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

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## Publications (48)

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, a...

This chapter explores connections between the Algebraic Limit Theorem for sequences, discussed in Abbott’s (Understanding analysis, 2nd ed. Springer, New York, NY, 2015) Section 2.3, and error accumulation when operating on rounded values. It considers implications for teaching and engaging with students around the use of actual versus approximated...

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 an...

This chapter employs ideas about rates of change and Cavalieri’s (2D) principle for conceptualizing area to offer an intuitive justification for the Fundamental Theorem of Calculus (analysis content discussed in Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Section 7.5).

This chapter considers several definitions for continuity, including the formal ε − δ definition common to courses in analysis, part of Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Sections 4.2–4.3. It also looks at the role of definitions more broadly in mathematics and in mathematics teaching.

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 (Understand...

This chapter explores connections between the Riemann integral, which is content in Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Sections 7.1–7.4, and the idea of area-preserving transformations as they relate to Cavalieri’s (2D) principle. It considers implications for justifying area formulas in school mathematics.

This chapter introduces the set of Teaching Principles (TPs) that guide our discussions of pedagogy. The principles introduced in this chapter are referenced throughout the textbook.

This chapter considers the role of inverse functions in solving equations. It also explores ways to characterize the existence of an inverse function in terms of properties like continuity and strict monotonicity. Proofs draw on the Intermediate Value Theorem. The content is related to ideas from Abbott’s (Understanding analysis (2nd ed.). New York...

This chapter explores the ε − N definition for convergence of a sequence; specifically, content similar to Abbott’s (Understanding analysis, 2nd edn. Springer, New York, NY, 2015) Section 2.2. It considers implications for teaching about processes for approximating real numbers—especially, irrational numbers.

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...

This chapter revolves around some preliminaries of the set of real numbers that are typically introduced in real analysis—the relationship of \(\mathbb {R}\) to \(\mathbb {N}\), \(\mathbb {Z}\), and \(\mathbb {Q}\), the Axiom of Completeness, an introduction to ‘ε’ in analysis, etc. Specifically, content similar to Abbott’s (Understanding analysis,...

This chapter explores the concept of differentiability and its relationship to continuity; content in Abbott’s (Understanding analysis (2nd ed.). New York, NY: Springer (2015)) Section 5.2. It explores the formal definition of the derivative as a limit involving what we refer to as secant slope functions. In the context of teaching, we draw a sharp...

Getting certified to teach high school mathematics typically requires completing a course in real analysis. Yet most teachers point out real analysis content bears little resemblance to secondary mathematics and report it does not influence their teaching in any significant way. This textbook is our attempt to change the narrative. It is our belief...

This chapter uses the examples of approximating a circle with regular polygons, and approximating the standard normal density function with Taylor polynomials, to illustrate a principle we call “modeling the complex with the simple.” This principle relates to real analysis (what we discuss in this chapter is from Abbott’s (Understanding analysis (2...

This article explores how, within formal mathematics, there are two conflicting definitions of "function".

This article explores secondary teachers’ opportunities to learn from an innovative real analysis course, as reflected in their actual classroom teaching. The course used cases of teaching as a site for applying mathematics and developing pedagogical mathematical practices. This article explores particular teaching moments in ( N = 6) secondary tea...

The authors would like to include the following changes in the published article.

In the mathematical community, two notions of "function" are used: the set-theoretic definition as a univalent set of ordered pairs, and the Bourbaki triple. These definitions entail different interpretations and answers to mathematical questions that even a secondary student might be prompted to answer. However, mathematicians and mathematics educ...

In this commentary, we make a case for the explicit inclusion of combinatorial topics in mathematics curricula, where it is currently essentially absent. We suggest ways in which researchers might inform the field’s understanding of combinatorics and its potential role in curricula. We reflect on five decades of research that has been conducted sin...

Most prospective secondary mathematics teachers in the United States complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically-motivated instructional model to design modules for a real analysis course that could inform secondary teachers’ actionable content knowledge and pedagogy....

Prospective secondary mathematics teachers are typically required to take advanced university mathematics courses. However, many prospective teachers see little value in completing these courses. In this paper, we present the instantiation of an innovative model that we have previously developed on how to teach advanced mathematics to prospective t...

A 2D version of Cavalieri's Principle is productive for the teaching of area. In this manuscript, we consider an area-preserving transformation, “segment-skewing,” which provides alternative justification methods for area formulas, conceptual insights into statements about area, and foreshadows transitions about area in calculus via the Riemann int...

Most prospective secondary mathematics teachers complete a course in real analysis, yet view the content as unrelated to their future teaching. We leveraged a theoretically-motivated instructional model to design modules for a real analysis course that could inform secondary teachers’ pedagogy, focusing on how this model was implemented in a single...

Prospective secondary mathematics teachers frequently take as many (or more) mathematics courses from a mathematics department as they do methods courses from an education department. Sadly, however, prospective secondary teachers frequently view their mathematical experiences in such courses as unrelated to their future teaching (e.g., Zazkis & Le...

This paper analyzes some of the ambiguities that arise among statements with the copular verb “is” in the mathematical language of textbooks as compared to day-to-day English language. We identify patterns in the construction and meaning of “is” statements using randomly selected examples from corpora representing the two linguistic registers. We c...

In this paper we report on a survey designed to test whether or not students differentiated between two different types of problems involving combinations - problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count ordered seq...

Research-based guidelines for learning variation exist (e.g., Franklin et al., 2007; Garfield, delMas, & Chance, 2007), but little is known about how teachers plan to teach standard deviation, or how these plans align with recent recommendations. In this article, we survey lesson plans designed by inservice and preservice secondary mathematical tea...

In the United States and elsewhere, prospective teachers of secondary mathematics are usually required to complete numerous advanced mathematics courses before obtaining certification. However, several research studies suggest that teachers' experiences in these advanced mathematics courses have little influence on their pedagogical practice and ef...

Secondary mathematics teachers are frequently required to take a large number of mathematics courses – including advanced mathematics courses such as abstract algebra – as part of their initial teacher preparation program and/or their continuing professional development. The content areas of advanced and secondary mathematics are closely connected....

Over the past century, mathematicians and mathematics educators have explored various ways in which abstract algebra is related to school mathematics. These have included Felix Klein's work as a mathematician—in which his synthesis of the study of geometry through abstract algebraic structures has proved influential on our approach to teaching seco...

In this paper, we elaborate on a theoretically motivated instructional model for designing modules for secondary mathematics teachers in an abstract algebra course. We illustrate this model by elaborating on two modules, Functions and k-Product Property, and report some findings from a small-scale study with two secondary mathematics teachers. Find...

The notion of practice-based models for mathematical knowledge for teaching has played a pivotal role in the conception of teacher knowledge. In this work, teachers' knowledge of mathematics that is outside the scope of what is being taught (nonlocal mathematics) is considered more explicitly. Drawing on a cognitive model for the development of mat...

Combination problems are a cornerstone of combinatorics courses, but little research has been done examining the ways that students perceive and differentiate among different combination problems. In this article, we investigate how mathematics education students (n = 18) in a discrete mathematics course view two categorically different combination...

This manuscript presents findings from a study about the knowledge for and planned teaching of standard deviation. We investigate how understanding variance as an unbiased (inferential) estimator – not just a descriptive statistic for the variation (spread) in data – is related to teachers’ instruction regarding standard deviation, particularly aro...

This article draws on semi-structured, task-based interviews to explore secondary teachers’ (N = 7) understandings of inverse functions in relation to abstract algebra. In particular, a concept map task is used to understand the degree to which participants, having recently taken an abstract algebra course, situated inverse functions within its gro...

In this paper, we explore in more detail why knowing advanced mathematics might be beneficial for teachers, specifically in relation to their classroom practice. Rather than by listing courses or specific advanced topics, as though those were the agents of change, we do so by considering advanced mathematical content for teachers in terms of more g...

Future teachers often claim that advanced undergraduate courses, even those that attempt to connect to school mathematics, are not useful for their teaching. This paper proposes a new way of designing advanced undergraduate content courses for secondary teachers. The model involves beginning with an analysis of the curriculum and practices of schoo...

In this paper we report on a survey study to determine whether or not students differentiated between two different categories of problems involving combinations – problems in which combinations are used to count unordered sets of distinct objects (a natural, common way to use combinations), and problems in which combinations are used to count orde...

In an undergraduate Calculus III class, we explore the effect of “flipping” the instructional delivery of content on both student performance and student perceptions. Two instructors collaborated to determine daily lecture notes, assigned the same homework problems, and gave identical exams; however, compared to a more traditional instructional app...

This paper explores the potential for aspects of abstract algebra to be influential for the teaching of school algebra (and early algebra). Using national standards for analysis, four primary areas common in school mathematics – and their progression across elementary, middle, and secondary mathematics – where teaching may be transformed by teacher...

The work that mathematics teachers do is frequently mathematical in nature and different from other professions. Understanding and describing common ways that teachers draw upon their content knowledge in the practice of teaching is important. Building on the descriptions by McCrory et al. (Journal for Research in Mathematics Education 43(5) 584–61...

The purpose of this study was to investigate teachers’ subject matter knowledge relevant to the teaching of informal line of best fit. Task-based interviews were conducted with nineteen pre-service and in-service mathematics teachers. The results include descriptions and categorizations of teachers’ conceptions, criteria for placement, accuracy of...

Studies exploring the role and impact on teaching of secondary teachers' knowledge of the mathematical horizon will be presented; topics include CCSSM, abstract algebra, real analysis, and statistics. Illustrations connected to classroom teaching practices will be examined and implications for mathematics teacher education discussed.

The recent trend to unite mathematically related disciplines (science, technology, engineering, and mathematics) under the broader umbrella of STEM education has advantages. In this new educational context of integration, however, STEM teachers need to be able to distinguish between sufficient proof and reasoning across different disciplines, parti...

The practice of problem posing is as important to develop as problem solving. The resulting explorations can be mathematically rich.

Contemporary technologies have impacted the teaching and learning of mathematics in significant ways, particularly through the incorporation of dynamic software and applets. Interactive geometry software such as Geometers Sketchpad (GSP) and GeoGebra has transformed students' ability to interact with the geometry of plane figures, helping visualize...

## Citations

... Currently, the research literature has focused on the content dimension through a cognitive lens, offering evidence of teachers' difficulties with connecting secondary and university mathematics (Wasserman et al., 2018;Zazkis & Leikin, 2010). To address this issue, teacher educators have developed teacher education materials that support teachers' construction of specific mathematical ideas from an advanced and coherent standpoint (e.g., Carlson et al., 2021;Izsák et al., 2022;Moore et al., 2019;Thompson et al., 2019;Wasserman et al., 2022). However, not much emphasis has been placed on the other two dimensions of double discontinuity, including how PSTs interpret their experience of secondary-university mathematics transition, as well as how this experience is constructive toward their identity and professional development. ...

... A literature survey shows that though there is yet little research on teaching and learning curve fitting in tertiary education and in particular in secondary education, it still covers individual students' conceptions (Casey, 2015;Casey & Nagle, 2016;Groth et al., 2018;Sorto et al., 2011); teachers' conceptions (Casey & Wasserman, 2015); students' group work (Dreyfus & Hillel, 1998); tertiary textbooks (Key, 2005;Yee, 1994); and secondary textbooks (Gea et al., 2015;Lesser, 1999). Yee (1994) investigated how statistics textbooks in tertiary education justify or prove the least-squares method. ...

... We know from studies with pre-service teachers that undergraduate mathematics students do not always develop the connections between mathematics content areas we may have expected (e.g., Wasserman, 2018). Research akin to this proposed study has the potential to help us understand the themes students actually come away with on not only a course-by-course basis but also across sequences of courses and the entire mathematics curriculum. ...

... It is well-established that discrete mathematics is an important component of the K-16 mathematics curriculum (Hart & Martin, 2018;Kapur, 1970;Lockwood et al., 2020;National Council of Teachers of Mathematics, 2000;Rosenstein et al., 1997). Many problems within enumerative combinatorics are accessible to a wide range of students and have the potential to generate rich mathematical discussions, promote collaborative problem-solving, and engage students in productive mathematical practices (National Governors Association Center for Best Practices & Council of Chief State School Officers, 2010)-qualities that can promote equitable mathematics learning (Boaler, 2016). ...

... As many students get to know mathematics as a formal and deductive discipline for the first time at university (Halverscheid & Pustelnik, 2013), it can take some time to develop joy and value and a positive image of its own abilities concerning this form of mathematics. Lecturers can support this development by highlighting the advantages of this form of mathematics and by building bridges to school mathematics (Weber et al., 2020) which could be helpful for students in teacher education programs, in particular. ...

... Auch international wird dieser Problematik Beachtung geschenkt (z. B. Fukawa-Connelly, Mejía-Ramos, Wasserman & Weber, 2020). ...

Reference: Theoretischer Rahmen

... Other researchers (e.g., Wasserman, 2018a;Wasserman et al., 2019) have addressed ways to develop mathematical knowledge for teaching in the preparation of secondary mathematics teachers. Wasserman et al. (2019) articulate four ways in which prospective teachers can engage in developing mathematical knowledge for teaching and describe these as existing along a spectrum spanning from "mathematical in nature" to "pedagogical in nature" (p. ...

... The key course objective was to bridge between university experiences of mathematics education majors and the realm of classroom teaching (c.f, Wasserman et al., 2019), with a specific focus on reasoning and proving. Thus, the course activities aimed to help PSTs enhance their content and pedagogical knowledge of reasoning and proving, connect it to the secondary school curriculum and apply this knowledge by designing and teaching four proof-oriented lessons in local schools. ...

... The question of the contribution of tertiary mathematics to school teaching was posed already more than a century ago by the renowned mathematician Felix Klein, who identified and drew attention to the discontinuity between teachers' experiences of studying mathematics at university and of teaching mathematics at school (Kilpatrick, 2019). Nevertheless, our knowledge and understanding of the actual contribution of tertiary courses in general, and of tertiary calculus in particular, are still far from satisfying (Even, 2011;Wasserman, 2018aWasserman, , 2018bZazkis, 2020;Zazkis & Leikin, 2010). Moreover, documented examples of how teachers draw on content or approaches from tertiary mathematics courses in teaching are fairly rare (Zazkis, 2020). ...

... However, simply because teachers draw great benefits from their field of work does not imply that these benefits also translate from teachers to students in the classroom. Indeed, it has been shown that there is no significant correlation between a teacher's knowledge in abstract algebra and students' achievement in school algebra [2][3][4][5][6][7][8][9]. This conflict raises the question of whether students themselves can benefit from learning abstract algebra if they Figure 1. ...