The Effect of Eliciting Repair of Mathematics Explanations of Students With Learning Disabilities

Abstract

Mathematical reasoning is important in conceptual understanding and problem solving. In current reform-based, discourse-oriented mathematics classrooms, students with learning disabilities (LD) encounter challenges articulating or explaining their reasoning processes. Enlightened by the concept of conversational repair borrowed from the field of linguistics, this study designed an intervention program to facilitate mathematical reasoning of students with LD. Conversational repair, an ability to repair communicative breakdowns or inaccuracies, was designed in an implicit–explicit continuum to elicit self-explanation from students with LD in the context of mathematics word problem solving. Using a multiple-baseline across participants design, the study found that the intervention was effective for improving students’ mathematical reasoning and problem-solving ability measured by their self-explanation and word problem–solving performance. It provided implications for future studies concerning the use of conversational repair in mathematics classroom discourse for individuals with LD.

Keywords

learning disabilities mathematical reasoning self-explanation conversational repair mathematics word problems

The Standards for Mathematical Practice (Common Core State Standards Initiative [CCSSI], 2012) state that mathematically proficient students should be able to “justify their conclusions, communicate them to others, and respond to the arguments of others.” Students are expected not only to solve problems on paper but also to express their thought processes/reasoning to others for the sake of discussion and argumentation. This activity entails self-explanation, defined as making utterances that “involve not only inference of new knowledge but also clarification of the problem and justification of activities that occur during the problem-solving process” (Neuman, Leibowitz, & Schwarz, 2000, p. 199). Self-explanation in mathematical reasoning is an inter-individual talk, which enables students to “both solidify their own understandings and potentially support their peers” (Kotsopoulos, 2010, p. 1052) in mathematical sense making.

In general education, numerous studies have shown that learners who do self-explanation perform better in various cognitive tasks (e.g., Aleven & Koedinger, 2002). This is known as the self-explanation effect. For example, Rittle-Johnson (2006) found that self-explanation led to greater learning and transfer (2-week delay) for third through fifth graders solving mathematical equivalence problems, regardless of instructional conditions (direct instruction or discovery learning).

Studies concerning self-explanation have not been widely conducted in special education. According to a meta-analysis of instructional components in mathematics instructions for students with learning disabilities (LD; Gersten et al., 2009), though there were intervention studies that contained student verbalization such as self-instruction and metacognitive strategy instruction, self-explanation was not specifically explored. According to Baxter, Woodward, and Olson (2001), students with LD are only minimally involved in mathematics classroom discussions. They rarely speak and are easily distracted. Their passivity may be attributed to cognitive overload from the reform-based curriculum and inadequate opportunity to speak.

However, existing research indicates that these students could also learn thinking behaviors such as asking questions, disagreeing, explaining, and suggesting solutions (Berry & Kim, 2008) if teachers used effective instructional strategies (Baxter et al., 2001). For example, Baxter, Woodward, and Olson’s (2005) study on four seventh graders whose mathematics achievements were in the lowest third of their class found that they were willing to explain their reasoning and feelings in journals, and that more than half of each student’s journal contained descriptions of their problem-solving steps.

Instructional Strategies to Scaffold Self-Explanation of Students Without Disabilities

To improve student self-explanation, we uphold the proposition raised by the National Mathematics Advisory Panel (NMAP; 2008) that exclusive use of either entirely “student centered” or “teacher directed” instruction is not effective for any student. This also aligns with social development theory (Vygotsky, 1978) and social constructivism (Cobb, 1994), both of which agree on the importance of communication in teaching-learning activities. In consonance, various instructional strategies have been used to scaffold self-explanation in both general and special education. In general education, for example, a series of studies has been conducted concerning intelligent instructional software called the Geometry Cognitive Tutor for learning geometry. One of the earlier versions of the software supported explanation by asking students to select from a reference list the principles that justified problem-solving steps. Students’ responses were followed by corresponding feedback from the tutor. It was found that students demonstrated better problem-solving performance and greater understanding after working with the software (Aleven & Koedinger, 2002). The software was then further developed to allow students to type in a self-explanation in their own words (Aleven, Ogan, Popescu, Torrey, & Koedinger, 2004). The researchers realized that using students’ own language as the interface would allow the learners to spend more time recalling and building on prior relevant knowledge. Based on student input, the software provided “a sequence of increasingly more directed feedback messages” (Aleven, Koedinger, & Popescu, 2003, p. 41), which ended with a “bottom-out hint” (Salden, Koedinger, Renkl, Aleven, & McLaren, 2010, p. 382). This experiment (Aleven et al., 2004) found that, students in the dialogue condition provided better explanations on geometry theorems and definitions.

Instructional Strategies to Scaffold Self-Explanation of Students With Disabilities

Concerning instructional strategies to scaffold inter-individual expressive reasoning for students with LD, of most relevance were the studies about coaching students with LD for explanations on the topic of animals (Scruggs, Mastropieri, & Sullivan, 1994; Scruggs, Mastropieri, Sullivan, & Hesser, 1993; Sullivan, Mastropieri, & Scruggs, 1995). For example, in Sullivan et al.’s (1995) study, 63 fourth- and fifth-grade students with LD were assigned to three conditions and provided with information about animals. In the coaching condition, students were given different levels of coaching after they heard information such as “The honey bear has a double layer of fur.” In Coaching 1, the question was “Why would that make sense?” If a student failed to produce an appropriate explanation, Coaching 2 was implemented, where the student was asked another question with relational information as a hint, such as “What would honey bears eat and where would they get their food?” If the student failed again, Coaching 3 gave a statement, which provided a clearer hint, and asked for the reason again, such as “remember, honey bears steal honey from bee hives, so why would it make sense that the honey bear has a double layer of fur?” If the student failed once again, in Coaching 4, the students were provided with the reason in the form of a question, and all they needed to give was a yes or no response, such as “since honey bears need protection from the bees when they steal their honey, would it make sense that the honey bear has a double layer of fur?” In the provided-explanation condition, the experimenter told the reasoning after the basic information and asked students to repeat what they had heard. In the no-explanation control condition, the experimenter provided only the basic information without explanation, and asked students to repeat it. The study showed that students in the coached condition performed the best for both the immediate and delayed recall as well as for the explanation tests. This indicated that when students with LD were coached to generate their own explanations, they remembered substantially and significantly more information. However, they needed a necessary prompt, such as the “why” question, as students with LD did not volunteer the strategy use.

Stacked Requests for Repair

Stacked requests is a technique to elicit conversational repair (Schegloff, Jefferson, & Sacks, 1977) that is originated from conversation analysis (Liddicoat, 2007). Essentially, the stacked requests refer to a sequence of requests from the listener that trigger the speaker to repair the initial utterance, which results in a successful conversation with mutual understanding (Schegloff et al., 1977). Both the feedback in the Geometry Cognitive Tutor and the coaching questions about animals reviewed above could be seen as stacked (i.e., a series of) requests that provided increased scaffolding to elicit proper reasoning or repair of self-explanation from students. We see potential for adapting this method into mathematics education as a scaffolding strategy to improve self-explanation for the purpose of enhancing mathematical reasoning. This is a worthwhile endeavor answering the urgent call of current mathematics education reform to involve all students in discourse-oriented instruction. Therefore, the present study proposed the use of repair requests with varied levels of scaffolding to elicit self-explanation repair of mathematics word problems from students with LD. The specific research questions are as follows:

Research Question 1: What was the quality of self-explanation of students with LD before intervention?

Research Question 2: What was the effect of repair requests on the quality of self-explanation of students with LD?

Research Question 3: What was the effect of repair requests on the word problem–solving performance of students with LD?

Method

Participants

We secured approval from the institutional review board prior to recruiting participants. The participants consisted of three fourth-grade students with LD from two Midwest, urban, public elementary schools in the United States. They met school district criteria for having a specific learning disability (SLD; see Table 1 for demographic information). The special education eligibility criteria used by the participating school defines SLD as a severe discrepancy between the student’s academic achievement and normal or near normal potential. Participants who were identified as SLD were recommended by the school for further testing. Those who performed below 70% correct in problem solving and in self-explanation on the criterion test were considered meeting the selection criteria, because a score of 70% or more correct could be viewed as an average grade or “C,” according to Montague and Bos (1986).

Table 1.

Participating Students’ Demographic Characteristics.

Variable	Amy	Bill	Carl
Gender	Female	Male	Male
Ethnicity	Caucasian	Caucasian	African American
Age	9 years 1 month	10 years 6 months	10 years 11 months
Grade	4	4	4
Classification	SLD	SLD	SLD
SES	Medium	DK	DK
Reduced/free lunch	Paid	Reduced	Reduced
Years in special education	1 year	2 years	3 years
Learning support classroom	Reading Spelling Writing	Reading Math	Reading Math
Percentage of time in general education class	65	40–79	80
IQ	WISC-IV	OLSAT	WISC-IV
Full scale	94	98	73
Verbal	100	97	83
Performance (non-verbal)	94	100	NA

Note. SLD = specific learning disability; SES = socioeconomic status; DK = do not know; WISC-IV = Wechsler Intelligence Scale for Children–Fourth Edition (Wechsler, 2003); OLSAT = Otis–Lennon School Ability Test (Otis & Lennon, 1995).

Dependent Measures

The participants took a criterion test (and its alternate forms) and a transfer test. Each test was analyzed for two dependent variables: problem-solving accuracy and quality of self-explanation (primary dependent measure).

Criterion test

The criterion test and its alternate forms were used in the pre-test, the intervention, the post-test, and the maintenance test phases. In each test, there were six one-step equal group (EG) problems (extracted from the problem database in Xin, Wiles, & Lin, 2008), two for each of the three variations as shown in the first two columns of Table 2. The order of the problems was randomized across the alternate test forms. We chose to use this criterion test because the test items were designed in alignment with the National Council of Teachers of Mathematics standards (NCTM, 2000), which emphasizes varying constructions of word problems for assessing conceptual understanding of mathematics problem solving (Cawley & Parmar, 2003). According to Xin and Zhang (2009), Cronbach’s alpha of the criterion test was .86, and the parallel-form reliability of the alternate forms of the criterion test was .85. Each problem was printed on the top of a sheet of 8½ inch by 11 inch unlined paper, leaving space below for participants to work on the solutions. The six sheets of paper for a test were stapled together.

Table 2.

Examples of Three Variations of EG Problems.

Problem type	Problem	Solution	Modeled explanation
“Product” (the number of total item[s]) is unknown.	It takes a sewing machine 2 hr to finish sewing a dress. If there are 6 dresses to sew, how long will it take the sewing machine to finish sewing all the dresses?	2 × 6 = 12	I did 2 times 6 is 12, so it takes 12 hr for the sewing machine to finish sewing all the dresses. The reason I did so is that the sewing machine takes 2 hr to sew one dress, another 2 hr to sew another dress, and another 2 hr for another dress. It is like many groups of 2 hr. There are 6 dresses, so it is like 6 groups of 2. To find the total number of hours, I used the number of hours for each dress, which is 2, times the number of dresses, which is 6.
“# of units” (number of group[s]) is unknown.	It takes a sewing machine 2 hr to finish sewing a dress. If the sewing machine worked for 6 hr without stopping, how many dresses can it finish sewing?	6 ÷ 2 = 3	I did “6 divided by 2 is 3,” so it can finish sewing 3 dresses. The reason I did so is that the sewing machine takes 2 hr to sew one dress, another 2 hr to sew another dress, and another 2 hr for another dress. It is like many groups of 2 hr. The total number of hours is 6, so it is like dividing 6 into groups of 2. Therefore, I divided 6 by 2.
“Unit rate” (the number of item[s] in each group) is unknown.	A sewing machine can sew a type of dress. The sewing machine worked for 6 hr without stopping and finished sewing 2 dresses. If it took the same amount of hours to sew each dress, how many hours did the sewing machine use to finish one dress?	6 ÷ 2 = 3	I did “6 divided by 2 is 3,” so it used 3 hr to finish each dress. The reason I did so is that the sewing machine worked for 6 hr in total to sew two dresses, and each dress took the same number of hours. It is like distributing the 6 hr equally into 2 groups, so I divided 6 by 2 to find out how many in each group.

Note. EG = equal group.

Transfer test

The transfer test was developed with the purpose to evaluate whether the participants could transfer what they learned in problem solving, along with reasoning, from simple context one-step problems to more complicated two-step word problems. To ensure the content validity of the transfer test, the test items were directly taken from school-adopted mathematics textbooks. The transfer test had the same format as the criterion test. It had four two-step EG word problems: three from enVision MATH Common Core Edition (enVision MATH, 4th Grade, Charles et al., 2012) and one from Harcourt Math Indiana Edition (Harcourt Math, 4th Grade, Maletsky, Andrews, Burton, Johnson, & Luckie, 2004). The two-step solution of each problem involved one of four possible combinations of the four basic operations (i.e., ××, ÷÷, ×÷, ÷×). The transfer test was conducted toward the end of the pre-test, post-test, and maintenance test phases, respectively.

Accuracy of problem solving

The accuracy of problem solving was measured by the total points the participant earned for correctly solving problems on each test. For the criterion test and its alternate forms, each problem was assigned 2 points, 1 point for the procedure and 1 point for the answer; the total possible score for each test was 12 points. Table 3 provides the scoring criteria and sample solutions for different scores.

Table 3.

Examples of Problem-Solving Performances Scored as 2 Points, 1 Point, and 0 Points.

Scoring	Sample solutions (in the context of solving the word problem: It takes a sewing machine 2 hr to finish sewing a dress. If the sewing machine worked for 6 hr without stopping, how many dresses can it finish sewing?)
2 points	1. The participant only provided the correct number. For example, “3.” 2. The participant provided a correct problem-solving process (no matter what strategy was used), and it led to the correct number as the answer, such as the following: a. “6 ÷ 2 = 3,” b. “6 − 2 − 2 − 2 = 0, so 3.”
1 point	1. The participant provided a problem-solving process that reflected the correct understanding and reasoning, but ended up without a number as the answer or with an incorrect number as the answer. For example, “6 ÷ 2 = 2.” 2. The participant had the correct number as the answer, but the problem-solving process was incorrect (reflecting incorrect understanding or reasoning of the problem context). For example, “6 − 2 − 2, so 3.”
0 points	1. Both the problem-solving process and the answer were incorrect. For example, “6 − 2 = 4.” 2. Only an incorrect number (e.g., “4”) or nothing was written as the answer.

Each problem in the transfer test had two steps. Each step was assigned 2 points, 1 point for the procedure and 1 point for the answer, the same as the scoring system for the criterion test. Thus, the total possible score for each problem was 4 points and the total possible score for the transfer test was 16 points.

Quality of self-explanation

The participants’ initial explanations (before any prompt from the teacher) for each problem in a test were scored, and these scores were added up to measure the quality of self-explanation on the test. For the criterion test and its alternate forms, each problem was assigned 2 points for self-explanation; the total possible score for each test was 12 points. For the transfer test, each problem was assigned 4 points for self-explanation, 2 points for the first step and 2 points for the second step; the total possible self-explanation points for the transfer test was 16. Table 4 illustrates the scoring rubric and sample explanations for 2-point, 1-point, and 0-point explanations. The scoring rubric was developed on the basis of the reasoning scoring rubric of the Test of Problem Solving Elementary (3rd edition; Bowers, Huisingh, & LoGiudice, 2005), which assesses a school-aged child’s thinking, reasoning, and problem-solving abilities through linguistic expressions. Specifically, the authors first identified the scenarios that anchor the following three levels of reasoning (a) no reasoning, (b) emerging reasoning, and (c) correct reasoning; then, we described them in reference to Bowers et al.’s (2005) levels of linguistic expressions.

Table 4.

Examples of Self-Explanations Scored as 2 Points, 1 Point, and 0 Points.

Scoring	Sample explanations (in the context of solving the word problem: It takes a sewing machine 2 hr to finish sewing a dress. If the sewing machine worked for 6 hr without stopping, how many dresses can it finish sewing?)
2 points	The statement expresses clearly and correctly the meaning of the numbers mentioned in the statement, and it shows a correct understanding of the multiplicative relationships between quantities. In other words, it correctly expresses the meaning of “multiple groups of something” such as the following: a. “It takes 2 hours to sew one, another 2 to sew another one, another 2 . . . so it is like a pattern, totaling 6 hours. So I need to find out how many 2s there are in 6, so I divided and got that it can sew 3 dresses.” b. “It is an equal group problem because it takes an equal number of hours to finish every dress. We know the total number of hours for sewing dresses is 6. We also know sewing one dress takes 2 hours. To answer how many dresses we do 6 ÷ 2.” c. “It is to find out if you put 6 in groups of 2, how many groups can there be.”
1 point	1. The statement contains less advanced reasoning or strategies (such as repeated addition or repeated subtraction) that could lead to the correct answer, but does not indicate an awareness of the “multiple groups of something” situation. e.g., “I minus 2 from 6 and keep going down and down.” 2. The statement contains important information, but is limited, reflecting vagueness, confusion, incompleteness, or immaturity in the understanding of multiplicative relationships between quantities (such as repeating the information given in the problem following a correct algorithm for solution, inadequate clarification on the meaning of the numbers in the explanation following a correct algorithm for solution, and mixing up the “unit rate” and the “number of units”, etc.). For example, a. “It takes 2 hours to do one, 6 hours in total, so 6 divided by 2.” (It is a repetition of the given information in the problem without understanding shown.) b. “Six divided by 2 is 3, so 3 dresses.” (Repetition of the algorithm) c. “Two groups of three will be six.” (Mixing up the “unit rate” and the “number of units”) d. “It is to put the 6 hours into 2.” (Inadequate clarification for the meaning of the number “2”, which can be “2 groups” or “groups of 2”)
0 points	The statement is irrelevant or gives incorrect understanding of the multiplicative relationships between quantities, such as a. “Two hours to sew a dress and now 6 hours, so I did 6 - 2.” b. “I do not know how to explain it.” c. “6 groups of 2.”

Design

A multiple-baseline design across participants (Horner et al., 2005) was used to evaluate the functional relationship between the intervention and participants’ self-explanation quality and word problem–solving accuracy. With the multiple-baseline design across participants, the intervention was introduced to different participants one at a time after consistent response patterns were observed in each individual baseline. When the participant who was receiving the intervention showed a clear change in behavior pattern, the second participant/baseline was introduced to the intervention. If the changes in each baseline occurred only when the intervention was introduced, a functional relationship was demonstrated (Kennedy, 2005).

Chronologically, the experiment included four phases: the pre-test, the intervention, the post-test, and the maintenance. The four phases belonged to two major conditions in terms of experimental design: the baseline condition (including pre-test, the post-test, and the maintenance) and the intervention condition.

Procedure

The participants were recruited from their classrooms to a quiet room in their respective schools five times a week to participate in the study. The first author conducted the study with each of the participating students one session a day, Monday through Friday. All sessions were videotaped. The participants’ names in the following description are all pseudonyms.

After all three participants completed one criterion test on the first day, the first author randomly selected one student, Amy, to continuously take alternate forms of the criterion test. Once Amy’s self-explanation scores showed a stable trend, the intervention was introduced to Amy. However, the rest of the participants continued with the baseline condition. When Amy’s self-explanation scores showed an accelerating trend upon receiving the intervention, a second randomly selected participant, Bill, would then enter the intervention phase after he had taken alternate forms of the criterion test and showed a stable trend. The same sequence was followed for the third participant, Carl.

Baseline condition (A)

The first author asked the participants to read each problem, write down all the steps involved in the problem-solving processes, including math sentences or equations, in the space provided, and then explain each of the problem-solving steps. During the post-test and the maintenance test phases, the participants followed the same procedure as described above.

Intervention condition (B)

As in the case of the baseline condition, the first author gave the participants one alternate form of the criterion test to work on with the same directions. For each problem, after the participant finished the initial explanation, the researcher implemented the intervention according to the quality of the explanation. That is, if the explanation indicated correct and clear reasoning, the participant would not receive any scaffolding for this problem, and would move on to the next problem. Otherwise, the intervention package was implemented. The intervention phase for a specific participant would complete when the participant’s problem-solving scores and initial self-explanation showed a stable trend with no less than 75% correct. Therefore, the number of sessions a participant needed for completing the intervention condition varied from one to another as it depended on each participant’s progress.

Intervention components

We adapted Weiner’s (2005) classification and developed four types of repair requests based on their functions: general requests, requests for revision, requests for specification/clarification, and direct teaching/modeling, which comprised the intervention package of this study. The components were applied on different occasions:

The general request “Do you want to try again to make it better?” was given after a participant produced a problematic initial explanation. A problematic explanation was defined as one of the two types: (a) one that was relevant but not precise, which indicated emerging reasoning; and (b) one that was incorrect or offered no response, which indicated faulty or no reasoning.

Following the general request, if the participant’s repaired explanation still reflected emerging reasoning, the first author then requested specification/clarification of the unclear parts. This was accomplished by repeating the repairable parts in the participant’s response and adding a wh - question, or by saying “And could you tell me more about why you did so?” After this, if the explanation was still relevant but not precise, the first author modeled a full-scored explanation to the participant (see the last column of Table 2 for modeled explanations for different problem types).

If the participant’s repaired explanation, following the general request, indicated a faulty reason or no reasoning, the first author would make a request for revision. That is, the first author would offer the meanings of the two known numbers in the problem by saying “This number means . . . and this number means . . . So do you want to revise your explanation?” Then, if the repaired explanation indicated an emerging reasoning, the request for specification/clarification, and the direct teaching/modeling if necessary, would follow, as it is described above. Otherwise, if the explanation indicated a faulty reason or no reasoning, the first author would provide direct teaching on how to solve and explain the problem.

Direct teaching

The direct teaching method was based on the conceptual model-based approach to teach word problem solving (Xin, 2012). Specifically, the word problem (WP) story grammar question cards developed by Xin et al. (2008) were adopted to teach students how to identify and represent the three elements in an EG problem, map the information in the equation, and use the equation to solve the problem (see Figure 1) following the teaching script of Xin (2012).

Figure 1.

Equal group problem prompt card.

The flowchart in Figure 2 shows the movement among the different treatment components as an intervention proceeded. The intervention components, the contexts in which each component was applied, and examples are demonstrated in Table 5.

Figure 2.

Flowchart of the movement among the different treatment components.

Table 5.

Intervention Components and Corresponding Application Contexts.

Intervention components	Examples (In the context of solving the word problem: It takes a sewing machine 8 hr to finish sewing a dress. If the sewing machine worked for 96 hr without stopping, how many dresses can it finish sewing?)
General request	Student: I drew a picture. Teacher: Do you want to try again to make it better?
Request for revision	Student: I got 8 times 96. Teacher: The number 8 means the number of hours the sewing machine needs to finish one dress. The number 96 means the total number of hours the sewing machine worked to sew dresses. So do you want to revise your explanation?
Request for specification/clarification a. Repeating the repairable part in the participant’s response and adding a wh-question. Or, b. “And could you tell me more about why you did so?”	a. Student: It takes 8 hours to sew one dress, another 8 to sew a second one . . . so it is to see how many in 96. Teacher: How many what? Student: How many 8s. b. Student: I did 96 ÷ 8. Teacher: You did 96 ÷ 8. And could you tell me more about why you did so?
Direct modeling of self-explanation	See the last column of Table 2.
Direct teaching	See Xin (2012) for the full teaching script.

Treatment Fidelity and Inter-Rater Reliability

One third of all the intervention sessions across the three participants were observed by an independent observer to check the treatment delivery. Treatment fidelity was calculated as the percentage of correctly implemented treatment components divided by the total possible treatment components. The treatment fidelity was calculated to be 96%. The first author transcribed all the sessions, and scored all the tests for problem-solving accuracy as well as the quality of self-explanation. A PhD student, who was majoring in special education at the time of the study and was blind to the purpose of the study, independently re-scored 30% of the test items. To calculate inter-rater reliability, the number of agreements was divided by the number of agreements and disagreements and multiplied by 100%. The inter-rater reliability for the problem-solving scores was 95%, and was 93% for the self-explanation scores.

Data Analysis

Each participant’s performance (on both the problem-solving measure as well as the self-explanation measure) during the baseline, intervention, and post-intervention phases was plotted in a graphic display. Visual inspection was used to evaluate the quantitative information of the graph (Kennedy, 2005). Specifically, the visual inspection focused on three within-phase dimensions: (a) the level of the data, typically represented by the mean or median; (b) the trend of the data; and (c) variability. It also focused on two between-phase dimensions: (a) immediacy of effect and (b) overlap of data points.

We used a nonparametric procedure, the percentage of non-overlapping data (PND; Scruggs, Mastropieri, & Casto, 1987), to estimate the effectiveness of the intervention. According to Scruggs et al. (1987), PND is calculated as “the number of treatment data points that exceeds the highest baseline data point in an expected direction” divided by “the total number of data points in the treatment phase” (p. 27). Scruggs and Mastropieri (1998) described PND values of 90%, 70%, and 50% as benchmarks for large, moderate, and low proportions of effect, respectively. We chose PND over parametric approaches for calculating effect size (e.g., the standardized mean difference) because single subject research designs typically incorporate a graphic display of results with a limited number of data points, which may restrict a reliable and valid use of parametric analysis approach (Scruggs & Mastropieri, 1998).

Results

Figure 3 shows each participant’s performance measured by problem-solving accuracy as well as the quality of self-explanation on the criterion test (including its alternate forms) across the baseline, intervention, post-test, and maintenance phases. Student performance on the transfer test was also marked on the same line graph for the baseline, post-test, and the maintenance phases.

Figure 3.

WP solving and self-explanation performance for the criterion tests and the transfer test during the baseline, intervention, and post-intervention conditions for the three participants.

Baseline Analysis

Amy

(a) Problem solving: Amy’s data showed a moderate degree of variability, with a median score of 5 points (ranging from 3 to 8 points) and a moderate downward trend. Amy’s strategy at this phase was repeated addition/subtraction on the calculator and the drawing of lines on her worksheets to indicate the number of times she had added or subtracted. She easily mismarked or miscounted the tallies. After three worksheets, Amy used multiplication to replace both repeated addition and repeated subtraction for the rest of the problems (Worksheets 4 and 5). This decreased her performance level. (b) Self-explanation: Amy’s data showed a low degree of variability, with a median score of 3 points (ranging from 0 to 5 points) and a moderate downward trend. During the baseline condition, most of the time Amy’s self-explanation showed emerging reasoning (see Example 1).

Example 1 (in Session 1)

(Task: Uncle Jim is a painter. He says that it takes 8 gallons of paint to paint one entire house. How many houses could he paint with 408 gallons of paint, if he uses the same amount of paint on each house?)

Amy (A):

If you have a calculator, you put 408 in the calculator and minus 8, minus 8 and keep on going. I put tally marks every time I did it. I got 50 8s to get to 0 from 408. What the calculator does is once you do minus every time it shows what you have and then goes down each time you do it.

Amy’s explanation in Example 1 showed that she used a less advanced strategy (repeated subtraction) to solve the problem (though she miscounted the number of times she subtracted 8). According to the coding scheme in Table 4, such explanations were scored as 1 point.

Bill

(a) Problem solving: Bill’s data showed a low to moderate degree of variability, with a median score of 4 points (ranging from 0 to 6 points) and a moderate upward trend because his first session was the only session in which he received 0 points for problem solving. Bill used all four operations (addition, subtraction, multiplication, and division). He often worked on the calculator first and then copied the numbers on paper. He only had a math equation as the answer; there were no drawings or tallies as in the case of Amy. (b) Self-explanation: Bill’s data showed a low degree of variability, with a median score of 0 points (ranging from 0 to 1 point) and nearly a flat trend.

The most distinct feature of Bill’s explanations was providing general statements, which made it difficult for the researchers to evaluate his reasoning or understanding of the problem. For instance, in nearly every case he would say, “I solved it this way because it would be better for me. It feels better and easier for me to do.” In another example, he stated, “I did this because it was very hard to add for me or use anything else. That’s why I used multiplication because multiplication is easy for me to do.” Explanations such as these were coded as 0 points according to the coding scheme.

Carl

(a) Problem solving: Carl’s data showed a low degree of variability, with a median score of 0 points (ranging from 0 to 1 point) and a nearly flat trend. Throughout the baseline condition, Carl used multiplication three times (for three division problems); he solved all other problems with either addition or subtraction. (b) Self-explanation: Carl’s data showed no variability as he obtained 0 points for each of the sessions. Carl’s most common explanation in the baseline condition focused on the calculation process (see Example 2).

Example 2 (in Session 1)

(Task: Gary made 41 buttons when running for class president. It takes 23 drops of glue to make each button. How many drops of glue did Gary use?)

Carl (C):

I add 41 and 23. I put the biggest one on top and I add the 1 and 3 together first, and then add 4 and 2. If I do times, 3 times 1 it will be back to 3. It will get me wrong.

Such explanations were coded as 0 points as they showed incorrect understanding of the multiplicative relationship. Another strategy Carl often used was the “keyword” strategy. He made statements such as “‘Buy’ means take away,” “‘Use’ means take away,” and “I times it because it said ‘how many’ so I times it in my head then on the calculator.” In these instances, Carl made the judgment based on one or two words in the problem but not on a conceptual understanding of the mathematical problem structure. Such explanations were scored as 0 points.

Intervention Analysis

Amy

(a) Problem solving: Amy’s problem-solving performance during the intervention phase overall showed a moderate degree of variability and a moderate upward trend. She had a substantial increase in the first session of the intervention condition (12 points, as compared with 4 points in the last session of the baseline condition). The median of Amy’s problem-solving scores during the intervention was 10 points (ranging from 8 points to 12 points), and the PND was 85.7%. (b) Self-explanation: Amy showed an improvement during the first intervention session (7 points, as compared with 0 points during the last session of the baseline). During the intervention phase, Amy’s self-explanation performance showed a moderate upward trend with a low degree of variability. Her self-explanation scores ranged from 5 points to 9 points, with a median of 7, and a PND of 85.7%. Figure 3 shows that Amy’s performance in both problem solving and self-explanation reversed from a decreasing trend during the baseline to an increasing trend during the intervention condition.

Starting from Session 4, Amy began to develop her own method of explanation. For the problems in which the product was unknown, she would say “do m n times” (m stands for the “unit rate” and n stands for the “number of units,” please refer to Figure 1). This is illustrated in Example 3 as shown below.

Example 3 (in Session 7)

(Task: There are 26 legs on a particular type of centipede. How many total legs would 31 centipedes have?)

So basically what you are doing is um . . . you have 26 legs on one centipede. One centipede has 26 legs, and they want 31 centipedes, so like centipedes, centipedes (drawing centipedes on paper) I am just pretending these are 26 legs, and pretending these are 31. So basically this is 31, so they want 31, they have 31, and 31 is how many times you do 26, so you do 26 31 times.

The above explanation earned 2 points.

Bill

(a) Problem solving: Bill’s problem-solving performance showed a moderate upward trend with a low to moderate degree of variability. His performance increased from 4 points during the last baseline session to 10 points during the first intervention session. His median performance during the intervention phase was 12 points (ranging from 8 to 12 points). The PND of Bill’s improvement in problem solving was 100%. (b) Self-explanation: Bill improved in his self-explanation performance from 0 points during the last baseline session to 5 points during the first intervention session. His self-explanation performance showed a moderate upward trend with a moderate degree of variability. His scores ranged from 4 points to 10 points during the intervention with a median of 8 points and a PND of 100%.

In the latter part of the intervention condition, Bill started to develop his own technique of explaining division problems (see Example 4). Compared with division word problems, multiplication word problems (product unknown) were more difficult for Bill. During the early stage of the intervention phase, his most common explanation for multiplication problems is illustrated in Example 5.

Example 4 (in Session 7)

(Task: One box of chocolates costs $16. If Erika has a total of $144, how many boxes of chocolates can she buy?)

B (Bill):

They want to see how many 16s are in 144.

Example 5 (in Session 3)

(Task: You can pick your own strawberries at the festival. If there are 16 pints of strawberries in a box, how many pints are in 5 boxes?)

You want to multiply 16 to 5 to get the answer because . . . they want us to see how many boxes they want . . . I mean, see how many 16s in 5 boxes.

The “how many 16s in 5 boxes” showed that he was mechanically but incorrectly applying his method of explaining division problems to multiplication problems. This type of explanation reflected a faulty understanding of the multiplicative relationship between numbers. Thus, it was scored as 0 points according to the scoring rubric. Toward the end of the intervention condition, he developed his approach for the explanation of multiplication problems: “there are n ms.”

Carl

(a) Problem solving: Carl made an improvement in problem-solving performance from 0 points during the baseline condition to 4 points during the first session of the intervention. His problem-solving performance during the intervention phase overall showed a slow upward trend with moderate degree of variability. The median of his performance during the intervention was 12 points (ranging from 4 to 12 points). The PND of Carl’s improvement in problem solving was 100%. (b) Self-explanation: Carl improved his self-explanation from 0 points during the last baseline session to 3 points during the first session of the intervention. During the intervention condition, Carl’s self-explanation performance showed a slow upward trend with a moderate degree of variability. His scores ranged from 1 point to 12 points, with a median of 7 points. The PND of Carl’s improvement in self-explanation was 100%.

As in the cases of both Amy and Bill, Carl gradually developed his explanation technique. He developed a strategy to explain multiplication problems earlier than division problems, with the “format” of “n groups of m” (n stands for the “number of units” and m stands for the “unit rate”; see Example 6).

Example 6 (in Session 2)

(Task: You can pick your own strawberries at the festival. If there are 16 pints of strawberries in a box, how many pints are in 5 boxes?)

There are 16 pints. They want to put 5 equal groups of 16.

Before learning a proper way of explaining division problems, Carl’s most common mistake was to explain division problems in a “multiplication way.” See Example 7 below for illustration.

Example 7 (in Session 3)

(Task: Edwin received a total of $374 to buy basketballs for the football team. Each basketball costs $34. How many basketballs can he buy?)

They want, there are 374 dollars, and they want 34 basketball, so they want 34 equal groups of 374.

In Example 7, Carl solved the problem correctly with the use of division, but his explanation reflected incorrect understanding of the multiplicative relationship among the quantities. Thus, explanations as such were scored as 0.

During the last sessions of the intervention condition, Carl developed his strategy for explaining division problems. See Example 8 for illustration.

Example 8 (in Session 13)

(Task: It takes a total of 195 lemons to make 15 gallons of lemonade. How many lemons does it take to make just one gallon?)

So 15 groups of 13 in 195.

Post-Test and Maintenance Test

Three data points were collected for the post-test and the maintenance test, respectively. Both the post-test and the maintenance test used the pre-test worksheets. However, the problems were ordered differently. During the post-test, which immediately followed each participant’s intervention condition, Amy and Carl got full points (12 points) in problem solving across three points of testing. Bill got 12 points in problem solving for the first two tests and 10 points for the last test. As to self-explanation performance, Amy’s scores for the three tests were 10, 10, and 6 points (median = 10); Bill’s were 9, 9, and 7 points (median = 9), and Carl’s were 10, 9, and 10 points (median = 10).

During the maintenance test administered 1 month after each participant’s post-test, all three participants received a full score of 12 points in problem solving across the three points of testing. As to self-explanation, Amy scored 8, 6, and 11 points for the three points of testing (median = 8); Bill scored 6, 9, and 10 correct (median = 9); and Carl scored 10, 7, and 8 (median = 8).

Transfer Test

Before the intervention, the three participants, Amy, Bill, and Carl, scored 2, 4, and 2 points, respectively, in problem solving. They scored 1, 0, and 2 points, respectively, in self-explanation. Following the intervention, these three participants scored 12, 2, and 4 points in problem solving, respectively. In contrast, they scored 10, 1, and 4 points, respectively, in self-explanation. In the maintenance, they scored 6, 4, and 2 points in problem solving, respectively, and scored 4, 4, and 1 point in self-explanation, respectively.

Discussion

This study designed an intervention package consisting of repair requests with different levels of scaffolding, and explored its effectiveness in promoting self-explanation and word problem–solving performance of students with LD. As reviewed earlier, the ability to explain one’s own thoughts and reasoning to others becomes increasingly important in mathematics classrooms (CCSSI, 2012). However, students with LD are marginally involved in classroom discussions (Baxter et al., 2001). Little has been done to examine how they articulate their reasoning and ways to promote such ability. Emerging from existing literature on “a sequence of feedbacks” (e.g., Aleven et al., 2003) and “coaching questions” (e.g., Sullivan et al., 1995), this study aimed to address this issue by designing an intervention that highlighted the critical role of communication in learning. Consistent with the intervention design in existing literature (e.g., Aleven et al., 2003; Sullivan et al., 1995), students were provided with opportunities to self-construct explanations rather than passively waiting for direct teaching from the instructor. Meanwhile, they were not left alone in the process; they were guided with different levels of scaffolding toward a meaningful explanation.

Intervention Effectiveness on Self-Explanation

To answer the first research question on how the participants’ self-explanations were like before the intervention, and the second research question on the effectiveness of the intervention on the quality of their self-explanation, we have seen from the results that all participants began with low-level performance in self-explanation on the criterion test. At the beginning of the experiment, Amy’s explanation primarily reiterated what she had done on the calculator. Bill began with general statements that failed to reflect his mathematical reasoning. Carl talked about how he added or subtracted two numbers. With the intervention, participants gradually needed less and less scaffolding from the researcher to produce satisfactory self-explanations. When a mistake occurred, they could recognize it and self-initiate the repair. For instance, during the late intervention phase, Carl made a self-initiated repair of the initial self-explanation by saying “They want 19 groups of 551. No. Nineteen groups of 29” in solving the problem: “It costs a total of $551 to buy 19 super-sized pizzas for a school party. How much did each pizza cost?”

As such, this study supports existing studies (e.g., Berry & Kim, 2008) that students with LD were able to learn thinking behaviors, self-explanation specifically, when provided with appropriately designed scaffolding. It extends research on intervention design for promoting self-explanation by using the stacked requests, specifically in the form of repair requests.

Intervention Effectiveness on Word Problem Solving

To answer the third research question concerning the participants’ problem-solving performance, similar to the self-explanation measure, all participants began with a low-level performance in problem solving on the criterion test. Once they entered the intervention condition, they demonstrated varying levels of immediate increases in their problem-solving performance. Compared with self-explanation scores, their problem-solving scores improved more quickly and steadily. For each of the participants, there were very few overlaps on the level of performance across the baseline and the intervention conditions as shown by the PNDs. The intervention effect was maintained during the post-test and the maintenance test for all three participants.

It should be noted that through communication with the participants’ schoolteachers, we learned that word problems were not a focus in any of their regularly scheduled classroom teaching. Therefore, the intervention likely contributed to the change in the participants’ performance on self-explanation and word problem solving. As for the transfer test, none of the participants seemed to make significant improvements, indicating that this transfer test may be too far of a transfer for them. According to existing literature (e.g., Fuchs et al., 2003; Xin & Zhang, 2009), students with disabilities need systematic and explicit instruction on skill transfer for them to solve more complex problems. In fact, the transfer test in this study involved two-step word problems, rather than one-step problems as in the case of the criterion tests.

Implications for Practice

Currently in schools, students’ ability to articulate their reasoning processes is still not given its due attention. This is particularly the case for students with LD. The results of this study suggest that teachers provide more opportunities, along with necessary scaffolding or support, to engage students with LD in reasoning and activities such as talking about how they solve the problem. As supported by the findings of is study, students’ enhanced ability in articulating the reasoning process positively impacted on their problem-solving performance. Hopefully, it will also promote their participation in classroom discourse.

One thing worth attention, as shown during the late intervention condition of this study, is that the participating students all developed their own techniques for explaining their reasoning process for solving different types of problems. Notably, Carl used the same linguistic expression (“n groups of m”) for all three types of problems. That may be because the uniform self-explanation allowed Carl, the lowest performer in the pre-test, to become less concerned with the linguistic aspects of his explanation, and more focused on determining the meaning of the numbers. Interestingly, none of the participants developed their expressions exactly the same as modeled by the researcher (see Table 2). Instead, they formed expressions that facilitated their mental organization of mathematical quantities and relationships. In response, sensitivity on the teachers’ part may be important in noting the student’s preferred method of expression, affirming it, securing its mathematical validity, and enhancing it. As such, teachers can better facilitate the participation of students with LD in classroom discourse.

Limitations and Future Directions

First, this study focused on EG multiplication/division word problems and did not teach the distinction between addition/subtraction and multiplication/division. Participants may have gradually realized that the tests were all about multiplication/division problems. That may have especially been an issue in the case of Carl who only used addition/subtraction in the pre-test phase. We also acknowledge that unlike the other two participants, Carl has an IQ (73) within the low average range (70–85). Practically, it is not uncommon for schools to identify students with low average IQs as having LD. The lower IQ score may have contributed to Carl’s slower progress compared with the other two participants. Nevertheless, the intervention was effective for Carl as well. Second, there is limited existing research concerning interventions targeting for self-explanation or articulation of mathematical reasoning involving students with LD. As such, the present intervention design was exploratory in nature. Future research could extend this study in various ways, such as applying it to different student populations or to a group setting among peers as a collaborative learning strategy.

Conclusion

Current reform in mathematics education emphasizes students’ ability to articulate their thinking. Students are expected not only to do mathematics but also to communicate their reasoning with mathematics languages. However, research has shown that students with LD have little quality participation due to possible challenges in mathematical conceptual understanding and articulation. More studies are needed on strategies to improve the reasoning and articulation of students with LD in mathematics classroom discourse. To fill this gap, this study designed an intervention package to scaffold the self-explanation of students with LD in the context of mathematics word problem solving. This strategy design provided students with opportunities, as well as necessary support, for their concept development and self-constructed explanations. The results of this study indicated that the intervention strategy promoted self-explanation as well as problem-solving performance of students with LD. The preliminary findings from this study suggest that, with an implicit–explicit continuum of support, students with LD could also benefit from a constructivist environment. The intervention program developed in this study is supported by Mayer (2004) who proposed that constructivist-oriented learning might be best supported by more structured/guided cognitive activity that provides both the experience and the structure necessary to facilitate students’ deep understanding of concepts and strategies.

Footnotes

Declaration of Conflicting Interests

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

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This study was partially supported by the Bilsland Dissertation Fellowship the first author received from Purdue University to support her dissertation study.

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