M. A. (Ken) Clements

Effects of classroom instruction on students’ understanding of quadratic equations

Two hundred and thirty-one students in six Grade 9 classes in two government secondary schools located near Chiang Mai, Thailand, attempted to solve the same 18 quadratic equations before and after participating in 11 lessons on quadratic equations. Data from the students’ written responses to the equations, together with data in the form of transcripts of 36 interviews with 18 interviewees (a high performer, a medium performer, and a low performer from each of the six classes), were analysed. Using a rubric for assessing students’ understanding, the analysis revealed that at the post-teaching stage students improved their performance on quadratic equations and had a better understanding of associated concepts than they had at the pre-teaching stage. However, many were still confused about the concepts of a variable and of a “solution” to a quadratic equation. After the lessons, most students had acquired neither an instrumental nor a relational understanding of the mathematics associated with solving elementary quadratic equations.

Prospective Middle-School Mathematics Teachers’ Knowledge of Equations and Inequalities

This chapter describes an investigation into the algebra content knowledge, in relation to elementary equations and inequalities, of 328 US teacher-education students who were seeking endorsement to become specialist middle-school mathematics teachers. Most of these prospective teachers had done well in high school mathematics and were taking their last algebra course before becoming fully qualified teachers of mathematics. After reviewing the scant literature on the teaching and learning of quadratic equations, and of linear inequalities, we summarize a pencil-and-paper instrument, developed specifically for the study, which included linear and non-linear equations and inequalities. The students were also asked to comment, in writing, on a “quadratic equation scenario” that featured four common errors in relation to quadratic equations. Data analysis revealed that hardly any of the 328 students knew as much about elementary equations or inequalities as might reasonably have been expected. Brief details of a successful intervention program aimed at improving the pre-service teachers’ knowledge, skills and concepts relating to quadratic equations and inequalities are given, and implications of the findings for mathematics teacher education and, more generally, for the teaching and learning of algebra, are discussed.

He would be Good: Abraham Lincoln’s Early Mathematics, 1819–1826

This chapter, jointly written by Nerida Ellerton, Valeria Aguirre Holguin, and Ken Clements, offers a comprehensive analysis of the earliest extant handwritten manuscript of Abraham Lincoln—22 pages (11 leaves), from a cyphering book that Lincoln prepared between 1819 and 1826. The pages are examined from historical, mathematical and educational perspectives, and an order in which they were written is conjectured, with justifications provided. This is the first time all 22 pages have been examined, it having become received tradition that there were only 20 surviving pages. The authors argue that if the corpus of all extant U.S. cyphering books were to be studied, then the future President’s cyphering book would not be particularly outstanding from calligraphic, penmanship, or abstract mathematical points of view. Analysis reveals, nevertheless, that Lincoln succeeded in his quest to prepare an attractive cyphering book in which almost all entries were arithmetically correct—despite his having to cope with rough frontier circumstances.

Postscript: Framing Research Aimed at Improving School Algebra

This final chapter is written as a guide to persons wishing to carry out research which aims to improve middle-school students’ understanding of school algebra to the point where not only will the students be able to generalize freely, but will also be able to apply the algebra that they learn. The first point made in the chapter is that mathematics education researchers need to take the history of school mathematics more seriously, because the six purposes of school algebra identified in the historical analysis presented in Chapter 2 of this book were important not only in helping the research team identify the importance of language factors in school algebra, but also in designing the study which would be carried out. The second point made is that in a design-research study the theoretical frame is likely to be not one single theory but a composite theory arising from a bundle of part-theories that are suggested by needs revealed in the historical analysis. The third, and final point is the need for mathematics education researchers to remember that, ultimately, the aim of school mathematics is to help students learn mathematics well, so that the students will be competent and confident to use it whenever they might need it in the future. Research designs should be such that tight assessments can be made with respect to whether the results of the studies will help educators improve the teaching and learning of algebra in schools. Suggestions for organization and teaching methods which will generate appropriate discourse patterns in algebra classrooms are made, and an invitation to replicate the main study is extended.

Review of Pertinent Literature

This chapter frames the main study described in this book in terms of the theoretical positions of Charles Sanders Peirce, Johann Friedrich Herbart, and Gina Del Campo and Ken Clements. Peirce’s tripartite position on semiotics (featuring signifiers, interpretants, and signifieds), Herbart’s theory of apperception, and Del Campo and Clements’s theory of complementary receptive and expressive modes of communication, were bundled together to form a hybrid theoretical position which gave direction to the study. The chapter closes with careful statements of six research questions which emerged not only from consideration of the various literatures, but also from a knowledge of practicalities associated with the research site, from our historical analysis of the purposes of school algebra, and from our review of the literature.

Research Design and Methodology

The main study featured a mixed-method design, with complementary quantitative and qualitative data being gathered and analyzed. Since random allocation of students to two groups occurred, it was legitimate for null and research hypotheses to be formulated for the quantitative analyses, and those hypotheses are carefully defined in this chapter. One of the important challenges was to identify the population to which inferences would be made. Details relating to the development of appropriate pencil-and-paper tests and an interview protocol are also given, as are details relating to the calculation of Cohen’s d effect sizes.

Identifying a Problem with School Algebra

The chapter begins by presenting data which suggest that there is a longstanding, and fundamental, problem with school algebra. The problem is that many students who try hard to understand the fundamental principles of algebra, fail to do so. But that statement raises an important question—Why do so many school students find it difficult to learn the subject well? The authors of this book answer that question from three different perspectives: the first relates to the question why students are asked to learn algebra. Adopting a historical method of analysis, we identify six purposes which have been offered as reasons for why school students should study algebra. The second perspective relates to theories which help explain why so many secondary-school students do not learn algebra well. And, the third perspective offers a set of principles which might begin to provide an answer to the fundamental problem. These principles were applied in an intervention study with seventh-grade students which is described in this book.

Historical Reflections on How Algebra Became a Vital Component of Middle- and Secondary-School Curricula

The chapter begins by identifying, and placing in their historical contexts, the main issues in a longstanding debate over the purposes of school algebra. The following six purposes for school algebra, recognized by various writers over the past three centuries, are then identified: (a) algebra as a body of knowledge essential to higher mathematical and scientific studies, (b) algebra as generalized arithmetic, (c) algebra as a prerequisite for entry to higher studies, (d) algebra as offering a language and set of procedures for modeling real-life problems, (e) algebra as an aid to describing structural properties in elementary mathematics, and (f) algebra as a study of variables. The question is then raised, and discussed, whether school algebra represents a unidimensional trait.

Quantitative Analyses of Data

Quantitative data from the main study are summarized and analyzed. Both the structure and modeling workshops generated statistically significant performance gains. Thus, after students had participated in both the workshops, their performances on both parts of the Algebra Test—that is to say, on the questions concerning structure and on the questions concerning modeling—were much improved. Cohen’s d effect sizes for each set of workshops (the structure workshops and the modeling workshops) were large. The chapter concludes by introducing two questions. First, although the performance gains were highly statistically significant, and the effect sizes large, were they educationally significant? And, second, “What was there about the interventions which generated such apparently impressive results?”

Framing a Classroom Intervention Study in a Middle-School Algebra Environment

It has become a tradition in the field of mathematics education that before a researcher outlines the research design for a study, he or she should outline a theoretical framework for the investigation which is about to be conducted. Then, after research questions are stated, and the design of the study is described, the investigation takes place. The data gathering, data analyses, and interpretation are guided by the theoretical framework and conclusions are couched in terms of, and seen in the light of, the theoretical framework. There are many mathematics education researchers who regard this theory-based process as sacrosanct, as absolutely essential for high-quality research. In the first part of this chapter it is argued that the traditional “theoretical-framework” process just described is flawed, that it can result in important aspects of data being overlooked, and that it can lead to incorrect, or inappropriate, conclusions being made. It is argued that the first thing that needs to be done in a mathematics education research investigation is to identify, in clearly stated terms, the problems for which solutions are to be sought. Having done that, historical frameworks—which have only occasionally been taken seriously by mathematics education researchers—should be provided. Then, having identified the problems and having provided a historical framework, a design-research approach ought to be adopted whereby a theory, or parts of a theory, or a combination of parts of different theories, are selected as most pertinent to the problems which are to be solved. This chapter identifies three main problems: (a) “Why do so many middle-school students experience difficulty in learning algebra?” (b) “What theoretical positions might be likely to throw light on how that problem might be best solved?” (c) “In the light of answers offered for (a) and (b), what are the specific research questions for which answers will be sought in subsequent chapters of this book?”