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A framework for integrating the history of mathematics into teaching in Shanghai

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Abstract

One major obstacle for integrating the history of mathematics in teaching (hereafter referred to as IHT) is how to help teachers, particularly those who lack experience in IHT, use historical materials in their teaching. Based on many IHT practices in Shanghai, this paper proposes an IHT framework including two parts: one is a triangular pyramid IHT model, and the other is a design-based IHT procedure. The pyramid model is composed of three different communities: teachers, researchers, and historians of mathematics. The design-based procedure, also called operating model, includes cyclic stages, which are investigation, design, implementation, assessment, and publication. The primary purpose of this framework is to build a bridge between theory and practice by combining the two models to address particular teaching concerns and improve teaching. In addition, this paper details the process of the dynamic pyramid model and the design-based IHT procedure, and illustrates an example of the application of this framework. Finally, this paper discusses the differences and similarities between this model and others and also discusses the challenge of applying this model in practice.

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Notes

  1. IHT and related words are defined as follows:

    Integrating the history of mathematics into teaching will be referred to as IHT; the actual practice of IHT will be referred to as IHT instruction; the models of IHT, such as historical parallelism, the genetic approach (see Furinghetti & Radford, 2008), and the guided reinvention (Freudenthal, 1973) will be referred to as IHT models; the design-based procedure for IHT will be referred to as the IHT design-based procedure; and an IHT case refers to a developed paper on a completed IHT instruction. All IHT cases should be evaluated and approved by teams.

  2. Paul Ernest and Morris Kline had similar expressions of historical parallelism. Ernest (1998) claimed, “The parallel between the historical evolution of mathematics and an individual’s learning of mathematics has been noted since the time of Darwin” (p. 25). Kline (1966) also stated, “In building mathematics constructively the genetic principle is enormously helpful. This principle says that the historical order is usually the right order and that the difficulties which mathematicians themselves have experienced are just the difficulties our students will experience”” (p. 324)

  3. “Expert teachers” refers to teachers with many years of teaching experience who try to address particular teaching concerns by reviewing research and conducting studies (e.g., action research and case study).

  4. In each team-project (e.g., mathematical induction), teachers have some particular teaching concerns, such as the reasons why most students do not understand the relationship between the two steps of mathematical induction and why some students forget the base step in using mathematical induction to solve problems.

  5. The Tower of Hanoi puzzle was invented by E. Lucas in 1883. It is also known as the Tower of Brahma puzzle. The puzzle states that there are three posts in a temple room. On one of the three posts are 64 golden disks. Acting out the command of an ancient prophecy, Brahmin priests have been moving these disks to another post in accordance with the immutable rules of Brahma since that time. According to the legend, when the last move of the puzzle is completed, the world will end.

    Regarding the reasons for selecting the Tower of Hanoi, we assumed that students can visualize recursive manipulation after they play the Tower of Hanoi. Also these manipulatives can cause all students to actively think about the process of manipulation.

  6. In 1640, Pierre de Fermat wrote to Frénicle de Bessy that the first five numbers of the form \( {2}^{2^n}+1 \) are all primes. This led him to propose the Fermat Prime Conjecture: that all numbers of this form are primes.

    In 1729, Christian Goldbach (1690–1764) asked Euler, “Is Fermat’s observation known to you, that all numbers \( {2}^{2^n}+1 \) are primes?” Euler showed that Fermat’s assertion was wrong by splitting the number \( {2}^{2^5}+1 \) into the product of 641 and 6,700,417 (Sandifer, 2006).

  7. For example, after students answered the teacher’s leading question of “What is the second step?” (e.g., students said, “We need to prove that the formula is also right for n = k + 1 when we know the formula for n = k”), teachers made leading comments like “Yes, we need to prove that for n = k + 1, the formula is also right. That is, we need to prove that the recursive relationship is right.”

  8. Let n be a positive integer. Show that any .2n × 2n. chessboard (n ≥ 2) with one square removed can be tiled using L-shaped pieces, where these pieces cover three squares at a time (L-shaped piece such as the following picture) (Harel, 2002).

    Two reasons for selecting the L-shaped problem are that this problem can be regarded as an activity (drawing pictures to solve the problem) and the nature of this problem requires students to use recursion to prove the conclusion.

  9. The control group teacher (James) employed a conventional method of teaching, following the instruction from teacher textbooks in his classroom. First, James asked students to complete two examples of finding patterns to explain the necessity of proving the guessed patterns. Second, students played a game with dominos, and James guided students in constructing the concept of mathematical induction. Third, James further explained the principle of mathematical induction. Finally, he and the students worked together to complete one example and two exercises.

  10. For example, Joe asked students to compute the results when n = 0, 1, 2, 3, 4, respectively. Next, he asked students to search for a pattern that existed between these results. Some students responded: “Primes”. Then, teachers said, “We admired a great mathematician, Euler, because after half a century, he discovered that the result for n = 5 is not a prime.” Meanwhile, the screen showed the big number that decomposed into the multiplication of two factors.

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Acknowledgements

We gratefully thank the editors and anonymous reviewers for their valuable comments and suggestions concerning an earlier version of this manuscript, thereby contributing to its improvement. We also acknowledge Drs. Kathleen M. Clark and Man Keung Siu for their suggestions on the first version of this manuscript.

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Authors and Affiliations

  1. Department of Teaching, Learning, & Culture, Texas A&M University, College Station, TX, 77840, USA

    Ke Wang, Yeping Li & Michael S. Rugh

  2. School of Teacher Education, East China Normal University, Shanghai, China

    Xiao-qin Wang

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  1. Ke Wang
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  2. Xiao-qin Wang
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Correspondence to Xiao-qin Wang.

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This article is based on first author's dissertation research at East China Normal University.

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Wang, K., Wang, Xq., Li, Y. et al. A framework for integrating the history of mathematics into teaching in Shanghai. Educ Stud Math 98, 135–155 (2018). https://doi.org/10.1007/s10649-018-9811-x

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