**“Focusing on the Efficacy of Teaching Advanced Forms of Patterning on
First Graders’ Improvements in Reading, Mathematics, and Reasoning Ability.”**

*Robert Pasnak,
Ph. D. and Julie K. Kidd, Ed. D. *

George Mason
University

Cognition and
Student Learning Research Grant Program R305A090353

Project
Officer:* Dr. Carol O’Donnell*

Institute of Education
Sciences

Dr. John Q. Easton

U. S. Department of
Education

Drs. Robert Pasnak, Department of Psychology, CHSS, and Julie K. Kidd,
College of Education and Human Development, are completing a three-year project
supported by an IES grant, R305A090353 from the Cognition and Student Learning
Research Grant Program. The project was designed to test the feasibility
of improving student's academic achievement by teaching them age-appropriate
concepts that some children are slow to develop. This project is being
conducted in first grade classrooms of the Alexandria City Public School
system, with the encouragement of Dr. Monte Dawson, Director of Monitoring and
Evaluation. The project officer for this project is Dr. Carol O’Donnell.

Under the capable
guidance of site managers Marinka K. Gadzichowski, Deborah Gallington,
and Katrina Schmerold, the project was successful in
raising both reading and mathematics scores.

An Efficacy Test of Patterning Instruction for First Grade Children

Robert Pasnak,
Julie K. Kidd, K. Marinka Gadzichowski,
Debbie A. Gallington, Patrick McKnight, Caroline E.
Boyer, and Abby Carlson

George Mason University

Author Note

The
research reported here was supported by the Institute of Education Sciences,
U.S. Department of Education, through Grant 305A090393 to George Mason
University. The opinions expressed are those of the authors and do not
represent views of the Institute or the U.S. Department of Education.

Correspondence should be addressed to Dr.
Robert Pasnak, 3F5, George Mason University, Fairfax,
VA 22030 rpasnak@gmu.edu, 703 674 9494, fax 703 993 1359

Abstract

First grade students were given lessons on patterning, reading, mathematics, or social studies for 15 minutes per day, three days per week, for most of the school year. “Patterning” is instruction designed to help young
children understand simple, repetitive patterns of shapes, colors, numbers, or
letters, such as red, blue, red, blue, red, blue. In the present research this instruction was
extended to include symmetrical patterns, patterns with increasing numbers of
elements, and patterns involving rotation of an object through six or eight
positions. In May the children were tested for their mastery of patterns, reading, and mathematics. Factor analyses showed large effects of
patterning instruction on both reading and mathematics which were fully
mediated. There were no differences on some individual scales, but on others the children who received the patterning instruction scored best, often by grade equivalents of four months or more. Extant explanations of the efficacy of patterning instruction were reviewed, and the potential importance and limitations of its role in early elementary school education were discussed.

* **Key** **Words:** *Patterning, Reading, Mathematics, Achievement, First Grade

An Efficacy Test of Patterning Instruction for First Grade Children

**Overview**** **

In the introductory section, we describe briefly the practice of teaching children to understand patterns in early elementary school. We point out the linkages
between patterning and early mathematics that previous writers have
hypothesized, and also suggest a potential link to reading. Such empirical evidence as exists for the effectiveness of patterning instruction in producing academic gains is described. This section ends with a description of the experimental design used to determine whether extensive patterning instruction produced academic gains superior to those produced by instruction in other subject matters.

**Introduction**

**The Place of Patterning in
Education**

“If you ask any kindergarten
teacher, he or she is likely to consider the study of patterns to be an
essential part of the mathematics program” (Economopolous, 1998). This instruction often begins with colored blocks and is referred to as “patterning”. In early elementary school (K-1) children are usually taught *simple** **alternations* - e.g., ababab, wherein two elements alternate, and *double** **alternations*, aabbaabb, wherein pairs of elements alternate – and more complicated patterns (e.g., abcabc) as the unit of instruction continues.
It was recommended (National Council of Teachers of Mathematics; 1993, p. 2) that “children should focus on regularity and repetition … (and) be involved in recognizing, describing, extending, transferring, translating, and creating patterns.” The ubiquity of patterning instruction in the USA - and whether it was
advisable - was recently addressed by the National Mathematics Advisory Panel (2008).
It has been claimed that instruction in patterning leads to learning and mastery of abstract cognitive skills, including identifying differences and similarities of objects (Papic, 2007), identifying elements that repeat or change in number (Economopolous, 1998), detecting regularity (Scandura, 1971), and generalizing and abstracting relationships (Threlfall, 2004).
Ultimately, having such skills would be expected to translate into better understanding of mathematics, especially early algebra (Papic, 2007; Warren, Cooper, & Lamb; 2006). After summarizing relevant literature, Clements and Sarama (2004 p.
187) concluded that “geometry
and patterning are foundational for mathematics learning.” This idea seems reasonable, as patterning incorporates many changes in reasoning ability, especially distinguishing and ordering items and discovering rules that relate them, that occur as children make significant advances in thinking from ages four to seven years. There is substantial correlational evidence that some of these changes, expressed in the development of various forms of classification and serial ordering, are correlated with reading and mathematics achievement (e.g. Arlin, 1981; Dudek, Strobel, & Thomas, 1987; Silliphant, 1983; Waller, 1977). There is also some empirical evidence that patterning
instruction can improve kindergarteners learning of mathematics (Herman, 1973)
and first grade students’ learning of both mathematics and reading (Hendricks
et al., 1999; Hendricks, Trueblood, & Pasnak, 2006).

** Patterning and
early mathematics .** Educators
have advocated teaching alternating patterns to as a foundation for early
mathematics ( e. g., Papic,
2007;
Clements & Sarama, 2004; Threlfall, 2004; Warren, Cooper, & Lamb, al. 2006),
especially prealgebra (White, Alexander, & Daugherty, 1998; Clements & Sarama,
2007a,b,c). White et al. theorized that
such patterning was a stepping stone toward a very early form of analogical
reasoning, which in turn was directly related to mathematical learning. They
posited that this step consists in part of recognizing the pattern of relations
between various symbols and is best understood as a series of component or
elemental cognitive or metacognitive processes or
functions described by Sternberg (1977, 1981). Their analysis of children’s
performance on a test based on the processes of encoding, inferring, mapping,
and applying (Sternberg, 1981) provided support for the relation between
patterning and a primitive form of analogical reasoning, which in turn
predicted children’s learning of mathematics. Children’s ability to discern
patterns accounted for more variance than other aspects of numeracy, such as
comparing quantities and recognizing numbers. Hence, the ability to understand
relations between a pattern’s elements may be a precursor to understanding more
complex relations between numerical terms and expressions.

A related idea is that the repetition involved
in alternation patterns causes children to focus on the predictability of the
relationship between elements in a pattern, which can generalize to predictive
relations between numbers. When
kindergartners count by fives, 5,10,15,20, they learn that in this counting
sequence the ones digit alternates, 0,5,0,5.
Economopoulos (1998) hypothesized that when examining this pattern, so
similar to the patterns of shapes or colors they have learned, they may link it
to the base 10 number system and generalize that any number which is a multiple
of 5 ends in 5 or 0. Clements, Sarama, and Dibiase (2004 p. 27)
developed this idea, pointing out that there are numerical patterns in decades,
teens, and hundreds that parallel the integers 1, 2, 3, etc. Clements et al. (2004) suggested that a key
element in object-counting readiness is learning such standard sequences of
number words, learning that is facilitated by discovering patterns.

Patterning may also contribute to prealgebra because it is an early, age-appropriate form of
instruction in rules and relations (Threlfall,
2004). Clements and Sarama
(2007c, pp.507) theorized that “algebra begins with a search for patterns. Identifying patterns helps bring order,
cohesion, and predictability to seemingly unorganized situations and allows one
to recognize relationships and make generalizations.” When learning to apply a pattern rule to
patterns made of new and different elements a child learns the “algebraic
insight” – the understanding that a relation is not tied to particular concrete
items. Hence, “recognition and analysis
of patterns are important components of the young child’s intellectual development
because they provide a foundation for the development of algebraic thinking”
(Clements & Sarama, 2007b, p. 524). This idea is supported by Barody’s
(1993) report that kindergarten children name patterns, (e. g. a child might
spontaneously say of an alternating pattern “it’s an ABAB”). Barody noted that
this is a step toward algebra because it is using variable names for patterns
with different physical elements.

** Patterning and early reading.** While educators have focused on
early mathematics, Hendricks et al. (2006) found that patterning instruction *also *improved first graders’
reading. If this result is replicable,
it indicates that the thinking involved in understanding patterns may be
helpful for learning to read. Although there
are no published studies directly addressing how understanding patterns could
improve reading, Manning, Manning, Long, and Kamil
(1995) described steps in the development of reading by young children that
implied a potential role for appreciating patterns. They posited that children manifest a progressive development of the ability to recognize spatial and
temporal patterns of correspondences between written and spoken words. When asked to identify words in written sentences used to replicate something that had been spoken, kindergarten children never changed the word order. At first they attended to only nouns and verbs and did not appear to recognize functor words (prepositions, auxiliaries, or articles) as meaningful words. They did, however, preserve the pattern of nouns and verbs. Next, children began to recognize correspondences between the left –right spatial sequence of written words and the temporal sequence of spoken words. They initially failed to do this, however, with sentences that had more than one functor word. Manning et al. pointed out that this is a limitation in applying concepts similar to one that occurs with numbers.
Thus, Greco, Grize, Papert, and Piaget, (1960 pp. 23-24) reported that children start out applying number rules to only single digit numbers and then gradually extend application of the same rule to larger numbers. Children make similar progress with increasing numbers of functor words. Next they develop the understanding that every spoken word in a sentence should be present when the sentence is written down. However, at this stage they do not identify specific segments of the written sentence with the spoken words consistently. The next advance, which may well be influenced by an improved ability to recognize patterns, takes place when children focus on the pattern of the spoken words and recognize correspondences between the spoken words and the written words in a sentence. Manning et al. proposed that children move on to a higher stage when they identify segments of written sentences by adding add letter-sound correspondences to the pattern of the words. Hence, although Manning et al. did not address patterning, they may have provided a rationale for relating patterning to reading.

* ***Empirical
studies of patterning instruction.*** *If there are relations between understanding patterns and early reading or mathematics, there remains the empirical question of whether it is causal and whether patterning instruction improves academic achievement. Good understanding of patterns may, like any aspect of intelligence, be correlated with analogical reasoning, prealgebra, or reading, but it is incumbent on researchers to show that improving understanding of patterns through instruction actually improves a child’s academic achievement. There seem to be only three empirical studies of the practical effects of patterning instruction *per se*. Herman’s (1973) dissertation research involved giving African American and Latino/Hispanic kindergartners from disadvantaged backgrounds 24 lessons on simple alternation and double alternation patterns. She found small but statistically significant gains on mathematics achievement measured by
the Metropolitan Readiness Test . These were limited to the children who spoke English as their native language. Much later, Hendricks et al. (1999) conducted a multiple baseline study using four boys (median age 79 months) who were English language learners. The boys were taught 30 class inclusion problems and 400 patterning problems. They showed gains on class inclusion and patterning which were accompanied by gains averaging 9.1 points on a measure of general cognitive functioning (the Slosson Intelligence Test, or SIT) and 30.8 points on total academic achievement as measured by the Diagnostic Achievement Battery-2 (DAB-2). However, the design of the study does not permit separation of the academic gains from the general educational experiences and maturation of these children. English language learners might well be expected to make significant gains over the course of a year on tests administered in English.

In a subsequent experiment with random assignment of 62 children to two groups, Hendricks, Trueblood, and Pasnak (2006) showed that instruction on 480 patterns did benefit first grade children (i.e. 7-year-olds). The patterns were presented in a variety of media to increase generalization – an approach Warren et al. (2006) has independently recommended
– and ranged from simple linear orderings on one dimension to multidimensional sequences presented as matrices of colors, letters, numbers, animal stickers, clock faces, and shapes. The instruction began with
simple alternations, but patterns became progressively more difficult in length, number of dimensions, number of items, and number of missing items. The first graders in the control group were taught academic material recommended by their teachers. When posttested the children in the groups did not differ significantly on the DAB-2. Hendricks et al. (2006) attributed the absence of a significant instruction effect to a 5-point superiority of the control group on the Slosson Intelligence Test (SIT), which was used as a dependent variable reflecting patterning
gains in Hendricks et al. (1999). Small but statistically significant advantages for the experimental group on the DAB-2 Reading, Mathematics, and Total Achievement measures emerged when SIT scores for the groups were equalized via MANCOVA.

**Justification
for the present study.** In sum, Herman (1973), Hendricks, et al. (1999), and Hendricks et al. (2006) produced only limited support for the effectiveness of patterning instruction. The multiple
baseline design used by Hendricks et al. (1999) was not a design that could
show that the mix of class inclusion and patterning they employed was what
produced the achievement gains observed at the end of the year. Neither Herman (1973) nor Hendricks et al.
(2006) found significant differences between the means for their patterning and
the control groups in their a priori analyses.
Herman (1973) was able to show a significant difference only after discarding
the data from more than half of her subjects.
Hendricks et al. (2006) were able to show a significant difference only
after magnifying the difference between patterning and control children via a
post hoc MANCOVA based on IQ. While it
is certainly plausible that an IQ difference diminished the difference in
achievement by the groups, that is unproven, and MANCOVA is a type of analysis
that has been termed “delicate”, *especially
*when groups differ systematically in the covariate (Myers & Well, 2003,
p. 432), as was the case for Hendricks at al. (2006). Even after the compromises in analysis that
Herman and Hendricks made to produce positive results, the effects they could
show were small.

Patterning is nevertheless widespread, strongly endorsed by teachers (e. g., Economopolous, 1998) and incorporated in carefully designed curricula (Clements & Sarama, 2004). However, patterning is not among the subject matters recommended by the National Mathematics Advisory Panel (2008). That panel has recommended streamlining mathematics instruction in the early grades to emphasize the most critical topics, which would reduce or even eliminate instruction in patterning (NMAP, 2008 p. xiii). Such a reduction may be justified because the empirical evidence for the effectiveness of patterning instruction provided by Herman (1973) and Hendricks et al. (2006) dissertations is rather meager. Hence, it is important to develop more conclusive evidence as to whether patterning instruction produces important academic gains.

The complexity of such an endeavor is daunting. Patterning is taught from preschool through first grade, and there are many ways of teaching it to children of widely varying backgrounds and abilities. There are many types of patterns, which may or may not involve more than one dimension, and which differ in the complexity of the pattern rule. No one research program will be able to
resolve all of the issues involved in patterning instruction.

As a start, we investigated the use of a teaching method that involves little skill and should be easily replicable. The approach was to teach children to fill in gaps in six types of one-dimensional patterns. These stimuli avoided the complexities of multiple dimensions while reducing the likelihood of nonconceptual solutions. Although patterning can be taught at earlier ages – the National Council of Teachers of Mathematics (1993) recommended that patterning be a focal point in preschool and kindergarten curricula - we targeted first graders who had not mastered these patterns as well as most of their classmates. Clements and Sarama
(2007c p. 504) suggested that without intervention kindergarteners who lagged
behind their peers in patterning would later have trouble in mathematics and
school in general. The same might well
be true – or even more true – of children who were *still *lagging in patterning when in
first grade. Hence
we
expected that children who were
behind their peers in patterning were those who might benefit most from special
instruction in it, that patterning would be well within these first graders’ zone of proximal development (Vygotsky, 1978), and that replicating this feature of the Hendricks et al. (2006)
study was valuable because those researchers reported both mathematics and
reading gains. These were largely executive decisions; a fuller understanding of the effects of teaching patterning awaits experiments with different parameters.

**Experimental Design***.* We randomly assigned children to experimental and control groups, conducted patterning and control instruction for most of the school year, and employed several standardized tests of reading and mathematics. A conventional approach would be to have an experimental group taught patterning and a control group taught nothing or experiencing ordinary classroom activities. However, Pasnak and Howe (1993, p. 232) argued that a better control is an “active” control group - one that receives as much attention and investment of resources as the experimental group. Such control groups, receiving the same number of special sessions as the experimental group, on constructive but different academic material, control for familiarity, Hawthorne, and expectancy effects better than conventional “business as usual” controls. Accordingly, we randomly assigned children to one experimental group to be taught patterning and three active control groups to be taught reading, mathematics, and social studies, respectively.

The first control group was to have sessions of instruction in reading, including phonics, matched in timing and extent with those of the experimental group. Because they received extra instruction specifically on reading, these children should score better on reading measures than those in the mathematics and social studies control groups. The experimental group should also improve in reading if the patterning instruction helped them develop a foundation for better understanding of reading in the course of regular classroom instruction. This is suggested by the analysis of early
reading by Manning et al. (1995), and is in line with the results of Hendricks et al. (2006). If similar results are found here, it is critical evidence that patterning facilitates progress in reading.

The second control group was to receive instruction in mathematics in sessions matched in timing and extent with those of the experimental group. This group should prove inferior to the experimental group and the reading control group on tests of reading if the patterning and reading instruction affect reading. However, year-end scores on mathematics measures should exceed those of the reading and social studies groups if the mathematics instruction is effective. Likewise, better understanding of patterns should result in better performance in mathematics according to White et al. (1998) and many educators (e.g., Clements & Sarama,
2007b; Economopolous, 1998; Papic,
2007).

The third control group was to be directed on social studies projects, which in the first grade of this school system consists primarily of cutting, pasting and drawing on worksheets and making collages. This control group gives a comparison for
the effectiveness of the patterning instruction in both reading and mathematics, as the social studies instruction conveys little advantage in patterning or in the verbal or quantitative spheres - certainly less than instruction directly on patterning, reading or mathematics. This is the control group that most researchers employ, and gives a baseline for measurement of the effects of the patterning instruction and both of the other two forms of control instruction.

In sum, these four groups give a comprehensive picture of the efficacy of the patterning instruction, while controlling for artifacts and equalizing investment of resources. The patterning
and reading instruction should produce better reading than mathematics or
social studies instruction, and the patterning and mathematics instruction
should produce better mathematics performance than reading and social studies
instruction. We hypothesized that the effects of
patterning instruction on reading and mathematics achievement would be mediated through the specific gains in patterning ability. Thus, we expected a mediation model whereby instruction led to proximal changes that resulted in more substantive performance changes. We also hypothesized
direct effects of the control reading instruction on reading and the control
mathematics instruction on mathematics.

**Method**

**Participants**

Parental consent to
take a screening test was obtained for 443 first-grade children enrolled in the public school system of an urban school district in a metropolitan area in the mid-Atlantic
region. Many of
the children in this system were from immigrant families, many lived in
subsidized housing, and 57% received free or reduced lunches. The screening test was administered,
and parents of the eight children who
scored lowest in each of 16 classrooms were asked to sign letters of informed consent allowing
their children to participate in the research. Ten families declined, so the parents of
ten additional low scoring children were asked, and agreed, supplying the
desired sample of eight children per class.
In following this selection procedure, we recognized that there was no
sensible absolute criterion for performance on our screening test, and that we
wanted to avoid imbalance by having each teacher have two children in each
instructional condition. Some classes would inevitably have better performing
children than other classes, and having the same number of children from each
class in each condition avoided confounding the design.

The
rationale for selection of children who scored poorly on the screening
test was
that patterning instruction was likely to be most
fruitful with children who did not already have a good mastery of patterning. Such
children might have more potential for improvement in patterning and hence for improvement in academic abilities supported by patterning, than those who already had a good mastery of patterning. After attrition 120 children remained, 64 boys and 56 girls. Of these, 52 (43%) were African American, 42 (35%) were Hispanic/Latino, 16 (13%) were Middle Eastern, 3 (2.5%) were Caucasian, and 7 (5.8%) were of an unspecified ethnicity. The
mean age
for these children was 6 years 5.19 months, SD = 3.36 months.

**Instructional Materials**

The patterns taught were single and double alternations,
symmetrical patterns, progressive patterns involving increasing numbers of elements, sizes, or values, rotation patterns, and random repeating patterns (See Figure 1). The elements of random patterns did not have an underlying functional relationship. The
patterns were presented on note cards, minicomputers, and whiteboards and with manipulatives.

Coins, manipulatives, minicomputers,
number cards, Whispy Readers (small curved tubes that a child held to his or her mouth and ear like an old fashioned telephone receiver while reading), children’s poems, puzzles, maps, mazes, cut-and-paste materials and various activity pages were used in instructing the children in control activities.

**Tests**

The screening test consisted of 12 patterns presented on flip charts. These patterns all consisted of 5-item sequences of letters, numbers, time (clock faces) or rotation of an object in which either one or two steps in rotation or value might be skipped; e.g. _?_ 14,
17, 20,
23, or 12
O’Clock, 2 O’Clock, 4 O’Clock, 6 O’Clock, ____? or D, G, _?_, M, P (See
Figure 2.) The
missing item
that the
child was
to identify was equally often the first, middle, or last one in the sequence. Patterns were presented in horizontal or vertical orientations,
and four
alternatives, also presented horizontally or vertically, were given from which the child was to select the item missing from the pattern.

A test for far generalization (administered
at the
end of
the year) had twelve pattern problems,
six using dice and
six using playing cards. The dice and the playing cards were laid out on a table top in two different types of patterns, in a counterbalanced
order. One type of pattern was a symmetrical pattern that skipped two numbers, e.g. a die showing two pips, a die showing four pips, a die showing six pips, a die showing six pips, a die showing four pips and a die showing two pips. The
second type
of problem featured two sequences combined into a single ascending pattern, e.g. 1, 3, 2, 4, 3, 5. In the case of the playing card patterns, a single suit was used in each problem.

The measures of school achievement were standardized tests. Three were tests of achievement in reading - the Test Of Word Reading Efficiency or TOWRE, the Gray Oral Reading Test 4,or GORT, the Test of Early Reading Ability-3 or TERA - and three were tests of mathematics achievement -the Woodcock-Johnson III Math Concepts scales A and B, (W-J 18A and W-J 18B) and the Key Math 3 test.

According to the manual, the TOWRE’s reliability is exemplary - coefficients
range from
.90 to
.99. Concurrent validity with another widely used reading test, the Woodcock Reading Mastery Tests – Revised was .85 to .89, and predictive validity correlated with the GORT- 3 scores in the .75 to .80 range.

GORT-4 reliabilities
are also
high, ranging from .85 to .95 on test retest comparisons and .91 to .97 on content sampling. Validity inferences can be drawn from comparisons with six other standardized tests; the median coefficient for the oral reading quotient is .63.

Similarly, TERA-3 reliabilities range from .83 to .95. Concurrent validity coefficients are lower but still respectable:
correlations with the SAT-9, WRMT-NU/R, and teacher judgments range from .40 to .66.

The Woodcock-Johnson
III (W-J) is perhaps the best regarded and most widely used test of young children’s academic achievement and has the largest standardization
sample of
any individually administered
achievement test. The Mathematics Concepts scales (18A and 18B) measure understanding of mathematics. These scales have shown significant differences with substantial power coefficients
(.53 to
1.43) for
smaller samples in local school systems. The manual (McGrew & Woodcock, 2001) gives a reliability coefficient of .84 for 7-year-olds. Mather and Gregg (2001) reported reliability coefficients over .80 for each scale. Convergent validity coefficients of .71 and .64 with the Diagnostic Assessment System, .68 and .70 with the Wechsler Individual Achievement Test, and .62 and .66 with the Kaufman Test of Educational Achievement are given for the verbal and quantitative scales in the manual.

Median reliabilities
for test-retest reliabilities
on the
Key Math
subscales is .86 for younger examinees and reliability of the Total Test score is .97. Convergent validity with the KTEA II was .67 to .75 in one study and ranged from .66 to .80 with the ITBS.

In sum, these are widely used standardized tests that are respected by educators, even though none is perfect.

**Procedure**

** Overview. **** **Children were given the screening test individually in October. The eight children in each classroom who scored
lowest were selected for the research. A random numbers table was used to assign two to patterning instruction, two to reading instruction, two to mathematics instruction, and two to social studies
instruction.
These children were taught whatever they were assigned to – either patterning or mathematics or reading or social studies – for 15 minutes three times per week during “centers time,” an hour or so devoted to individualized
or small group activities, from November through April. The order of instruction was counterbalanced
except as
interrupted by absences or special events, so that teachers engaged in each form of instruction first, second, third, or fourth equally often.

In May, school psychologists,
who were
blind to
the condition to which children had been assigned, re-administered
the original screening test, and gave seven more tests to each child - the GORT, the TOWRE, the TERA, the W-J Math Concepts scales A and B, the Key Math test, and the “far generalization”
patterning test.

** Patterning Instruction.** Patterns, which were single and double alternations, symmetrical patterns, progressions
with increasing numbers of elements, sizes, or values, rotations, and random repeating patterns, were displayed on note cards,
white boards, table tops, or minicomputers.
Each pattern had a missing element in the beginning, middle, or end of the pattern. Each
problem displayed four options for completing the pattern,
and the children were to
identify the option that completed the pattern. Performance was scaffolded through explanation and repetition until each child was able to demonstrate mastery of each pattern by
selecting the correct option on their first attempt on three consecutive sessions.

In addition to identifying the missing element in a pattern, children were taught to use manipulatives (small objects) to extend patterns. Teachers would start a pattern, provide the children with more manipulatives, and request that they complete or extend it. Children were also asked to create patterns to be completed by the teacher or another child. White boards were also employed for these purposes. The
only difference was that patterns were drawn on the boards instead of being made from manipulatives.

**Mathematics**** ****Instruction.** Each mathematics lesson featured a different kind of activity, such as counting by fives and tens, addition, recognizing and naming shapes, and understanding
simple fractions. First the teacher did a brief assessment of whether the children had
the fundamental abilities needed to do the chosen task. If
needed, there were fall-back or jump ahead options so that the teacher could match the activity to the best starting point for that day. After the day’s instruction had been accomplished,
it was
concluded with a task or question addressing the overarching point of the activity.

There were necessarily many math activities during the school year, and they were very variable. An example of an activity is counting. A session began with children quickly counting to 100 as a review. This
was followed by a task wherein the children were to pick up in order cards numbered 1 to 100, which had been spread out in front of them in a scattered, disorganized array. The
activity could be made easier by reducing the number of cards to 25 or even ten, or extended by asking the child to pick up the cards in reverse (decreasing) order. The
teacher would direct and scaffold as necessary. If
this was
too difficult, the number cards 1 through 20 were used. The final activity was to put the cards away in deciles, i. e, first collecting all those between 0 and 10, then all those in the teens, then all those in the 20s, etc.

**Reading**** ****Instruction.** A
brief children’s poem with a targeted end rhyme (e.g. -own) was the focus for each week’s three sessions. The sessions began with a minute or so of discussion to put the children at ease and improve conversational skills. Then
each child would read aloud the poem that had been the focus the previous week, using a Whispy Reader to avoid disturbing the other children. This
reading of
material already covered was designed to improve comprehension and fluency and to teach sight words and decoding. The
teachers helped the children
as much
as needed, and queried them with questions about the familiar poem they were reading to solidify their comprehension of it. This took about three minutes.

The next six minutes of the session were devoted to the week’s new poem, and varied according to the three sessions for that week. On
the first day, the teacher read the poem aloud and talked to the children about what they had just heard, trying to improve comprehension. On the second day, the teacher and children read the poem together. The
teacher emphasized fluency and discussed unfamiliar words in the poem to improve the children’s vocabulary. On the third day, the children read the poem alone as well as they could and brief discussion and questioning were employed to improve comprehension, fluency, and vocabulary. Rhyming word flashcards which had
the same
end sound as the week’s poem were then
used for
four minutes in a phonics activity. The children were helped to recognize the identity between the end sounds in the poem and the end sounds of the words on the cards.

Each session ended with a minute spent in by the teacher and child summarizing what had been attempted and accomplished
during the
session.

**Social**** ****Studies**** ****Instruction.**** **Social studies activities changed daily and featured a variety of activities that highlighted civics, geography, and important people and events in history. The
instructor would join the children in different activities such as coloring activity sheets, making collages, and so forth.

**Results **

There were no significant differences among the four groups on the number of items correct on the screening test -*F*(3,117) = 0.24, *p* >.05 – indicating that random assignment created an equivalent groups design. ANOVA showed that differences
between the dimensions were trivial, *F*(3,357) = .02, *p*
>.05. This echoed the findings of Gadzichowski, Kidd, Pasnak, &
Boyer (2010) for similar patterns. There
were also no significant differences for the orientation of the patterns,* F(* 1,119 )= .04, *p*
> .05, or the position of the missing item, *F*( 2,238 )= 1.69, *p*
> .05.* * Hence these variables were
collapsed. We subsequently assessed
performance on the individual scales, followed by a factor analysis for the
achievement variables and by mediation analyses to determine whether
achievement was mediated by patterning.

**Individual
Scales **

Results for individual test scales
foreshadowed the results for the analyses of the more powerful composite
measures. The patterning group was
significantly better than each of the other groups on the patterning posttest
and the patterning far generalization test (See Table 1). On the TOWRE Word and TERA
measures of reading, the patterning group made the highest scores in an
absolute sense, but the patterning and reading groups did not differ
significantly. Both made significantly
higher scores than the other two groups.
On the GORT, the only differences were that the patterning group was
superior to the mathematics and social studies groups. There were no significant differences on the
TOWRE phonemics scale (See Table 2).

The
patterning and mathematics groups were both significantly better than the other
groups on the W-J Mathematics Concepts Scales (See Table 3). Mean scores for the patterning group were
always higher than for the other groups on the Key Math scales, and the
differences on all except Geometry and Multiplication were significant (See
Table 4).

**Composite Scores**

We opted to reduce the achievement variables to two common factors – mathematics and reading outcomes - by using a standard unit-weighted factor score (Morris, 1979). These scores tend to maintain predictive validity found in standard exploratory factor analyses but they improve generalizability of the findings (Grice & Harris, 1998). We
grouped test
scores according to their relevance on either mathematics or reading, standardized
the scores (z-scores), and then computed the mean standardized
scores to
serve as
factor score estimates for the two outcomes. The
internal consistency of each factor – as indicated by Cronbach's alpha – was 0.91 and 0.52 for mathematics and reading outcomes, respectively. The
low internal consistency for the reading outcomes might be attributable to the fact that there were fewer reading-specific outcome scores (4) compared to mathematics-specific outcome scores (12). Nevertheless, the factor scores reflected the common factor underlying both constructs and reduced the data sufficiently
to permit efficient hypothesis tests. These factors ought to produce more generalizable results than separate tests on each
mathematics and reading scale while also increasing statistical power and creating a simplified set of findings.

** **Figure 3
shows mean differences on the
patterning pretest, mediator (total
number correct on the patterning posttest and the far generalization test, i.e., Patterning Posttest Scores) and outcome variables (i.e., Mathematics Performance and Reading Performance). Each group is shown with their associated standard error bars for all four measures.

**Mediation**** ****Analysis**

We conducted two separate mediation analyses for the factors previously identified, one analysis for the
reading factor and one for the mathematics factor, using standard bootstrapped
regression models (1000 samples). Due to restrictions in regression on handling nominal-level
predictors, we recoded group assignment to reflect patterning instruction (1) or other instruction (0). Additionally, we chose to bootstrap the results because that additional step provides more robust hypothesis tests when computing indirect effects (Preacher & Hayes, 2008). Reported parameter estimates for all mediation model results reflect mean estimates from the bootstrapping
procedure; thus, the standard errors are more akin to population estimates rather than to corrected estimates. Population estimates, therefore, allow us to use z-scores and z-tests (i.e., z = b/se). Results from the bootstrapped
procedure are reported with the customary
95% critical z-score value of 1.96. Indirect (i. e., mediated) effects were computed using the standard Baron and Kenny (1986) method along with the Sobel test (Sobel, 1982); all results reflect the bootstrapped estimates. In addition to the mediation models, we conducted
several tests of simple effects that were specified *a priori* based upon
the rationale behind our design. We
hypothesized that both patterning and mathematics instruction would produce
significant differences in mathematics outcomes compared to reading and social
studies instruction. We also
hypothesized that patterning and reading instruction would produce significant
differences on the reading compared to the mathematics and social studies
instruction. These specific, simple
effects were tested using Bonferroni protected
t-tests. All analyses were conducted in the statistical package R (R Development Core Team, 2011).

**Mediation**** ****Models**

The two mediation models produced largely similar results. First, both models showed a significant indirect effect,
supporting the mediation
hypothesis. The effect between patterning instruction and mathematics achievement showed a significant indirect effect through the patterning posttest variable (ab = 0.45, SE_{ab} = 0.13, *t *=
4.1, *p* < .0001); the same significant effect was evident for reading achievement (ab=0.43, SE_{ab}= 0.12, *t *= 4.0, *p* < .0001). Figures 4 and 5 show the significant direct and indirect effects for both models. Both
models show
that patterning instruction was associated with a positive change in patterning posttest performance. Additionally, the patterning posttest
performance was positively associated with both mathematics and reading achievement. In both cases, the direct effect was nonsignificant, indicating that the effect of patterning was fully mediated.

**Simple Effects** The Bonferonni-protected t-tests produced significant results
for our simple effects. We report exact *p*-values but the critical value for 95%
confidence intervals was .005 (.05/10) based upon the previous mediation models
and additional simple effects tests. The
patterning and mathematics treatments produced a significant difference in
mathematics-specific outcomes compared to the reading and social studies
treatments, *t*(118)
= 4.1, *p* < .00001. The means for the instruction effect for each
group were .25 (SD =.69) for the patterning and mathematics children and -.25
(SD =.65) for the reading and social studies children, respectively. Those significant mean differences translate
to a large effect size (d=0.75).
Additionally, patterning and reading instruction produced a significant
difference in reading-specific outcomes compared to the math and social studies
instruction, *t*(118)
= 4.6, *p* < .00001. For this second instruction effect, means and
standard deviations were .24 (.60) and -.25 (.58) for the patterning and
reading children and mathematics and social studies children,
respectively. Those significant mean
differences translate to an even larger effect (d=0.83). The direct effects of instruction, therefore,
were significant and had large effect sizes.

**Discussion**

This
study indicates that year-long instruction of first-graders on patterning,
beginning with alternations and extending to more complex patterns, can improve
academic achievement. There were
substantial mediation effects for patterning on composite measures of reading
and mathematics. The mediation analysis
shows that not only were the composite reading and mathematics scores higher
for the children taught patterning, they were predictable from how well the
children learned patterning. Many of the gains were large enough – 2 to 8 months in grade equivalencies
– to be
very welcome to educators charged with improving academic achievement. Tables 1-3 show that the children receiving patterning
instruction fared well on each individual scale. This is the best evidence to date that
patterning has a place in the instruction of young children. Consequently, we looked very
closely at
the experimental design to determine whether the advantage of the patterning instruction could be due to some difference between the groups other than the instruction they received.

In the first place, children were randomly assigned to the four conditions, which should equalize the children in these conditions in all characteristics, *known** **and** **unknown*. The
scores of
the children on the screening test show that the randomization left the four groups very equivalent
in patterning. We did not pretest on reading or mathematics,
which would have increased the power of the analysis at the cost of introducing the test-treatment confound (Kerlinger, 1986, p. 311). However, the statistical analyses have
sufficient power to show that the probability
is very
small that
a failure of randomization accounted for the differences in the scores obtained by failing to equalize the groups in reading, mathematics or any other variable, known or unknown, at the outset. That
is a
second, scientifically accepted reason for discarding the idea that the any particular group was inherently a better group of children.

We note that the same number of children in each classroom was assigned to each group, equalizing classroom and teacher effects, and that the testers were blind to the type of instruction each child received. Hence, it seems reasonable to conclude that, as
shown by both mediation analyses, the superiority the patterning children showed on the reading and mathematics measures was due to their ability to recognize and interpret patterns.
Hence, what
has been
a consensual practice among educators was supported by the empirical data from this study. Instructing first graders in recognizing patterns led to improved understanding
of novel patterns they had not been taught, as shown on the far generalization test. This is an
indication that this aspect of thinking had been improved. More
pertinent, perhaps, is the finding that mastery of patterning mediated performance on standardized tests of reading and mathematics. It
would be
the fact
that patterning improves performance in both spheres, reading and mathematics, that makes it a valuable form of instruction.

It
is critical to remember that all of the children, including those in the patterning and social studies groups, experienced a curriculum rich in mathematics and reading instruction all day long throughout the school year. In these children’s classes, there were four
critical areas for instruction in mathematics.
These were developing their understanding of (1) addition, subtraction,
and strategies for addition and subtraction; (2) whole number relationships and
place value, including grouping in tens and ones; (3)
linear measurement and measuring lengths as iterating length units; and (4)
attributes of geometric shapes. The grade one content placed emphasis on number
sense with counting, sorting, and comparing sets of up to 100 objects.
Fractional concepts were expanded from halves and fourths to thirds. Students used nonstandard
units to measure length, weight, mass, and volume. They also investigated data collections and
organized and interpreted the data.

All children were also immersed in an environment rich with high-quality print of all kinds (posters, signs, notes, songs, poems, books from different genres and a listening center/computer station). They were helped to develop oral language skills, phonetic skills, phonemic awareness, and comprehension skills and to significantly increase their reading fluency and vocabulary knowledge. The year progressed from narrative fiction to nonfiction to fiction, and ended with persuasive texts and poetry. The children studied fiction to develop a thorough understanding of the elements of story including character, setting, problem and solution, as well as the ability to retell a story. Nonfiction instruction emphasized understanding of the differences between fiction and nonfiction and a basic knowledge base regarding nonfiction text features including the title, table of contents, captions, headings, pictures and photographs, diagrams, charts, and captions. Children were also helped to develop the ability to use pre-reading strategies such as previewing, predicting, and setting a purpose for reading.

Inasmuch
as there is little direct connection
of patterning with the goals of the classroom instruction described above, we think it unlikely that patterning instruction in isolation
could have
produced mathematics or reading
gains. However, it appears that an improved ability to recognize and understand patterns – i.e., what follows what – if
embedded in ongoing classroom instruction,
improves understanding of that classroom instruction.

The description of the early development of reading offered by Manning et al. (1995) could be adapted to explain the effects of patterning instruction on reading. Understanding a pattern requires understanding the sequence of items in the pattern, i.e., what follows what. When the beginning readers described by Manning et al. came to understand that the order of spoken words in a sentence is the same as the order of the written words, so that they knew what follows what, they showed the kind of understanding involved in patterning. The fact that such children never change the word order of the written words from that of the spoken words suggests that they indeed hew to the pattern of the spoken words. The account provided by Manning et al. was not tested in this research; it remains a theory, or a mini-theory. However, some explanation is needed for the improvements in reading shown here and foreshadowed by Hendricks et al. (2006). The account