Benjamin Dickman

The Hewitt School · Mathematics

In the first part of this paper, we provide an example of a project designed to foster mathematical creativity among students at an independent, all-girls school in the Northeastern United States. The mathematical motivator for the project is a polyomino proof by induction first formulated by Solomon Golomb. We explain how the project has been impl...

Although many well-intentioned organizations and institutions emphasize the importance of antiracism and social justice in mathematics education, there remains a lack of practitioner-oriented curricular materials exhibiting how these long-overdue shifts can manifest in classrooms. This article describes how modifying a project on absolute value fun...

We present in this paper a pair of approaches to support mathematics educators and learners in formulating original tasks. In particular, we facilitate the posing of rich mathematical problems by using two novel methods that were created by a mathematics department at a K-12 school in the United States, and further developed alongside our students...

Explore three original problems, the thinking behind their formulation, how they can be solved, and related extensions.

The William Lowell Putnam Mathematical Competition, often referred to as the Putnam Competition or the Putnam, is considered the premiere mathematical competition for undergraduate students in the United States and Canada. Despite a paucity of female participants and high-scoring women across its history, an all-female trio from Mississippi Woman's...

In this article, a K-12 mathematics educator and a recent (2020) high school graduate discuss curricular work related to math trails, which are based around the idea of mathematizing potential discoveries along a physical walk. The intersection of math trails with the realities of schooling amid the COVID pandemic is described, along with ways in w...

Noticing Humans refers to the importance of students being aware that they are seen by others – including, but not limited to, their math teachers – and that we, as teachers, would do well to interrogate how our external perspectives match or don’t match students’ internal perspectives. Noticing Wonders refers to a specific in-class activity relat...

As digital spaces evolve, mathematics educators must develop an awareness of the ways in which these environments can facilitate discussion and foster creativity. Question and Answer (Q&A) sites such as Mathematics Educators Stack Exchange (MESE) provide a platform through which those interested in the teaching and learning of mathematics can harne...

In this paper, the role of mathematical pathologies as a means of fostering creativity in the classroom is discussed. In particular, it delves into what constitutes a mathematical pathology, examines historical mathematical pathologies as well as pathologies in contemporary classrooms, and indicates how the Lakatosian heuristic can be used to formu...

In creating rich mathematical tasks, one technique is to require routine propositions be proved in multiple ways. We provide an example using a statement about divisibility, and include seven proofs along with commentaries about their presentation. Moreover, we investigate how our proposition can be generalized and connected to higher-level mathema...

Guessing is typically incorporated into mathematical problem solving by drawing from intuition while seeking a solution. We advocate for a form of guessing in which the answer is already known, and our goal is to engage in Pólya's “Looking Back” stage by using the known solution to develop mathematical intuition.

Sunken Treasure (pp. 99-106); Viral Marketing (pp. 167-174); Picking a Painting (pp. 225-232). Three lesson plans on mathematical modeling from the COMAP Mathematical Modeling Handbook.

In mathematical modeling, the topic of apportionment provides powerful tools for the design of games. In particular, letter frequency based modeling can be used to compute appropriate letter distributions for a variety of word games. This paper will provide possible letter distributions for two different games: Scrabble and Boggle.

Mathematical problem posing is an important skill for teachers of mathematics, and relates readily to mathematical creativity. This article gives a bit of background information on mathematical problem posing, lists further references to connect problem posing and creativity, and then provides 20 problems based on the multiplication table to be use...

Mathematical problem posing and creativity are important areas within mathematics education, and have been connected by mathematicians, mathematics educators, and creativity theorists. However, the relationship between the two remains unclear, which is complicated by the absence of a formal definition of creativity. For this study, the Consensual A...

Festschrift for Dr. Henry Otto Pollak. Contributors: Benjamin Dickman & Andrew Sanfratello; Henry Pollak & Sol Garfunkel; Alan H. Schoenfeld; Rita Borromeo Ferri; Hugh Burkhardt; Werner Blum.

The Evolving Systems approach to case studies due initially to Piaget-contemporary Howard Gruber, and complemented by subsequent work on sociocultural factors developed by Mihaly Csikszentmihalyi and others, provides an inroad for examining creative achievements in a variety of domains. This paper provides a proof of concept for how one might begin...

George Polya's seminal work "How to Solve it" describes general methods for approaching and solving mathematical problems. The book begins with Polya's four principles for solving a problem: understand the problem, make a plan, carry out the plan, and look back. Much of the remainder of the book is devoted to an encyclopedic listing of different he...

Let f in Q[z] be a polynomial of degree d at least two. The associated canonical height \hat{h}_f is a certain real-valued function on Q that returns zero precisely at preperiodic rational points of f. Morton and Silverman conjectured in 1994 that the number of such points is bounded above by a constant depending only on d. A related conjecture cla...

This thesis concerns the number of zeros of a multivariable polynomial $f$ over a finite field. More specifically, the zeta-function of $f$ is defined in terms of a certain power series with coefficients determined by the number of zeros of $f$ over various finite fields. Our main result is Dwork's Theorem, stating that the zeta-function of $f$ is...