Nerida F. Ellerton

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.

International Collaborative Studies in Mathematics Education

This chapter focusses on the concept of “collaboration,” with particular reference to mathematics education research in which the participating scholars are from different nations. After commenting that collaboration involves more than sharing, uniting, or cooperating, the concept is discussed in the light of the work of ICMI, IEA, PISA, RECSAM, MERGA, PME, LPS, and international aid programs. After providing summaries of the work undertaken in these programs and organizations, the following seven dimensions that influence the quality of “collaboration” within a program or organization were informally identified: (a) clear statement of raison d’etre, (b) consistency of actions with raison d’etre, (c) level of democratic governance, (d) whether wider international discussion is stimulated, (e) the extent of influence on policies, (f) the extent of influence on practices, and (g) the extent of influence on research directions. Using these dimensions as criteria, we assessed the quality of collaboration in the work of each of the above-named programs or organizations. Our conclusion is that, whereas the early work of ICMI did not feature high-quality collaboration, the ongoing work of most aspects of the other programs and organizations does feature high-quality collaboration.

The need for balance

I feel honoured to have been selected by the Executive of the Mathematics Education Research Group of Australasia to be Editor of the Mathematics Education Research Journal. I see MERJ as an International Mathematics Education Research Journal with an International Editorial Board. As Editor of MERJ I will not rest until I believe the Journal ranks among the best mathematics education research journals in the world. I am the second Editor of the Mathematics Education Research Journal, following in the steps of the foundation Editor, Dr Philip Clarkson. Mathematics educators owe Phil a debt of thanks for his tireless and creative efforts in establishing MERJ ás a quality education research journal.

Problem Posing in Mathematics: Reflecting on the Past, Energizing the Present, and Foreshadowing the Future

Five themes from the first 25 chapters of this book are identified: (a) the object of mathematical investigation as the construction of the problem itself and not just as finding the solution to a problem; (b) problem posing as an agent of change in the mathematics classroom; (c) integrating problem posing into mathematics classrooms; (d) problem posing as a conduit between formal mathematics instruction, problem solving, and the world outside the classroom; and (e) the need for appropriate theoretical frameworks for reflecting on problem posing. The fact that the chapters were prepared by a total of 52 authors from 16 countries is used to justify the claims that problem posing is not merely a local phenomenon, and that its place in school mathematics is gaining increasing recognition. Several imperatives for the field are set out, with mathematics educators urged to find ways and means of translating the obvious authenticity and enthusiasm displayed in this book into active research and practice in mathematics classrooms around the world.

Problem Posing as an Integral Component of the Mathematics Curriculum: A Study with Prospective and Practicing Middle-School Teachers

Details are presented of a study in which problem posing was an integral component of a mathematical modelling class for preservice and practicing middle-school teachers. One of the activities involved a project in which students individually planned and drafted mathematical modelling problems. Students then shared their draft problems with their peers before developing and presenting final versions of their problems to the class. Their personal reflections on the project formed an important part of the activity. Results are discussed in terms of an Active Learning Framework, and characteristics of a pedagogy for problem posing are proposed.

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.

The Need to establish a research base

A recent article in the Weekend Australian claimed that accel~~ated learning techniques now make it possible for children to cover a 12-month curriculum in one month Oones, 1994). The same article stated that these approaches to accelerated learning have been endorsed by the New South Wales Department of School Education. Although broad reference was made to other approaches such as those used by Edward de Bono, no reference was made to any specific research data which might support the claims made.

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.