Students and the Struggle with Mathematical Quantifiers

Abstract

The purpose of this work is to study student understanding of quantifiers. We compare three groups: freshmen that have no background with quantifiers, mathematical logic students who have learned about quantifiers, and seniors who have studied and used quantifiers in their previous courses. Most of the previous studies on this topic were based on questionnaires with both formal symbolic quantifiers (the universal quantifier “∀” and the existential quantifier “∃”) and informal language-based quantifiers. Most questions asked the participants to
decide if a statement given was True or False and to justify their reasoning. Often participants were asked to rewrite a formal statement informally or vice versa. The analysis of data showed that students at every level struggle with quantifiers, and some methods were proposed to aid the learning of these concepts.
Our research began with a pilot study analyzing answers given by college freshmen in a mathematical logic course. Each question had a formal (or informal) statement, and the students were asked to rewrite it informally (or formally). The results depended strongly on the number of quantifiers included and the majority of students had problems with the proper order of two or more quantifiers. Next, we prepared a more in-depth survey that was administered to three groups of students at different levels of their academic development. We also compared two different ways of teaching quantifiers at the introductory level (active learning vs. traditional style). We analyzed data using a statistical T-test applied to different samples and presented the results. Our overall goal is to study the obstacles to comprehending mathematical quantifiers and improve the methodology about teaching assessing the retention of this knowledge a few years after the initial learning happened.

Presentation