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META Math Webinar May 2020


James Tanton (PhD, Princeton 1994, mathematics) is the MAA’s Mathematician-at-Large. He writes books and video courses, advises on curriculum, consults with educators, and gives demonstration classes and professional development sessions across the globe. He has taught mathematics both at university and high-school institutions and is absolutely committed to promoting effective and joyful mathematics thinking, learning, and doing at all levels of the education spectrum.

James will be presenting his paper, Seriously: Why is Negative Times Negative Positive? Reflections on a META Math Annotated Lesson Plan.


Here's an age-old question: Why is negative times negative deemed positive? The universal "inner workings'' of arithmetic in school mathematics are often overlooked. School students are usually presented with different ad hoc models to motivate various arithmetical operations---the product of negative numbers, the distributive property of multiplication over addition, for instance---and the idea that there are fundamental common structures to these models is buried. It is not until an advanced undergraduate course---perhaps an Abstract Algebra or a Proofs course---that students are invited to explore the questions of why arithmetic works the way it does. But, can our undergraduates, masters of ring and field operations, explain why the product of two negative numbers is sure to be positive in the real number system? Can they themselves answer the age-old question? In this presentation we outline a draft annotated lesson plan developed for MAA'sMETA Math, the "Mathematical Education of Teachers as an Application of Undergraduate Mathematics'' project.


Cody L. Patterson is an Assistant Professor of Mathematics at Texas State University. His research investigates secondary students’ and teachers’ mathematical meanings for concepts and procedures in high school algebra, such as solving equations and graphing quantitative relationships. As a teacher educator, Dr. Patterson works with preservice and inservice teachers to enhance opportunities for middle and high school students to engage in reasoning-rich mathematics tasks in the classroom.

Cody will be presenting his paper, Building Connections to Secondary Algebra Teaching in a Course on Ring and Field Theory, co-authored by Christopher Duffer. 


With its focus on the theory of polynomial equations, an undergraduate course in ring and field theory seems, on the surface, to be an ideal venue for connecting advanced mathematics to secondary mathematics teaching. However, some complications arise in practice. First, when connections focus on exploration of thematically related extensions of high school algebra (such as solving the general cubic or proving that certain geometric objects are not constructible), preservice teachers (PSTs) are often left believing that the purpose of their advanced undergraduate education is to understand ideas at the horizon of secondary mathematics, not to support their own teaching practice. Second, when connections focus on establishing rigorous foundations for high school algebra, PSTs often sense that they are engaged in a discourse that is thematically consistent but not epistemically consistent with the mathematical activity they will conduct in their classrooms. Therefore, one challenge for us as mathematics teacher educators is to link not only mathematics content but discourses, and work together with PSTs to forge a common vision for what conceptually grounded algebraic discourse rooted in deductive reasoning can look like for high school students. In my talk, I will present an example of how this challenge arose in a cross-listed abstract algebra course for undergraduate students and master’s students in mathematics education (including preservice high school and college teachers), and a glimpse of a vision for how to address the challenge.


Scott Zinzer is an Assistant Professor of Mathematics at Aurora University. Many of his favorite teaching assignments include courses in mathematics for preservice and inservice K-12 educators.

Scott will be presenting his paper, Explicit Teaching Connections in Elementary Number Theory.


A course in Elementary Number Theory offers many rich connections to the K-12 mathematics curriculum. In this presentation, I describe the inclusion of explicit teaching tasks, reflections, and assignments newly added to my Elementary Number Theory course for mathematics majors. These activities are designed to highlight applications to the work of high school mathematics teachers and to encourage future secondary teachers to seek connections to the elementary mathematics curriculum. At the same time, the activities require an added depth of mathematical thinking that benefits every student. In addition to sharing sample activities, I will give an overview of student reactions to the activities and the impacts on student learning and engagement.


Elizabeth Burroughs is Department Head and Professor in the Department of Mathematical Sciences at Montana State University in Bozeman, Montana. She has served on the Mathematical Association of America’s Congress as the Representative for Teacher Education and as chair of the MAA Committee on the Mathematical Education of Teachers. She was a lead writer for the MAA’s Instructional Practices Guide (2018) and was a member of the writing team for the Association of Mathematics Teacher Educators’ Standards for Preparing Teachers of Mathematics (2017). She has co-led, alongside knowledgeable and perceptive colleagues, NSF-funded projects focused on the creation and use of materials for teacher preparation in undergraduate mathematics courses; grades K-8 classroom coaching; and mathematical modeling in elementary grades.

Elizabeth will be presenting her paper, Five Types of Connections Addressing Mathematical Knowledge for Teaching Secondary Mathematics, co-authored by Elizabeth G. Arnold and Elizabeth W. Fulton.


Robust preparation of future secondary mathematics teachers requires attention to the acquisition of mathematical knowledge for teaching. Many future teachers learn mathematics content primarily through mathematics major courses that are taught by mathematics professors who do not specialize in teacher preparation. What curricular supports could exist to assist mathematics professors in making connections between the content of undergraduate mathematics courses and the content of secondary mathematics? This articulation of five types of connections that can be used as part of secondary mathematics teacher preparation offers one perspective on how to support mathematicians who teach future teachers.


Andrew Kercher is a Mathematics Education Ph.D. candidate at the University of Texas in Arlington. His primary research interest is in the development of mathematical problem solving strategies in post-secondary mathematics students, which is the topic of his dissertation. After graduation he plans to pursue a job in academia.

Andrew will be presenting his paper, On the Development and Effectiveness of Tasks Focused on Analyzing Student Thinking as an Application for Teaching in Abstract Algebra, co-authored by James A. Mendoza Álvarez and Kyle Turner.


The Mathematical Education of Teachers as an Application of Undergraduate Mathematics (META Math) project developed lessons connecting content from an undergraduate abstract algebra course to related secondary school content that prospective secondary mathematics teachers (PSMTs) may eventually teach. This explicit integration of applications to teaching responds to recommendations from the Conference Board of the Mathematical Sciences in its publications on the mathematical education of teachers. These recommendations call for PSMTs to experience and understand the underlying mathematical connections between advanced mathematical content required for their undergraduate degrees and content they will teach. META Math lessons, which have been field tested at several universities, integrate student-thinking tasks (STT) that require undergraduates to analyze and explain student work presented in the task. I will outline the development and refinement of these types of tasks for an undergraduate abstract algebra course based upon participants’ work and instructor and participant interviews. I will also discuss the effectiveness of the STTs as perceived by undergraduates who engaged in these tasks.