Developing Historical Awareness Through the Use of Primary Sources in the Teaching and Learning of Mathematics

Abstract

This chapter explores the notion of historical awareness in relation to using primary historical sources in the teaching and learning of mathematics. It does so through the analysis of two teaching experiments originally designed from different, yet related perspectives. We present an analytic framework for identifying historical awareness and use it to analyze two cases in order to explore and discuss how and in what sense using primary historical sources in the mathematics classroom have potential to develop students’ historical awareness. Finally, we point toward more general challenges in mathematics education that historical awareness may be suited to address and articulate.

Appendices

Appendix 1

Primary Source Project Tasks 1–7 (Ruch, 2019).

Task 1. Write out the algebraic details of Newton’s fluxion method for n = 3 using modern algebraic notation to find the fluxion of x³. Sketch the curve y = x³ and label the key quantities for Newton’s fluxion method.

Task 2. Convert Newton’s argument that $\left( {x^n } \right)^{\prime} = nx^{n - 1}$ to one with modern limit notation for the case where n is a positive integer. You may use modern limit laws.

Task 3. What do you think of Berkeley’s criticisms? Is the value of o in Newton’s text both an increment and “not an increment”? How might Newton have responded?

Task 4. Use L’Hˆopital’s definition of a differential, some algebra, and your own words to explain why the differential of xy is $ydx + xdy + dx \cdot dy.$

Task 5. What do you think of L’Hôpital’s argument that he could eventually ignore dx dy because it is “a quantity infinitely small, in respect of the other terms y dx and x dy”?

Task 6. Comment on Cauchy’s definition of limit and his proof that $\frac{sinsin \alpha }{\alpha } = 1.$ What adjustments, if any, are needed to conform to the modern definition of limit?

Task 7. We have seen two explanations of the derivative rule $\left( {x^n } \right)^{\prime} = nx^{n - 1} .$

• (a) Compare and contrast the arguments of Cauchy and Newton.

• (b) Rewrite Cauchy’s argument in your own words with modern terminology, using modern properties of limits.

Appendix 2

Examples of essay assignments from the HAPh-module on “Boolean Algebra and Shannon Circuits” (Jankvist, 2011c).

• a. According to Hamming, what does it mean that a piece of mathematics is effective?

• b. Do a comparison of the relative effectiveness of Boole’s and Shannon’s works (systems) distinguishing between effectiveness in terms of philosophy and effectiveness in terms of applications.

• c. Based on your answers to the above questions, discuss different types of “the effectiveness of mathematics”. Recapitulate what Hamming means by the title of his paper “The Unreasonable Effectiveness of Mathematics”, and why it may be seen as “unreasonable”.

• d. Do you consider Boole’s introduction and Shannon’s application of the idea of an algebra operating only on the elements 0 and 1 along with the mathematical interpretation of “and” and “or” as an example of Hamming’s ‘the unreasonable effectiveness of mathematics’?