Introducing Students to the Role of Assumptions in Mathematical Activity

Abstract

Assumptions play a fundamental role in disciplinary mathematical practice, especially concerning the relativity of truth. However, much is still unclear about ways to help students recognize key aspects of this role. In this paper, we propose a set of principles for task design to introduce students to the role of assumptions in mathematical activity, with particular attention to the following two learning goals: recognize that (1) a conclusion depends on the assumption(s) underlying the argument that led to it; and (2) making the underlying assumption(s) explicit is crucial to reaching consensus on the conclusion. In the context of a 3-year design research study, we first used existing literature to construct an initial version of task design principles which we then empirically tested and refined by designing and implementing two tasks in Japanese school classrooms. One of the tasks was in the area of functions at the secondary level and the other in the area of geometry at the elementary level. We analyze two classroom episodes to discuss the promise and evolution of our proposed task design principles. In addition, our analysis sheds light on the role of the teacher’s instructional moves and the students’ mathematical knowledge during the implementation of the designed tasks.

Notes

Acknowledgement

We would like to thank the teachers, Kiyokuni Wada, Masashi Honma, Keiji Horiuchi, and Kosuke Ide, for their collaboration in implementing the tasks in their classrooms. We are also grateful to the editor and anonymous reviewers for their insightful comments and suggestions on our paper. An earlier version of this paper was presented at the 11th Congress of European Research in Mathematics Education (Komatsu et al., 2019).

Notes

1 Assumptions have also been examined in research on mathematical modeling, where mathematics is employed to solve problems and explain phenomena in situations outside of mathematics (see Stylianides & Stylianides, 2023, for a related discussion). Our study differs from mathematical modeling studies as it is concerned with students’ activity within mathematics.

2 For example, as we discussed earlier, special types of quadrilaterals, such as trapezoids and rectangles, can be defined in a hierarchical system whereby one type of quadrilateral is regarded as a particular case of another type.

3 For instance, an axiom related to congruent triangles in Hilbert theory can be described as “in ΔABC and ΔA’B’C’, if AB $\cong$ A’B’, AC $\cong$ A’C’, and $\angle$BAC $\cong \angle$B’A’C’, then $\angle$ABC $\cong \angle$A’B’C’.” Regarding the conditions of tasks, for example, one possible answer to the task asking the number of different addition sentences for 5 is “if the task refers to the sum of two numbers within the set of natural numbers and commutative expressions are regarded as different, then there are four sentences,” and the if-part is conditions of the task.

4 The task investigated by Komatsu (2017) was about proving a property of the perpendicular lines drawn from the vertices of the base of an isosceles triangle to the opposite sides of the triangle. The implicit condition in this task was the intersection of these perpendicular lines and sides.

5 The left column of Table 4 is identical across the two tasks as these tasks aim at common learning goals.

6 Our judgement of the secondary students’ mathematical attainment as being above average is based on the fact that the school imposes an entrance examination, and thus has a selective admissions system. In the next paragraph, we mention that the elementary students in our study had mixed mathematical attainments. Our judgement in this case is based on the fact that the participating elementary schools are public and do not impose entrance examinations to select students before they admit them. Also, the students studied mathematics with their regular classmates without any grouping by attainment.

7 There are some exceptions; for example, music is often taught by music teachers.

8 Textbook publishers need to get approval for publication from the Ministry. Publishers submit their textbooks to the Ministry, which closely reviews whether the textbooks are in accordance with the national curriculum.

9 The description in Komatsu et al. (2019) paper was briefer, and we provide a more detailed description and analysis in this paper.

10 One illustrative task in the student textbook is: “A person stands on each of the four vertices of a rectangle with 10m length and 6m width, and people line up on its four edges, 1m apart. How many people are there?”

Additional information

Funding

This work was supported by the Japan Society for the Promotion of Science, Grant Numbers 18K18636 and 19H01668.