Examining Patterning Abilities in First Grade Children: A Comparison of Dimension, Orientation, Number of Items Skipped and Position of the Missing Item

doi:10.4236/psych.2012.312A174

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Psychology

2012. Vol.3, No.12A, 1177-1182

Published Online December 2012 in SciRes (http://www.SciRP.org/journal/psych) http://dx.doi.org/10.4236/psych.2012.312A174

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

Email: [email protected]

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

Introduction

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

K. M. GADZICHOWSKI

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

measure.

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

complicated.

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.

1178

K. M. GADZICHOWSKI

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.

Method

Participants

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.

Materials

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).

Procedure

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.

Results

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.

Discussion

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

K. M. GADZICHOWSKI

1180

(a) (b)

(c)

5 7 9 11 ?

12 13 15 10

(d)

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

K. M. GADZICHOWSKI

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

K. M. GADZICHOWSKI

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

Dawson and the participation of the Alexandria City Public

Schools.

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