“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
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 firstname.lastname@example.org, 703 674 9494, fax 703 993 1359
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
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.
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.
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.
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.
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.
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.
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.
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).
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.
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).
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, SEab = 0.13, t = 4.1, p < .0001); the same significant effect was evident for reading achievement (ab=0.43, SEab= 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.
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