Transition From Intuitive to Advanced Strategies in Multiplicative Reasoning for Students With Math Difficulties

Abstract

This study explored how a teaching experiment, which taught double counting with a constructive task assignment according to an ongoing assessment, helped students with mathematics difficulties develop their multiplicative reasoning skills. The participants were two fifth graders with math learning disabilities and one at risk. A micro-genetic approach with a single-participant design was used. Investigators coded and analyzed four strategies the children used. Results showed that unitary counting was dominant during the baseline sessions. During the teaching experiment, a great increase in the use of double counting and an obvious decrease in the use of unitary counting was observed. The participants’ improvement in problem-solving accuracy on the posttest validated their strategic development. Implications are discussed in the context of employing strategic changes as a measure for assessing students’ academic achievement and interventions for helping children adopt advanced strategies.

Keywords

strategic development multiplicative reasoning mathematical difficulties

Approximately 5% to 8% of school-aged children experience difficulties in learning mathematics (math difficulties, or MD), as characterized by their poor performance in class and poor standardized test scores (Geary, Hoard, Byrd-Craven, & DeSoto, 2004). Geary refers to children as MD if their achievement scores in mathematics fall below a specific cutoff, which may range from lenient (30th percentile) to restrictive (5th or 10th percentile; Geary, Hoard, Byrd-Craven, Nugent, & Numtee, 2007). Individual variability is one of the most striking features of children’s reasoning (Siegler, 2007). Generally speaking, low achieving students usually utilize 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). Understanding how children with MD differentially develop their problem-solving strategies is critical for educational practitioners to provide suitable interventions.

We first address the importance of studying children’s strategic development in multiplicative reasoning. Then, we review the literature of how normal-achieving students develop their strategies of solving multiplicative reasoning problems and the lack of literature for strategic development of students with MD. Next, we review the existing interventions for multiplicative reasoning from both the math education and special education fields. Finally, we introduce the micro-genetic approach, which may address the potential pitfalls of the current research methods in special education and math education.

Significance of Studying Children’s Strategic Development of Multiplication

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). Children usually use multiple ways or strategies to solve problems. Generally speaking, children’s strategic development involves four aspects: the variety of strategies children use to solve a problem, the frequency of using advanced strategies, the accuracy of executing strategies, and the flexibility of children’ strategic choices when solving different problems (Lemaire & Siegler, 1995).

Children with MD were found to have difficulties in transitioning from intuitive strategies to advanced strategies (Geary, 1990). While developing additive reasoning, for example, although both students with and without MD develop various counting strategies during the 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; Goldman, Pellegrino, & Mertz, 1988; Jordan, Levine, & Huttenlocher, 1994). Unfortunately, children’s strategic development for multiplication is still much less understood than it is for addition (Kouba, 1989; Lemaire & Siegler, 1995), despite the fact that developing students’ multiplicative reasoning skills is a major instruction objective in elementary curricula from third grade through fifth grade (National Council of Teachers of Mathematics, 2000).

Strategic Development in Multiplicative Reasoning

In multiplicative reasoning, one composite unit is distributed across the other, and children need to coordinate the two quantities (Nunes & Bryant, 1996; Park & Nunes, 2000; Piaget, 1965; Steffe, 1994; Vergnaud, 1983). For example, in the statement “She earns 40 dollars per day,” the dollars (one quantity) are distributed across the days (another quantity). Multiplicative reasoning is a type of quantitative reasoning (i.e., a mental operation between a set of quantities and the relationships between them, such as the constitution of speed as a new quantity derived from distance and time) rather than a numerical reasoning (i.e., reasoning about particular numbers to evaluate a quantity, such as hours plus hours equals hours to evaluate the same quantity of time in the additive reasoning; Thompson, 1994). Therefore, it is critical for children to realize the existence of a set of quantities instead of merely a set of numbers in multiplicative reasoning.

Children use multiple strategies to solve multiplicative reasoning problems, and they make decisions regarding which strategy to use according to their current cognitive capacity, experiences, and specific problem situations (Lemaire & Siegler, 1995; Siegler, 1988). Early studies (Anghileri, 1989; Brown, 1992; Kouba, 1989; Mulligan, 1992; Mulligan & Michelmore, 1997; Oliver, Murray, & Human, 1991; Steffe, 1988) identified a variety of strategies normal-achieving students use for multiplicative reasoning. These studies also provided evidence that children’s solution strategies begin generally with direct modeling and unitary counting; progress to skip counting, double counting, and repeated addition or subtraction; and then progress to the use of known multiplication or division facts (Downton, 2008). With learning and maturing, the relative frequencies of use of these strategies change (Siegler, 2007). The general strategic developmental pattern is demonstrated in Figure 1 according to Kouba’s (1989) data.

Figure 1.

Normal-achieving students’ multiplicative strategic development based on Kouba’s (1989) data.

Direct representation and skip counting are two strategies of unitary counting, as they address only one number or 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). 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, 1989). However, skip counting does not suggest a child is able to coordinate the two quantities. Children often do not know where to stop counting, which indicates that students are not coordinating the two quantities (Kouba, 1989). Afterward, 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, “three plus three plus three plus three” to solve “three times four.”

It is noteworthy that children’s shifting to double counting (DC) is a milestone of their development of multiplicative reasoning. DC 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, when a child was asked to do “20 × 20,” he sequentially put up fingers 20 times while saying “20, 40, 60, 80,—340, 380, 400” (Steffe & Cobb, 1998). DC is “an advance over the more basic direct representation because it requires more abstract processing and involves integrating two counting sequences” (Kouba, 1989, p. 152), and DC indicates “the development from uniformly counting in ones to rhythmic counting in groups” (Steffe, 1986, as cited by Anghileri, 1989, p. 374). Teaching DC has been documented to be critical for improving children’s multiplicative reasoning development (Steffe & Cobb, 1988, 1998; Tzur, 2004).

And eventually normal-achieving students shift to recalled number facts, which is the highest strategy people use. Kouba (1989) explained that this strategy was being used when “the child obtained the answer by remembering the appropriate multiplication or division combination” (p. 153). Unfortunately, little research has explored if children with MD have the same problems in direct retrieval in multiplication as in addition.

In all, in spite of considerable research on normal-achieving students’ strategic development on multiplicative reasoning skills, there is a dearth of known research on how students with MD make their transition from less advanced strategies to advanced strategies in multiplicative reasoning, and sparse research has been done regarding how to improve the strategic development of students with MD.

Existing Interventions of Multiplicative Reasoning

In the field of math education, research emphasizes a deep understanding of the nature of multiplicative reasoning from the constructivist perspective (Steffe & Thompson, 2000; von Glasersfeld & Steffe, 1991). Teaching experiments with one-on-one clinical interviews (Cobb & Steffe, 1983; Steffe, Thompson, & von Glasersfeld, 2000) is widely used and is considered the most powerful methodology to study multiplicative reasoning (Confrey & Harel, 1994; Simon, 1995). A significant characteristic of the constructive teaching experiment is the use of task design to promote students’ thinking: The selection of high-level cognitively complex tasks promotes children’s capacity to think, reason, and problem solve (Smith & Stein, 1998; Steffe & Thompson, 2000). Experimenters make decisions concerning situations to create and critical questions to ask, based on the teacher or experimenter’s current interpretations of the child’s ongoing performance (Steffe, 1991, 1994). Constructivists believed that if students are challenged at an appropriate level with nonrountine tasks, they develop their cognitive abilities by reflecting the relationship between their goals and the actual effects of their current activities or strategies (Simon & Tzur, 2004; Simon, Tzur, Heinz, & Kinzel, 2004). Such task-design effects function through two types of reflections of an activity (strategy)–effects relationship (Simon et al., 2004). Type I reflection consists of comparison between the learner’s goal and the actual effect of the activity (strategy); Type II reflection consists of comparison across records of experiences in which the learner uses various strategies to solve problems with similar situations.

Another characteristic of constructive teaching experiments is their emphasis of children’s construction of the concept of DC (Steffe, 1994; Steffe & Cobb, 1988). DC enables a child to quantify, in the absence of objects (i.e., in anticipation), the total number of 1’s that are embedded in a given number of same-size composite units without having to count each and every singleton. In addition, Tzur and Simon (2004) distinguished two stages during the transition from students’ available concepts to intended strategy in division of fraction and hence provided appropriate tasks and prompts to lead students to construct the target strategy. Specially, in the first, participatory stage, the learner forms a provisional anticipation of the activity–effect relationship that he or she cannot access directly from his or her available strategies but that has to be prompted for the target strategy; in the second, anticipatory stage, the learner forms a robust activity–effect relationship that he or she can independently and spontaneously call up and use in new situations.

The advantage of such teaching experiments is that students’ ability in multiplicative reasoning is not only measured “right” or “wrong” but is measured with an in-depth analysis of how they solve the problems (Confrey & Harel, 1994). However, the teaching experiments usually lack standardized, quantitative, overt measures for the improvement (Steffe & Thompson, 2000). In addition, whether or not students with MD are able to easily self-construct the concept of composite units remains an unanswered question.

On the other hand, special education interventions primarily emphasize improving children’s overt performance (i.e., problem-solving accuracy) on predetermined tasks with an explicit instruction approach. A considerable number of interventions have been developed to address multiplication facts. For example, some interventions (i.e., mnemonics, constant time delay) have been developed to teach children with MD the direct retrieval of multiplication facts from a behaviorist perspective (Cybriwsky & Schuster, 1990; Greene, 1999; Koscinski & Gast, 1993; Mattingly & Bott, 1990; Morton & Flynt, 1997; Wood, Frank, & Wacker, 1998). Some other interventions teach students with MD concrete-representation-abstract representations (Harris, Miller, & Mercer, 1995; S. P. Miller & Mercer, 1993) or teach a series of strategies (e.g., doubling, repetitive addition, referring to relevant problems, math facts retrieval) to help automaticity (Kroesbergen & Van Luit, 2002; Van Luit & Naglieri, 1999; Woodward, 2006). In addition, across interventions for solving multiplicative word problems, the schema-based intervention (SBI) has been evidenced to effectively teach students with MD (Fuchs, Fuchs, Finelli, Courey, & Hamlett, 2004; Fuchs, Fuchs, Hamlett, & Appleton, 2002; Fuchs, Fuchs, Prentice, et al., 2004; Griffin & Jitendra, 2009; Jitendra, Dipipi, & Nora, 2002). During the SBI, the teacher explicitly teaches children how to map the problem features to the corresponding schema and then teaches solution rules to solve the problems. Xin and colleagues (Xin, 2008; Xin, Jitendra, & Deatline-Buchman, 2005; Xin, Wiles, & Lin, 2008; Xin & Zhang, 2009) transformed SBI and later developed conceptual model based problem solving, which involves symbolic or algebraic expressions of mathematical relations in generalizable conceptual models in equations (e.g., unit rate × # of units = total or product).

These interventions are primarily measurable with an overt performance outcome and have adopted an explicit intervention approach. However, they have rarely addressed how children develop their conceptual understanding of multiplicative reasoning. Moreover, the explicit approach rarely considers tasks as an instructional component to trigger students’ self-construction.

Therefore, further research is needed to explore how to integrate both the advantages from constructive teaching experiments in math education and explicit interventions in special education. It is critical to develop a research method that not only emphasizes the fine-grained analysis of conceptual changes of the nature of multiplicative reasoning but also provides a measurable outcome for the intervention with effective instructional components (e.g., explicit instruction, constructive tasks assignment) to students with MD.

The Micro-Genetic Approach

The micro-genetic approach has provided a possibility to bridge the current research approaches of math education teaching experiments and the special education interventions. Currently, the micro-genetic approach is widely used for investigating children’s strategic development during learning, reasoning, and problem solving (Siegler, 2006). The micro-genetic approach is defined by three characteristics: “Observations span the whole period of rapidly changing competence; the density of observations within this period is high, relative to the rate of change; observations of changing performance are analyzed intensively to indicate the processes that gave rise to them” (Siegler & Svetina, 2002, p. 793). With measurement of both performance and percentage of strategy use, Siegler and colleagues have conducted a series of studies on children’s learning and reasoning and successfully revealed how changes occur with or without interventions. Furthermore, the micro-genetic approach created a unique frame for analyzing data from five perspectives (Siegler, 2006): source of change (what leads children to adopt new strategies, path of change (the sequence of strategies children use while gaining competence), rate of change (the amount of time or experience from the initial use of a strategy to consistent use of it), breadth of change (how widely the new strategy is generalized to other problems), and variability of change (differences among children in the previous four dimensions). The fine-grained and multiple-angled analyses provide rich information for understanding the underlying cognitive mechanisms.

A particular advantage of the micro-genetic approach is its revealing of the strategies that children develop in response to an instruction, and thus this approach helps under- stand how instruction exercises its effects (Siegler, 2002). The micro-genetic approach not only answers whether the changes happen after an intervention but also answers how changes happen during the intervention (Siegler, 2006). In particular, micro-genetic studies have successfully revealed how task assignment facilitated children’s reasoning and helped children adopt new strategies (Siegler & Chen, 1998; Siegler & Shrager, 1984; Siegler & Stern, 1998). Moreover, a number of studies have successfully employed this approach to research students’ mathematics learning for normal-achieving students (Opfer & Siegler, 2007; Siegler & Shrager, 1984; Siegler & Stern, 1998). However, this research approach has not been applied on special education populations.

Research Questions

The purpose of this article was to explore how a teaching experiment, comprising constructive task assignment with ongoing assessment and explicit instruction of DC, affected the multiplicative reasoning strategic development for students with MD, with the micro-genetic framework. Specifically, we explored (a) what strategies students with MD or students at risk used during baseline sessions and (b) how the teaching experiment facilitated the participants’ strategic development in solving multiplicative problems.

Method

Design

To provide a fine-grained analysis for individuals’ learning process during intervention, micro-genetic studies usually focus on a relatively small number of individuals (Siegler, 2006) and often employ single-participant designs (Agre & Shrager, 1990; Johnson & Mervis, 1994; Schoenfeld, Smith, & Arcavi, 1993). 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. With the multiple-probe design, data are collected intermittently during the preintervention probe condition. Specifically, when a stable baseline (i.e., the problem-solving accuracy and strategy choice) was observed for one student, treatment was introduced to this student while other participants remained in the baseline condition. When improvement (i.e., strategic changes) for Child A was observed, Child B was introduced to the teaching experiment. And when improvement for Child B was observed, Child C was introduced to the teaching experiment. In this design, replication of treatment effects from the second and third participants is demonstrated if changes in performance occur only when treatment is introduced.

Different from traditional assessment that usually takes place after an intervention, the trial-by-trial assessments were conducted within an intervention in a micro-genetic approach and thus allowed a precise assessment of strategy learning or discoveries (Siegler, 2006). Each trial of the students’ problem solving was observed to provide a fine-grained analysis of how the participants learned during the intervention. Such trial-by-trial strategy assessments are fundamental to micro-genetic studies (Siegler, 2006). As the micro-genetic approach requires dense observations throughout intervention, videotaping and coding were employed (Siegler, 2006).

Participants and Setting

This study was conducted within the larger context of an National Science Foundation–funded project (Xin, Tzur, & Si, 2008). Two 5th grade students with math learning disabilities (MLD), Andy and Barb, and a student at risk for MLD, Cathy, from an urban elementary school participated in this study. Participant recruitment was based on teachers’ recommendations of those students experiencing difficulties in learning mathematics and failure on the Indiana Statewide Testing for Educational Progress–Plus (ISTEP+) in mathematics (Indiana Department of Education, 2007). All three students failed the ISTEP+ test. Andy and Barb were diagnosed with MLD by school psychologists because there was a discrepancy between their IQ and their achievement on the standard test of mathematics (which was below the 30th percentile), and they had received special education services for 3 years. Cathy was not officially diagnosed with MLD, but she failed the ISTEP+. Students’ demographic information is provided in Table 1.

Table 1.

Student Demographic Information.

Variable	Andy	Barb	Cathy
Gender	M	F	F
Ethnicity	Caucasian	African American	Multiracial
Age	10 years	10 years	10 years
Grade	4	5	4
Classification	MLD	MLD	At risk
SES	Low	Med-low	Med-low
Reduced or free lunch	Free	Free	Free
Years in special education	3	3	—
Learning support classroom placement	Math/ED	Math/ED
% in general education class	60	50	100
IQ	82	85	77
ISTEP+ Math	341 fail PR < 30%	164 fail PR < 30%	373 fail
Math test PR	27.0% (WIAT-2)	26.3% (SAT)	NA
Grade equivalent	3rd graders	NA	NA

MLD = math learning disabilities; SES = socioeconomic status; ED = emotional disorder; GE = grade equivalent; ISTEP+ = Indiana Statewide Testing for Educational Progress–Plus; PR = percentile rank; WIAT-2 = Wechsler Individual Achievement Test–Second Edition; SAT = Stanford Achievement Test; NA = not available.

All three students were reported as struggling with learning mathematics according to their teachers. Cathy’s teacher reported that she displayed difficulties with completing her regular class assignments including operations of proportions, fractions, decimals, and finding the greatest common factor. However, Cathy did not receive any special education services. Andy was studying prealgebra, time, and data in his regular class, and his individualized education program annual goals primarily targeted tasks relevant to multiplicative reasoning. Similarly, Barb, a fifth grader with MLD, also displayed difficulties with tasks involving multiplicative reasoning during her regular class. In the resource room, Andy and Barb were doing STAR Math, a daily progress-monitoring software tool that monitors and manages mathematics skills practice.

Researchers decided to give the participants a preliminary assessment to evaluate their understanding of math concepts and then provided appropriate teaching experiments to intervene on the children’s reasoning skills. An assessment of additive reasoning showed that these three students already had certain number sense (e.g., the sense of number magnitude, the concepts of cardinality and ordinality) and additive reasoning skills (e.g., using counting all and counting on and counting large strategies to solve addition problems, the concept of the base 10 number system and placing value). However, their insufficiency in solving multiplicative problems was obvious. Therefore, the researchers decided to start from additive reasoning strategies (i.e., unitary counting) to build up children’s advanced strategies for multiplicative reasoning.

All instruction and testing were conducted in vacant teacher conference rooms during the daytime school hours. The rooms were equipped with white boards, tables or desks, and chairs. Pencils, scratch paper, and cubes were provided for the participants. Each session lasted about 40 min. The testing and teaching were videotaped and transcribed with the exception of Pretest 3 and the posttest because of technical issues.

Experimenter

A professor in math education conducted the teaching experiment with the children. This professor has 25 years of K–12 school teaching experience for teaching students with MLD and about 20 years of research experience in math education at higher education institutions. The instructor implemented the assessment during the pretest (1 and 2) and posttest sessions, in addition to the teaching experiment. Pretest 3 was implemented by other research team members as the experimenter was out of town.

Tasks

All assessment sessions and intervention sessions targeted solving multiplicative reasoning problems. Tasks involved multiplicative, partitive, and quotitive division problems. The baseline phase included three probes. The first two probes were developed by the experimenter and the research team, including both multiplication facts and story 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 probe was adopted from a published study (Xin, Wiles, et al., 2008) involving story problems. The posttest probe comprised items on Pretest 2. Similarly, tasks used during the teaching experiment were multiplicative reasoning problems, including easy problems (e.g., one-step problems with multiplicands not bigger than 5) and complicated problems (e.g., two-step problems, problems with various contexts, problems with multiplicands larger than 5).

Procedures

Baseline

During both baseline sessions and teaching experiments, trial-by-trial strategy assessments were conducted. During baseline, students completed three researcher-developed multiplicative probes. The students were requested to complete the problems independently. When children said they did not know what to do or could not give an answer in 1 min, the experimenter said “OK” and then moved to the next problem.

Teaching experiment

The teaching experiment was based on Tzur and Simon’s (2004) stage distinction theory and the activity–effects relationship theory (Simon et al., 2004) as previously mentioned. The intervention was designed to help students learn DC through the participatory and anticipatory stages by the students’ reflections of the activity (strategy)–effects relationship during a game titled Please Go and Bring Me (PGBM). An ongoing trial-by-trial assessment, similar to the pretest, was implemented at the beginning of each trial for the micro-genetic observation and analysis. The experimenter orally presented each problem to the children and asked the children to speak aloud to explain how they solved the problem and observed how children solved the problems. Students’ independent response and explanations were recorded for each trial. After the children completed their first attempt to solve problems, the experimenter gave right or wrong feedback to the children. When the children said they did not know what to do or could not give an answer in 1 min, or when the children had solved the problem incorrectly, the instructor explicitly demonstrated DC to students during the teaching experiment. When children demonstrated an attempt to double count but failed in execution, the instructor provided only prompts to help the students complete the DC strategy, rather than giving explicit instruction. Examples of teaching experimental instruction are provided in Table 2, and a sample of explicit teaching of DC is provided in the excerpt accompanying Table 2.

Table 2.

Examples of Teaching Experimental Trials and Explicit Teaching of Double Counting (DC; Session 1, Andy).

Task	Student’s first trial answer	Accuracy	Student’s explanation	Strategy use	Following intervention
3 towers of 5 cubes	15	1	I added 5 and 5 and another (5) is 15	Repeated addition	NA (Move to next question)
3 towers of 4 cubes	13	0	4 + 4 is 8 and then 4 more is 12	Repeated addition	Explicit teaching DC^a
5 towers of 5 cubes	25	1	5, 10, 15, 20, 25	Skip counting	NA
2 towers of 7 cubes	15	0	6, 7, 8, 9, 10, 11, 12, 13, 14, oh, 14	Counting by ones	Explicit teaching DC^a

NA = not available.

An example of explicit teaching DC is below (Session 1, 7 towers of 4 cubes, Cathy).

T: Let me suggest the following. I’ll give you my fingers for every tower. Okay? Every time we have a tower that’s [holds up one finger] one tower of four—how many do we now have? You can use your fingers for [counting] the four [cubes].

C: So 4.

T: What if I brought another tower [raises a second finger]?

C: 8.

T: What if I brought another tower [raises a third finger]?

C: 12? [doesn’t look confident] No wait.

T: You can use your fingers to figure it out.

C: [counts under her breath] 16? No!

T: So we had 4—now let’s use your fingers [shows counting on from 4 on his other hand] 5–6–7–8. And then, 9–10–11–12.

C: Yeah.

T: So now we have 3 [towers], what if we added another one [raises a fourth finger]?

C: 16.

T: OK another one.

C: [counts on her fingers under the table] 17–18–19–20.

T: So with 5 we have 20. We still have to go [bring towers] two more times.

C: [counts on her own fingers] 21–22–23–24–28.

During the PGBM game, different tasks were assigned to individual students according to the ongoing strategy assessment on each session. The ongoing assessment was conducted after each session by analyzing the students’ strategy use to determine at which stage the children were regarding developing their DC strategy. At the beginning, the experimenter asked, “Please go and bring me M towers of N cubes” and then observed how children solved the problems. In such tasks the student often went to a box containing individual cubes and created a tower N cubes high and then returned the tower to the table, and the process continued repeatedly until the child brought M towers of N. Such tasks were believed to trigger the learner’s understanding of finding the total number of 1s (cubes) embedded within a given number of composite units (towers of cubes), which was a foundation for understanding DC.

Once such situations are recognized, the teacher asked questions such as, “Pretend you have M towers of N cubes; how many cubes do you have?” In such questions students may choose not to go and bring real cubes but to use DC. These tasks were designed to help children move into the participatory stage of DC. At this stage, students may demonstrate an attempt to use DC but still need prompts to complete the execution of DC. According to the activity–effect relationship theory (Simon et al., 2004), the teacher’s statement “What if I brought N towers of M cubes?” was expected to orient the students’ Type I reflections between the accruing effects of the DC activity (strategy) and the goal of finding the total number of cubes. These tasks hence help students to assimilate into using two hands to keep track of each of the two number sequences (DC) rather than physically bringing towers of cubes.

When the experimenter and the research team found that the students had reached the participatory stage of DC, they used complicated problems for promoting transition to the anticipatory stage through students’ Type II reflection. Large number tasks requiring more than two hands or two-step problems with common unit composites were chosen, for example, “pretend you have 7 towers of 6” or “pretend you have 3 towers of 4 and I have 2 towers of 4; how many do we have in all?” Students may have struggled with the problems because each multiplicand was larger than 5 (fingers of a hand) or there were more than 2 numbers in the problem. However, this brought about the students’ Type 2 reflection of comparing the effects by using DC and the effects by unitary counting through “go and bring” real cubes—DC was apparently more efficient than unitary counting for such complicated problems. Such reflection of activity (strategy)–effect relationship was expected to make the child easily see the advantages (accuracy and speed) of DC over physically bringing cubes multiple times and counting dozens of 1s. Hence, the students were expected to adopt DC and pay more attention to conceptually distinguish and coordinate two number sequences.

Each teaching experiment session lasted approximately 40 min and involved 6 to 10 problems, depending on the speed of the children’s problem solving and the amount of explicit instructions presented. The teaching experiment sessions ended when the students demonstrated an ability to independently and accurately use DC to solve complicated problems consistently. After the teaching experiment, a posttest parallel to Pretest 2 was conducted to assess the children’s progress. Students were asked to complete the posttest independently. In the present study, teaching experiment sessions in addition to two baseline sessions were videotaped, transcribed, and analyzed.

Dependent Measures

As this study was aimed at investigating how children with MD changed their reasoning during the teaching experiment, the primary dependent variable was students’ strategy use across the sessions. Such a dependent variable is commonly used in existing micro-genetic studies. Students’ performance on multiplicative problem solving was also recorded as validation data.

Students’ strategic use frequency

The 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. Four general types of strategies and DK (don’t know) were coded. The four strategies involved unitary counting strategies (counting by ones and skip counting), repetitive addition or subtraction, DC, and direct retrieval. The percentage of time each particular strategy used was calculated as the number of times a particular strategy was used divided by the total frequency of all strategies used. Children’s strategic data included only their first independent attempt.

Students’ problem-solving accuracy

The percentage correct of each participant’s performance on solving multiplicative problems was calculated for baseline, teaching experiment, and posttest sessions. The measure of students’ performance on solving multiplicative problems served as a validation for the effectiveness of the teaching experiment. The accuracy data were based on the children’s first independent attempt.

Treatment Fidelity

The experimenter developed a teaching plan before each session of the teaching experiment. The teaching plan involved the teacher’s reflection and analyses of the child’s current strategic development and following tasks to be assigned according to the student’s current level. The teaching plans were discussed and modified with the members of the research team before implementation. After the implementation, a fidelity checklist was developed to examine the instructor’s adherence to the lesson plan, including following the lesson plans to choose the proper tasks, asking children to independently solve the problem, explicitly demonstrating DC at the correct time and under the proper conditions, and prompting the children’s correct execution of DC at the correct time and under the proper conditions. Of the sessions, 30% were checked with the fidelity checklist. Treatment fidelity ranged from 85% to 100%, with a mean of 91% across all observed sessions.

Interrater Reliability

A team of graduate students who were unaware of the purpose of this study recoded 30% of the transcripts to check interrater reliability. Data on students’ performance and strategy were checked. The interrater reliability was 89.02%.

Results

Problem-Solving Accuracy

All of the participants increased their percentage of accuracy in solving multiplicative problems from baseline to the posttest session. Specifically, Andy improved his percentage correct from 0% on average on the pretests to 50% at the posttest session, and Barb improved from 0% to 46%; Cathy increased from 20% (median) to 40%. The accuracy data are presented in Figure 2.

Figure 2.

Three participants’ multiplicative strategic development.

It is noteworthy that we could not claim a rigorous functional relationship between the teaching experiment and students’ accuracy improvement: Although all tasks were multiplicative reasoning problems throughout the study, the task difficulty changed from easy to complicated during the teaching experiment. As an instructional component of the teaching experiment, such task assignment was unavoidable—the present study sought to examine how the children responded to such an instructional component. More or less, the accuracy data provided some validation for the measure of strategic development: All three participants’ accuracy was enhanced from parallel pretest (Probe 2) to posttest, and the participants’ high accuracy in solving complicated problems during the teaching experiment suggested the children’s improvement.

Students’ Strategic Development

This study focused on investigating the relationship between the teaching experiment and the students’ strategic changes to reveal how the participants changed their inner reasoning skills. Data were analyzed from multiple perspectives of the micro-genetic approach. To identify the source of changes, a staggered graph (Figure 2) for the multiple-probe design was used to establish a functional relationship between the teaching experiment and targeted strategic changes. For the path of changes, we conducted a session-by-session and trial-by-trial analysis to reveal how children first used the targeted DC strategy (see Table 3). Rates of the changes were revealed with the slopes of the strategy changing curves in the graph, in addition to analyzing how many trials students experienced from the first use to consistent use of the targeted DC strategy. The variety of changes was revealed by analyzing the variety of the above four dimensions.

Table 3.

Strategies Children Used on Trials Immediately Before and After Their First Use of Double Counting (DC).

Trials	Andy	Barb	Cathy
−5	DK	Repeated addition	DK
−4	Repeated addition	DK	Skip counting
−3	Repeated addition	Count by ones	Skip counting
−2	Skip counting	Count by ones	Count by ones
−1	Count by ones	Count by ones	Repeated addition
0	DC	DC	DC
1	Direct retrieval	Count by ones	DK
2	Count by ones	Repeated addition	DK
3	Count by ones	Count by ones	DC
4	Count by ones	Count by ones	Skip counting
5	Count by ones	Direct retrieval	DK

0 = the trial when the student used DC for the first time; 1 = the first trial immediately after the student’s first use of DC; 2 = the second trial immediately after the student’s first use of DC; –1 = the first trial immediately before the student’s first use of DC; –2 = the second trial immediately before the student’s first use of DC; DK = don’t know.

Baseline

The baseline data across three students consistently demonstrated that the students with MD used very few strategies, and the most frequently used strategy was unitary counting. Andy used unitary counting strategies for 75.30% (mean) of the trials during pretest sessions, and Barb used unitary counting during the baseline sessions for 73.75% (mean) of the trials. Barb also used direct retrieval strategy (M = 8.33%) and “don’t know” for 8.33%. Cathy used more types of strategies than Andy and Barb, but unitary counting was also the dominant strategy for her. She used unitary counting strategies for 55% of all trials; she also used repeated addition or subtraction strategies (M = 18.75%) and direct retrieval strategies (M = 17.14%) and replied with “don’t know” for 10% (mean) of the problems. None of the three students utilized the DC strategy.

Teaching experiment

We found a functional relationship between the teaching experiment and students’ strategic changes. 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, DC strategies appeared during the first session of the teaching experiment for all three participants (Andy = 12.5%, Barb = 12.5%, Cathy = 18.18%). DC became the most frequently used strategies in the last session across the three students (Cathy = 57.14%, Andy = 66.67%, Barb = 66.67%). It is noteworthy that all three children displayed a rapid increase in the use of DC during the transition from the second to the third session (Cathy = 27.27% to 57.14%, Andy = 22.22% to 42.85%, Barb = 0.0% to 42.85%) when they were requested to shift from solving easy to complicated multiplicative problems.

On the other hand, the graph suggested that the participants decreased their frequency of using unitary counting strategies. Andy decreased from 75.30% during the pretest sessions to 0.0% during the last session. Barb decreased from 73.75% on pretests to 0.0% during the last session. Cathy decreased from 55.0% to 26.36% but showed a regression to more than 40.0% at the end.

The frequency of the participants’ use of the repeated addition or subtraction strategy and the direct retrieval strategy did not change greatly. Only Cathy used the repeated addition or subtraction strategy at the baseline sessions. All three students slightly increased their use of this strategy at the first and second teaching experimental sessions; however, they did not use it during the last session. Andy seemed to increase his use of direct retrieval from 0% during pretesting to 33% at the last session, whereas the other two children maintained a relatively low frequency (no more than 20%) for all sessions.

We conducted a trial-by-trial analysis to explore how children first used the DC strategy. Table 3 records the five trials before and five trials immediately after each child’s first use of DC. Data suggested that children used multiple strategies to solve problems before their first use of DC. None of the participants demonstrated their first use of DC immediately after the first explicit teaching of that strategy. As a matter of fact, our trial-by-trial analysis revealed that even after the first explicit modeling of DC, all three participants still chose other strategies for a few following trials (Cathy = 4 trials, Barb = 3 trials, Andy = 2 trials) prior to their first use of DC. Data also showed that they used multiple strategies after their first use of DC. If we defined a consistent use of DC as using the targeted strategy two out of three trials successively, further analysis found that Andy experienced 13 trials between his first use and consistent use of DC, Cathy experienced 6 trials, and Barb experienced 11 trials.

Discussion

One purpose of this study was a preliminary exploration of how students with MD or at risk for failure in mathematics differed from normal-achieving students in multiplicative strategic choices. The present study was also interested in how these students shifted from the use of intuitive strategies to advanced strategies while involved in a teaching experiment of DC to improve their multiplicative reasoning within a micro-genetic framework.

MD Students’ Strategic Choices on the Baseline Sessions

The baseline data described the natural status of the multiplicative reasoning strategic choices among the three students with MD before the teaching experiment. According to the data in Kouba’s (1989) study, normal-achieving students dominantly use unitary counting strategies only in first grade (97.23%), whereas they gradually decrease their frequency of employing the unitary counting to 66.16% of all strategies used in second grade and 30.11% in third grade (Figure 1). The pretest data in this study showed that the three participants used an extremely high percentage of unitary counting strategies. The baseline data suggested a significant gap between students with and without MD.

Specifically, unitary counting was the most dominant strategy of all three participants during the baseline sessions. They did not use DC at all and used direct retrieval very rarely. Their dominant reliance on unitary counting suggested that the three participants could keep track of only one number sequence and could not use DC to keep track of two number sequences. Although 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 seemed to have problems with both conceptual understanding and retrieval skills in multiplicative reasoning.

Participants’ Strategic Development During the Teaching Experiment

Five dimensions of change are discussed below according to the framework of micro-genetic studies (Siegler, 2006), from the perspectives of both the participants’ performance and strategic development. We also made a preliminary comparison between students with and without MD according to the literature and the current limited data.

Source of change

An adapted multiple-probe design across participants established a functional relationship between the teaching experiment and students’ strategic development. We observed an insistence on using unitary counting at the first session for all three students, and we did not introduce the teaching experiment to the next participant until the previous participant demonstrated a change of strategic choices.

The instructional package (i.e., explicit teaching or prompting DC and constructive task assignment according to ongoing assessment) was effective for facilitating strategic development. On one hand, the participants’ learning through direct instruction supported the former conclusion that explicit instruction benefits students with learning disabilities (Baxter, Woodward, & Olsen, 2001; Kroesbergen & Van Luit, 2003, 2005; Kroesbergen, Van Luit, & Maas, 2004). On the other hand, results indicated that appropriate task assignment would be effective for the participants’ adoption of the targeted strategy. The literature (Siegler & Shrager, 1984; Siegler & Stern, 1998; Simon et al., 2004) suggested that challenging tasks, which increase the discrepancy of advantages of using less advanced strategies and advanced strategies, trigger students’ reflection on the activity (strategy)–effect relationship. Although the present study did not ask the participants to report their feelings of various strategies, further research may explore if participants found DC was more effective for solving such complicated problems than other strategies and hence increased their use of DC.

Path of change

The path of change refers to the sequence of strategies children use while gaining competence (Siegler, 2006). The three participants demonstrated sequences similar to those normal-achieving students go through during multiplicative reasoning strategic development: some strategies become less frequent (unitary counting), some become more frequent (DC for students with MD), and some become more frequent and then less frequent (repeated addition).

Our trial-by-trial analysis revealed that students increased the variety of strategies used before their first use of DC. They began to use multiple strategies, including counting by ones, skip counting, repeated addition, and direct retrieval. According to the research literature, children’s transition from one to multiple strategies indicated their underlying reasoning (Siegler, 2007) and indicated students’ exploration of new strategies. A further trial-by-trial analysis revealed that none of the participants first used the targeted DC strategy immediately after the experimenter explicitly taught it. In other words, explicit modeling did not lead to students’ immediate learning. Future research may explore how to help children with MD to effectively internalize the explicit taught contents and to decrease the latency from students’ “seeing it” to “using it.”

Rate of change

Rate of change refers to the amount of time or experience between the initial use of a strategy and consistent use of it (Siegler, 2006). Based on the limited data, the participants seemed to demonstrate a period of resistance to adopting the new strategy. Counterintuitively, the participants’ use of the DC strategy did not show a rapid increase immediately after the occurrence of this strategy during the teaching experiment. This trend persisted until the participants were asked to solve complicated problems during the third teaching session. All three participants experienced more than six trials of problem solving from their first use of the DC strategy to their consistent use of DC. It seemed that it took participants a long time to be able to consistently adopt the targeted strategy even after their first use of the strategy.

The current limited data support the overlapping wave theory (Siegler, 2007) that the first use of a strategy is only the first step in children’s strategic development; instead, adopting a strategy is progressive or even regressive. Only when children accumulate information about the advantages of the particular strategy in solving a particular type of problem will they increase their use of the strategy for that problem type (Siegler, 2007).

Breadth of change

Breadth of change refers to how widely the new strategy is generalized to other problems (Siegler, 2006). However, the limited data of this study did not provide a formal assessment to measure how far the participants could transfer the advanced strategies to various situated problems. This is a limitation of the current study, and further research will be needed to address this issue. Another issue we need to note is that although the students learned and adopted the DC strategy, they achieved only around 40% correct even at the end of the intervention. Although DC was an indicator of students’ conceptual understanding of multiplicative reasoning, it is not the most efficient strategy to solve all problems. Students may need more advanced strategies (e.g., direct retrieval) to solve problems more efficiently.

Variability of change

Variability of change refers to the differences among children in the previous four dimensions. It was interesting to find quite a variety of children’s strategic choices. Children used more types of strategies during the teaching experiment than during the baseline sessions, although the teaching experiment sought to teach only DC. The variety of strategic choices was considered an indicator of children’s reasoning development (Siegler, 2007).

It is noteworthy that the participants did not consistently increase or decrease the use of a certain strategy as we intuitively assumed. Children did not use the targeted strategy (i.e., DC) immediately after the instructor’s explicit instruction, nor did they consistently use the targeted strategy immediately after their first use. Even in the teaching experiment when the instructor explicitly taught students DC, the students spontaneously employed other strategies on the subsequent problems. This result is consistent with previous studies claiming that human thinking is much more variable than we had assumed (Adolph, 1997; Alibali, 1999; P. H. Miller & Aloise-Young, 1995; Siegler, 2007; Tunteler & Resing, 2002).

In addition, interindividual differences were also found. For example, Barb demonstrated a regression in her use of DC: She utilized some DC during the first session, whereas she ceased using it during the second session. Barb and Andy experienced more trials than Cathy from their first use of DC to a consist use of it. Cathy’s magnitude of increasing her percentage of using DC was the smallest among the three students.

Conclusions, Limitations, and Implications

This study revealed how three students with MD or at risk for MD progressed in their strategic choices during the multiplicative reasoning teaching experiment. We found, first, that the three participants with MD utilized fewer strategies than normal-achieving students described in the research literature; unitary counting was the dominant strategy and led to their low performance in solving multiplicative problems in the baseline sessions. Second, the teaching experiment, which involved both explicit teaching of DC and constructive task assignment, was effective in helping students with MD develop their strategic choices. Third, students may show a latency between the explicit modeling and their self-use of the targeted strategy; constructive task assignment may help students internalize the modeled strategy and use it more consistently. Finally, children’ strategy changes could be a valid measure, in addition to accuracy, to assess children’s academic achievement.

Limitations of this study included the following: First, the single-participant design restrains the generalization of the findings to the overall population with MD. Second, for the data on normal-achieving students, we relied on the former literature. Third, on the posttest, the improvement was not that significant (e.g., to 40%, 46%, and 50%) and fell below a common criterion of 80% or greater. Finally, the third pretest and the posttest were neither one on one nor videotaped, so it was hard to judge children’s strategic choices on these two probes.

Regarding the implications for teachers, DC instruction is applicable for classroom teachers. Moreover, this study suggests that teachers must pay attention to how children solve the problems and choosing appropriate tasks to help students’ reasoning according to assessments. During the teaching experiment, the teachers acted not only as instructors to distribute knowledge to students but also as researchers investigating children’s internal reasoning and then used this information to make appropriate instructional decisions.

Further research may employ a large group comparison to explore how students with MD fall behind their normal-achieving peers. In addition, although the micro-genetic approach has been widely used in psychology, it still needs certain refinement when employed for students with disabilities; in particular, the methodological issues regarding how to integrate micro-genetic analysis with a single-participant design would be of interest to future researchers.

Footnotes

Acknowledgements

The authors thank Dr. Ron Tzur for implementing the teaching experiment and his valuable feedback during manuscript preparation and revision. The authors also thank Lafayette School Cooperation (LSC; particularly Dr. Edward Eiler) for their support of this research project. In particular, we thank the administrators (particularly the principal, Mrs. Pat Miller), teachers and staff, and students at Earhart Elementary School as well as Casey Hord at Purdue University and Laura Hord at LSC who facilitated this study.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interests with respect to the authorship and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research and/or authorship of this article: This research was partially supported by the National Science Foundation, under Grant DRL 0822296. The opinions expressed do not necessarily reflect the views of the foundation.

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