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Dubinsky & Wilson’s research on students’ understanding of the function concept

Ed Dubinsky and Robin Wilson’ article, “High School Students’ Understanding of the Function Concept,”
will be published in March 2013 issue of the Journal of Mathematical Behavior 32 (2013) 83 101. For a pre-publication draft PDF, click here


In the United States and throughout the world, there is a cacophonous debate about the quality of mathematics education from pre-school through collegiate levels.  We have local, national and international exams, many curriculum projects and a plethora of rhetoric about many issues.  One of those issues concerns the mathematics education of students from under-represented and under-served populations.  This means, generally, students at the lowest level of academic achievement and socio-economic status.  It has been well documented that, while it is not the only factor, there is a strong correlation between socio-economic status and achievement in mathematics for elementary and secondary school students (Jordan, 2007 and Jordan and Levine, 2009).  Anyon (1981) points out that even with a standardized curriculum, social stratification of knowledge along socio-economic lines takes place.  There are also often references in the literature that algebra serves as a “gatekeeper” subject and that mathematics literacy is rapidly becoming a pre-requisite for economic access (Kamii, 1990, Moses et. al, 1989).  In addition, there is evidence of a strong overlap between socio-economic status and students in the bottom quartile of mathematics achievement (National Mathematics Advisory Panel, 2008).  At the same time there has been little research on the needs of students in these groups in relation to specific mathematics concepts (Lubienski and Bowen, 2000).  The authors of this paper chose to focus on the students in the bottom quartile of mathematics achievement because of the general lack of attention given to this group, even in the conversation around providing mathematics for all.   For a while, the debate was about whether the education of students in this target population should focus on skills and procedural facility or on conceptual understanding.  Then it seemed that people began to realize that both were essential, but still there are arguments about which should come first, which of the two potential kinds of learning depends on and is based on the other.  There is considerable “data” that buttresses these arguments, but almost all of it comes from scores on mandated tests of various kinds.   Unfortunately, although these tests may be effective in measuring, in some sense, skills and procedural knowledge, they tell us very little about conceptual understanding which is much harder to evaluate.  Indeed, there is very little data about the conceptual understanding of students from the aforementioned target populations.

This paper reports the results of one example of an overall approach, the Algebra Project, designed to begin to fill that lacuna.  We are not claiming that the results of this study are anything like a complete solution to the problems described in the previous paragraph.  We are not even addressing the full range of difficulties.  Rather, we are focusing on one very narrow part of the overall problem.  The goals of the Algebra Project are: to help high school students from the target population in one country (the United States) to pass all state and federally mandated examinations; graduate from high school “on time”; do well enough on college admissions examinations to be accepted into college; and to have knowledge and understanding of high school mathematics sufficient to place into, and succeed, in credit-bearing college mathematics courses.  We give a single example, an existence proof if you will, of one class that used a curriculum and pedagogy developed by Bob Moses’ Algebra Project.  As a result of this approach, most of the students developed sufficient procedural knowledge to achieve most of these goals.  Moreover, using qualitative research techniques from Mathematics Education research, it was possible to see that they made a strong start on developing an understanding of one of the most important topics in high school mathematics — the function concept.  In particular, we will consider several difficulties that, according to the literature, high school and college students have in developing their understanding of the function concept and see what effect our curriculum and pedagogical strategy for functions had in helping our participants overcome these difficulties.

In Section 1 we review some of the literature on learning and teaching the concept of functions, focusing in particular on some specific conceptual difficulties that are reported.  We will also refer to APOS Theory, which gives a description of a possible process by which the function concept can be learned.  APOS Theory can be used, and in many studies has been used, successfully, as a strictly developmental perspective (e.g., Breidenbach et al. 1992), or as a strictly analytical evaluative tool (e.g., Dubinsky et al., to appear), or as both (e.g., Weller et al., 2011).  In this study, it is used as a strictly analytical evaluation tool.  APOS Theory focuses on models of what might be going on in the mind of an individual when he or she is trying to learn a mathematical concept and uses these models to evaluate student successes and failures in dealing with mathematical problem situations.   We chose APOS Theory for this study because of its effectiveness in previous studies over the past 28 years (see Weller et al., 2003).   In Section 2 we describe the Algebra Project, which is the national program under which this research was conducted.  In Section 3 we give a detailed description of the study including the formal research question, the participants, the instructional treatment, and the research methodology.  Our results will be presented in Section 4.  Here we will consider five of the aspects of the function concept studied in the literature as reported in Section 1.  Finally, in Section 5 we compare results on our participants’ apparent understanding of aspects of the function concept as reported in Section 4 with that of other, often more mathematically sophisticated, participants as reported in the literature review in Section 1.  In this last section, we will also draw some conclusions, discuss the limitations of the study, consider what might be some topics for future research in this area, and mention some possible Implications for teaching practice.

***...continued in pre-publication draft PDF: click here
also available at

***and, posted earlier this Fall:

AP Curriculum author Ed Dubinsky advising international colleagues

Ed Dubinsky, author of Algebra Project curriculum and retired professor of mathematics and mathematics education, is engaged in two exciting international partnerships.

An international team of seven researchers in mathematics education residing in four different countries are coming together to write a book on APOS Theory.  Dubinsky and his colleagues have written that APOS “begins with the hypothesis that mathematical knowledge consists in an individual’s tendency to deal with perceived mathematical problem situations by constructing mental actions, processes, and objects and organizing them in schemas to make sense of the situations and solve the problems.” Springer will publish the book in the fall of 2013. Meanwhile, this fall Dubinsky is consulting with the Center for Successful Teaching and Learning of Hochschule Braunschweig/Wolfenbutel in Germany.  He will advise the institution on the revision of their teaching of mathematics.

Dubinsky is co-author, with Bob Moses, of “Philosophy, Math Research, Math Ed Research, K–16 Education, and the Civil Rights Movement: A Synthesis”, which appeared in the March 2011 issue of the AMS Notices, available for PDF download at

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