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Mathematics Research for the Beginning Student

Eli E. Goldwyn, Sandy Ganzell, Aaron Wootton
Publication Date: 
Number of Pages: 
Foundations for Undergraduate Research in Mathematics
[Reviewed by
Duane Graysay
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One of the well-worn challenges of mathematics instruction is supporting students in their early course experience to develop an understanding of mathematical inquiry that might lead them to major in mathematics or to pursue advanced study in mathematics. This excellent two-volume set will be a useful resource for any instructor of undergraduate or advanced high school mathematics who is interested in providing opportunities for motivated students to develop such an understanding by pursuing independent inquiry in mathematics. As the editors explain in their preface, such experiences are valuable for helping students at the beginning of their college career to discover interests and talents in mathematics that their classroom experiences may not have revealed, for recruiting and retaining those students in STEM majors, for supporting their success in upper division courses, and for stimulating their pursuit of STEM fields in their postgraduate careers.
The volumes are an interesting blend of resource and textbook. Taken as a whole, the primary audience for each volume is mathematics instructors who are looking for a resource for independent inquiry projects for students. Each chapter, however, is written to function like a textbook, with students as the intended audience. The individual chapters of each volume are written to pique students’ interests in topics outside or beyond those that students typically encounter in their early undergraduate courses. Each set of authors have written their chapter to introduce fundamental concepts and definitions from some area of mathematics, then to introduce students to interesting and progressively more challenging problems in that area, including – in some cases—open problems. The chapters span a variety of topics, each chosen to represent an area of mathematics in which mathematicians are actively pursuing open questions. The chapters do not direct instructors or mentors to particular teaching strategies or modes of engagement to use with students, though the chapters do provide a structure and sets of challenge problems for independent investigation by students.
According to the volume titles, topics in Volume 1 are meant to be accessible to students who have not yet completed a first course in Calculus, while authors of chapters in Volume 2 assume that the reader has previously familiarity with Calculus. Each volume is well-aligned to their stated audience and purpose, both in content and in style and tone. The chapters of Volume 1 are written toward an audience of mathematically interested students, regardless of major area and with minimal expectations for prior mathematical knowledge, often in a conversational style suitable for the intended audience. Volume 2, in contrast, is written in a style that is more suited for students with experience and proficiency in topics from Calculus, with each chapter written in a more formal style.
An instructor who is interested in creating opportunities for secondary and early undergraduate students to engage in self-directed exploration and inquiry in mathematics will find the volumes of this set a strong addition to their professional resources.
Duane Graysay is an Assistant Professor in the Department of Mathematics and in the School of Education at Syracuse University, where he researches methods for teaching mathematics and for preparing prospective secondary mathematics teachers. He can be contacted at