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HOW DO CHANGES HAPPEN? TRANSITION FROM INTUITIVE TO ADVANCED STRATEGIES IN MULTIPLICATIVE REASONING FOR STUDENTS WITH MATH DISABILITIES Dake Zhang Purdue University [email protected] Casey Hord Purdue University [email protected] Yan Ping Xin Purdue University [email protected] Luo Si Purdue University [email protected] Ron Tzur Purdue University [email protected] Suleyman Cetintas Purdue University [email protected]

This study investigated how students with mathematics learning disabilities (MD) or at-risk for MD developed their multiplicative reasoning skills from intuitive strategies to advanced strategies through a teaching experiment. The participants consisted of two fifth graders with MD and one at-risk. A micro-genetic approach with a single subject design was used. Investigators coded and analyzed five strategies children used. Results showed that the participants had fewer strategies than normal-achieving students, but they improved their performance throughout the teaching experiment. The participants increased their use of double counting and direct retrieval, and decreased their use of unitary counting during the intervention. Approximately 5% to 8% of school aged children have math disabilities (MD), as defined by poor performance in class and poor standardized test scores (Geary, 1990). Individual variability is one of the most striking features of children's reasoning (Siegler, 2007). Low achieving students usually have fewer strategies than high achieving students, and use less advanced strategies more frequently than high achieving peers on a variety of reasoning tasks (Siegler, 2007). The purpose of this paper is to explore how students with MD develop their strategies of multiplicative reasoning through a teaching experiment. Framework Studies on children's strategic development help people understand how students with MD gradually fall behind their peers by comparing the strategic development of children with and without MD (Geary, 1990). While developing additive reasoning, for example, children normally progress from "count all" (i.e., they simply count the two addends from 1) to "counting on" (i.e., they start counting from the first addend) and "count large" (i.e., they count from the larger addend), and eventually to verbal retrieval (Siegler & Shrager, 1984); however, although both students with and without MD equivalently develop various counting strategies at first and second grades, only students with MD continue to have difficulty in shifting to retrieving correct answers after third grade (Geary, 1990; Geary & Brown, 1991). However, children's strategic development for multiplication is still much less understood than for addition (Lemaire & Sielger, 1985; Kouba, 1989), especially for students with MD. In multiplicative reasoning, one composite unit is distributed across the other, and children need to be able to coordinate the two quantities (Steffe, 1994). Although normal achieving children in

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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kindergarten can solve some multiplicative problems by directly modeling the problem context and counting all the items one by one (Downton, 2008; Kouba, 1989; Mulligan & Mitchelmore, 1997), it takes time for them to establish a real conceptual understanding of multiplicative reasoning, which is an invariant relationship between two quantities (Piaget, 1965; Vergnaud, 1983). Early studies (Anghileri, 1989; Brown, 1992; Kouba, 1989; Mulligan & Mitchelmore, 1997; Steffe, 1988) have identified a variety of strategies normal-achieving students use for multiplicative reasoning. These studies have also provided evidence that children's solution strategies begin generally with direct modeling and unitary counting; progress to skip counting, double counting, repeated addition or subtraction; and then, to the use of known multiplication or division facts (Downton, 2008). The general strategic developmental pattern is demonstrated in Figure 1 according to Kouba's data.

Figure 1. Normal achieving students' multiplicative strategic development (Kouba, 1989). Direct representation and skip counting are two strategies of unitary counting, as they address only one number/counting sequence. Direct representation is at the most basic level (Anghileri, 1989). Kouba (1989) described direct representation as an activity where "children used physical materials to model the problem and some form of one-by-one counting in calculating the answer" (p.152, Kouba). Skip counting is a strategy in which children count by multiples, such as counting "five, ten, fifteen, twenty, twenty five, thirty" for solving "six groups of five" (Kouba). However, skip counting does not suggest a child is able to coordinates the two quantities by tracking two counting sequences. An indicator that students are not coordinating the two quantities is that children often do not know where to stop counting (Kouba). Children's shifting to double counting is a milestone of their development of multiplicative reasoning. Double counting indicates the transition from a unitary counting stage to a binary counting stage (Vergnaud, 1983), where children explicitly keep track of two quantities while counting two number sequences. For example, double counting occurs when a child count "1, 2, 3, 4, 5" with one hand, then counts "1" with the second hand; and then the child continues count "6, 7, 8, 9, 10" with the first hand, then counts "2" with the second hand. Double counting is "an advance over the more basic direct representation because it requires more abstract processing and involves integrating two counting sequences" (p.152, Kouba, 1989). However, no study has investigated whether or not children with MD have double counting strategies. Afterwards, additive or subtractive strategy occurs when the child exhibits use of repeated addition or subtraction to solve a problem (Kouba, 1989); for example, a child clearly states:

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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"three plus three plus three plus three" to solve "three times four." And eventually, normal achieving students shift to recalled number facts, which is the highest strategy people use. Kouba explained that this strategy is being used when "the child obtained the answer by remembering the appropriate multiplication or division combination" (p.153, Kouba). Unfortunately, little research has explored if children with MD have the same problems in direct retrieval in multiplication as in addition. Little is known about how students with MD make their transition from less advanced strategies to advanced strategies in multiplicative reasoning. As such, the purpose of this paper was to explore how a teaching experiment effected on the multiplicative reasoning strategic development for students with MD. Specifically, (1) what strategies students with MD or at-risk used in pretests; (2) did the teaching experiment improve students' performance in solving multiplicative problems; (3) how did strategic development occur during the teaching experiment? Method Design Micro-genetic studies often employ single subject designs (Siegler, 2006). An adapted multiple probe design (Horner & Baer, 1978) across participants was employed in this study to establish a functional relationship between the teaching experiment and students' performance and strategic changes. Specifically, when a stable baseline was observed for one student, treatment was introduced. When improvement for Child A was observed, Child B was introduced to treatment. And when improvement for Child B was observed, Child C was introduced to treatment. In this design, replication of treatment effects is demonstrated if changes in performance occur only when treatment is introduced. The independent variable was the sessions (the pretests and the teaching experiment). The primary dependent variable was students' strategy use across the sessions. Students' performance in solving multiplicative reasoning problems was also assessed before and during the last session of the teaching experiment. Procedure This study was conducted within the larger context of the NSF-funded, Nurturing Multiplicative Reasoning in Students with Learning Disabilities project (Xin, Si, & Tzur, 2008). Two fifth grade students with MD (Chad and Tina) and a student at-risk for MD (Megan) from an urban elementary school participated in this study. The pretest sessions involved five to six multiplicative problems such as "A platoon must have exactly 7 spaceships. The player received 21 spaceships to begin the first game. How many full platoons can be made?" The third author, a professor in math education conducted the teaching experiment to the children. The activity was "Towers of Cubes." The goal was for students to figure out the relationships between three quantities: the number of towers, the number of cubes in each tower, and the total number of cubes. Tasks involved multiplicative, partitive, and quotitive division questions such as "Please go and bring me 5 towers of 6 cubes. How many cubes do you have in all?" and "I have 12 cubes in 3 towers. How many cubes are there in each tower?" Although they varied from session to session based on an on-going assessment of students' performance, all tasks shared the common nature of multiplicative reasoning. The instructor explicitly demonstrated double counting to students during the teaching experiment. During the last session of the teaching experiment, the instructor asked students to solve questions similar to those in the

Swars, S. L., Stinson, D. W., & Lemons-Smith, S. (Eds.). (2009). Proceedings of the 31 st annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education. Atlanta, GA: Georgia State University.

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pretests, such as "I have 30 cookies in 6 boxes. How many cookies do I have in each box?" Each session lasted 30-50 minutes. The discourse of the teaching experiment was videotaped and transcribed. Data Coding & Analysis Students' strategies were coded according to a coding scheme investigators developed based on existing theoretical and empirical literature on solving multiplicative problems and the nature of students' activities. Five general types of strategies were involved: unitary counting strategies, repetitive addition or subtraction, double counting, direct retrieval, and DK (don't know). Interrater reliability was checked by a team of graduate students who were unaware of the purpose of this study recoding 33% of the transcripts. The inter-rater reliability was 92%. Graphical presentation of data is important for micro-genetic studies (Sielger, 2006). By visually analyzing data, five dimensions of change were investigated: (1) source of change (what leads children to adopt new strategies); (2) path of change (the sequence of strategies children use while gaining competence); (3) rate of change (the amount of time or experience from the initial use of a strategy to consistent use of it); (4) breadth of change (how widely the new strategy is generalized to other problems); as well as (5) the variability of change (differences among children in the previous four dimensions). Results Students' Multiplicative Problem Solving Performance All the participants increased their percent of accuracy in solving multiplicative problems from baseline to the last session of the intervention. Specifically, Chad improved his percentage correct from 44% on average of the pretests to 83.33 % during the last session of the teaching experiment; Megan improved her percentage correct from 55% to 87.5 %; and Tina improved from 40 % to 83.33%. Students' Strategic Development Figure 2 represented the three students' multiplicative strategies used during baseline and intervention. Generally, participants increased the types of strategies they used, and they also increased the frequency of advanced strategies used during intervention. Baseline. The baseline data across three students consistently demonstrated the students with MD used very few strategies, and the most frequently used strategy was unitary counting. Chad only used unitary counting strategies for 91.93% of the trials during pretest sessions, and he replied with not knowing how to solve the problem or what strategy to use for 8.19% of all trials. Similarly, unitary counting was the dominant strategy Tina used during the pretest sessions (77.77%); Tina also used repetitive addition strategy (11.11%) and of direct retrieval strategy (11.11%). Megan used more types of strategies than Tina and Chad, but unitary counting was also the dominant strategy for her. She used unitary counting strategies for 41.67% of all trials; she also used repeated addition/subtraction strategies (33.33%), direct retrieval strategies (16.67%), and replied with "don't know" for 8.33% of the problems. Double counting strategy was not found in any of the three students. Compared to Figure 1 for normal-achieving students (Kouba, 1989), the participants had fewer strategies and more heavily rely on unitary counting. Teaching Experiment. The data from the teaching experiment suggested that students increased the variety of strategies they used; in particular, they increasingly used more advanced strategies. Specifically, double counting strategies appeared in the first session of the teaching experiment for Chad (14.28%) and Megan (7.14%), and in the third session for Tina with

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40.91%. Double counting strategies consistently increased and became the most frequently used strategies in the last session across the three students (50% for Megan, 60% for Chad, and 33% for Tina). The appearance and increasing frequency of double counting strategies indicated that children explicitly mastered how to coordinate two quantities in multiplicative reasoning. In addition, all three children increased their frequency of the direct retrieval strategy used. Chad did not use this strategy during the pretests, but he began to employ it during the second session in teaching experiment (11.11%) and used it with 20% of the tasks during the last session. Similarly, Megan's usage of the direct retrieval strategy increased to 21.43% in the last session of intervention and Tina increased to 16.67%.

Figure 2. Three participants' multiplicative strategic development

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On the other hand, the graph suggested that the participants decreased their frequency of using the unitary counting strategies. Chad consistently decreased from 91.93% on the pretest sessions to 20% during the last session. Tina and Megan increased their frequencies of using unitary counting strategies at the beginning of the intervention (Tina increased to 90% in Session 2, Megan increased to 68.75% in Session 2), but decreased the frequency of their usage of unitary counting strategies during the later sessions of the intervention (Tina decreased to 16.67% and Megan decreased to 21.42% in the last session). Similarly, the students also decreased their frequency of using repeated addition/subtraction strategy during the teaching experiment. Discussion The purpose of this study was to explore how students with MD or at risk for failure in mathematics differed from normal achieving students in multiplicative strategy choice, and how these students shift from intuitive strategies to multiplicative reasoning strategies while receiving instruction to improve their multiplicative reasoning. Based on the limited data from the three students, the results suggested the students with MD and those at risk seemed to have fewer strategies and their strategies were less advanced than normal achieving students on the pretest sessions. Nevertheless, after the intervention, students improved their percentage correct in solving multiplicative problems; the frequencies of strategies the participants used changed as well. Five dimensions of change were discussed below according to the framework of microgenetic studies (Siegler, 2006). Regarding the path of change, the three participants demonstrated the similar pattern with which normal achieving students go through during the multiplicative reasoning strategic development; that is, beginning generally with counting by ones; then, transitioning to double counting, repeated addition or subtraction; and then, to recall of math facts. A short-term increase of their use of unitary counting was found before the participants consistently faded it out. Double counting increased robustly; direct retrieval also increased on a limited basis. Specifically, unitary counting was the most dominant strategy of all three participants during the baseline sessions (97.93% for Chad, 71.77% for Tina, and 41.67% for Megan). They did not use double counting at all, and used direct retrieval very rarely. These results suggest that the three participants could only keep track of one number sequence while in the fifth grade and that these students did not use double counting to keep track of two number sequences. Whereas early studies (Geary, 1990; Geary & Brown, 1991) found that the major problem for children with MD in additive reasoning was direct retrieval, the current results indicated that the participants with MD seem to have problems both in conceptual understanding and retrieval in multiplicative reasoning. The baseline data indicated a significant gap between students with and without MD. According to the data in Kouba's (1989) study, normal-achieving students dominantly use unitary counting strategies at only grade one (97.23%); while they gradually decrease their frequency of employing the unitary counting to 66.16% of all strategies used at grade two and 30.11% at the grade three (Figure 1). The pretest data in this study showed that the three participants used an extremely high percentage of unitary counting strategies. It seemed like that Chad was equivalent to the first-grade level normal students in strategic development; Tina seemed to be equivalent to the second-grade level and Megan was equivalent to the third-grade level. The differential strategy choice may explain why students with MD or at-risk for failure in

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mathematics have lower performance in solving multiplicative problems than normal achieving students. The appearance of double counting strategy is an indicator that children conceptually understand the nature of multiplicative reasoning. Children coordinate two quantities by keeping track of two number sequences with double counting. The three participants consistently increased their use of double counting throughout the teaching experiment (i.e., up to 50% for Megan, 60% for Chad, and 33% for Tina). It is noteworthy that normal-achieving students usually increase the frequency of using double counting to 8.74% at Grade 2 and then decrease the use to 3.98% at Grade 3 (Figure 1, Kouba, 1989). This study suggested that double counting appears to be especially useful for children with MD. The explicit demonstrations of keeping track of two quantities strengthened the students' conceptual understanding of multiplication and reduce their cognitive load when processing the problems. In addition, as the students with MD have difficulties in direct retrieving (Geary, 1990; Geary & Brown, 1991); double counting makes it especially suitable for them to explicitly demonstrate the coordination of two quantities, which is the core meaning of multiplication. Although the three participants increased their use of direct retrieval strategy, they did not get to the level at which normal-achieving fifth grader students perform. Normal-achieving third graders use direct retrieval strategy for 59.66% of all trials (data from Kouba, 1989), but Chad, Megan and Tina only used direct retrieval strategy for 20%, 21.43% and 16.67% of the last session of intervention, respectively. The limited increase indicated that students with MD may need special interventions to help them shift from counting into verbal retrieval For the source of change, an adapted multiple probe design across participants established a functional relationship between the teaching experiment and students' strategic development. Thus it is the "towers and cubes" activity that helps students with MD to adopt new strategies. This activity provides students with manipulatives to solve multiplicative problems. The teacher's demonstration of double counting seems to be effective in teaching students how to coordinate two number sequences, and it may explain the appearance of double counting strategy. During the teaching experiment, the instructor explicitly demonstrated the double counting in finding out the total number of cubes across a specific number of towers. The intervention also emphasized making distinctions between the unit of one (1's) (e.g., total number of cubes) and the composite unit (e.g., a tower of 6). In all, through the "cubes and towers" game, children seem to conceptually develop multiplicative reasoning. As for the rate of change, the participants' use of double counting strategy increased saliently as soon as the occurrence of first use of this strategy during the teaching experiment. The participants' gaining of direct retrieval was slower than the change of double counting. For the breadth of change, the participants could solve problems with various semantic structures; the students were also able to successfully associate the real life problem context in the last session with the multiplicative scheme they learned in the "cubes and towers" problems, and use the advanced strategies (e.g., double counting) to solve problems within new contexts. And for the variability of change, the participants demonstrated some differential patterns from normal achieving students regarding strategic developmental level and transition pattern. Results also demonstrated individual differences among the three participants. In sum, this study revealed how three students with MD or at-risk for MD in mathematics progress in strategic choices during multiplicative reasoning instruction. Due to the limited generalization of single subject design, a group design study is underway.

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Endnotes 1. This research was supported by the National Science Foundation, under grant DRL 0822296. The opinions expressed do not necessarily reflect the views of the Foundation. References Anghileri, J. (1989). An investigation of young children's understanding of multiplication. Educational Studies in Mathematics, 20, 367-385. Brown, S. (1992). Second grade children's understanding of the division process. School Science and Mathematics, 92(2), 92-95. Downton, A. (2008). Links between children's understanding of multiplication and solution strategies for division. In M. Goos, R., & K. Makar (Eds.). Proceedings of the 31st Annual Conference of the Mathematics Education Research Group of Australia. (pp.171-178). Sydney, Australia: MERGA. Geary, D. C. (1990). A componential analysis of an early learning deficit in mathematics. Journal of Experimental Child Psychology, 49, 363-383. Geary, D. C., & Brown, S. C. (1991). Cognitive addition: Strategy choice and speed-ofprocessing differences in gifted, normal, and mathematically disabled children. Developmental Psychology, 27, 398-406. Horner, R. D., & Baer, D. M. (1978). Multi-probe technique: A variation of multiple baselines. Journal of Applied Behavior Analysis, 11, 189-196. Kouba, V. L. (1989). Children's solution strategies for equivalent set multiplication and division word problems. Journal for Research in Mathematics Education, 20(2), 147-158. Lemaire, P., & Siegler, R. S. (1995). Four aspects of strategic change: Contributions to children's learning of multiplication. Journal of Experimental Psychology: General, 124, 83-97. Mulligan, J. T., & Michelmore, M. C. (1997). Identification of multiplicative thinking in children in Grades 1-5. Journal for Research in Mathematics Education, 28(3), 309-331. Piaget, J. (1965). The child's concept of number. New York, Norton. Siegler, R. S. (2006). Microgenetic analyses of learning. In W. Damon & R. M. Lerner (Series Eds.) & D. Kuhn & R. S. Siegler (Vol. Eds.), Handbook of child psychology: Vol. 2. Cognition, perception, and language (6th ed., pp. 464-510). Hoboken, NJ: Wiley. Siegler, R. S. (2007). Cognitive variability. Developmental Science, 10, 104-109. Siegler, R. S., & Shrager, J. (1984). Strategy choices in addition and subtraction: How do children know what to do? In C. Sophian (Ed.), The origins of cognitive skills (pp. 229-293). Hillsdale, NJ: Erlbaum. Steffe, L. P. (1988). Children's construction of number sequences and multiplying schemes. In J. Hiebert & M. Behr (Eds.), Number concepts and operation in the middle grades. Vol. 2 (pp.119-140). Hillsdale, NJ: Lawrence Erlbaum. Steffe, L. P. (1994). Children's multiplying schemes. In G. Harel & J. Confrey (Eds.), The Development of multiplicative reasoning in the learning of mathematics. Albany: State University of New York Press. Vergnaud, G. (1983). Multiplicative structure. In R. Lesh, & M. Laudau (Eds.), Acquisition of mathematics concepts and processes (pp.128 -175). London, Academic Press. Xin, Y. P., Tzur, R., & Si, L. (Aug. 2008). Nurturing Multiplicative Reasoning in Students with Learning Disabilities in a Computerized Conceptual-Modeling Environment (NMRSD). National Science Foundation (NSF), $2,969,894 (2008 to 2013).

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