Instrumental Orchestration of the Use of Programming Technology for Authentic Mathematics Investigation Projects

Abstract

This chapter focuses on teaching the use of programming technology for pure or applied mathematics investigation projects to university mathematics students and future mathematics teachers. We investigate how the theoretical frame of instrumental orchestration contributes to our understanding of this teaching. Our case study is situated within the implementation of three undergraduate mathematics courses offered at Brock University (Canada) over the past 20 years, whose main activities include investigation projects. The study examines an instructor’s actions and decision-making in relation to potential students’ schemes that might have been promoted, implicitly or explicitly, by the instructor. The analysis also focuses on two student schemes, namely, the scheme of articulating a mathematics concept within the programming language and the scheme of validating the programmed mathematics. The case study led us to develop an orchestration and genesis alignment (OGA) model that associates different elements of the instructor’s orchestration with the intended student development of specific schemes. Our findings highlight the instructors’ dual role as policy makers and as teachers responsible for orchestrating students’ instrumental geneses (i.e., their web of schemes development). Findings also highlight the integration of projects as a key element of the exploitation mode.

Appendix: Bill’s MICA II Assignment 1 Guidelines, Winter 2019

• Note: All of your code should be carefully structured and very easy to read with all variables, functions and subroutines labeled in a helpful way. In addition, the interface should be user friendly and attractive.

• 1. Suppose that a needle of length 1/2 is dropped onto a plane of parallel lines that are 1 unit apart. By modifying the Buffon Needle program given in class, find the probability that the needle touches a line. Hand in your finished program which should look like the one given in class but with the appropriate modifications. Label this program as “Buffon Needle Problem”. (25 marks)

• 2. Consider the region R in [0,2] x [0,2] for which

$$ \sin \left(\ {\textrm{a}}^{\ast }{\textrm{x}}^{\ast}\textrm{x}+{\textrm{b}}^{\ast }{\textrm{y}}^{\ast }{\textrm{y}}^{\ast}\textrm{y}\ \right)>\sin \left(\ {\textrm{c}}^{\ast }{\textrm{x}}^{\ast}\textrm{x}+{\textrm{d}}^{\ast }{\textrm{y}}^{\ast}\textrm{y}\ \right) $$

Hand in a program that makes this area appear on the screen for different values of a,b,c and d and estimates its area using n points chosen at random in R. The user should be able to input a,b,c,d and n. Label this program as “Area In A Square”. (25 marks)

• 3. By choosing n points at random inside [−1,1]⁴, write a program to estimate the hypervolume of the unit hypersphere in R⁴ which is the set of points for which x² + y² + z² + w² < = 1. The user should be able to input the sample size n and the number of samples w. The output should show the mean and standard deviation of the w samples. Estimate the hypervolume accurate to one decimal place and use your observations to explain why you are confident that your first decimal place is correct. Also hand in a printout of your code which should be in the simplest possible form. Do not hand in the working program. (25 marks)

Do either question (4) or question (5). Your choice!

• 4. Suppose that a needle of length 1 is dropped onto a plane of parallel horizontal and vertical lines that are 1 unit apart. By modifying the code for the Buffon Needle program given in class, find the probability that the needle touches any of the lines. Hand in your written explanation of the mathematics behind your method as well as your working program. This program does not have to have a graphical component (unless you’d enjoy giving it one). Label this program “Buffon-Laplace Problem”. (25 marks)

• 5. Suppose that two numbers a and b are chosen at random from {1, 2, ..., n}. Let Pn be the probability that they are relatively prime. As n goes to infinity, does the limit of Pn exist? Hand in the program you write to investigate this question and a discussion of what you observed. Can you guess the exact limit? (25 marks)