Bad Foundations and Manipulable Objects

TL;DR

This paper proposes using Maxima to help students with poor mathematical foundations visualize operations, addressing challenges in teaching calculus to students who rely on memorization and superficial understanding.

Contribution

It introduces a method leveraging Maxima to improve visualization and understanding for students with bad mathematical foundations.

Findings

Maxima can be used to visualize mathematical operations.

The approach helps students with poor foundations understand calculus concepts.

Visualization tools can improve learning outcomes for extreme cases of bad foundations.

Abstract

Imagine a student -- let's call him `E', and make him a "he" -- that is enrolled in Calculus 2, and who believes that to pass in maths courses he only needs to memorize methods and apply them quickly and without errors. Let's imagine that `E' is an `E'xtreme case of "bad foundations" and that he knows how to solve $x+2=5$ by doing $x=5-2=3$, but he doesn't know how to substitute the $x$ in $x+2=5$ by 3, and the only way that he knows of "testing the solution" is to apply the same method again and check that he got the same result. When we are teaching Calculus to classes that have many students that are extreme cases of bad foundations we need new strategies and tools; for example, we can't pretend that "taking a particular case" is an obvious operation anymore -- instead we need ways to make these operations easy to visualize. This article shows a way to do that using Maxima.