Secondary school students’ levels of understanding in computing exponents

doi:10.1016/j.jmathb.2007.11.003

The Journal of Mathematical Behavior

Volume 26, Issue 4, 2007, Pages 301-311

https://doi.org/10.1016/j.jmathb.2007.11.003 Get rights and content

Abstract

The aim of this study is to describe and analyze students’ levels of understanding of exponents within the context of procedural and conceptual learning via the conceptual change and prototypes’ theory. The study was conducted with 202 secondary school students with the use of a questionnaire and semi-structured interviews. The results suggest that three levels of understanding can be identified. At the first level students’ interpretation of exponents is based upon exponents that symbolize natural numbers. At Level 2, students’ knowledge acquisition process is a process of enrichment of the existing conceptual structures. Students at this level are able to compute exponents with negative numbers by extending the application of prototype examples. Finally, at Level 3 students not only extend the prototype examples but also reorganize their thinking in order to compute and compare exponents with roots, a concept which is quite different from the concept of exponents with natural numbers.

Introduction

In recent years, there has been an increased interest in educational research in the understanding of advanced mathematics topics such as functions, limits, infinity (Dubinsky, 1991; Lakoff & Núñez, 2000; Sfard, 1991). However, there has been comparatively little research focused on students’ learning and understanding of exponents, which are important mathematical concepts and central to many collegiate mathematics courses, including calculus, differential equations and complex analysis. Much of the published research on this topic consists of researchers’ suggestions and descriptions of instructional designs to be used in teaching students about exponents (Barnes, 2006, Weber, 2002), while as Confrey and Smith (1995) note, there have been relatively few studies about students’ understanding of exponential numerals. Moreover, the research focusing directly on exponents as an autonomous mathematical object is very limited, with more studies considering exponents in conjunction with functions and concatenations (Confrey & Smith, 1995; Lee & Messner, 2000). In particular, little is known about students’ mental constructions and the way in which they develop a meaningful understanding of exponents or logarithms.This article deals with students’ understanding of exponents and is part of a large research and development project, initiated 5 years ago, that investigated a number of calculus concepts (Christou, Pitta-Pantazi, Souyoul, & Zachariades, 2005). The main purpose of this study is to describe the students’ levels understanding of exponents and to analyze students’ understanding within the context of procedural and conceptual learning via the conceptual change and prototypes’ theory.

Section snippets

Theoretical framework

In this section, we first refer to the approaches that have been developed to explain the mechanisms governing concept learning by focusing on the theory of conceptual change and the role of prototypes. Second, we discuss previous research on exponents.

Purpose of the study

The purpose of this study was to investigate the present situation concerning the knowledge of exponents by high school students in Cyprus. It has been found that the concept of exponents is not clearly defined in the students’ minds (Sastre & Mullet, 1998; Weber, 2002). As a matter of fact, the concept of exponents is taught mainly through a few examples with little attention being placed on the conceptual development of exponents.

In addition, research has shown that the concept of exponents

Participants

The sample of the study comprised 202 high school students. These students were approximately equally divided between boys and girls and came from middle class public schools. All students had some familiarity with the concept and notational conventions of exponential expressions because these were introduced in the curriculum 2 months prior the administration of the test. Specifically, during the instruction students initially assumed that the exponents were positive and looked at zero (a = 1)

Results

The results are presented within the context of prototypes and conceptual theory and according to the aim of the study which focuses on students’ understanding of exponents. To this end, many of the presented interviews excerpts purported to clarify the way in which students think. Much less attention was paid to their difficulties and misconceptions.

Discussion of the results—the proposed model

Based on the results of this study and the abilities exhibited by students in each group, a model of students’ exponential reasoning can be proposed. This model is presented in Table 3. In the first column of Table 3, we describe the levels of students’ progress as they develop their understanding of exponents. The levels are based on (a) students’ thinking while comparing exponential expression, and (b) the conceptual theory, the theory of conceptual change and the prototypes ideas of

Conclusions

Conceptual change (Merenluoto & Lehtinen, 2002), as the process through which students fundamentally expand, revise or change their existing knowledge, provides the means for tracing how students with initial knowledge of calculating simple exponential processes (such as a^x where a and b are positive integers—the prototype concept) can arrive at a conceptual understanding of exponents involving not only real but irrational numbers as well. The understanding of exponent expressions is studied in

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View full text

Secondary school students’ levels of understanding in computing exponents

Abstract

Introduction

Section snippets

Theoretical framework

Purpose of the study

Participants

Results

Discussion of the results—the proposed model

Conclusions

School Science and Mathematics

Contemporary Educational Psychology

Journal of Mathematical Behavior

Journal of Mathematical Behavior

EQS structural equations program manual

The logic of concept expansion

The embodied, proceptual and formal worlds in the context of functions

The Canadian Journal of Science, Mathematics and Technology Education

Splitting, covariation, and their role in the development of exponential functions

Journal for Research in Mathematics Education