Abstract
Mathematics education continues to emphasize explorative proving, wherein proving involves producing statements, producing proofs, looking back (examining, improving, and advancing) at these products, and the interactions among these aspects. This study aims to develop an intended explorative proving mathematics curriculum by focusing on students’ ability to plan and construct proofs. We first set Levels 1 and 2 of “planning a proof” and “constructing a proof,” respectively, and Level 0 as the starting point of the learning progression where there is no differentiation between planning and constructing. Next, we combined them, and set nine learning levels in addition to “looking back” as the characteristics of explorative proving. Then, we elucidated two learning progressions of explorative proving as a curriculum framework considering the relationship between planning and constructing a proof. To develop the curriculum based on these learning progressions, we made corresponding tables of units with these learning progressions according to the units of Japan’s national Course of Study, and then showed an example of localizing one of the progressions and its effects by the implemented curriculum. By adopting the method of lesson study and a design experiment, we implemented geometry lessons for 8th graders that aim to shift the progression through the learning levels. The results clarify the advantages and limitations of the developed curriculum, which enabled us to refine a more robust curriculum. Finally, we identify the characteristics of this approach to curriculum development of explorative proving and the necessity of the method of lesson study and design experiment as a realistic dimension of curriculum development and improvement.
Highlights
Understanding in mathematics develops through justification and discovery, with proof and proving at the center
Current Japanese students learn deductive proofs in Grades 8 and 9, mainly through geometry, but one of the issues, which has not been fully addressed, is how to support students who have difficulties in many activities related to proof and proving: constructing deductive proofs and why they are necessary (e.g., MEXT, 2018), understanding logical structures of deductive proofs (e.g., Miyazaki et al, 2017), and realizing explorative proving by students (Miyazaki and Fujita, 2015)
We consider that students’ difficulties in Japan might be improved if we could re-organize a curriculum from a wider view of proof and proving. In contrast to this rigid view of proof, we regard proving as a flexible, dynamic, and productive activity (Komatsu, 2017). To teach this view of proving, we are currently undertaking a study based on a design experiment (Cobb et al, 2003), for developing a new explorative proving curriculum in Japanese lower secondary school (Grades 7–9, 13–15 years old)
Summary
Understanding in mathematics develops through justification and discovery, with proof and proving at the center. In producing the proof “if AB=AC in ABC, ABD= ACD,” students at Level 0, where they have no differentiation between planning and constructing a proof, cannot imagine the deductive connection of the premise “AB=AC in ABC” and conclusion “ ABD= ACD” required to make level P1 planning They first understand this connection in the Level C1 proof (Figure 3), and form and express by themselves by focusing on hypothetical syllogism. In passing through Level (P2, C1), the learning of P2 is difficult to realize because students cannot learn to distinguish thinking forward from the conclusion, from thinking forward from the premises, and how to connect these processes (P2) without the chance to form and express the deductive connection based on universal instantiation and hypothetical syllogism (C2). C1)→(P1, C2)+EIA” instead of “(P1, C1)→(P1, C2),” and the unit of “Congruence and properties of congruent triangles” does not need to shift to “(P1, C2)+EIA.”
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