
of Data
or
and
1
120
Classroom
Instruction
Data on pupil
and teacher
observation
coding
method used
to gather
the project
staff
Cookson,
student
at different
sequence
in which
(Romberg,
were taking relevant observation
collected
and the
in the manual produced Small,
six
A time-sampling
by
&
Carnahan,
(a) (c)
behavior
time a target students
a coding
cycle.
to observe
the
to observe
student
The behavior and pupil
one-minute
time
interval.
moments is minimized.
sampling
a description
the target
student
was obtained for
that
The coding
and the
gave a running
a particular
In this
teacher.
account
observation
way,
(b) to
of the
the
of the
bias
were used one instant
a series
of what took place period.
to
one
that
activities
observer
at that
six
teacher
at the beginning
categories
of what was occurring
The
behavior,
of those
process,
of
on the observation
categories
in precisely
Through this
the
aspects
organizational
only
observations
was observed.
student's
target
to the
instances
It was estimated
to be coded consisted were involved
for
with
(along
The
period.
while
was coded
in
in a particular
prior
and was invariant
period
and (d) to code the appropriate
classroom,
was fixed
in
was used
procedure
the observation
were observed
to eight
were observed
was observed
students
target
each
represented
the teacher,
behaviors
moments throughout
behavior
was needed:
classroom
verbal
The teacher's
of all
times)
teacher
of days.
the students
place.
verbal
observe
and teacher
of the observation
beginning
record
observers
an
using
of the data
fully
to eight
sequence
form.
nature
are described
to train
each of six
minute
it
on a sample
each class
eight
The exact
form.
were gathered
1979).
In brief,
which
behavior
in to for both
of codings in the
Instruction
Classroom
for
The observation
session
did not
The beginning
ended. coincide
always
with
instruction
as scheduled.
mathematics
class
Actual
mathematics
occurred.
Data aggregation
each behavior aggregated
of interpretation, on total
(based
for
separately
give
an overall
yield
estimates
directions; Teacher group,
engagement
of engaged
time;
grouping; behaviors
speaking
of explanations. were summarized
for
were summarized on content
in terms
whether and if
of teacher
of whether
occurrence
is used. period.
of
Data were The data
in each class
engagement
and
rates.
of three
and teacher actions it
were summarized
was on content
or
with whom.
interacting,
in terms of interactions,
or directions,
Interactions
category
in terms
Pupil
engaged,
interactions;
each behavior
affect
behaviors,
interactions. if
data were
of mathematics
factors
teacher
time.
observational
the total
data were summarized actions,
pupil
the amount of time
instances)
teaching
of how instructional
behavior-pupil in terms
of the
picture
The observational categories:
each class
the
the amount of time
the proportional
observed
in
was to take
of available
for
counts
period
involved
represented
represented
The basic
form of frequency
of time
instruction
the measure
and analysis.
For purposes
coded.
time
In most cases,
with
closely
in the
hand,
that
of mathematics
two measures
Available
on the other
instruction
aggregated
and ending
in which mathematics
time,
coincided
observing
As a result,
for
instruction
of the observation
and ending
the beginning
were obtained.
time period
scheduled place.
began when the mathematics
session
began and ended when the mathematics
instruction class
a class
121
speaking
to
questions,
feedback
behaviors
and pupil
engagement
were engaged
when the
or not pupils
and type
122
Instruction
Classroom
teacher
was speaking,
interactions,
the fact
for
there
the analysis
were three
minutes.
Thus,
in this
analysis with
pupils
examined
in five
for
we present
the data
various
content
in terms
data
different
the data
by class.
level
of cognitive
we
Third, group.
within level
all
classrooms
to cognitive
respect
for
grade.
within
class.
and Discussion
The emphasis
In grade
emphasized is
emphasis percentages solutions
is
2, basic
facts
are the little
not
of time spent
time
teachers
the curricular
of the
time
varies
on addition
is
for both addition
on addition In
grades.
numerousness
facts,
3, most of the
algorithms.
The only
disappointing
problems.
on writing
However,
child-related. on counting
this
at grade
sentences
differential
For example,
and
and subtraction
In grade
time spent
on
emphasis
spent
across
spent
skills.
as are counting
to verbal
reflect
obviously
percentage
on computational
program-related,
data
of total
Almost half
grades.
1, the highest
counting.
These
areas.
and subtraction.
percent
with
the percentage
43 presents
common in these
still
are aggregated
Covered
Table
grade
are observed
of time
three
the data by cognitive
Results Content
since
on
grade,
we have aggregated
Second,
students
in the study: The raw data
3, we have examined all
data was based
and percent
First,
to grade.
in grade
the data
Finally,
dimensions
primary
ways.
we have examined
Fourth,
of the observational
the number of minutes
respect
were observed
information.
group of the pupils.
and cognitive
class,
no teacher
listening,
and providing
questioning,
The plan
to groups,
speaking
are
and finding emphasis
the reduction
3 was not matched
in
by the
is
Table Percentage
of Time Spent
43
on Mathematical
by Grade--Teacher
Area
Numerousness Ordering Basic
facts
(Add) (Subtract) Problem solving Sentence
(24 days,
50-60
14.3
Grade 2 min/day)
(25 days,
50-55
6.4
5.2
5.6
15.5
13.3
(14.7)
(6.8)
(.8)
(6.5)
2.6
1.4
.8
.8
writing
Algorithms
0
3.1
(Add)
(0)
(3.1)
(Subtract)
(0) 9.3
Counting TOTAL TOTAL and subtraction
(0) 12.4
47.7
50.2
Other arithmetic
13.2
16.8
Other math
39.1
33.0
Addition
Area
Log Data
Grade 1 Content
Content
min/day)
124
Classroom
children's
Instruction
to use
failure
can be argued
that
presumes
prerequisites
(like
about
at odds with
have mastered
and basic
numbers
(are
it
3 (emphasis
on
most of the
and have acquired
facts)
in Cognitive
on these
the data
In fact,
problems.
of the program at grade
children
counting
of reasoning
of course,
that
to solve
technique
the structure
algorithms)
level
this
a high This
Group 5-6).
students
is,
in the last
presented
two chapters. this
Also,
the mathematics
curricula
important
structural
Bay Infant
School
etc.,
stations, grades
in terms
was filled
with
text
paper-and-pencil
seatwork.
text
to capture
materials,
or in small
was used.
a single
fails
in
included
The program in Sandy
programs.
manipulative
independently
and no basal
of the content but it
schools,
of those
features
at Waimea Heights,
involved
many
groups,
However,
was followed
learning
in the third and most activities
Actions
Pupil
The data
Grade. Table
44.
in the time
structure
spent
on pupil
Significant across
apparent
grades.
station
the
spent
time
the text-based is
interesting
rate
approach
to note
due in part
in the schools.
consistent
activities with
instruction used
that
are
to the differences The high
amount of
in grades
1 and 2
the manipulative-based,
at grade
Similarly, 3 is
at the Waimea Heights the largest
in
differences
at the Sandy Bay School.
in large-group instruction
and grouping
and individual is
are presented
by grade
Both are likely
of the curriculum
on small-group
learning
actions
engagement
(85% and 68%, respectively)
it
fair
in these
to explore
opportunities
is
description
difference
70% of
consistent
School.
with
However,
in engagement
is
Classroom
Table Observed
of Pupil
Actions
Grade 1
125
44
and Percent
Minutes
Instruction
of Time
by Grade
Grade 2
Grade 3
Action
Pupil
Minutes
%
Minutes
%
Minutes
%
Engagement 559
55
771
71
1369
77
449
45
317
29
403
23
488
89
656
86
1149
88
62
11
107
14
164
12
Individual
302
30
165
15
11
1
Small
group
553
55
583
53
524
29
group
156
15
343
31
1259
70
105
6
time
Engaged
time
Off-task Types of
engagement
Content Directions Grouping
Large
Interactions Target
speaking
62
6
51
5
Target
listening
91
9
163
15
279
15
80
1427
79
858
85
880
Teacher
99
65
161
76
296
78
Pupil
48
31
36
17
77
20
6
4
16
8
6
2
None Interaction
Other adult
other
party
Classroom
126
between
Instruction
curriculum
the same school).
(in
1, many children
when they
behavior
was observed
the next
task
with
due to increased
less
as shown in Table
grade
1 (class
in grade
two classes.
Pupils
in receiving
directions, with
since
the five
about
Many students
now proceeded
of this
is
change
with
2, this to
probably
behavior
due to the particular
probably
group. pupil
steadily
increases
grouping
are striking,
across
more of the time.
they were more likely
to be engaged
in grouping classes
to be
are not likely on
are similar and interactions
in engagement
teacher.
46.
percent
for
of time
all
Overall, groups. of time
from 21% for Group 1 to 68% for in percentage
3, 4,
from the other
different
were off-task
third-grade
by
for
Classes
Differences
cognitive
with
the data
they were more likely
categories
in Table
are presented
actions
pupil
purposes,
clearly
The number of minutes
action
into
interacting
all
of the
a function
subdivided
differences
The large
groups
differences
and if
of curriculum,
Cognitive
4 is
class
pupils.
dimension.
varying
Class
3.
other
are probably
instructions
2) are shown again.
2 (class
they were engaged,
if
for
is
in grade
By grade
For comparative
45.
in that
Furthermore,
that
and part
1) and grade
and 5 are all
function
for
that
had.
class
interacting
was observed
or familiarity
maturity
the same
an activity.
Part
hesitation.
Grade 3 data were further
Class.
it
time waiting
frequently.
little
student
a student
In fact,
had completed
of the system
expectations
who are following
considerable
spent
what to do next
teacher
2 students,
1 and grade
grade
and percent students
of engaged
instruction
in large-group
the pupil
time
in
differences
Group 6 children.
coded for
coded
in the cognitive
the percent Also,
of time
All
action
other categories
a
Table Observed
Minutes
Grade 1-Class
1
and Percent
45
of Time of Pupil
Grade 2-Class
2
Class
Actions
by C
3
Action
Pupil
Minutes
%
Minutes
Engaged time
559
55
Off-task
449
%
Minutes
%
Min
771
71
402
98
6
45
317
29
8
2
3
488
89
656
86
364
95
4
62
11
107
14
21
5
1
Engagement time
Types of encouragement Content Directions Grouping Individual
302
30
165
15
6
1
Small group
553
55
583
53
101
24
2
Large group
156
15
343
31
317
75
7
Interactions Target
speaking
62
6
51
5
24
6
Target
listening
91
9
163
15
112
26
1
858
85
880
80
289
68
8
Teacher
99
65
161
76
122
92
1
Pupil
48
31
36
17
10
8
6
4
16
8
1
1
None Interaction:other
Other adult
party
46
Table Observed Minutes
Cognitive Pupil
Actions
and Percent
Group 1
of Time of Pupil
Actions
by Cognitive
Cognitive
Group 2
Cognitive
Group 3
Cogn
Minutes
%
Min
-..-
Minutes
%
Minutes
%
Engaged time
420
64
850
65
721
70
3
Off-task
237
36
460
35
310
30
1
361
86
690
83
634
90
2
57
14
140
17
68
10
Engagement time
Types of engagement Content Directions Grouping Individual
167
25
201
15
104
10
Small group
356
54
593
45
444
43
1
Large group
135
21
510
39
496
48
3
Interactions Target
speaking
Target listening None Interaction: other
37
6
61
5
63
6
76
12
545
83
164 1090
12 83
162 825
15 79
party
Teacher
80
71
167
74
162
73
Pupil
24
21
46
20
55
25
9
8
12
5
6
3
Other adult
3
Classroom
are not
in engagement and teacher 1,
grade
or of practical
striking
at grade
and grouping effects
five
3, 12 of 21 children
with
instruction,
the data
for
are presented with
other
is
is
for
in pupil
49,
50,
at
Groups 1 and 2;
of whether
received
each class levels
between
The
1 (grade
Class
in time pupils
and then only
or
different
are presented.
within
Only the difference
5 show high
differences
between
Group 2 children Overall,
for
with
1)
interact
Group 1 and Group
in different
cognitive
1, the only
grade
other
pupils
observable
(17% for
to 46% for these
2 children
with
Group 5-6
data
suggest
(structure often
virtually
for
hand,
all
Group 1
children Class
in 3
no exhibits
students.
vary by cognitive
pupils
for
classes.
4, on the other
on directions
other
data
third-grade
on content Class
more time with
the within-class
the three
engagement
are due to grade
Grade 1 and grade
As with
48.
students.
interactions
Summary.
2) children
interactions
groups
with
engagement
2 (grade
and 51 contain
and Class
of students
that
Group 3 children).
cognitive
for
within
cognitive
in Table
different
pupil
fact
the question
capacities
children
47.
Class
and 32% for
Tables
only
due to the
class,
(24% to 45%).
difference
lower
To answer
of interest
are presented
children
for
in Table
The data groups
by the grade,
were in cognitive
cognitive
of different
pupils
3 children
class.
different
children
is
differences
were in Groups 1 and 2; and at grade
children
within
not children
data
This
129
were in Groups 4, 5, and 6.
level
Cognitive
confounded
observed
these
However,
earlier.
children
of eight
2, six
are clearly
described
of six
interest.
Instruction
level
much Again, (31%
children). that
differences
of the curriculum)
worked in small
groups
in grouping or teacher. and
130
Classroom
Instruction
Table Observed
and Percent
Minutes
47 of Time of Pupil
Group Within
by Cognitive
Cognitive
Group 1
Class
Actions
1, Grade 1
Cognitive
Group 2
Cognitive
Group 3
Action
Pupil
Minutes
%
Minutes
%
Minutes
%
Engagement time
Engaged
time
Off-task
260
60
189
51
110
54
174
40
181
49
94
46
230
89
159
87
99
90
28
11
23
13
11
10
Types of engagement Content Directions Grouping Individual
129
30
119
32
54
26
group
235
54
197
53
121
59
Large group
70
16
56
15
30
15
26
6
24
6
13
6
Small
Interactions Target
speaking
Target
listening
41
9
30
8
20
10
368
85
318
85
172
84
Teacher
46
70
36
67
17
52
Pupil
16
24
17
31
15
45
4
6
1
2
1
3
None Interaction
Other adult
other party
Classroom
Table Observed
Minutes
and Percent
of Time of Pupil Class
Actions
2, Grade 2
Cognitive
Group 1
Cognitive
Group 2
Cognitive
Group 3
Minutes
%
Minutes
%
Minutes
%
Action
Pupil
131
48
Group Within
by Cognitive
Instruction
Engagement time
160
72
399
72
212
69
63
28
158
28
96
31
131
82
336
85
189
90
29
18
57
15
21
10
38
17
82
15
45
14
Small group
121
54
294
53
168
54
Large group
65
29
179
32
99
32
Engaged
time
Off-task
Types of engagement Content Directions Grouping Individual
Interactions Target Target
speaking
12
5
20
4
19
6
listening
35
16
84
15
44
14
177
79
454
81
249
80
34
84
40
65
None Interaction: Teacher
other party 72
87
Pupil
8
17
8
8
20
32
Other adult
5
11
9
9
2
3
132
Classroom
Instruction
Table Observed
and Percent
Minutes
49 of Time of Pupil
Group Within
by Cognitive
Cognitive
Group 3
Actions
3, Grade 3
Class
Cognitive
Group 4
Cognitive
Group 5,6
Action
Pupil
Minutes
%
Plinutes
%
Minutes
%
Engagement Engaged time Off-task
time
144
98
80
96
178
99
3
2
3
4
2
1
127
93
76
96
161
95
10
7
3
4
8
5
Types of engagement Content Directions Grouping 5
3
0
0
1
0
Small group
33
22
20
23
48
26
Large group
114
75
67
77
136
74
5
2
2
14
8
Individual
Interactions Target
speaking
8
Target
listening
47
31
16
18
49
29
98
67
69
79
122
66
52
95
16
94
54
89
None Interaction:other Teacher
party
Pupil
3
5
1
6
6
10
Other adult
0
0
0
0
1
1
Table Minutes
Observed
and Percent
by Cognitive
50 Actions
of Time of Pupil
Group Within
Class
4,
Grade 3
Cognitive
Group 2
Cognitive
Group 3
Cognitive
Group 4
Minutes
%
Minutes
%
Minutes
%
262
68
151
69
148
62
121
32
101
40
92
38
195
76
119
83
112
77
60
24
24
17
33
23
0
0
0
0
0
0
Small group
102
27
62
25
51
21
Large group
275
73
187
75
187
79
17
4
14
6
8
3
Pupil
Action
Engagement Engaged time time
Off-task
Types of engagement Content Directions Grouping Individual
Interactions Target
speaking
Target
listening
50
13
36
14
30
12
318
83
204
80
203
84
Teacher
44
66
33
67
29
76
Pupil
21
31
16
33
9
24
2
3
0
0
0
0
None Interaction:
Other adult
other
party
Co
134
Classroom
Instruction
Table Observed
Minutes
and Percent
51 of Time of Pupil
Group Within
by Cognitive
Cognitive
Group 3
Class
5,
Cognitive
Action
Pupil
Minutes
%
Minutes
Action
Grade 3
Group 4 %
Cognitive
Group 5,6
Minutes
%
Engagement Engaged time Off-task
time
104
87
103
90
110
92
16
13
11
10
10
8
100
98
94
99
95
95
2
2
1
1
5
5
Types of engagement Content Directions Grouping 0
0
0
0
5
4
Small group
60
48
58
48
58
46
Large group
66
52
63
52
63
50
9
7
9
7
11
9
Individual
Interactions Target
speaking
Target
listening
None Interaction: Teacher
other
15
12
16
13
9
7
102
81
95
79
106
84
20
83
22
85
13
72
party
Pupil
1
4
4
15
5
28
Other adult
3
13
0
0
0
0
Instruction
Classroom
for mathematics
individually
common in grade
Differences
3.
or to students'
teachers Only pupil
interactions
childrens'
cognitive
to interact
with
interactions
Teacher
but this
for
number of minutes
levels
grade the
teacher
and by cognitive
more likely
groups
where such
infrequent.
behaviors
are presented
group/class
interactions.
The data
52.
The differential
directions
related
shown in Table be teacher teacher
to grade
The data
Class.
varies
earlier.
or class
on teacher
the
speaking
on content
content.
Another
content.
However,
of class
grade
effect
by class
teachers
1 (grade
3 teacher
the percent
to be a grade
group,
are presented
and is
in speaking
the teacher
while
first.
by cognitive
in
or giving
explaining
Time spent
The differences
specific.
teacher
level
behaviors
and two of the third-grade
example,
at
consistent
with
on directions
is
level.
The differences
53.
spent
grade
teachers
coded are discussed
by grade
teachers
with
of time
by class,
behaviors
time
discussed
program expectations inversely
on teacher
content
vs.
and percent
in the behaviors
engaged
Grade.
appears
due to
occurs
only
pattern.
Behaviors
various
Table
is
due to
instructional
in higher
behavior
and even then
are likely
are plausibly
pupils children
(with
others),
the
group work was
large
time
with
other
with level
while
in engaged
familiarity
are allowed
The data
Then,
instruction
135
of time
about
on content
in class
the
3 are
appear
3 (grade
teachers
are large.
effect,
3) spent
57% of the explain since
to
first-grade For
51% of the observed
4) spent
or curriculum
grade
content
between
1) spent
(class
within
time
82% on
time on
directions all
three
grade
136
Classroom
Instruction
Table Observed
and Percent
Minutes
of Teacher
Behaviors
Grade 1 Teacher
52 of Time
by Grade
Grade 2
Grade 3
Behavior Minutes
%
Minutes
%
Minutes
%
Interaction Listening
187
17
206
18
216
11
Speaking
640
58
677
59
1238
64
None
276
25
254
22
485
25 25
Speaking/large
group
91
14
209
31
313
Speaking/small
group
82
13
65
10
227
18
Speaking/individual
467
73
402
59
697
56
Speaking/content
367
57
404
60
823
66
Speaking/directions
268
42
256
38
347
28
Low-level
135
12
157
14
338
17
33
3
29
3
199
10
1006
91
1035
91
1819
94
79
90
89
94
109
92
97
100
101
98
115
93
0
0
2
2
9
7
questions
Direction-related
questions
No feedback Feedback/individual Low-information
feedback
High-information Explaining Explaining
feedback
content
130
12
117
10
323
17
directions
235
21
228
20
165
9
Table Observed
Minutes
Grade 1-Class
B r Teacher Teacher Behavior
Minutes
1 %
and Percent
of Time of Teacher
Grade 2-Class Minutes
53
2 %
Class Minutes
Behaviors
by
3 %
Minu
Interaction Listening
187
17
206
18
55
12
129
Speaking
640
58
677
59
290
63
681
None
276
25
254
22
116
25
294
group
91
14
209
31
128
44
134
group
82
13
65
10
41
14
107
467
73
402
59
121
42
439
Speaking/content
367
57
404
60
239
82
391
Speaking/directions
268
42
256
38
45
15
240
Low-level
135
12
157
14
94
20
172
33
3
29
3
22
5
125
1006
91
1035
91
434
94
1025
Speaking/large Speaking/small Speaking/individual
questions
Direction-related
questions
No feedback Feedback/ individual Low-information
feedback
High-information
feedback
79
90
89
94
24
92
71
97
100
101
98
23
82
77
0
0
2
2
5
18
Explaining
content
130
12
117
10
96
21
139
Explaining
directions
235
21
228
20
26
6
126
Instruction
Classroom
138
3 teachers
three
teacher
behavior
of time
to 53% for
spent
of time
speaking
to 27% for
Group 1 children
confounded
undoubtedly
behaviors
were observed
different
cognitive
classes time
(1, spent
difference content
2,
interacting
with
was found.
In class
or grouping
treatment treat
cognitive
capacity
of these
in relation
the percent to
are
differences
Groups 1-5/6 teachers
with
various
teacher
in of the in terms
of
Only one important
that
the teacher
on
spoke
from 82% to 66%.
varied
considerably
cognitive
for
in their
or individual
teaching
than to differential
classes
from others,
the basis
For four differences
children.
5, the time
of time
to students
here.
were no striking
within
is not
of
the percent
Similarly,
Data on the percent
different
different
the
level.
patterns
of students
some students
all
seem due more to grade,
differences
Overall,
from 39% for
shifts
are not presented there
coded
from 22% for Group 1 children
decreases
class.
groups
In summary, although
style,
1
from 67% for
decreases Second,
directions
in each class
across
children
54.
First,
groups.
Group 5-6 children.
by grade
3, and 4),
decreased
behavior,
about
However,
group within
Cognitive
cognitive
of time
in Table
Group 5-6 children.
directions
explaining
Group 5-6 children.
6% for
across
to individual
speaking
Group 1 children time teachers
the grade
and percent
are presented
categories
are striking
differences
percent
The number of minutes
group.
Cognitive six
(6%, 11%, and 3%) than either
2 (20%) teachers.
(21%) or grade
for
time
less
spend
Teachers
capacities. but these
data
such differentiation.
suggest
may that
Table Observed
Teacher
Minutes
and Percent
54
of Time of Teacher
Cognitive
Group 1
Cognitive
Group 2
Cognitive
Minutes
%
Minutes
%
Minutes
Behaviors
Group 3
by Co
Cognit
Behavior %
Minu
Interaction Listening
127
19
231
16
139
13
5
Speaking
394
60
837
58
722
66
31
None
141
21
380
26
235
21
15
190
23
189
26
9
Speaking/large
group
83
21
Speaking/small
group
47
12
116
14
100
14
5
63
433
60
18
Speaking/individual
264
67
529
Speaking/content
233
59
479
57
475
66
20
Speaking/directions
155
39
327
39
228
31
8
84
13
178
12
196
18
8
12
2
78
5
68
6
5
592
83
1334
92
1018
93
48
60
91
98
93
70
95
2
94
3
Low-level
questions
Direction-related
questions
No feedback Feedback/individual Low-information
feedback
High-information
feedback
Explaining
content
Explaining
directions
69
99
114
100
75
1
1
0
0
5
6
72
11
166
11
182
17
7
143
22
254
18
164
15
3
140
Instruction
Classroom
Teacher
Engagement
Behavior/Pupil
The number of minutes
and percent
performed
various
actions
section.
As with
the previous
for
children
finally,
differing
increases
speaking
when teachers pupil
when there
and providing
certain
actions
3 and 5 in grade than rates
categories
level
effect
noted
Cognitive cognitive is
across
level
groups
is
and even when the
for
teacher
teacher
and
3.
increases
rates
actions
when teachers
from 50% to 76%.
3.
five
related
by
are
Engagement Similarly, from 42% to
to teacher
in the other
3 if
a teacher
performed As would from
different
4 are lower
in
the grade
and certainly
effect,
4 were omitted. on time pupils
data
were doing
Many of the differences
is
56.
In fact,
two classes.
when teachers
children
regardless
3 is
in Class
rates
in part
Class
in Table
4 in grade
Class
Engagement
is
when teachers
classes
increase
from 65% of the time
similar
for various
engagement
all
analyses,
increases,
are speaking
group,
engagement
engagement
The overall
57.
in this
aggregated
cognitive
1 to 78% in grade
on pupil
were engaged
in Table
as cognitive teachers
grade
group.
groups
reported
Pupil
increases
previously
for
would be higher
are reported
information.
presented
from previous
be expected Classes
is
class,
are no interactions
The information
Class.
teachers
class.
55.
as do all
that
the data were first
engagement
are not speaking
grades,
questioning
then
from 59% in grade
engagement
78% across
all
in Table
are presented
grade
on pupil
time
were engaged
sections,
by grade,
The data
Grade.
of coded
and children
group within
cognitive
Interactions
in differing different
are striking. in engagement
to 86%.
Second,
of whom the teacher
not speaking
(62% engagement
is
things First, when
the pattern speaking to 89%).
to,
Classroom
Table Observed
Minutes
Behaviors
55
and Percent
and Pupil
of Time of Teacher
Engagement
by Grade
Grade 2
Grade 1
141
Instruction
Grade 3
Interaction Minutes Teacher
%
Minutes
%
Minutes
%
speaking/
Pupil
engaged
356
59
463
70
919
78
Pupil
off-task
245
41
197
30
259
22
253
57
265
68
502
74
Small group
51
67
45
69
160
80
Large group
52
63
153
75
256
86
203
50
308
72
449
76
104
61
151
76
152
72
99
42
157
69
295
78
75
60
108
71
263
81
when teacher
Pupil engaged to: speaking Individual
Not speaking Pupil
when teacher:
engaged
Listening Pupil
when:
engaged
No interactions Pupil asks:
engaged
Low-level High-level Questions
when teacher
questions questions about
directions
Pupil engaged when teacher provides: feedback Low-information Positive Information Explains
feedback about directions
content
8
67
19
83
42
95
15
52
20
69
133
70
44
48
67
68
90
80
32
54
56
72
52
87
83
68
89
77
255
82
131
58
149
67
114
72
Table Observed Minutes
and Percent
Grade 1-Class
1
56
of Time of Teacher
Grade 2-Class
2
Behaviors
Class
and Pupil
En
3
Interaction Minutes Teacher Pupil Pupil
speaking/ engaged off-task
%
Minutes
%
Minutes
%
Minu
356 245
59 41
463 197
70 30
263 2
99 1
43 22
253 51 52 203
57 67 63 50
265 45 153 308
68 68 75 72
109 31 123 138
99 100 99 96
27 6 9 21
104
61
51
76
51
98
7
99
42
157
69
88
95
14
Pupil engaged when teacher asks: Low-level questions High-l evel questions Questions about directions
75 8 15
60 67 52
108 19 20
71 83 69
86 23 21
99 100 100
11
Pupil engaged when teacher provides: feedback Low-information Positive feedback Information about content Explains directions
44 32 83 131
48 54 68 58
67 56 89 149
68 72 77 67
22 20 86 24
100 100 100 96
5 2 9 8
Pupil engaged when teacher speaking to: Individual Small group Large group Not speaking Pupil engaged when teacher: Listening Pupil engaged when: No interactions
7
57
Table Observed
of Time of Pupil
and Percent
Minutes
for Various
Interactions,
Cognitive
Group 1
Cognitive
Minutes
%
Minutes
by Cognitive
Group 2
Engagement Group
Cognitive
Group 3
Cognit
Minutes
%
Minu
Interaction
Teacher Pupil Pupil
speaking/ engaged off-task
%
254 137
64 35
518 267
66 34
509 188
73 27
21 6
174 26 54 166
67 55 65 62
306 78 133 332
61 71 77 63
289 75 145 211
69 82 78 63
12 3 5 11
94
75
145
68
81
60
3
72
51
188
60
130
66
7
Pupil engaged when teacher asks: Low-level questions High-level questions about directions Questions
51 7 8
62 88 67
114 15 41
68 75 59
141 17 46
75 85 70
6 1 3
Pupil engaged when teacher provides: feedback Low-information feedback Positive about content Information Explains directions
43 32 53 87
62 65 74 61
68 46 106 159
62 66 68 67
50 37 147 101
70 77 82 63
2 1 5 2
Pupil engaged when teacher speaking to: Individual Small group Large group Not speaking Pupil engaged when teacher: Listening Pupil e,ngaged when: No interactions
144
Classroom
Instruction
in the same manner,
Third,
1 children
the
Finally,
apparent
when teachers
However,
as in the previous
found across
different
level
same pattern
of increase
students
or provide
Group
were no teacher in engagement
is
information.
are the
these
analyses,
within
Although
there
children
of differing
within
same differences
tables
data
was also
for
of
for
of differences
these
summarizing
children
calculated.
in each class
no pattern
levels,
cognitive Thus,
each class in engagement
some variation
was apparent.
The engagement
class.
levels
cognitive is
when there
children,
question
from 51% for
levels.
grade
Cognitive
increases
engagement
to 91% for Group 5-6
interactions.
class
pupil
data
in any
are not
presented. In summary, the data behavior
and not
style within
that
suggest
relating
differences
to differences
pupil
are due to grade
in cognitive
to type
engagement
capacity
level
of teacher
and teacher
among the students
each class.
Summary and Conclusions Data from the sample this
not receive
different
in how the five these
effects.
organize
and teach
in the five
classes
in cognitive
who differed
There were some overall with
the study
with
provides
both
First,
teachers
based
on school
traditions.
and patterns
of grouping
did
capacity differences
but
grade
and
some interesting
instruction.
mathematics
in
observed
Group 1 and Group 5-6 children,
and are confounded
Nevertheless,
about mathematics
emphasis
dealt
are slight
insights
in content
children
instruction.
teachers
differences
teacher
that
indicate
study
of students
students
tended
to
Differences
were based
on
school
curricula.
teacher
actions
In particular,
1 and 2 to grade
from grades
of activities.
and organization
emphasis
1 and 2) has an open,
(grades
on the other
Heights, formal
and direct.
actions,
teacher
Second,
Third,
there
others
who used
reflected
that
the other
hand,
while
Classes
following
in large
and in
School Waimea is more on pupil
effect
was to be expected. was not related
capacity
to reason.
one teacher
between
was following
were on task
Children
approach.
differences
level
schools
or their
the same curriculum.
good teaching
a shift
instruction
grade
engagement
work problems
were important
actions
program.
in which
program within
the mathematics
to how students
a school
and pupil
actions,
3 reflected
oriented
the overwhelming
Hence,
in pupil
Sandy Bay Infant
activity is
hand,
either
the differences
145
Instruction
Classroom
and two
3 and 5 in grade
a direct or small
the same program,
3 clearly
instruction Class
groups.
4, on
was not a successful
class. Fourth, related groups
the only
to cognitive to interact
opportunity of grade,
interesting capacity
with
to interact. school,
other This
and teacher
pupil was the pupils
behavior tendency
more often
effect, variables.
however,
we found that for
children
when there may also
was
in higher was an
be a function
6
Chapter
SUMMARY,CONCLUSIONS,AND IMPLICATIONS
The question who differ
children
that
each study
Then,
the basic
This
studies
is
both
addition
Cognitive
what we learned
to of
series
so
of the question. we hoped to answer
that
studies.
Finally,
but not at all
five
problems,
the influence
performance,
and final
and
are suggested We
and to teachers.
headings:
clear.
the strengths
implications
developers,
under
from
has evolved
and specifies
and subtraction
and subtraction
cognitive using
capacity,
the concepts
and
on
of instruction reflections.
Capacity
The original
to deal
capacity
available
psychometric test
scores.
that
assumed
question
cognitive
similar
both
perspectives
aspect
the picture
and subtraction,
of addition
addition
This
intellectual
and provocative,
discussion
this
verbal
that
to curriculum
researchers,
have organized
skills
was related
from each together,
interesting
of each of the
weaknesses
solving
children's
that
instruction.
on a different
we believe
summarizes
chapter
to other
Do
was,
question.
In retrospect, these
different
information
the
by putting
problems
and to classroom
studies
to add and subtract
we assumed
question,
shed light
would
learn
capacity
from four
was reported
of five
set
and subtraction
capacity
cognitive
studies
this
on addition
performance their
in cognitive
In asking
differently?
in these
investigated
with
techniques First,
young children
mathematical would yield
a set 146
of tests
differ
information groups measuring
in their
and that
of students short-term
with memory
147
Conclusions
was administered.
(M-space)
capacity
were used
tests
M-space
tests
The developmental between
differences
only
with
deal
were then used
to assist
have limited
groups
of students.
in describing
comparisons
qualitative
(M-space
memory capacity
most quantitative
of handling
but have no sophisticated
count
serially
six
derive
the groups.
are incapable
1),
Data from the
same children.
to empirically
Group 1 children
Cognitive level
to the
was given
tests
psychological
of developmental
a set
Second,
can
tests,
and can
strategies,
counting
at a moderate
and transformations
level.
level
distractions),
perceptual if
can count
These
sets
or taken
of these
four
The final
groups.
in that
3 or 4 and have sophisticated
on measures
Group 3 children
Group 6 children
only
differ on the
counting
strategies
all
four
They and are
and distinct
to each other
than they
members have a memory capacity strategies.
from Cognitive
from Cognitive inclusion
sets.
both
groups,
Group 4 children
under rearrangement.
and transitivity
class
or
problems.
counting
differ
of transitivity
Groups 3 and 4 children
are limited.
are more similar
from each other
Cognitive
from particular
from each other
are distinct
are larger
children
and rearrangement
and logically,
psychometrically
only
whether
two groups
from the remaining
of level
and can determine
transitivity
first
are different
and overcome
but have no sophisticated
sets,
to solve
unable
of sets
skills
the quantitative
However,
can
(they
comparisons
qualitative
has been put with
an object
(M-space
memory capacities
rearrangement
after
correspondence
preserve
smaller
with
have no difficulty
2),
have larger
Group 2 children
Cognitive
Group 5 and Cognitive test.
Groups 5 and 6
148
Conclusions
children
differ
we combined
six
in the early
thinking
school
in the
of 1 enables
M-space
level
level
sequence:
to solve
enough for
the development
simple
and probably
is
to early
an M-space
Fourth, tasks.
an
Fifth, and is
correspondence
a
And sixth,
of most number skills.
necessary
reasoning
comparison--qualitative--
comparison
qualitative
seems to be
Third,
operations.
of 3 seems to be necessary
tasks
counting
tests.
capacity
mathematical
level
M-space
in the preschool
sequences
a child
of 2 is for
prerequisite M-space
analyses
qualitative
in children's
Second,
following
a global
First,
apparent
correspondence--quantitative--logical level
in all
to cognitive
respect
in mathematically-related
years
to develop
appears
is years.
to the developmental
elementary
with
propositions.
distinction
quantitative
related
and analyzed
gathered
the following
suggested
of memory capacity;
Groups 5 and 6.
The data
versus
on one measure
only
for
success
for
the development
an
in sophisticated of addition
and subtraction. Problems differ
significantly
tasks.
However,
not based
in their
The next
model of cognitive Brown (1978), of cognition and "executive" in long-term
that
in research
(memory capacity, aspects
memory, etc.).
Using
simple
purely
children
mathematical It
empirical.
is
mathematical
proposed
by Campione and
the "architectural"
automaticity,
of cognitive
that
would be to use a theoretic
between
distinguishes
is
process
such as that
processing
which
we took
of how children step
indicate
to perform
ability
the approach
on any theory
information.
The data
and recommendations.
processing
speed
features
of processing,
(metacognition,
such a model would enhance
etc.) schema
our
149
Conclusions
of cognitive
understanding
in this
followed
sets
especially
important.
identifying
groups
the instruments
on).
of tests First,
with
a smaller
known in the
M-space.
Cucui,
Digit
are convinced
that
capacity
The second forward
set
important
verbal
Also,
skills
(simple
counting,
should
also
The relationship
when children
indicate
Span, Mr.
that
other
levels
M-space levels
memory capacity
an important
from
chunking
(Counting
Span)
use
phenomenon is well
we
Nevertheless,
of cognitive
feature
researchers
of class
groups
Sophisticated
counting
and subtraction
We recommend that
other
class
distinguished
addition
3.
be developed the
that
back tests.
research.
tests
and so
head,
measure
subjects.
of tests
other
can organize
of the body,
may be due to chunking.
suggest
much to be
children
to separate
However,
and strongly
in Chapter
Finally,
the four
is
leave
This
in
Unfortunately,
are indicated
difficult
is
memory capacity
in solving
demonstrated
levels
accurately.
and counting
side
and Backward Digit
of their
memory capacity
trait
information.
that
3 in many cases
processing
left
M-space
We believe
Placement,
level
capacities.
underlying
(e.g.,
but it
1, 2, and 3 relatively above
it
higher
literature,
actual
this
of memory to store
part
was most useful
on the Mr. Cucui Test,
by "chunking"
As a result,
cognitive
to be
processing
memory capacity
to assess
used
of cognitive
different
In particular,
information
approach
study.
We found three
desired.
more than the psychometric
capacity
tests
counting
that on,
different back,
skills
problems,
such tests
measure counting
were the counting are as
be used
in
counting
counting
all,
etc.)
and used. inclusion inclusion
test
distinguished
skills
groups
to how children
of students. work certain
150
Conclusions
the part-part-whole
(particularly
problems
We recommend the development
clear. assess
the way in which
Addition
Verbal
Solving
One indication that
can write
and the
strategies
subtraction
size
occasions
at two or three
student
or skills
to solve
a variety
in which of four
six
levels
to find
both
they were able
The data were gathered
problems.
on several
child
used
they
the problem
about
3, we examined
number of questions
(the
phenomena.
to add and subtract
sentence
concepts
In Chapter
answer.
appropriate students
or subtraction
about
that
For such problems,
problems.
or subtraction
tests
Problems
have learned
verbal
simple
an addition addition
learned
students
not at all
of other reason
logically
and Subtraction
that
can solve
they
and study
individuals
is
problems)
children and use
the
the performance
to answer
and
in interviews
of each
were given
of difficulty,
of
correctly)
of addition
problems
is
to each
determined
by the
of numbers in the problem. The results
considerable verbal
described
The overall
and in the
on the tasks
the SN level
problems
answered
72% correctly;
answered
correctly.
performance
problem,
strategies
with
that
to solve
used
different
there
the semantics
this
of the problem,
operation
the size
in the problem,
processing 72% of
students 67% were
variability
was influenced
in both
by several
of the numbers in the the grade
of
problems.
answered
problems,
was considerable
variation
a variety
those
problems,
and of the NR and R level However,
was
cognitive
Students
high.
of the LN level
correctly;
there
to solve
ability
they
was relatively
and strategies;
the implied
3 indicate
children's
of students
performance
capacities
factors:
in the
variability
problems
in Chapter
level
of the
Conclusions
and the
child, level
of performance
types
of tasks.
problems,
across
which
in turn
children
that
strategies the following answered
change
cognitive
two-thirds
particular
group used,
percentages 20% on the
The summary information for
the
children
group of children tasks--the The strategies
that
when students
level,
this
Overall, children.
counting Also,
SN tasks
skills,
for
SN and LN level
only
used,
reflects
clearly
They had low memory capacity, and were only which
the children; compare tasks
able
they
used
a
the
by direct
modeling. were either 6 at the SN
time.
cognitive
lacked
capacity
of
systematic
model the problems.
more memory capacity, inappropriate
83% and 93% of the
This
of the 12
was on task
to directly
requires
used
59.
one exception,
on" 37% of the
"counting
in each
students
in Table
on three
The exception
modeling.
category
and use of strategies
be solved with
children
students if
on
of problems.
set
only
can easily
if
that
presented
satisfactorily
behavior
the compare task,
impossible
same semantic
and
a semantic
are indicated
students
used
within
were
are summarized
only
strategies
Group 1 is
that
these
or direct
inappropriate
these
performed
three
is noted
tasks the
and strategies
group used
on the performance
in cognitive
of
than combine
The performance
groups.
of the
to highlight
the
semantic
the conclusion
are easier
problems
correct
Similarly,
on at least
different
support
in each cognitive
correctly.
strategy
the six
58 summarizes
than compare problems.
The percent
pages.
Table
on both performance
findings
children
at least
results
are easier
in different
for
items
the
that
(1981),
The most striking for
all
In general,
Greeno and Riley
of the child.
capacity
cognitive
151
time,
strategies
was on the
respectively.
Table
Frequency
and Percent
Correct
for
Level
SN
58
Each Task on Different
Level
Items
for All
Level
LN
Task Frequency 1 Change/join
(+)
2 Change/separate 3 Combine/part 4 Combine/whole 5 Compare
(-) unknown
(-)
unknown
(+)
(-)
6 Change/join,
change
set
unknown
(-)
%
Frequency
%
Frequency
85
85
201
78
125
79
79
184
71
89
68
68
190
74
88
80
80
194
75
115
41
41
157
58
90
77
77
191
74
105
N
Table Performance
59
and Common Use of Strategies
Task
Percent Correct
Direct Modeling
for
Cognitive
Counting Sequences
$N Level 1 Change/join
2 Change/separate 3 Combine/part 4 Combine/whole
77
(+)
53
70
(-)
57
unknown (-)
40
unknown (+)
70
57
5 Compare (-) 6 Change/join,
change
set
unknown (-)
37 LN Level
1 Change/join (+) 2 Change/separate 3 Combine/part 4 Combine/whole
37 40
(-)
unknown (-)
37
unknown (+)
5 Compare (-) 6 Change/join,
change
set
unknown (-)
37
Group 1
Routine Mental Operation
154
Conclusions
on the performance
The summary information for
could
children
find
with
problems,
answers
group 2 students
numbers),
SN set,
the pattern
solve
only
task
of
(large
1 at a satisfactory
of performance.
level
their strategy
for
routine
mental
of problems
SN set.
all
LN problems.
used
they most frequently Only on this
1.
for
for
The summary information in Table
61.
on all
satisfactory some difficulty
with
and 6 was performance strategy, strategies,
strategies
inappropriate did these
used
only
larger
children
in over half
of
systematic
and task
numbers on all
4 on the
(NR and R),
tasks
except and they
use an algorithm,
in the use of an algorithm.
made errors
appears
were coded
with
number
the small
not work most of the
could
SN and LN levels
the problems
task
the use of
although
Group 2 students 6 at both
on task
Finally,
used
strategies
problems.
strategies
counting
was the most frequently
commonplace with
The students
Inappropriate
compare problems.
with
problems,
was becoming
operations
over
modeling
the SN and LN level
both
in the
the trials
Direct
capacity.
cognitive
used was consistent
Group 2 students
information
The strategy
often
could
of
their
on the NR and R sets
However,
group of
This
Although
on the LN than the
lower
60.
the SN and LN sets
on both
of compare tasks.
was similar.
the performance
shown in Table
satisfactorily
the exception
was slightly
performance
task
Group 2 is
in cognitive
students
and use of strategies
particularly however,
Their tasks task
the
overall
children level
in cognitive
of performance
at the SN and LN levels, 5.
only
Direct
is
on the
small-number
and routine
mental
modeling
SN problems.
operations
is
although
For the NR and R set,
satisfactory.
Group 3 quite they
had
on tasks
1, 4,
a reasonable Counting
were also
being
used
60
Table Performance
Task
and Common Use of Strategies
Percent Correct
Direct Modeling
for
Counting Sequences
Group
Cognitive
Routine Mental Operation
SN Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
85 81 77 83
33 31 27 27
44 52 50 48 36
85 LN Level
1 2 3 4 5 6
Change/j oin (+) (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
72 68 71 68
42 43 42 45
24
70
28
26
NR,R 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) Change/join, change set unknown (-)
67
21
Level
38
Table Performance
Task
61
and Common Use of Strategies
Percent Correct
Direct Modeling
for
Counting Sequences
Group
Cognitive
Routine Mental Operation
SN Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
100 89 89 89 67 94
33 56 50 56 44
39 33 28 33 44 39
LN Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
72 68 71 80 74 82
23 27
35 26 26 42 36 32
29 26 26 24 35
NR,R Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
34
89 34 29
23 27
70 66
36 36
Conclusions
with
number problems.
small
on all
LN level
fairly
high
of inappropriate
The Group 3 students
less
problems
The summary information 62.
these
children
for
on the LN problems
counting
and direct
and counting
modeling
Group 4 students
on the two addition
The summary information
used
These
counting
solutions. tasks
2 and 3, they
routine
mental
operations
operations on the NR
strategies
all
in Cognitive problems
mental
by
satisfactorily.
direct
and algorithms
only
most
problems, They used
modeling. with
shown They
to find
subtraction
employed
is
Group 5-6
operations
on the NR and R simple frequently
of
problems.
and routine
strategies However,
solved
mental
in the use of algorithms
of children
children
in
appears
of the Group 3 students.
and routine
strategies
There was some increase
63.
on the NR
of strategies
and choice
from those
and R problems.
in Table
a
Also,
the NR and R
Group 4 children
Cognitive
little
very
used
Group 4 students
for
algorithms
the performance
Not surprisingly, differ
tasks.
was apparent
strategies
used
were used
strategies
than did the Group 2 children.
frequently
Table
counting
of the NR and R level
and on three
tasks
frequency
and R tasks.
Sophisticated
157
the three
easiest
tasks. More important, particular
these
strategy
of the problem, Furthermore,
data
depends
the
size
show that
on several
of the numbers,
the availability
a child's
factors,
including
and the implied
or use of a strategy
to use a
decision
appears
the semantics operation. to depend on
memory capacity. Five
general
of how children persistent
observations learn
to solve
use of inappropriate
from the data such problems. strategies
relate First,
implies
to our understanding the frequent either
an
and
Table Performance
Task
62
and Common Use of Strategies
Percent Correct
Direct Modeling
for
Counting Sequences
Cognitive
Group
Routine Mental Operation
LN Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) Change/join, change set unknown (-)
91 77 75 86 86 84
20
34 39 32 25 27 23
41 27 25 39 32 45
NR,R Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) Change/join, change set unknown (-)
78
38 31 31
74 67
21 29
31 31
Table Performance
Task
63
and Common Use of Strategies
Percent Correct
Direct Modeling
for
Counting Sequences
Cognitive
Group
Routine Mental Operation
LN Level 1 2 3 4 5 6
Change/j oin (+) (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
33 48 31 38 48 43
95 95 98 95 93 98
50 24 48 48 36 45
NR,R Level 1 2 3 4 5 6
(+) Change/join (-) Change/separate Combine/part unknown (-) Combine/whole unknown (+) Compare (-) change set unknown (-) Change/join,
88 74 83 93 71 90
32 31 33
40 26 60 62
160
Conclusions
of some students
unwillingness
to use a particular
memory capacity Verschaffel to find
an answer
some students
to a particular to solve
try
available
strategies try,
across
difficulties
some children direct
Second, is
combination
for
the important
The strategy is
is
for
could
strategy. model"
to find
familiar
with
exhibited
becomes
strategy
efficient
sophisticated
This
for
Also,
We
of the
to represent It
use.
is
direct
or
modeling
data need not be remembered. 3 and 6, but additional and the original
appear solving
comparison
strategies,
but it
problems,
modeling
problems is
still
an
a "when in doubt
many problems.
Even
many of the large-number
the importance
although
memory part.
large-number
to follow
modeled
suggests
procedures; counting
or fingers
more cumbersome,
Many students
answers.
when
the task.
understanding
students
Even with
in Group 5-6 physically
problems
a better
be used with
even more memory storage. modeling
For
2, and 4, where the change
1,
to remember the whole
modeling
to complete
that
prior tasks
requires
third-graders
of information.
use of chips
data;
direct
one can always
track
that
Thus,
represented.
Finally,
appropriate
we believe
of the use of inappropriate
SN tasks
appropriate
required
where physical
However,
many of the problems.
strategy
can be physically all
preserves
are
they
counting
to obtain
(the
modeling
appropriate
that
and
have.
and easiest
the first
particularly
storage
in order
DeCorte
do not have systematic
investigation
tasks
with
to understand
but lose
to them to solve
recommend a more careful
sets)
fail
problem.
problems
or an inadequate
We agree
may get mixed up and be unable
they
strategies
in the task,
strategy.
Group 1 and Group 2 children
example,
they
that
(1981)
most students
to engage
children knew basic
of being in Group 5-6 facts,
and could
Conclusions
and subtraction
addition
perform
some problems.
modeled
directly
Third,
the data
with
modeling
also
either Counting
operations;
the choice
children
solved
to be solved
likely
Furthermore, mental
only
on task
frequently,
comfortable
to write
students solve
problems
addition
for
use
routine
expression
on the NR and R tasks.
for many answers.
finding
of
knowledge
used
algorithms
at higher
cognitive
but know of and are
The children's
strategies.
algorithms,
strategies.
in their
are appropriate
the mathematical
and subtraction
and use
teachers
expected to
the algorithms had been on
Most instruction and the children's
performance
was
good.
reasonably Fifth, problems
who were limited
algorithms
was
were more
algorithms
Students
other
operations
6 at the SN level
and subtraction
and they made many errors.
in using
groups,
problems.
operations,
that
of the numbers
mental
task
mental
may see
levels
mental
did children
1 (combine/join)
or routine
strategies
routine
cognitive
counting
sophisticated
large-number
Only the Group 2 children, counting
all
as a cumbersome procedure
was perceived
mental
The LN, NR, and R problems
the use of addition
Fourth, children
by using
or routine
before
routine
Only for
strategy.
with
operations
by using
strategies.
the dominant
for
levels,
direct
replaced
on the size
depends
At all
the SN problems
than counting
strategies
may be learned
of strategy
in the problem.
counting
counting
strategies
involved
many children
that
indicate
systematic
operations.
rather
still
they
efficiently,
algorithms
161
it
is
apparent
was not directly
most instruction
that
the way in which
related
to classroom
was on addition
and subtraction
solved
students
instruction. facts
(use
the
In grade of routine
2,
162
Conclusions
mental
operations),
skills
to solve
on algorithms,
and recommendations.
at each of the four
tasks
and subtraction
instructional
did not use them to solve
levels--does Nesher,
problems.
better
of the overall
picture
the small
Second,
the
in order.
Third,
cross-sectional
duration
should
between
be carried
Finally,
proposed
by Nesher,
decision
sequence
theory
Since verbal
and Skills
we also
of addition 4.
for
categories
tasks. for are there
the three-month studies
is
period.
of longer
an obvious
in light addition Our data
(1982).
to select
use
most mathematics
problems,
and skills chapter
children
across
confounding
capacity.
and Riley
Greeno,
us a
would give
of the
theory
and subtraction suggest
that
is more complex
a strategy
the than
suggests.
the Concepts
Using
over
there
Fourth,
of semantic
of the development
14
data were gathered,
data need to be re-examined
these
list
number of subjects
changes,
and cognitive
level)
age (grade
a larger
with
indicate
out.
(1982)
and the method of selection
in performance
data
of addition
types
of strategies
some longitudinal
change
Although
of problems
development
Studies
although
little
was relatively
set
all
and Riley
Greeno,
of items--six
sample
not include
number of students
are limiting.
study
the
First,
Use of a more comprehensive
types.
this
3, most of the
but students
and counting
modeling
problems. Problems
this
used direct
In grade
the problems.
time was spent verbal
but most students
A set
of Addition textbooks
examined
do not emphasize
students'
and subtraction.
of achievement
and Subtraction
performance This
monitoring
study
tests
is
that
the solution
of
on the concepts reported
in
measured
a
Conclusions
of mathematics
variety
at each grade
Instruction
the autumn term. were unable addition
level
In grade
to solve
be said
for
effect,
increases
most problems;
objectives.
In particular, improved
some aspects
the overall
level
Instruction
did not
value,
those
trouble
addition
with
open sentences
in these
groups
improved
cognitive
facts.
differed
in performance
groups.
and
to the
overall
level on
performance and writing
facts, for
instruction
most students and knowledge
Yet almost
on
were still of basic
no time was allocated
group within
by cognitive at all
groups
over
all
performed
Overall,
grade,
Group 1 children
grades.
many of the objectives,
cognitive
However,
systematically that
on
for
areas.
were represented with
struggled
higher
writing
grade,
each grade
was not high.
time was not allocated
in third
and subtraction
and skills,
concepts
and subtraction
For example,
Performance all
of addition
and subtraction
instruction
of the fact
had an
clear.
on many objectives
in spite
was not high,
was very
very
not
on many of the
growth within
different
seem to be related
knowledge
topics.
having
of performance
Thus,
number sentences
across
their
for many
on the addition
and subtraction
was not uniform
place
improved
Thus,
dramatically.
improvement
of performance.
were not as apparent
clearly
over students
year,
instruction
2, although
level.
the same could
although
performance
of addition
on some objectives
of the school
dramatically,
in performance
Grade 3 students
start
at each grade
by the end of the term,
In grade
subtraction.
objectives.
algorithms
had an effect
1, at the
had improved
skills
was administered
objectives
163
while
children
not
in grade
1
the Group 2 students
of the objectives. better
although
The children
than children
who differ
in the
in lower
in cognitive
processing
164
Conclusions
performed
capacity
instructional
although
the differences
Influence
received
series
whether
or not
children
about
addition facts,
study
be related
to grade
time are likely
engaged
with
the
instructional
with
other
in each grade
pattern.
is possibly Teacher
is
emphasis
spent
for
did in these
classrooms
reflect
grade
1 and 2,
for mathematics
and individually
due to the cognitive
seemed to
In grades
structure.
3.
or student
level
Differences familiarity interactions
groups
on
both
3, computational
Only the number of pupil
behaviors
is
on addition
facts
2, basic
due to teachers
is
out in many schools.
group work was common in grade
large
in pupil
belong.
groups
portrayed
And in grade
and curriculum
in small
time
The
behavior.
what is
carried
In grade
pupil
to pupil
1, primary
What pupils
time,
However, is
are taught.
level
were working
pupils
capacities
cognitive
on allocated
mathematics
and counting.
are stressed.
while
monograph attempted
different
in relationship
In grade
and subtraction
instruction
with
are not dramatic.
50% of the total
algorithms
children
in this
reported
of how instruction
numerousness,
children
actions
and subtraction.
addition
to be consistent
appears
and Subtraction
data were collected
too typical
perhaps
some of
instruction.
and teacher
of this
findings
on Addition
in the
Observational engagement,
for
accounted
experience
performance
study
different
objectives,
capacity.
of Instruction
to determine
First,
children,
processing
The final
and pupil
teaching
between
cognitive
of specific
regardless
or grade.
emphasis,
Thus,
with
differently
to which
and individual
165
Conclusions
teaching
The data
be on certain
routine
such as sentence
others
or change
performance
they
learn
to
the
and with
seems to
classrooms
but not on of
modeling
regardless
teach
appears
and algorithms)
the semantics
of individual
that
students
seems to proceed
of the verbal
of various
instruction use
without
to solve
to build
attempts verbal
consideration
problems. of the
on In
level
of
appears
to
children.
Reflections this
In concluding 1.
solve
provide
a basis
related
concepts
clustering
monograph,
seven
The information-processing
children
initial
that
the strategies
instruction
answers
performance.
in arithmetic. is no evidence
there
Finally,
within
or direct
counting,
children
classrooms,
processing
facts
(basic
done to relate
to instruction
problems
that
What children
on finding
is
is
Nothing
procedure.
Final
writing,
The emphasis
problems.
but within
The emphasis
procedures
of
the basis
indicate
of cognitive
level
not
and algorithmic
children.
in each grade.
covered
content
their
with
be consistent
facts
varies,
the same way to all
basically
is
classrooms.
clearly
on basic
do in classrooms
What teachers
fact,
two studies
in the last
due to instruction
improve
in these
students
between
differentiation
capacity
cognitive
Certainly,
style.
of addition
a variety for
a better
and skills
of children approximation
into
thoughts
to the study
approach
and subtraction
understanding
them to solve
cognitive
groups
should
description
of how
problems
of the process
and using
of a more refined
come to mind.
of acquiring
problems. be viewed of capacity.
Our as a rough
166
Conclusions
For students
2.
Groups 1 and 2), needs
a more careful
basic
ideas
in our
(students
of inappropriate
analysis
strategies
to be done. The most interesting
3. children
should
To be more effective,
differently. addition
sentences
alternative
would be to teach
subtraction
facts
until
build
from verbal
the bridge Students
such problems
procedure
of modeling
is
important Although
efficient 7.
and feel Children
mathematical Teachers
throughout
we believe
that
in the eyes
use to solve
differ
take
in their
and build our intent
to study
with
routine
Students
later.
verbal
problems This sentence
are important,
when they
see
them to solve
problems.
to solve
a variety
should
begin
they
them as
of
where children
are.
and procedures
children
capacities.
was to incorporate learn
could
a mathematical
the strategies
how children
to relate
expressions.
procedures
of children
upon those
and
mathematics.
capacity
account
them.
mathematical
in using
Instruction into
problems
In conclusion, perspectives
confident
trying
to work with
all
on
One possible
for.
to use of algorithms
situation
skill
problems.
should
with
teaching
such as addition
have mastered
a problem
become important
called
without
procedures
need more opportunities
and to represent
6.
is
for
more instruction
clear,
routines
problems
of
and sequenced
and sequence
not yet
specific
students
on change
must be organized
strategies
or algorithmic
them to problems
a very
is
that
strategies
data
organization
skills
and counting
on the
be gathered.
curricula
the ideal
Although and subtraction
5.
are those Longitudinal
children
by specific
4.
writing
data
not on performance.
use,
strategies
only
with
struggling
data
from different
to add and subtract.
The
Conclusions
picture
that
concepts
important capacity
to handle
problems
by using
dismiss
is
emerges
or fail
information. invented
to see
The capacity
procedures
students
challenge instruction they
use.
of the
of children
invent
instruction
that
to solve
in schools
in the future
is
compatible
with
is
children's
this
by their a variety
procedures
processing
carried
of
of
have not been taught.
a variety
to change
to solve
taught
for
a variety
are limited
Most are able
strategies
the value
to learn
struggling
Some children
and skills.
problems.
in which
of children
for
solving the
information,
of problems,
They
and the way
out are not consonant. fact.
capacities
Our goal
is
The
to make
and the strategies
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