*Written by Mireille Saboya, Université du Québec à Montréal, Alexandre Ducharme-Rivard, Educational Consultant, Centre de services scolaire Marguerite-Bourgeoys (CSSMB), Annie Beauchamp & Lou-Anne Denis-Masson, Teachers (CSSMB)*

Problem-solving is an important skill in math curricula around the world, however, it presents many challenges for students, especially those with learning disabilities (LDs). Hanin and Van Nieuwohoven (2018) report several research studies highlighting poor problem-solving performance among elementary students in Grades 5 and 6 (students aged 10 to 12 ). Inspired by a teacher, who** implemented supports** and participated in research conducted by Theis et al. (2014), we set a goal to support students struggling with solving problems. We hoped that student anxiety would diminish and that they would develop problem-solving skills, through thoughtful engagement (Saboya et al., 2015).

## Pre-Teaching to prepare students for problem-solving

**The key is to meet with students for about twenty minutes two days before tackling problem-solving in class.** The students discuss the information presented, what they are being asked to determine and possible strategies. Next, each student individually writes what they think they need to do in class. The teacher assembles the students’ observations in order to highlight background knowledge and the objective of the problem. Giving students extra time to anticipate what needs to be done prepares them to engage in problem-solving along with their peers, as the class pace is no longer too fast for them. While Theis et al. reported that this aid had positive effects on engaging exceptional students, they observed, however, that during class, some students who used this aid had a tendency to bypass the problem-solving process by wanting to explain to the other students what needs to be done.

To counter this effect, we designed preparatory activities intended for exceptional students prior to solving problems in class; these activities were aimed at preparing students to use mathematical reasoning when they engage in problem-solving, without directly addressing the problem that would be given to the whole class. We based our approach on a problem that frequently presents a challenge for the students, who often engage in faulty reasoning that is hard to undo. Figure 1 presents the problem that we worked on; it is drawn from a bank of problems designed by educational consultants and teachers from the Centre de services scolaire Marguerite Bourgeoys, in Quebec, and used in Elementary Cycle Three, aged 10 to 12 ).

**Table 1. Chocolate box problem statement.**

Chocolate boxes
Ludovic is selling chocolate boxes to finance his winter camp with the Scouts. He has to sell 295 chocolate boxes in the shape of a prism with a square base and that have the following dimensions: He wants to put his chocolate boxes into one large box to make them easier to transport. He finds the following two large boxes at his home: In order to avoid breaking the chocolates, he cannot put his boxes upside down or sideways. He must place them flat in the large box. However, he can stack them. Ludovic thinks that he can choose either one of the two boxes to transport all of his chocolate boxes in one trip. Is he right? Justify your answer. |

Three lines of reasoning are usually leveraged by the students, and the first two of them are faulty: 1) Calculate the volume of the chocolate box and the volume of the two large boxes, and compare them; 2) Calculate the lateral areas of the chocolate box and the lateral areas of the two large boxes, and compare them, and 3) Compare the corresponding measurements of the boxes. The line of reasoning that the students use the most often is the first one, which is a comparison between the volumes of the boxes. In the case of the first two lines of reasoning, the conclusion that can be drawn is that Ludovic is right; he can put the 295 chocolate boxes in the two boxes, whereas when the lengths of the boxes are compared, Ludovic can put the 295 chocolate boxes only in the second box (which is the expected reasoning).

This problem requires the use of spatial awareness to visualize each of the boxes in space. Indeed, it is necessary to be able to *see* the positioning of the chocolate boxes in relation to the large boxes to compare the corresponding dimensions. Note that because the chocolate boxes are prisms with a square base, the problem is easier to solve. It is enough to compare the side of the chocolate box with both the length and the width of each of the large boxes[1]. In order to prepare exceptional students to solve this problem, we designed three preparatory activities; these build on work pertaining to the concept of volume and the development of a spatial sense, skills that are called for in solving the chocolate box problem.

*[1] If the chocolate box was a prism with a rectangular base, it would be necessary to proceed by making more comparisons: the length of the chocolate box would have to be compared with the length and the width of the large boxes, and then the width of the chocolate box with the length and the width of the large boxes.*

## Three preparatory activities for the chocolate box problem

Supplies for each activity (see Table 2).

**Table 2. Presentation of the three preparatory activities for the chocolate box problem**

Activity 1: Box of base 10 materials (with 1-cm cubes) and a transparent block measuring one cubic decimetre. |
Activity 2: Marbles, dried peas or macaroni, and a transparent block measuring one cubic decimetre. |
Activity 3: A series of identical novels and a cardboard box. |

Different questions were planned, and the teacher observed the reasoning used by the students and asked them to explain to the other students the process that they used.

In your opinion, how many centicubes/marbles/dried peas/pieces of macaroni/novels will be able to fit in the transparent cube?

Follow-up question: Fewer than 100, between 100 and 500, between 500 and 1,000, more than 1,000, more than 2,000?How many centicubes/marbles/dried peas/pieces of macaroni/novels can I put in the box?

What can I do to find the answer? Which mathematical expression will you use?

For the past three years now, we have introduced these preparatory activities to our exceptional students in Grades 3 and 4 who are integrated into our classrooms. These students need to appropriate the concepts in question using manipulatives. This enables them to move on to the abstract level of thinking and to transfer what they learned through experimentation. The manipulatives provide direct and efficient feedback. We meet with the students for 15 minutes per activity, outside of classroom hours. Once the empirical approach has been carried out, we support the students in shifting towards mathematical operations. The day after each activity, it is taken up again in a large group. Students who were seen in subgroups can then experience the activity again with a feeling of competence and assurance. In addition, they are then more capable of understanding other approaches presented by their classmates. Sometimes we even notice that they speak up to correct the faulty reasoning of another student.

The first activity involves leading the students to observe that the number of centicubes fits perfectly into the block measuring one cubic decimetre and to find that this number is 1,000 cubes. When this activity was tested, some students stated that 100 cubes filled up the cubic decimetre block. Using manipulatives, they succeeded in detecting the error that they had made and in changing their approach.

In the second activity, the students analyzed the irregular shape of the pieces of macaroni, and they used new lines of reasoning to evaluate the necessary total number needed to fill the cubic decimetre block. This time, since the pieces of macaroni were not cube-shaped, there were gaps between them, and the space was not completely filled. In order to estimate the number of pieces of macaroni, for example, some students covered the bottom of the cubic decimetre block with a single layer of pieces of macaroni. They then assessed this layer as corresponding approximately to one sheet of 100 cubes. Subsequently, this number of pieces of macaroni was multiplied by 10 to obtain the number sought, since there are 10 sheets that fit into one cubic decimetre block. **We observed that manipulation did indeed help the students to move to a level of abstract thought.** We saw that, in the end, the students understood that by making a full layer of pieces of macaroni and multiplying the number of these pieces by 10, it was possible to arrive at a good estimate of the number of pieces of macaroni. Other students estimated that 2 or 3 pieces of macaroni could be placed in the space of a centicube, and then they multiplied this number by 1,000 since there are 1,000 centicubes in one cubic decimetre.

In the third and last activity, we came closer and closer to the context that would be experienced in solving the chocolate box problem. Two variables were changed: the content (novels) and the container (a cardboard box). The students observed that, like for the pieces of macaroni, there were gaps, or empty spaces, in the box. Through manipulation, they came to observe that different configurations were possible for placing the novels in the box. The teacher guided the students towards comparing the corresponding dimensions of the novel and the box to find the number of novels that fit into the box, which is the expected line of reasoning in the chocolate box problem.

## Reflection on the results

At the end of the three activities, the different contexts and the variety of material used enabled exceptional students to form an idea of the different possible situations that could occur and thus develop flexibility in their thinking. Here the material plays a central role; it fosters the development of meanings and can support lines of reasoning (Corriveau and Jeannotte, 2015). However, we were able to observe that students had difficulty seeing the multiplicative relationship between the object (cubes, pieces of macaroni, novels) and the receiving container (the cubic decimetre block or the box). The teacher played an essential role, keeping the pedagogical aim in mind and questioning the students about their reasoning, without giving them the answers.

The faulty reasoning pertaining to the volume came to light during the activity with the novels. We think that this reasoning should be followed through so that, by manipulating the material, the students themselves can become aware of how incongruous this line of reasoning is. Thus, the students measured the dimensions of the large box and calculated the volume, and then measured the dimensions of the novels to calculate their volume. Requiring students to measure the dimensions of prisms with a rectangular base allows them to give meaning to these dimensions, identify them, and recognize them. The teacher needed to support some of the exceptional students to identify one of the three dimensions; they could easily identify two of them but found it hard to see the third dimension. By dividing the volume of the large box by the volume of one novel, the students found the number of novels that *should* fit in the box. However, when they tried this, they noticed that they could place far fewer novels in the large box than what they had calculated. At that point, they were stymied, had trouble understanding, and became uncomfortable with the task. The teacher had to announce that this reasoning does not work, but why doesn’t it? In order for the students to properly understand, she used the definition of volume, which is the measurement of the amount of space that a substance or object occupies without any empty space or gaps. In calculating the volumes, we gave ourselves permission to cut up the novels into small pieces in order to fill all of the available space in the box. With this calculation of the volumes, we obtained a number of novels that includes a certain number of whole novels and pieces of novels to fill up the entire box. But we did not want to do this, we did not want to cut up our novels! The teacher then guided the students to compare the different dimensions of the box and the novels.

These preparatory activities allowed exceptional students to realize that the formula for volume maximizes the gaps, leaving no unoccupied space. But when can we use the formula for volume? Three possible situations stand out: when the objects fit exactly into the box, when a liquid is put into the box, and when the box is filled with sand. Therefore, when we focus on the volume, it is a matter of choosing objects or substances that can fill all of the available space:

- with cubes whose sides measured one centimetre, there was no extra space;
- with pieces of macaroni, it became evident that there was unoccupied space; and
- with the novels, the need to compare the corresponding dimensions of the box and the novels emerged.

These preparatory activities were significant for exceptional students. Comparing the corresponding dimensions of the box and the novels became a tangible activity. When the chocolate box problem was presented to all of the students in the classroom, exceptional students made connections with what had been done in the preparatory activities. They engaged in solving the problem at the same pace as the other students in the class; some of them became actively involved when they were part of a team, and they were no longer spectators. Their anxiety with regard to problem-solving had diminished considerably.

Executive functions make it possible to address challenges that are continually changing. Varying the context of the preparatory workshops helps to develop cognitive flexibility (see Figure 1). Doing the tasks first in subgroups has a positive effect on working memory, in particular, because the teacher requires the sustained attention of the students in the subgroup by constantly interacting with them.

**Figure 1. The executive functions (taken from Perreau-Linck, 2021)**

## Bibliography

CORRIVEAU, C. et JEANNOTTE, D. (2015). L’utilisation de matériel en classe de mathématiques au primaire: quelques réflexions sur les apports possibles. *Bulletin AMQ, Textes du 58e congrès*, LV(3), 32-49.

HANIN, V., et VAN NIEUWENHOVEN, C. (2018). Évaluation d’un dispositif d’enseignement- apprentissage en résolution de problèmes mathématiques : Évolution des comportements cognitifs, métacognitifs, motivationnels et émotionnels d’un résolveur novice et expert. Évaluer. Journal international de recherche en éducation et formation, 4(1), 37-66.

PERREAU-LINCK, É. (2021). Le point sur les fonctions exécutives : que sont-elles et que faire lorsqu’elles font défaut ? Institut des troubles d’apprentissage, https://institutta.com/mediatheque/fonctions-executives

SABOYA, M., BEDNARZ, N. et HITT, F. (2015). Le contrôle exercé en algèbre : conceptualisation et analyses en résolution de problèmes. *Annales de didactique et de sciences cognitives*, 20, 61-100.

THEIS, L., ASSUDE, T., TAMBONE, J., MORIN, M. P., KOUDOGBO, J. et MARCHAND, P. (2014). Quelles fonctions potentielles d’un dispositif d’aide pour soutenir la résolution d’une situation-problème mathématique chez des élèves en difficulté du primaire? *Éducation et francophonie, 42*(2), 158-172.

## About the Authors:

*Mireille Saboya** is a professor at the Université du Québec à Montréal. **She holds a doctorate in education specializing in didactics of mathematics. She is interested in the development of controlled action by mathematics students, and more particularly by students with learning disabilities. This involves, for instance, fostering reflective engagement by the students in an activity, being able to identify a winning procedure and to discard other procedures. In collaboration with people working in the school setting, interventions were constructed to help students in this regard. *

*Alexandre Ducharme-Rivard** is an educational consultant in mathematics at the elementary level at the Commission scolaire Marguerite-Bourgeoys. He holds a master’s degree in mathematics specializing in didactics of mathematics. He has worked for many years with teachers and remedial teachers at schools in the Montreal area, in contexts that are ethnoculturally diverse and have varied deprivation indexes. He has extensive expertise in supporting teachers at all elementary levels and has collaborated with different researchers on serious problems experienced in the school setting. *

*Annie Beauchamp** and Lou-Anne Denis-Masson have been elementary teachers for many years. After working on the project for two years, they have undertaken, with the researcher and the educational consultant, to continue reflecting on the interventions that were developed jointly. They opened the door of their classrooms and offered to have their interventions with the students filmed. *