Designing Problems for Improved Instruction and Learning -- Linear Algebra

TL;DR

This paper develops algorithms to help instructors craft linear algebra problems with specific properties, improving educational effectiveness and student confidence by controlling problem parameters through matrix reverse engineering.

Contribution

It introduces novel algorithms for reverse engineering matrices in linear algebra, enabling targeted problem design for enhanced instruction and learning.

Findings

Algorithms successfully generate matrices with desired properties.

Enhanced control over problem parameters improves instructional quality.

Students' confidence increases with tailored problem sets.

Abstract

One of the grand challenges of Mathematics instruction is to provide students with problems that are both accessible and have a reasonably elegant solution. Instructors commonly resort to resources like course textbooks, online-learning platforms, or other automated problem-generating software to select problems for exams and assignments. However, reliance on such tools may result in limited control over problem parameters, potentially yielding intricate solutions that impede students' understanding. This article centers on Linear Algebra, wherein we devise algorithms for reverse engineering matrices of integers with integer outcomes through operations such as the inverse, LU decomposition, and QR decomposition. The focus is on empowering instructors to manipulate matrix properties deliberately, ensuring the creation of problems that enrich instruction and foster student confidence. The intellectual endeavor of reverse engineering such problems, grounded in both theory and matrix properties, proves mutually beneficial for both students and instructors alike.