2.1 Strategies for solving division problems

Research about strategies with students with Mathematics difficulties has usually focused on the additive structure, and there is a shortage of research that analyzes how the multiplicative reasoning develops (^{Zhang et al., 2011}). Furthermore, there are no studies addressing these aspects that consider students with ASD (^{Gevarter et al., 2016}; ^{Hart & Cleary, 2015}; ^{King et al., 2016}). We are particularly interested in those studies that focus on the representations used and on the steps taken by the students. Taking these facets into account, the following three levels for solving a partitive division problem are described (^{Downton, 2008}; ^{Mulligan, 1992}):

Typically-developing students exhibit some of the above strategies more often than others, and, in many cases, they follow a progressive pattern (^{Downton, 2008}). For example, the majority of studies agree in their findings that the sharing one by one strategy is the least frequent, and that in those cases where this strategy is exhibited, the students quickly progress to more efficient strategies based on estimates and on one-to-many correspondence (^{Mulligan, 1992}). It has also been shown that low achieving students usually use fewer strategies than high achieving students, and that they show difficulties in transitioning from intuitive strategies to advanced strategies (^{Siegler, 1988}). Our study aims to contribute by examining what peculiarities the strategies described above have in a student with ASD.

2.2 Conceptual structure in partitive division problems

In order to solve an arithmetic word problem, students have to use reading comprehension skills and must be able to transform the word and numbers into the correct operations. Persons with ASD tend to process auditory and/or linguistic information more slowly than typically-developing individuals, albeit within a high range of variability. They may also find it complex to process verbal and visual information simultaneously (^{American Psychiatric Association [APA], 2000})

The underlying conceptual structure of partitive division includes the idea of partitioning a quantity into parts (the total is divided into two pairwise disjoint parts), along with the ideas of equity (all of the split parts are equal) and the representativeness of each part (any one part is representative of the rest) (^{Correa, Nunes, & Bryant, 1998}). Thus, in order to successfully solve a partitive division problem, the students: (1) have to separate the total number of objects (dividend) into containers (divisor) with no reminder; (2) equally; and (3) interpreting each container as being representative of the rest, with the number of objects in said container (quotient) being the solution to the question posed in the problem. When students have difficulties with understanding the conceptual structure of division, they take the wrong actions, which might be revealed by the following errors:

The above errors can be combined when solving a problem. For example, the objects may be distributed unequally, leaving empty groups (errors 2.1 and 3.1). This situation would indicate that the students have difficulties with the idea of both equity and the representativeness of each container. We will use this error categorization just presented to analyze the data from the study we conducted, the objectives, methodology and results of which are presented next.

4.1 Participant

We followed a descriptive case study methodology involving a student with ASD who will be referred to as Tom. The subject of the investigation is a male of 11 years and 8 months old at the beginning of the study. He was diagnosed with ASD at the age of 6 by a child psychiatrist, based on clinical evaluation and following diagnostic criteria of the Diagnostic and Statistical Manual of Mental Disorders (DSM-IV, ^{APA, 2000}). In the clinical evaluation no other comorbidity was found other than the diagnostic of ASD. According to the Global Assessment of Functioning Scale (GAF), the subject presents a level of affection between 41 and 50, characterized by severe symptoms with severe impairment of social, work or school activity.

The student presents a broad repertoire of stereotypical behaviors and exhibits repetitive conduct and an obsession for certain subjects. He has a full scale IQ of 54, as measured by the Wechsler intelligence scale for children WISC V. At the beginning of the study, he scored 56/72 (35/41 in informal thinking and 18/31 in formal thinking) in the test of early math achievement TEMA 3 (^{Ginsburg & Baroody, 2003}), which situates his mathematical knowledge in the second year of Primary school. Tom employs a functional language with a high level of both oral and written comprehension. As a student, he is disciplined and shows a special interest in manual tasks and drawing. Tom follows an adapted mathematics curriculum and takes supplementary language and mathematics classes. The numerical content he had learned prior to our research was: addition and subtraction with regrouping (he has not memorized number facts), solving addition and subtraction problems in writing; multiplication problems, though he had not memorized the times tables.

4.2 Design

In order to evaluate the knowledge of the student about the division operation, an initial test was applied which consisted in partitive and quotative division problems. The student was also able to solve all quotative division problems with drawings, but he was unable to find an answer in any of the partitive division problems, which he solved using multiplicative incorrect strategies through drawings. For this reason, a teaching sequence with partitive division problems was designed. The instruction was provided during supplementary classes in a weekly work session typically lasting one hour. Tom was given two to five problems in each session, depending on the receptiveness he demonstrated during the session. Over the course of three months and 15 sessions, he worked on a total of 49 problems. A Special Education teacher with previous experience with students with this disorder led all the sessions. The data collected in this study correspond to the records of all spontaneous student resolutions when solving problems.

4.3 Instruction

Different studies conducted with students with ASD have shown the benefits of using tangible and visual aids to teach arithmetic operations (^{Hart & Cleary, 2015}; ^{Rockwell et al., 2011}). The use of augmentative devices, such as pictograms, also enhances and facilitates communications with individuals with this disorder (^{Mirenda, 2003}). Both conclusions led us to craft a tangible device that resembles a pictogram, and which we call “pictomaterial”. This material is a posterboard with empty rectangular boxes drawn on it and tokens that have to be distributed among the boxes. Also drawn on the cardboard are arrows representing the distribution task (see Figure 3). This material is similar to the schemes used in previous studies conducted on students with ASD (^{Rockwell et al., 2011}), which utilized the Schema-Based Instruction methodology, with the added possibility of being able to manipulate the objects involved in the distribution. Different pieces of plasterboard were made, varying the number of boxes drawn based on the divisor. In the problems where this material was provided, the student was given the sheet of posterboard associated with the divisor in the problem and as many tokens as required for the dividend.

Figure 3 Error 1.3- Distributing fewer objects than the total (18:6 division). 

A teaching sequence using partitive division problems was designed. This sequence included problems in two different formats: with and without support material, as follows:

All problems used in the study were partitive division problems, where the total and the number of recipients were known quantities. Problems included names and contexts familiar to the student to make them interesting to him. The size of the total amount (dividend) ranged from 8 to 28. An example of such a problem is: Peter wants to share 20 pencils equally into 4 boxes. How many pencils does he put in each box? Each problem was provided written on a sheet of paper containing the problem statement and a box for Tom to write the final result, thus allowing us to observe if he was able to interpret if the number he obtained was the solution to the problem.

In the first session, the method for solving the problems was explained to Tom, with the pictomaterial being used to explain the distribution task. This way, Tom started to associate this action with the meaning of the word “distribute”. In successive sessions, and following teaching methodologies of previous studies of informal strategies (^{Mulligan, 1992}; ^{Zhang et al., 2011}), the tutor always let Tom solve each problem by himself so as to observe his strategies without intervening in them. When Tom encountered difficulties that he was unable to solve by himself, the tutor made a demonstration with pictomaterial showing how to share the total amount equally into the boxes. An example of such a situation is shown below.

As shown in the extract of the instruction, the student first shows an error of not separating the total into parts (error 1.1) and repeating the total amount in each container. With the help of the pictomaterial, he selects the total (8 tokens) and makes a distribution, first leaving empty containers, and finally solving it correctly. Table 2 shows the type and sequence of the 49 problems presented over the course of the 15 sessions.

4.4 Data collection

The data were collected during the sessions described. The answers to the problems that Tom solved by hand were gathered and all of the sessions were video recorded. This material was analyzed independently by three researchers, who for each problem identified: (1) The type of strategy used, based on the types specified above: sharing one by one, one-to-many correspondence and trial and error grouping; (2) the errors associated with the meaning of partitive division, based on the error types identified above, and related with the notions of partition, equity and the representativeness of each container; and (3) the correctness of the final answer. The analysis conducted by the three researchers were then compared, and they identified those cases where their interpretations did not match and analyzed them again together. The student tried to solve some of the problems multiple times. The information on each of these attempts was collected. The tutor also noted aspects of the student’s attitude and behavior prior to and during the session. This information was used to omit some of the sessions in which the student seemed to have a lack of concentration. The omitted sessions were repeated when the student was more receptive to them. The inter-rater reliability (obtained by the formula agreements/(agreements + disagreements) x 100%) across all sessions was 97%.

5.1 Strategy characteristics and types of errors when solving problems with material support

The student used the materials offered for every problem presented in this format, both the concrete material and the pictomaterial. The three distribution strategies identified by ^{Mulligan (1992)} were exhibited: (1) one-to-many correspondence, (2) sharing one by one, and (3) trial and error grouping, described above. Table 3 shows the progression in the types of strategies used over the course of the 15 sessions with material aids, distinguishing between correct and incorrect answers.

(1) As it is shown in the table, the one-to-many correspondence (OMC) strategy was preferred by the student for the problems with concrete materials. In the first three sessions, he had some problems, though he eventually overcame them. He did not make any subsequent adjustments in any of the cases, meaning he directly selected the quotient as the initial group. This was interpreted to mean he was doing some kind of preliminary mental summation calculation that allowed him to obtain the quotient in a simple manner when the numbers involved were small. When this strategy yielded incorrect results, it was either because he picked an initial quantity that was smaller than the result, leaving undistributed elements, or because he picked an initial quantity that was larger than the result, leaving some containers empty. We associate the first case with a difficulty with the meaning of partition, thought to stem from the fact that he has not yet developed the idea that this type of distribution requires not leaving any items undistributed (error 1.3). In these cases, the student began the grouping and stopped before all the objects were distributed. For example, when distributing 18 tokens in six boxes with pictomaterial, Tom began grouping them by twos and stopped after distributing 12 tokens, regarding the problem as solved (see Figure 3).

We associate the second case, leaving containers empty, with a difficulty understanding each container’s representativeness (error 3.1), since he divides the total and places an equal number of objects in some containers, which he fills, while ignoring one or more of the remaining containers. This indicates that he is cognizant of the need to distribute the total number of objects equally, but without realizing that any container is representative of any other container.

(2) The next most common strategy used in the problems presented with material format was sharing one-by-one (SOO). This strategy yielded the correct solution in every case, with one notable exception that took place in one of the early sessions, where the student correctly used it to distribute six items in three containers, but wrote as the answer “There are three in each box”. We interpret that the error made in this case resulted from giving the number of groups as the answer instead of the contents of any container (error 3.3).

(3) Only rarely did the student use the trial and error grouping (TEG) strategy with material aids (3 out of 24). In another of these cases, Tom made a subsequent adjustment, correctly distributing six objects in three boxes, but he wrote the final answer as “There are six objects in each box”. In this case, the error resulted from giving the initial number as the answer instead of the contents of any container (error 3.2).

The progress made in the problems with material aids during the sessions reveals an initial stage in which Tom used all three strategies with some difficulties, preferring the use of the one to many correspondence strategy. From session 6 onwards, he correctly solved all the problems with materials, with two distinguishable patterns: the continued use of one-to-many correspondence through session 11, and from there until session 15, the use of sharing one-by-one. As we will see next, he systematically adopted this latter strategy, probably as a result of the success he was having in problems with no material aids.

5.2 Strategy characteristics and types of errors when solving problems without support material

The problems presented in a format with no material aids were introduced in the fourth session, and Tom also employed the three distribution strategies: (1) one to many correspondence, (2) sharing one-by-one, and (3) trial and error grouping. Table 4 shows the progression in the strategies used by Tom for the problems with no material aids, and the number of attempts for each strategy.

(1) Initially, the student used the one to many correspondence strategy (OMC), but by way of a new multiplicative procedure in which he drew the total amount to distribute in each container instead of separating the total into groups (see Figure 4). We interpret this error as a manifestation of the difficulty understanding the notion of partition: the student does not associate the idea of distribution with the complete allocation into groups (error 1.1). Logically, this difficulty could only be exhibited in formats with no material aids due to the possibility of being able to draw as many objects as desired.

Figure 4 One to many correspondence strategy. Error 1.1 - Not separating the whole amount into parts (10:5 division). 

To overcome this difficulty, the tutor invited him to use the pictomaterial and to select the total amount of tokens to be shared (see Table 1 for an example of such a teaching situation) and, later, to replicate the resolution in the worksheet. Using this one to many correspondence strategy, supplemented with a preliminary drawing of the amount to distribute, the student made several mistakes. Figure 5 shows three attempts at solving the same 12:4 division problem. In the first attempt, he distributes more objects than there are in the quotient (four instead of three), filling every container and increasing the total number of items to distribute (error 1.2). In the second attempt, he also selects a larger number of objects that is higher than the quotient, but adheres to the total amount, as a result leaving a container empty (error 3.1). The last attempt shows the correct solution.

Figure 5 One to many correspondence strategy. Error 1.2 - Distributing more objects than the total (left). Error 3.1 - Leaving groups empty (center). Right answer (right) (12:4 division). 

(2) The trial and error grouping strategy (TEG) in problems with no material aids was rarely used, and was combined with the one to many correspondence strategy. For example, Figure 6 shows two attempts at solving an 8:4 division problem. In the first attempt, the student drew 11 pencils before erasing two of them and distributing the remaining nine into three of the glasses using a trial and error strategy. Two errors are present: distributing more objects than the total (error 1.2) and leaving groups empty (error 3.1). In the second attempt, he correctly solved the problem by using a one to many correspondence strategy. Note the variation in the level of detail in the representation of the objects in the first attempt (concrete representation) versus the second attempt (schematic representation).

Figure 6 Trial and error strategy. Error 1.2 and 3.1.(distributing more objects and leaving empty groups) (left). One to many correspondence strategy (correct answer) (right) (8:4 division). 

Another example in which the trial and error strategy was used allowed us to identify a problem with the idea of equity (error 2.1). In Figure 7, we can see that in a 30:5 distribution, Tom places the tokens arbitrarily in the containers until there are no more left, without verifying that every container has the same amount of tokens.

Figure 7 Trial and error strategy. Error 2.1 - Not distributing the objects equally (30:5 division). 

(3) The student finally adopted a sharing one-by-one strategy (SOO), which helped him overcome the errors he had made. He successfully implemented this strategy in all problems with and without support material of the last two sessions.

As Table 4 shows, many of the initial attempts were unsuccessful, and are associated primarily with the one to many correspondencestrategy. From session 11, he mainly resorted to the sharing one-by-one strategy, which proved successful and which he applied to problems with material aids as well, as noted in the previous section (see Table 3).

5.3 Types of errors identified in terms of the solution strategy

From the standpoint of the conceptual structure underlying the partitive division operation, Table 5 shows the frequency of the errors encountered and the strategies associated with the format of the problem (with and without support material).

The student exhibited different errors in all of the strategies utilized (see Table 5), though the majority occurred when using the one to many correspondence strategy. In the problems with material aids, almost all of the errors were associated with ignoring the representativeness of any container as the solution to the problem, while in the problems with no material aids, the difficulties stemmed from ignoring the requirement to make the distribution equal and from two aspects involving the idea of partition (not separating the whole amount into parts and distributing more objects than the total).