Curriculum Development for Explorative Proving in Lower Secondary School Geometry: Focusing on the Levels of Planning and Constructing a Proof

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.

[1]  D. Herrmann.  Proofs And Refutations The Logic Of Mathematical Discovery , 2016 .

[2]  Annemarie Sullivan Palincsar,et al.  Designing Educative Curriculum Materials: A Theoretically and Empirically Driven Process , 2014 .

[3]  Kotaro Komatsu.  Fostering empirical examination after proof construction in secondary school geometry , 2017 .

[4]  Yosuke Tsujiyama,et al.  Using Examples of Unsuccessful Arguments to Facilitate Students’ Reflection on Their Processes of Proving , 2018 .

[5]  Stefan Ufer,et al.  Strategies to foster students’ competencies in constructing multi-steps geometric proofs: teaching experiments in Taiwan and Germany , 2008 .

[6]  Patricio Herbst,et al.  Proving and Doing Proofs in High School Geometry Classes: What Is It That Is Going On for Students? , 2006 .

[7]  J. Sarama,et al.  Learning Trajectories in Mathematics Education , 2004 .

[8]  Michael De Villiers,et al.  Aspects of Proof in Mathematics Education , 2011 .

[9]  L. Schauble,et al.  Design Experiments in Educational Research , 2003 .

[10]  C. Hoyles,et al.  A Study of Proof Conceptions in Algebra , 2000 .

[11]  Terri Germain-Williams,et al.  Construct Viable Arguments and Critique the Reasoning of Others , 2013 .

[12]  Catherine Lewis,et al.  How Should Research Contribute to Instructional Improvement? The Case of Lesson Study , 2006 .

[13]  Michael De Villiers,et al.  Proof and Proving in Mathematics Education , 2012 .

[14]  Harold Pascoe Fawcett.  The nature of proof : a description and evaluation of certain procedures used in a senior high school to develop an understanding of the nature of proof , 1938 .

[15]  Susan B. Empson.  On the idea of Learning Trajectories: Promises and Pitfalls , 2011, The Mathematics Enthusiast.

[16]  Keith Jones,et al.  Students’ understanding of the structure of deductive proof , 2017 .

[17]  M. Villiers.  The role and function of proof in mathematics , 1990 .

[18]  I. Lakatos,et al.  Proofs and Refutations: Frontmatter , 1976 .