Investigations in Mathematics Learning
Official Journal of The Research Council on Mathematics Learning
TABLE OF CONTENTS
Volume 6, Number 2 - - Winter 2013 - 2014
The Concept of Slope: Comparing Teachers’ Concept
Images and Instructional Content. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1-18
Courtney Nagle, The Behrend College
Deborah Moore-Russo, University at Buffalo, SUNY
Abstract
In the field of mathematics education, understanding teachers’ content knowledge and
studying the relationship between content knowledge and instructional are both crucial.
Teachers need a robust understanding of key mathematical topics and connections to make
informed choices about which instruction tasks will be assigned and how the content will be
represented. Ma (1999) described this profound understanding of fundamental mathematics
as how accomplished teachers conceptualize key ideas in mathematics with a deep and
flexible understanding so that they are able to represent those ideas in multiple ways and to
recognize how those ideas fit into the preK-16 curriculum.
Slope is a fundamental topic in the secondary mathematics curricula. Unit rate and
proportional relationship introduced in sixth grade prepare students for interpreting equations
such as y = 2x-3 as functions with particular, linear behavior in eight grade. The focus on
relationships with constant rate of change leads to distinctions between linear and non-linear
functions and the idea of average rate of change in high school. Ultimately, these ideas
prepare students for instantaneous rates of change and the concept of a derivative in calculus.
The diversity of conceptualizations and representations of slope across the secondary
mathematics curriculum presents a challenge for secondary teachers. These teachers must
work flexibly and fluently with various representations in the many contexts in order for their
students to build a coherent, connected conceptualization of slope. Since secondary
mathematics teachers need a deep understanding of slope to mediate students’ conceptual
development of this key topic, the study reported here investigates both how teachers think
about and present slope.
The Mathematical Development in Number and
Operation of Struggling First Graders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19-47
John Lannin, University of Missouri
Delinda van Garderen, University of Missouri
J. Matthew Switzer, Texas Christian University
Kelley Buchheister, University of South Carolina
Tiffany Hill, University of Missouri
Christa Jackson, University of Kentucky
Abstract
Number and operations serve as the “cornerstone” of the K-12 mathematics curriculum in
many countries. Solving problems in the mathematical domains of algebra, geometry,
Investigations in Mathematics Learning
Official Journal of The Research Council on Mathematics Learning
measurement, and statistics is often closely connected to student knowledge of number and
operation (Griffin, 2005). Although considerable knowledge exists regarding the development of
number and operation for typically developing children, less is known about the development of
children who struggle in mathematics. Moreover, children enter school with considerable
differences in their understanding of number and operation.
While most children, through exposure to various informal and formal tasks, develop a
deeper understanding of number and operation, this development is delayed for some children.
These children do not achieve levels of proficiency required for higher mathematics . . . .
Therefore it is critical that difficulties in mathematics are addressed before they become chronic,
pervasive, severe, and difficult to remediate” (Fuchs, 2005, p. 351) . . .
In some studies, concerns regarding student retention of learned concepts, and success
generalizing and transferring mathematical ideas to other mathematical situations or domains
were noted. Yet it is unclear why these mixed results occurred . . . . Therefore, they strongly
recommend that, “future research should be directed at the role of individual differences in the
development of early numeracy and the characteristics of children’s learning responsible for
these differences” (Van Luit & Schopmann, 2000 p. 35).
Calculus Students’ Understanding of Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48- 68
Allison Dorko, University of Maine
Natasha M. Speer, University of Maine
Abstract
Researchers have documented difficulties that elementary school student have in
understanding volume. Despite its importance in higher mathematics, we know little about
college students’ understanding of volume. This study investigated calculus student’
understanding of volume. Clinical interview transcripts and written responses to volume
problems were analyzed. One finding is that some calculus student, when asked to find volume,
find surface area instead and others blend volume and surface area elements. We found that some
of these students believe adding the areas of an objects’ faces measures three-dimensional space.
Finding from interviews also revealed that understanding volume as an array of cubes is
connected to successfully solving volume problems. This finding and other are compared to what
has been documented for elementary school students. Implications for calculus teaching and
learning are discussed.