Nicholas H. Wasserman’s research while affiliated with Columbia University and other places

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Publications (52)


Making Advanced Mathematics Work in Secondary Teacher Education
  • Chapter

July 2023

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100 Reads

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3 Citations

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Nicholas Wasserman

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We argue that it is essential to provide opportunities for prospective secondary mathematics teachers to connect advanced mathematics content to secondary mathematics teaching practice (in addition to connections only to secondary mathematics), if advanced mathematics courses are to be useful to these teachers. In light of this argument, we review a selection of curricular materials satisfying two criteria: first, they are written for use in advanced mathematics courses that prospective teachers may take; and second, they feature explicit connections to secondary teaching practice. We use this review to highlight essential questions for the future of research and practice in advanced mathematics coursework. We suggest there is much unknown about how teachers can successfully integrate their experiences in advanced mathematics and pedagogical methods.


Fig. 1 An alignment between the common structure and outcomes of university teacher education
Fig. 2 Theoretical distinctions and relations for university mathematics in secondary teacher education
Making university mathematics matter for secondary teacher preparation
  • Article
  • Full-text available

May 2023

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244 Reads

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17 Citations

ZDM: the international journal on mathematics education

Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers—by which we mean teachers of students with ages, approximately, of 12–18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.

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Mathematical Challenge in Connecting Advanced and Secondary Mathematics: Recognizing Binary Operations as Functions

March 2023

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53 Reads

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2 Citations

For secondary mathematics teachers, it is important that their mathematical coursework helps deepen their understanding of the school mathematics they will teach. That is, making connections between advanced and secondary mathematics is vital for practicing and prospective teachers (PPTs). However, forming these connections poses significant mathematical hurdles. In this chapter, I explore the mathematical challenges that arise when PPTs are asked to make connections by recognizing ideas in advanced mathematics as being an instance of an idea studied in secondary mathematics. In particular, I look at the mathematical challenges faced by two PPTs as they tried to reconcile the definition of a binary operation in abstract algebra (i.e., ∗ : A × A → A) in terms of it being a function – something studied in secondary school. In this example, mathematical challenge is evident through the conceptual stages and shifts these two PPTs went through as they came to understand a binary operation as a function itself. I use this example to ground the discussion of mathematical challenges faced, more broadly, as PPTs develop connections from their advanced mathematical coursework. I also elaborate on the purposes such connections might serve, and why, for PPTs, these connections merit the mathematical challenges encountered to develop them.KeywordsMathematical challengeConnectionsSecondary teacher educationFunctionsBinary operationsAdvanced mathematics


Investigating a teacher-perspective on pedagogical mathematical practices: possibilities for using mathematical practice to develop pedagogy in mathematical coursework

February 2023

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71 Reads

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5 Citations

ZDM: the international journal on mathematics education

One of the challenges of university mathematics courses in secondary teacher preparation is incorporating pedagogical discussions. The focus in a mathematics course is—and should be—on mathematics. But research also suggests that without addressing pedagogical implications these content courses are not meaningful to secondary teachers’ future classroom practice. The thrust of this paper is exploring ideas for how to leverage mathematical practice in university mathematics courses—and, in particular, what have been described as Pedagogical Mathematical Practices (PMPs). The paper reports on a study of (n = 10) pre- and in-service mathematics teachers that explored the viability of the PMP construct, with the intent of specifying particular PMPs. Drawing on interviews with teacher participants who had recent experiences in an inquiry-oriented discrete mathematics course, the study reports on the ways in which they identified a set of mathematical practices as being productive pedagogically. The study contributes a teacher-perspective on the construct of PMPs, including the identification of four PMPs from the study data: explicit visualization; multiple approaches; concrete exemplification; and informal justification. Implications for their potential use in university mathematics courses with regard to teacher education are discussed.


Unpacking foreshadowing in mathematics teachers’ planned practices

Educational Studies in Mathematics

This paper provides an empirical exploration of mathematics teachers’ planned practices. Specifically, it explores the practice of foreshadowing, which was one of Wasserman’s (2015) four mathematical teaching practices. The study analyzed n = 16 lessons that were planned by pairs of highly qualified and experienced secondary mathematics teachers, as well as the dialogue that transpired, to identify the considerations the teachers made during this planning process. The paper provides empirical evidence that teachers engage in foreshadowing as they plan lessons, and it exemplifies four ways teachers engaged in this practice: foreshadowing concepts, foreshadowing techniques, foregrounding concepts, and foregrounding techniques. Implications for mathematics teacher education are discussed.


Divergence Criteria and Logic in Communication

January 2022

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17 Reads

This chapter is connected to the theory of convergent sequences in real analysis—in particular, theorems about properties of, and criterion for, convergence and divergence (content from Abbott (Understanding analysis (2nd ed.), New York, NY: Springer (2015)) Sections 2.3–2.5). It considers implications for teaching, particularly around the role of logic in mathematical communication and the challenges of communicating logic in teaching.


The Intermediate Value Theorem and Implicit Assumptions

January 2022

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18 Reads

This chapter engages with the content and proof of the Intermediate Value Theorem; similar to Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Section 4.5. Using this theorem as a case study, we explore the different ways explicit and implicit assumptions play a role in the back and forth communications between teachers and students in the classroom.




Differentiation Rules and Attention to Scope

January 2022

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9 Reads

This chapter explores a concept we refer to as “attention to scope” by examining the validity of a host of mathematical statements and proofs with respect to different domains over which they might be considered. As a case study, we undertake an extended proof of the power rule for differentiation, which is content connected to Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Sections 5.2–5.3, and reflect on lessons learned for the secondary school mathematics classroom.


Citations (32)


... Algebra is not merely about mathematical operations but also trains students to think abstractly, recognize patterns, and make generalizations (Newton et al., 2020;Trigueros & Wawro, 2020). Early understanding of algebraic concepts prepares students for more advanced mathematical topics in subsequent educational levels, such as calculus and geometry, which heavily rely on algebraic understanding (Frank & Thompson, 2021;Wasserman et al., 2023). Insufficient algebraic competence can hinder the development of critical thinking, making it difficult to master complex concepts and limiting opportunities in STEM fields and careers (Dolapcioglu & Doğanay, 2022;Spiller et al., 2023). ...

Reference:

Integration of item response theory in the development of PhET-based graphing lines worksheets for optimizing student algebra competence
Making university mathematics matter for secondary teacher preparation

ZDM: the international journal on mathematics education

... But functions need not (and often do not) have the same domain and codomain and may involve multiple variables in either their domain or codomain. For example, a binary operation on two elements, such as addition, multiplication, or the operation from any algebraic group could also be considered a function, f: X 2 → X (Wasserman, 2023). According to APOS theory (Asiala et al., 1996;, one of the most commonly adopted theoretical frameworks for conceptualizing mathematicians' and students' perspectives about function(s), a function could be conceptualized as either an action which can be evaluated at one input in the domain, a process which could be carried out on any input in the domain set to output an element of the codomain set, or an object in its own right that could itself be acted on by another mathematical operation (e.g., taking the derivative or integral of a function). ...

Mathematical Challenge in Connecting Advanced and Secondary Mathematics: Recognizing Binary Operations as Functions
  • Citing Chapter
  • March 2023

... This practice is pedagogical because teachers do this activity when they teach rules and procedures to students, and it is mathematical in that mathematicians engage in this activity as they justify and explain the mathematical rules they use. Wasserman (2022Wasserman ( , 2023 identified several PMPs that teachers use in their classrooms, but there is a need for researchers to identify and characterize other PMPs that teachers use. ...

Investigating a teacher-perspective on pedagogical mathematical practices: possibilities for using mathematical practice to develop pedagogy in mathematical coursework
  • Citing Article
  • February 2023

ZDM: the international journal on mathematics education

... Our investigation is embedded into the teaching and learning of real numbers and functions in (upper-level secondary) school and connects to the intricacies related to the set of real numbers (e.g., Barquero & Winsløw, 2022;Durand-Guerrier, 2016;Wasserman et al., 2022). In Germany, real numbers are often introduced in grade 10 (out of usually 13) as an extension of the set of rational numbers. ...

Understanding Analysis and its Connections to Secondary Mathematics Teaching
  • Citing Book
  • January 2022

... However, sometimes, different definitions for the same term exist, but do not define the same class of objects, introducing ambiguity into mathematical tasks. For example, in their recent work, Mirin et al. (2021) discuss two different definitions of a function, both accepted in the mathematical community, but which lead to different conclusions in the case of function invertibility. ...

On Two Conflicting Definitions of "Function"

For the Learning of Mathematics

... Researchers pointing out that the reasons for the difficulties experienced by teachers / PTs should be examined in depth state that the lack of knowledge of PTs about SPS has the potential to affect the structure of the tasks they prepare (Bakogianni, 2015;Casey et al., 2020;Chick & Pierce, 2008). PTs' design statistical question (Burgess, 2007;Leavy & Frischemeier, 2022), data collection (Hannigan et al., 2013;Lovett & Lee, 2018), data representation or interpretation (Casey & Wasserman, 2015;Hannigan et al., 2013) may also affect the prepared tasks. In other words, if PTs have difficulties in carrying out the SPS, it is likely that this will affect the statistical tasks they prepare (Casey et al., 2020;. ...

TEACHERS’ KNOWLEDGE ABOUT INFORMAL LINE OF BEST FIT
  • Citing Article
  • May 2015

Statistics Education Research Journal

... needed to examine how they can apply PMPs in actual classroom teaching (Wasserman & McGuffey, 2021). Examining how teachers in actual classrooms use PMPs for each of the five practices of orchestrating productive classroom discussion (Smith & Stein, 2018) can be useful for mathematics teacher educators by showing how advanced mathematics can inform specific practices (e.g., selecting, sequencing, and combining student responses) can contribute to productive discussions. ...

Opportunities to Learn From (Advanced) Mathematical Coursework: A Teacher Perspective on Observed Classroom Practice
  • Citing Article
  • July 2021

Journal for Research in Mathematics Education

... The mathematical ideas are developed based on some mathematical system. The knowledge helps to understand how mathematical propositions evolved or changed (Wasserman, 2018a). This knowledge focuses on how mathematical knowledge is developed and it is essential for mathematics teachers. ...

Connecting Abstract Algebra to Secondary Mathematics, for Secondary Mathematics Teachers
  • Citing Book
  • January 2018

... For the solutions in which items were combined in systematic or unsystematic manners (or partly systematic manners), we expected to see representations of different kinds; drawn ice creams that resemble real ice-cream cones or drawn jeans and sweatshirts to make outfits, lists with two-letter codes, and lists with written names of flavors or colors to make combinations. Previous research has shown that middle-school students can symbolize the set of outcomes with written lists (Lockwood et al., 2020;Maher & Yankelewitz, 2011). Listing elements of the set of outcomes unsystematically is found to be more common (Melusova & Vidermanova, 2015) than listing items systematically, and we expected more student solutions that list items without any organizational principle. ...

A case for combinatorics: A research commentary
  • Citing Article
  • September 2020

The Journal of Mathematical Behavior

... In mathematics, the definition of function has evolved over time with a focus beginning on quantities, but eventually arriving at a modern Dirichlet-Bourbaki definition that is common in most textbooks (Cha, 1999;Gök et al., 2019;Mesa, 2004). Weber et al. (2020) explained that this definition of function is in terms of "a domain, a codomain, and a correspondence between the domain and the codomain such that each member of the domain is assigned exactly one element of the codomain" (p. 2). ...

Connecting the learning of advanced mathematics with the teaching of secondary mathematics: Inverse functions, domain restrictions, and the arcsine function

The Journal of Mathematical Behavior