2012. Vol.3, No.12A, 1177-1182
Published Online December 2012 in SciRes (
Copyright © 2012 SciRes. 1177
Examining Patterning Abilities in First Grade Children: A
Comparison of Dimension, Orientation, Number of Items Skipped
and Position of the Missing Item
K. Marinka Gadzichowski
Department of Psychology, George Mason University, Fairfax, USA
Received July 18th, 2012; revised August 20th, 2012; accept e d S eptember 16th, 2012
Current curricula in most school districts in the United States include some instruction on the recognition
of patterns in kindergarten and continuing into the early elementary school years. Despite the fact that
patterning is so common in school cirricua, very few reports of what types of patterns are easy or difficult
for children to learn have been published. In an effort to address this issue, 121 first grade children from
an urban school district were tested with 48 patterns that varied in dimension, orientation, position of
missing items, and magnitude of the gap between items. An ANOVA for completely correlated factors
was conducted. Results indicated that only the magnitude of gaps (i.e., “skips”) made a significant differ-
ence. There were indications of an interaction between that factor (skips) and the position of a missing
item. Implications were discussed.
Keywords: Patterning; Oddity; Transitivity
Instruction in recognizing patterns constructed from small
objects, shapes, colors, and sometimes letters and numbers,
begins in kindergarten and continues into the early elementary
grades for most children enrolled in school in the United States.
Several instruction manuals exist concerning how to conduct
this instruction, which is called “patterning” (Burton, 1982;
Ducolon, 2000; Jarboe & Sadler, 2003). Patterning instruction
occurs nationwide but is not founded on empirical research;
rather, it is based on a consensus of educators who are of the
belief that the ability to recognize such patterns holds some
educational value (National Councils of Teachers of Mathe-
matics, 1993). It is also commonplace in parts of Europe
(Threlfall, 1999; Lilijedahl, 2004). The present way of thinking
is that instruction in patterning improves abstract cognitive
abilities (Economopolous, 1998; Papic, 2007) that ultimately
will lead to improved academic performance.
Empirical investigation of this proposition is quite limited.
There are only three experimental studies that address how this
instruction impacts other cognition or achievement. In Her-
man’s (1973) dissertation, 24 lessons on patterns made from
shape, size and color were given to 71 kindergarten children
from an impoverished background. The patterns were alterna-
tions of two or three items—AABAAB, ABBABB, ABABAB,
AABBAABB, and ABCABC. Numeracy ability was subse-
quently measured by two subtests of the Metropolitan Readi-
ness Test. Her findings were that English-speaking children
who received this instruction made significant gains in nu-
meracy skills following the instruction, although Spanish-
speakers did not.
A multiple baseline study by Hendricks et al. (1999) used
only four participants, all of whom were males in first grade
who had begun to lag behind their peers. All four of the par-
ticipants were English Language Learners. They were given
instruction in both patterning and class-inclusion. Two of the
four boys showed significant gains on the Slosson Intelligence
Test (SIT), one made gains approaching statistical significance
and one did not show a gain. The children all had low scores on
each of the Diagnostic Achievement Battery-2 (DAB-2) sub-
scales, which measured total achievement, listening, reading,
speaking, writing, math, written and spoken language. Al-
though individual gains were different there was an overall
pattern showing that the gains on patterning and class inclusion
were accompanied by significant gains on the DAB-2 measures
of academic achievement. The shortcoming of this multiple
baseline study is that while it shows that the children improved
on what they were taught (patterning and class inclusion), it
does not prove that gains on the SIT and DAB-2 was because of
the instruction they had received. There are a number of rea-
sons why an ESL child could make such gains. ESL children
may come into the school system with very low capacity for
English. Some ESL children are unable to communicate even a
simple thought when they begin the school year. After working
with their ESL teacher and spending seven hours a day hearing
and attempting to speak English, it is quite easy to see that they
would improve rapidly. Intelligence tests and measures of aca-
demic achievement rely on the ability to communicate in Eng-
lish. Children who have a poor vocabulary in English or do not
yet understand fully what is being said are going to score lower
as a result of their lack of language. As they learn new vocabu-
lary during the school year and become comfortable expressing
themselves in English it is only natural that they would score
higher on such tests. Additionally, a child who does not com-
prehend what is being asked of him may choose to randomly
select answers without even thinking about the choice, simply
as an avoidance tactic. Once they have attained mastery of the
English language they no longer feel insecure and are likely to
contemplate the answer choices without being self-conscious.
In 2006, Hendricks, Trueblood, and Pasnak expanded the
depth of patterning instruction to include 480 color, size, num-
ber, letter, and time patterns ranging from alternations similar
to those of Herman (1973) to complex patterns with varying
numbers of steps between items presented in matrices. The
results from their study indicated that the children who received
this type of instruction made greater academic gains than chil-
dren in their control groups, who had received instruction fo-
cused specifically on academic material that had been devel-
oped upon the recommendation of teachers. The children’s
scores on the DAB Total Achievement measure were correlated
with the patterning test scores, r(60) = .40 p < .01. Correlations
for the DAB’s subscales were similar: ma thematics r(60) = .42,
p < .01, written language, r(60) = .35, p < .01, spoken language,
r(60) = .38, p < .01 These are considered to be medium effect
sizes (Cohen, 1992).
There were no group differences on the DAB scale. However,
in spite of random assignment of participants to groups, there
was a 5 point difference between the groups on average intelli-
gence, favoring the control group, and unequal variances. Be-
cause of this, a MANCOVA was conducted and the SIT-R
scores were used as a covariate. Once the groups were equated
statistically on the SIT-R, the patterning group did display im-
provement that was significantly more than the improvement
made by the control group on the DAB Total Achievement
These three studies constitute the entire set of empirical evi-
dence that instruction in patterning has an effect beyond help-
ing children become better at patterning per se. However, it is
clear that patterning instruction has become a fixture in educa-
tion, and for that reason deserves investigation.
To date, no research has been published in the professional
literature that was designed to determine which types of pat-
terns are the easiest or hardest for children to learn. However,
Boyer, Sweeting, Pasnak and Kidd (2010) presented a poster
indicating that the number of items a pattern “skipped” was
relevant. First grade children were most accurate on patterns
that skipped a single item, followed by patterns that skipped
two items and finally, patterns that skipped three items.
For younger (preschool) children, research involving oddity
problems showed that the dimension (color, form, size, and
orientation) in which it was presented was relevant in deter-
mining ease or difficulty of an oddity problem. Color oddity
problems were easier than form oddity and orientation oddity
problems (Gadzichowski & Pasnak, 2010). Since the dimension
of the oddity problem did impact how easy the problem was for
the younger children (age 4), it is an area of interest when as-
sessing patterning ability in older children. Perhaps the dimen-
sion in which a pattern is presented affects how difficult it is to
understand the pattern. Boyer, Sweeting, Kidd, and Pasnak
(2010) reported that patterns of letters were easier for children
than patterns of time as represented by clock faces, and patterns
of numbers were harder than patterns of rotation, time and let-
ters. However, their design was nonfactorial and incomplete,
and Gadzichowski, Kidd, Pasnak, and Boyer (2010) reported
that these dimensions did not produce significant differences,
so the issue is only partially resolved.
Developmental psychologists have neglected patterning in
their study of cognition. There is no theory that puts patterning
among the cognitive tasks mastered during a child’s develop-
ment. Furthermore, there is no understanding or explanation of
how failure to master this cognitive ability would detract from a
child’s ability to master more abstract cognitive concepts. How-
ever, children transition from one type of thinking in preschool
to a more complex cognition in early elementary school—the
transition from preoperational to concrete operational thought
(Piaget, 1963/1936)—and patterning incorporates many of the
changes in reasoning ability made during this time. Patterning
may be closely related to transitivity, an ability in which some-
one is capable of deducing the relationship between two items
by examining how those items relate to a third item. Clements
and Sarama (2007) found that having the ability to make indi-
rect comparisons, along with the ability to order objects ac-
cording to their size (seriation) are cognitive skills that are nec-
essary for successful performance in elementary school. “In-
deed, research indicates that mastering these skills is one of the
strongest predictors of school success or failure” (Clements &
Sarama, 2008: p. 365). In seriation, a child must understand
how an item relates to the items that come before or after it in
the sequence. When looking at patterns, this same rule applies.
In seriation, the relation is a simple one. The item in question is
either smaller or larger than the neighboring items on some
dimension (height, width, weight, length, or overall size). In
patterning the relation is more complex; the relation might in-
volve size, color, size and color, shape, or other dimensions that
are more abstract, and the relationships can be much more
Transitivity is the understanding that if item A is related to
item B in some way, and item B is related to item C in some
way, then the relationship between A and C can be determined
by comparing these items to item B. The relationship between
A and C (the key items) is not directly observed but rather de-
duced by comparing the key items to another item. Both transi-
tivity and patterning incorporate the idea that an item is defined
by, and simultaneously defines, properties of items that follow
or precede it. The primary difference between transitivity and
patterning is that patterning does not require an individual to
utilize the relations of A to B and C to B in order to determine
the relation of A to C. Transitivity does require that one use
those relations. Since a child could make use of the transitive
relation or alternatively make use of the simultaneous presenta-
tion of all the items and compare A to C directly, perhaps pat-
terning is a precursor to transitivity.
Examining the differences between patterns wherein the
missing item occurs at the beginning, middle or end may offer
some insight into the relationship between patterning and tran-
sitivity. Children who are capable of solving patterns that are
missing the beginning or end item may have begun to master
transitivity, because they must infer the nature of the missing
item from one that just precedes or follows it. When children
can rely on the presence of items on both sides of a missing
item (patterns missing a middle object) to determine the nature
of the missing item, they may be reversing the transitivity op-
eration. Thus, when a child understands patterns of items in
which some items follow and precede other items, based on the
rule of that pattern, then the child can make inferences about a
neighboring item by looking at any one item in the pattern. A
more advanced extrapolation would be to use a single item to
make inferences about both of the neighboring items succes-
sively or simultaneously. The ability to compare the two infer-
ences about the neighboring items in order to relate the neigh-
boring items to one another would be considered transitivity.
Copyright © 2012 SciRes.
Hence, making inferences about neighboring items from pattern
rules may be a step in the process of developing transitivity.
A first step in understanding patterning as an aspect of cog-
nitive development is to determine what kinds of patterns are
easy for children and what kinds are difficult. For this reason
the research questions are as follows: Does the dimension—
colors, shapes, letters, etc.—make a difference, or do children
abstract the pattern rule independently of the dimension in
which it is presented? A second question involves the orienta-
tion of the pattern. Is a pattern presented horizontally, easier
than a similar pattern presented vertically? Intuitively, it seems
that there would be a preference for a particular orientation.
Children read from left to right, but number problems are most
often presented in a vertical orientation. Also of interest is the
number of items “skipped” in a pattern. It seems likely that
children find patterns that skip a single item to be easier than
those patterns that skip multiple items. Finally, when a pattern
is incomplete, does the position of the missing item make a
difference? Answers to these questions, which have never been
asked by psychologists or educators, can inform investigations
of how patterning relates to children's cognitive development,
and also aid educators in determining what types of patterns can
form the basis for the most fruitful classroom instruction.
After approval from an internal review board, parental con-
sent was obtained for 121 first-grade children from an urban
school district. Children took permission forms outlining the
research home to their parents and parents had the option to
return the permission form with their signature, allowing their
child to participate in the study. There were 67 females and 54
male participants. Of those participants 52 (43%) were African
American, 43 (36%) were Hispanic/Latino, 16 (13%) were
Middle Eastern, 3 (2%) were Caucasian and 7 (6%) were of an
ethnicity other than those listed. In the school district from
which participants were drawn, 57% of children receive free or
reduced cost lunch.
The patterns used consisted of four items that were shown in
a vertical or horizontal array, and one space where a missing
item should be placed to complete the pattern consisting. The
child was to select from four options the item which would
properly complete the pattern.
There were 12 rotation patterns, 12 time patterns, 12 letter
patterns and 12 number patterns. The rotation patterns consisted
of an object such as a bird that was rotated clockwise into one
of eight possible positions. One type of rotation problem fea-
tured a pattern that involved skipping single positions, e.g. the
bird was rotated from right side up to being horizontal, having
skipped being slanted at a 45 degree angle to the right. The
second type of rotation problem featured the object in question
skipping two positions, e.g. the bird started out right side up,
then skipped the 45 degree angle as well as being horizontal
and was presented at a 45 degree angle with its head facing the
lower right hand side of the page.
The time problems were constructed using clock faces. One
type of time problem showed clocks on which the time changed
by one hour and the other type of problem showed clocks on
which the time c hanged by two hours.
Letters were used to construct a third type of problem. The
letters used were all capital letters and as with the previous
problems there were letter problems that skipped a single letter,
e.g., P, R, T etc., and problems in which two letters were
skipped, e.g., B, E, H, etc.
Finally, the fourth type of problem was comprised of num-
bers 1 through 30. The number patterns included problems that
skipped a single number, e.g. 7, 9, 11 etc., as well as patterns
that skipped two numbers, e.g., 12, 15, 18, etc.
All of the aforementioned patterns were presented with the
missing item at the beginning, middle or end of the pattern. All
patterns were presented horizontally as well as vertically (see
Figure 1).
The patterns were presented to participants using a three ring
binder which held patterns that had been printed out in color, on
paper, and then placed into plastic protectors. Participants sat
on one side of the table and the researcher sat opposite them.
The patterns were presented to each child individually, and
each child had as much time as he or she needed to time to
select the correct answer from four the possible options given.
Because there was some concern about the attention span of the
children participating, the 48 pattern problems were given over
the course of two days; 24 problems were given the first day
and 24 problems were given on the second day. The order of
the presentation was counterbalanced by dimension, orientation
and the number of items the pattern skipped. The position of the
missing item was randomized within these constraints.
Descriptive statistics are presented in Table 1 and the analy-
sis is presented in Table 2. An ANOVA for completely
correlated factors was conducted, as every subject provided
data for every cell. There was a significant difference between
patterns that skipped one item and patterns that skipped two
items. There were no significant differences between the posi-
tions of the missing item in the patterns, and no significant
differences between the four dimensions, or the two orienta-
tions. The interactions of these characteristics were also non-
significant. There was a significant difference between patterns
that skipped one item and patterns that skipped two items.
However, the interaction of the number of items skipped and
the position of the missing item approached statistical sig-
nificance, making acceptance of the null hypothesis for this
comparison especially risky.
The finding that the four dimensions of time, rotation, letters
and numbers did not differ significantly was not surprising and
in fact supports a previous finding (Gadzichowski et al., 2010).
In that research, overall structure of the pattern proved to more
important than the type of items used in presentation. Although
one can never accept the null hypothesis in complete confi-
dence, in the case of patterning, it seems that the dimension in
which it is presented does not really play a primary role in how
easy or difficult a pattern is.
The fact that the orientation of the pattern (vertical pre-
sentation or horizontal presentation) was not significant is of
Copyright © 2012 SciRes. 1179
Copyright © 2012 SciRes.
(a) (b)
5 7 9 11 ?
12 13 15 10
Figure 1.
The examples are (a) a letters pattern, presented vertically, skipping a single letter, the middle item is
missing; (b) a rotation problem, presented vertically, skipping two positions, the missing item at be-
ginning; (c) a time problem, presented horizontally, skipping one “item” (one hour of time), the middle
item is missing; and (d) a numbers pattern presented horizontally, skipping a single number, the end
item is missing. The layout for this figure required some distortions; the patterns shown the children
were all equal in size.
Table 1.
Descriptive statistics for dimension, orientation, skip, and position.
Dimension Rotation Time Letters Numbers
Mean 1.528 1.534 1.529 1.529
SE .013 .013 .013 .013
Position First Middle End
Mean 1.528 1.545 1.516
SE .011 .011 .011
Orientation Horizontal Vertical
Mean 1.528 1.531
SE .009 .009
Skip One Two
Mean 1.544 1.516
SE .009 .009
Table 2.
ANOVA for dimension, orientation, skip, and position.
Factor df MS F p Partial Eta Squared
Individuals (I) 120 360.09
Dimension (D) 3 .01 .04 >. 05 .00
D × I 360 .24
Orientatio n (O) 1 . 01 .06 >.05 .00
O × I 120 .18
Skip (S) 1 1.38 4.76 <.05 .03
S × I 120 .29
Position (P) 2 .34 1.70 >.05 .02
P × I 120 .20
D × O 3 .15 .88 >.05 .01
D× O × I 360 .17
D × S 3 .16 .80 >.05 .01
D × S × I 360 .20
D × P 6 .11 .55 >.05 .01
D × P × I 720 .20
O × S 2 .11 .92 >.05 .01
O × S × I 240 .12
O × P 1 .15 .88 >.05 .01
O × P × I 360 .17
S × P 2 .44 2.59 <.10 .02
S × P × I 240 .17
D × O × S 3 .13 .72 >.05 .01
D × O × S × I 360 .18
D × O × P 6 .45 2.04 >.05 .02
D × O × P × I 720 .22
D × S × P 6 .29 1.61 >.05 .01
D × S × P × I 720 .18
O × S × P 2 .35 . 37 >.05 .01
O × S × P × I 240 .19
D × O × S × P 6 .51 1.11 >.05 .02
D × O × S × P × I 720 .46
interest since it would seem intuitive that patterns presented
horizontally would be easier than patterns presented vertically
because the children were all able to read in English and
English is read from left to right, rather than top to bottom. This
difference in orientation might at least be expected to play a
role in recognizing letter patterns, producing an interaction. But,
there was no interaction as well as no main effect. Perhaps this
is an area that merits further investigation. Significant diffe-
rences may not have emerged because there were only six letter
patterns presented vertically and six presented horizontally.
Those problems were not purely about vertical or horizontal
presentation but included other parameters such as different
position of the missing item as well as different numbers of
items skipped in the pattern. If there is in fact a significant
difference it may have been obscured. It also seems intuitive
that patterns comprised of numbers would be easier in the
vertical presentation since mathematics problems are usually
taught presented vertically as in the case of a typical addition or
subtraction problem. Again, the lack of a significant difference
may be due to having only six number patterns in each orien-
tation and the number of other factors involved. It may also be
true that, because opposite effects of orientation might be
expected for letter and number patterns, any differences may
have offset each other. Maybe future research should compare
letters and numbers presentated both vertically and horizontally
without any ext raneous factors.
The differences between patterns that skipped one item and
patterns that skipped two items were significant. This finding
may support the idea that patterning develops after the ability
for seriation but preceeds or occurs concurrently with the
Copyright © 2012 SciRes. 1181
mastery of transitivity. A child who is capable of solving
patterns that skip one item would be able to think beyond the
comparison needed in a seriation problem and may be able to
make the leap of comparison needed in a simple transitivity
problem. Problems that skip two items require an individual to
understand the relationship between the items presented and
mentally be able to fill in not just what would come im-
mediately after the last item but what would come even further
along in the sequence.
The position of the missing item (beginning, middle or end)
was not shown to be significant. However when the pattern
involved skipping two items rather than just one, the difference
between the positions of the missing item approached sig-
nificance (.10/p/.05). In the cases of problems skipping two
items scores were higher in absolute values for the middle
position than for the other two positions. It would be premature
to accept the null hypothesis in this case. The fact that these
findings approached significance makes them worth a second
look. Perhaps problems in which dimension as well as orien-
tation were not factors, allowing the children to focus solely on
the number of items skipped and the position of the missing
item would yield a significant difference.
In summation, the dimension in which a pattern is presented
is of no consequence, but the number of items skipped in a
pattern does make a difference. Patterning instruction should
therefore use any dimension and first teach children patterns
that skip a single item and then patterns that skip two or more
items. Further research should be conducted with patterns that
differ in only one or two characteristics rather than designs
which involve all factors. One possibility would be to vary only
the number of items skipped and the position of the missing
item, to determine how the number of items skipped in a pattern
interacts with the position of the missing item. Differences due
to orientation may emerge if patterns involving only letters or
only numbers are compared. Finally, differences due to such
factors might be compared with patterns that followed different
rules than those employed in this research.
Author Note
This research was conducted in partial completion of re-
quirements for the doctoral degree by K. Marinka Gadzicho-
wski. It was supported by grant R305A90353 from the Cogni-
tion and Student Learning Branch of the Institute of Education
Sciences, US Department of Education. The opinions ex-
pressed are those of the author and do not represent views of
the Institute of the US Department of Education.
The author express her appreciation for the assistance of Dr.
Robert Pasnak and the gracious cooperation of Dr. Monte
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