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Blue Skies Above the Horizon
Abstract
In this commentary chapter, we draw on ideas from Baldinger and Murray; Cuoco; Wasserman and Galarza; and Zazkis and Marmur to articulate our views on the importance of mathematical structure and its relevance in secondary mathematics teachers’ disciplinary knowledge. In particular, we organize our discussion around two related questions—about the connections between abstract algebra and secondary school mathematics, and about how these connections can support the development of teachers’ disciplinary knowledge.
... To address the discontinuity within teacher education, there is a need to foster a general awareness of mathematics as a connected landscape (Bass, 2022), in addition to fostering specific connections within and outside of mathematics, termed intra-and extra-mathematical connections, respectively (De Gamboa, et al., 2022). In our research, we conceptualize this awareness as part of teachers' knowledge at the mathematical horizon (KMH), which is characterized by an understanding of mathematical structure, practices, and values that allow for an interconnected view of the mathematical world and how to be within it (Zazkis & Mamolo, 2011;Mamolo & Taylor, 2018). Ami Mamolo ami.mamolo@ontariotechu.ca recounts, the Modellkammer, a collection of mathematical models introduced by J.A. Segner that emphasizes applications of mathematics to modern technologies, was an early feature of the progressive teaching approach at Göttingen University. ...
... The outer horizon represents the "greater mathematical world", including the general structures, values, and sensibilities, which are relevant to, and exemplified by, the particular object. Mamolo and Taylor (2018) articulated connections across conceptualizations of KMH, as depicted in Table 1. ...
... With these links, Mamolo and Taylor (2018) analyze how group theoretic ideas are a valuable component of mathematical knowledge for teaching. They articulate pathways from specific "locations" of the mathematical landscape of the moment, through the greater mathematical world, and back again, analysing examples that connect group structures to inverse functions (Zazkis & Marmur, 2018), isomorphisms to functions (Wasserman & Galarza, 2018), and Pythagorean triples to ring structures (Cuoco, 2018). ...
We report on a group theory activity in which learners explored dihedral symmetries through a tangible geometric model. This approach has historical roots in the work of Felix Klein’s Erlangen Program and his Elementary Mathematics from an Advanced Standpoint. We situate our study with respect to this history as well as current educational research in spatial visual reasoning, abstract algebra, and teacher knowledge. Our findings highlight opportunities that tangible geometric models can provide for fostering structural and interconnected understanding characteristic of teachers’ knowledge at the mathematical horizon.
Dans cet article, nous présentons différentes dimensions au cœur de la vision de l’enseignement des mathématiques véhiculée par Seymour Papert, une vision ancrée dans la nature et la spécificité des mathématiques
The secondary-school mathematics curriculum is narrow in scope and technical in character; this is quite different from the nature of the discipline itself. As a result, it offers little inspiration to both students and teachers, and provides students with poor preparation for university mathematics courses and indeed for life. Over the past century, recently more than ever, there have been calls for change, for a curriculum that is true to the subject of mathematics as the creation and study of patterns and structures. While there are hopeful responses to this at the elementary level, there is almost nothing at the secondary level. Ironically, it is felt that in order to prepare students for university calculus, the secondary curriculum simply has to be what it is. This is a special case of a myth that needs to be destroyed.
When public schooling was first introduced in the United States, early proponents emphasized the need for mathematics as critical for an informed citizenry in a democracy. Half a century later, this purpose of mathematics has been almost entirely overshadowed by the push for mathematics to maintain technological and economic advantages. The belief that preparation for technological careers is, and has historically been, the only purpose for school mathematics in the US has become a myth widely believed by the public and policymakers alike. As this myth took hold, mathematics curricula were narrowed, incorporating only the mathematics, and applications of mathematics, that supported this specific purpose. Not only does this narrowing of school mathematics negatively impact the development of informed citizens, but it limits the extent to which mathematics can be studied in ways that engage all learners. The emergence of mathematics for social justice is thus—in part—an attempt to recapture the broader purposes of school mathematics.
This article extends the notion of 'knowledge at the mathematical horizon' or 'horizon knowledge' introduced by Ball and colleagues as a part of teachers' subject matter knowledge. Our focus is on teachers' mathematical knowledge beyond the school curriculum, that is, on mathematics learnt during undergraduate college or university studies. We explore and exemplify what horizon knowledge may contribute to teaching elementary and secondary school mathematics in terms of relevant subject matter, as well as how such knowledge can shape interaction with students. We introduce a new perspective to the on-going discussions around mathematics for teaching by considering what may lie at the horizon of teachers' mathematical knowledge.
This article examines pre-service secondary school teachers’ responses to a learning situation that presented a student's struggle with determining the area of an irregular hexagon. Responses were analyzed in terms of participants’ evoked concept images as related to their knowledge at the mathematical horizon, with attention paid toward the influence of one on the other. Specifically, our analysis attends to common features in participants’ understanding of the mathematical task, and explores the interplay between participants’ personal solving strategies and approaches and their identified preferences when advising a student. We conclude with implications for mathematics teacher education research and pedagogy.
The conceptualisation of mathematical knowledge for teaching (MKT) has recently received much attention. Scholars have provided examples, studied effects, and debated importance. However, from among the MKT domains, horizon content knowledge (HCK) has received less attention. In particular, the nature of the knowledge as it is related to teaching is unclear. We argue for efforts to clarify definitions and to test and refine those definitions with the use of realistic and vetted examples of professional work. To advance this agenda, we provide a working definition of HCK and use it to discuss a vignette involving irrational numbers.
While teacher content knowledge is crucially important to the improvement of teaching and learning, attention to its development and study has been uneven. Historically, researchers have focused on many aspects of teaching, but more often than not scant attention has been given to how teachers need to understand the subjects they teach. Further, when researchers, educators and policy makers have turned attention to teacher subject matter knowledge the assumption has often been that advanced study in the subject is what matters. Debates have focused on how much preparation teachers need in the content strands rather than on what type of content they need to learn. In the mid-1980s, a major breakthrough initiated a new wave of interest in the conceptualization of teacher content knowledge. In his 1985 AERA presidential address, Lee Shulman identified a special domain of teacher knowledge, which he referred to as pedagogical content knowledge. He distinguished between content as it is studied and learned in disciplinary settings and the "special amalgam of content and pedagogy" needed for teaching the subject. These ideas had a major impact on the research community, immediately focusing attention on the foundational importance of content knowledge in teaching and on pedagogical content knowledge in particular. This paper provides a brief overview of research on content knowledge and pedagogical content knowledge, describes how we have approached the problem, and reports on our efforts to define the domain of mathematical knowledge for teaching and to refine its sub- domains.
Mathematics learning is a continuous process in which students face some abrupt episodes involving many changes of different natures. This work is focused on one of those episodes, transition from primary to secondary school, and targets teachers and their mathematical knowledge. By characterising the mathematical knowledge that teachers of mathematics need to smooth transition processes, we aim to highlight teachers’ impact in the continuity of mathematics education. The concept of Mathematical Knowledge for Teaching (MKT) developed by Ball among others (Ball, Thames & Phelps, 2008) and, within this framework, the construct Horizon Content Knowledge (HCK) emerge as our theoretical response to the knowledge needed for teaching mathematics in a continuous way, particularly relevant during transition to secondary school. The enrichment of the idea of HCK and the characterisation of its expression in the teaching practice intends to develop a theoretical tool to approach transition from teachers’ mathematical knowledge perspective.
Edmund Husserl (1859-1938) was one of the most influential philosophers of the twentieth century. Inventor of the 'phenomenological method' for the analysis of physical and mental phenomena, with 'intentionality' at its center, and initiator of the 'phenomenological' movement of contemporary human sciences, he became the great critic of naive 'naturalism' in philosophy and psychology ('psychologism'), and of what he regarded as the loss of orientation, certainty, and thereby of credibility of modern science. His appeal for 'going back to the things themselves,' and for overcoming the crisis of modern science by a foundation of all science and philosophy in a clearly understood 'life-world' (. Lebenswelt) inspired the work of a great number of philosophers, including Gadamer, Heidegger, Ingarden, Levinas, Merleau-Ponty, Ricoeur, and Sartre, as well as social scientists such as Buytendijk, Kurt Lewin, Luckmann, and Alfred Schütz. Husserl's impact on the social sciences was mainly indirect, through (1) revealing the deficiencies of a merely naturalistic approach in science, (2) supplying very detailed methods for the description of phenomena, relations and situations, and (3) stimulating applications of the phenomenological method in many areas of social science, e.g., in sociology as brought into effect by Schütz.
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 school mathematics and its teaching, and then using those to build up to the advanced mathematics – in this case, real analysis. After developing definitions, examples, theorems, and proofs, the model then reconnects to practice, asking the teachers to translate ideas from real analysis in ways that are appropriate for teaching high school content to students. To illustrate the model, we provide and discuss two example tasks.
Mathematics learning is a continuous process in which students face some abrupt episodes involving many changes of different natures. This work is focused on one of those episodes, transition from primary to secondary school, and targets teachers and their mathematical knowledge. By characterising the mathematical knowledge that teachers of mathematics need to smooth transition processes, we aim to highlight teachers’ impact in the continuity of mathematics education. The concept of Mathematical Knowledge for Teaching (MKT) developed by Ball among others (Ball, Thames & Phelps, 2008) and, within this framework, the construct Horizon Content Knowledge (HCK) emerge as our theoretical response to the knowledge needed for teaching mathematics in a continuous way, particularly relevant during transition to secondary school. The enrichment of the idea of HCK and the characterisation of its expression in the teaching practice intends to develop a theoretical tool to approach transition from teachers’ mathematical knowledge perspective.
The important difference between the work of a child in an elementary mathematics class and that of a mathematician is not in the subject matter (old fashioned numbers versus groups or categories or whatever) but in the fact that the mathematician is creatively engaged in the pursuit of a personally meaningful project. In this respect a child's work in an art class is often close to that of a grown‐up artist. The paper presents the results of some mathematical research guided by the goal of producing mathematical concepts and topics to close this gap. The prime example used here is ‘Turtle Geometry’, which is concerned with programming a moving point to generate geometric forms. By embodying the moving point as a ‘cybernetic turtle’ controlled by an actual computer, the constructive aspects of the theory come out sufficiently to capture the minds and imaginations of almost all the elementary school children with whom we have worked—including some at the lowest levels of previous mathematical performance.* This paper was given at the CEMREL Conference on Algebra in the pre‐College Curriculum, Carbondale, Illinois, November 1970.
In this article we offer a theoretical discussion of teachers' mathematics-for-teaching, using complexity science as a framework
for interpretation. We illustrate the discussion with some teachers' interactions around mathematics that arose in the context
of an in-service session. We use the events from that session to illustrate four intertwining aspects of teachers' mathematics-for-teaching.
We label these aspects “mathematical objects,” “curriculum structures,” “classroom collectivity,” and “subjective understanding”.
Drawing on complexity science, we argue that these phenomena are nested in one another and that they obey similar dynamics,
albeit on very different time scales. We conjecture (1) that a particular fluency with these four aspects is important for
mathematics teaching and (2) that these aspects might serve as appropriate emphases for courses in mathematics intended for
teachers.
Central to caring professions such as teaching is the need to notice and be sensitive to the experiences of pupils and teachers. Starting from this position, Researching Your Own Practice demonstrates that in order to develop your professional practice you must first develop your own sensitivities and awareness. One must be attuned to fresh possibilities when they are needed and be alert to such a need through awareness of what is happening at any given time. By giving a full explanation of this theory and a guide to its implementation, this book provides a practical approach to becoming more methodical and systematic in professional development. It also gives the reader a basis for turning professional development into practitioner research, as well as giving advice on how noticing can be used to improve any research, or be used as a research paradigm in its own right. The discipline of noticing is a groundbreaking approach to professional development and research, based upon noticing a possibility for the future, noticing a possibility in the present moment and reflecting back on what has been noticed before in order to prepare for the future. John Mason, one of the discipline's most authoritative exponents, provides us here with a clear, persuasive and practical guide to its understanding and implementation.
With an eye on the mathematical horizon: Knowing mathematics for teaching to learners’ mathematical futures
- D L Ball
- H Bass
Ball, D. L., & Bass, H. (2009). With an eye on the mathematical horizon: Knowing mathematics
for teaching to learners' mathematical futures. Paper presented at the 43rd Jahrestagung
der Gelleschaft fur Didaktic der Mathematik. Oldenburg,.Germany. Retrieved May 15,
2011, from: www.mathematik.uni-dortmund.de/ieem/BzMU/BzMU2009/BzMU2009-Inhaltfuer-Homepage.htm
Designing aesthetic experiences for young mathematicians: A model for mathematics education reform
- G Gadanidis
- M Borba
- J Hughes
- H Lacerda
Gadanidis, G., Borba, M., Hughes, J., & Lacerda, H. (2016). Designing aesthetic experiences for
young mathematicians: A model for mathematics education reform. International Journal for
Research in Mathematics Education, 6(2), 225-244.
Leveraging real analysis to foster pedagogical practices
- N Wasserman
- K Weber
- W McGuffey
- A Weinberg
- C Rasmussen
- J Rabin
- M Wawro
- S Brown
Wasserman, N., Weber, K., & McGuffey, W. (2017). Leveraging real analysis to foster pedagogical
practices. In A. Weinberg, C. Rasmussen, J. Rabin, M. Wawro, & S. Brown (Eds.), Proceedings
of the 20th annual conference on Research in Undergraduate Mathematics Education (RUME)
(pp. 1-15). San Diego, CA: RUME.
Knowing and using mathematics in teaching
Adler, J., & Ball, D. (Eds.). (2009). Knowing and using mathematics in teaching. For the Learning
of Mathematics, 29(3), 1-56.
The mathematical education of teachers II
Conference Board of Mathematical Sciences (CBMS). (2012). The mathematical education of
teachers II. Washington, DC: American Mathematical Society and Mathematical Association
of America.
Mathematics and beauty
- N Sinclair
Sinclair, N. (2006). Mathematics and beauty. New York, NY: Teachers College Press.