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In particular, we analyze the strategies that he displays and the observed errors associated to the difficulties he encounters that are related to the conceptual structure of division. We hope that the findings might provide some insight into future instruction designs for students with similar characteristics to those of the participant of the study. Learning an arithmetic operation is a complex process that spans across a long period of time.

To construct this process of learning arithmetic operations, different learning methodologies rely on giving students arithmetic word problems that describe a contextualized situation in a verbal information format. How many bags do we need? How many pieces of candy go in each bag?

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In this study, we will focus on partitive division problems. There is little research about the way in which persons with ASD learn division. One exception is the work by Levingston et al. That work, however, does not examine the meaning that the students assign to these problems in terms of their structure partitive division or quotative division , nor does it describe the strategies they used, which underscores the need to carry out studies in this direction. 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.

Furthermore, there are no studies addressing these aspects that consider students with ASD Gevarter et al. 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, ; Mulligan, :.

Direct modeling with counting. Concrete objects or drawings are used to represent the action described in the problem.

The students form equal sets from the quantity given in the problem and arrive at the solution by counting. Different strategies may be used at this level: 1 Sharing one by one : the student distributes objects one by one in the containers and counts the number of objects in each container at the end.

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If they have not, he repeats the process using a different initial estimate for the number of objects. No direct modeling. The same steps as in the previous level are taken, but objects are not used; instead, the student develops the verbal aspects and solving processes of the problem. Known or derived facts. Typically-developing students exhibit some of the above strategies more often than others, and, in many cases, they follow a progressive pattern Downton, 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, 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, Our study aims to contribute by examining what peculiarities the strategies described above have in a student with ASD.

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. They may also find it complex to process verbal and visual information simultaneously American Psychiatric Association [APA], 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:. Error 1. Not separating the whole into parts. The students reiterate the total number of objects in each of the containers. Figure 1 left shows an example of this error in a distribution. Distributing more objects than the total.

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In situations in which the students have available to them more objects than those indicated in the statement either because they have a large collection of objects or because they are able to draw them , they distribute them equally but place more objects than they should into the boxes see Figure 1 center. Distributing fewer objects than the total. The students distribute the objects equally without using all of the objects available see Figure 1 right.

Error 2. Not distributing the objects equally. The students distribute the total quantity among the containers available, but do not ensure that the resulting groups are identical. Figure 2 left shows this error in a distribution. Error 3. The students distribute the objects available equally, but ignore one or more containers. This reveals that they do not interpret each container as being representative of the others, with any container being identical to the others. Figure 2 right shows this error in a distribution. We interpret this to mean that they do not realize that the contents of any one of the containers is the solution to the problem.

As with error 3. The above errors can be combined when solving a problem. For example, the objects may be distributed unequally, leaving empty groups errors 2.

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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. In this exploratory study we consider how a student with ASD initially learns division by solving partitive division problems.

The problems of the study are presented to the participant in two different formats: with and without support material. In this paper we analyze the development of the strategies distinguishing between these formats, which are intended to develop an understanding of the meanings of this operation. The goal of this study is to describe the key procedures that the student carries out, more specifically:. The characteristics of the strategies and external representations used, in reference to the format in which the problems are presented with and without support material.

The observed errors associated to difficulties with the conceptual structure of partitive division the notions of partition , equity and representativeness. 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.

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. 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. 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.

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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. The use of augmentative devices, such as pictograms, also enhances and facilitates communications with individuals with this disorder Mirenda, 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.


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This material is similar to the schemes used in previous studies conducted on students with ASD Rockwell et al. Different pieces of plasterboard were made, varying the number of boxes drawn based on the divisor.

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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. A teaching sequence using partitive division problems was designed. This sequence included problems in two different formats: with and without support material, as follows:. With support material : The problem is presented to the student in written format accompanied by the concrete material mentioned in the problem such as lollipops and bags or by the pictomaterial.