Academic journal article
*The Mathematics Enthusiast*

# University Mathematics Teachers' Views on the Required Reasoning in Calculus Exams

## Article excerpt

Abstract:

Students often use imitative reasoning, i.e. copy algorithms or recall facts, when solving mathematical tasks. Research shows that this type of imitative reasoning might weaken the students' understanding of the underlying mathematical concepts. In a previous study, the author classified tasks from 16 final exams from introductory calculus courses at Swedish universities. The results showed that it was possible to pass 15 of the exams, and solve most of the tasks, using imitative reasoning. This study examines the teachers' views on the reasoning that students are expected to perform during their own and others mathematics exams. The results indicate that the exams demand mostly imitative reasoning since the teachers think that the exams otherwise would be too difficult and lead to too low passing rates.

Keywords: reasoning; creative vs. imitative; Calculus; University Calculus courses; Swedish exams

Introduction

The purpose of this study is to better understand university teachers' rationale when they create calculus exams, especially concerning the reasoning that the students are expected to perform in order to pass the exams. Earlier research indicates that students often use imitative reasoning when they solve mathematical tasks (Schoenfeld, 1991; Tall, 1996; Palm, 2002; Lithner, 2003). Imitative reasoning is a type of reasoning that is founded on copying task solutions, for example by looking at a textbook example or by remembering an algorithm or an answer. The students seem to choose1 imitative reasoning even when the tasks require creative reasoning, i.e. during problem solving when imitative reasoning is not a successful method (the concepts of "imitative" and "creative" reasoning are thoroughly defined in Section The Reasoning Framework). The use of algorithms saves time for the reasoner and minimizes the risk for miscalculations, since the strategy implementation only consists of carrying out trivial calculations. Thus using algorithms is in it self not a sign of lack of understanding, but several researchers have shown how students that work with algorithms seem to focus solely on remembering the steps, and some argue that this focus weakens the students' understanding of the underlying mathematics (e.g. Leinwand, 1994; McNeal, 1995) and that it might eventually limit their resources when it comes to other parts of mathematics, e.g. problem solving (Lithner, 2004). This is an important observation because it might be one of the reasons for students' general difficulties when learning mathematics.

An important question related to this situation is why the students so often choose imitative reasoning instead of creative reasoning. Several possible explanations are indicated by research. Hiebert (2003) states that students learn what they are given the opportunity to learn. He argues that the students' learning is connected to the activities and processes they are engaged in. It is therefore important to examine the different types of reasoning that the students perform during their studies. In a previous study (henceforth referred to as "the classification study') more than 200 items from 16 calculus exams produced at 4 different universities were analysed and classified (Bergqvist, 2007). The results showed that only one of the exams required the students to perform creative reasoning in order to pass the exam.

The general question in this study is therefore: Why are introductory level calculus exams designed the way they are, with respect to required reasoning? This question is examined through interviews with the teachers that constructed the exams analysed in the classification study (Bergqvist, 2007). The same conceptual framework (Lithner, 2008) used to classify the items is used in this study to analyze the teachers' statements.

Background

Students that are never engaged (by the teacher) in practising creative reasoning are not given the opportunity to learn (Hiebert, 2003) creative reasoning. …