Journal Articles
Using conceptual analyses to resolve the tension between advanced and secondary mathematics: The cases of equivalence and inverse (2023)
John Paul Cook, April Richardson, Zackery Reed, & Elise Lockwood
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Advanced mathematics is seen as an integral component of secondary teacher preparation, and thus most secondary teacher preparation programs require their students to complete an array of advanced mathematics courses. In recent years, though, researchers have questioned the utility of proposed connections between advanced and secondary mathematics. It is simply not clear in many cases—to researchers, teacher educators, and teachers themselves—exactly how advanced mathematics content is related to secondary content. In this paper, we propose using a conceptual analysis—a form of theory in which one explicitly describes ways of reasoning about a particular mathematical idea—to address this issue. Specifically, we use conceptual analyses for the foundational notions of equivalence and inverse to illustrate how the ways of reasoning needed to support productive engagement with tasks in advanced mathematics can mirror and reinforce those that are similarly productive in school mathematics. To do so, we propose conceptual analyses for the key concepts of equivalence and inverse and show how researchers can use these conceptual analyses to identify connections to school mathematics in advanced mathematical tasks that might otherwise be obscured and overlooked. We conclude by suggesting ways in which conceptual analyses might be productively used by both teacher educators and future teachers.
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Cook, J. P., Richardson, A., Reed, Z., & Lockwood, E. (2023). Using conceptual analyses to resolve the tension between advanced and secondary mathematics: The cases of equivalence and inverse. ZDM Mathematics Education. https://doi.org/10.1007/s11858-023-01495-2.
An initial framework for analyzing students’ reasoning with equivalence across mathematical domains (2022)
John Paul Cook, Zackery Reed, & Elise Lockwood
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The concept of equivalence is foundational in mathematics and is pervasive in the K-16 curriculum. Though much research has focused on equivalence, nearly all of it is domain-specific, and it is therefore unclear how students’ reasoning about equivalence in one domain might influence their reasoning about it in another, if at all. This highlights a need for increased theoretical unity and coherence. In this theoretical paper, we propose an initial framework for analyzing students’ reasoning about equivalence across domains. We use the framework to highlight commonalities amongst the ways in which equivalence is interpreted with respect to fractions, K-12 algebra, modular arithmetic, and linear algebra. We demonstrate the framework’s strength as an analytical tool by using it to conduct detailed analyses of student data from already-published studies in combinatorics and abstract algebra. We conclude by suggesting ways in which this framework lays a rich foundation for future research.
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Cook, J. P., Reed, Z., & Lockwood, E. (2022). An initial framework for analyzing students’ reasoning with equivalence across mathematical domains. Journal of Mathematical Behavior, 66(100935), 1-15. https://doi.org/10.1016/j.jmathb.2022.100935.
Explicating interpretations of equivalence in measurement contexts (2021)
John Paul Cook, Paul Dawkins, & Zackery Reed
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In this paper we analyze common solutions that students often produce to isomorphic tasks involving proportional situations. We highlight some key distinctions across the tasks and between the different equations students write within each task to help elaborate the different interpretations of equivalence at play: numerical, transformational, and descriptive. We use this opportunity to further explore the value of operationalizing these interpretations in both research and instruction.
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Cook, J. P., Dawkins, P. C., & Reed, Z. (2021). Explicating interpretations of equivalence in measurement contexts. For the Learning of Mathematics, 41(3), 36-41. https://www.jstor.org/stable/27091219.
Conference Proceedings
Analyzing students’ reasoning about equivalence: The importance of transformational activity (2024)
John Paul Cook, April Richardson, Zackery Reed, & Elise Lockwood
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Equivalence is a foundational idea across the mathematics curriculum. There is considerable evidence, however, that students at all levels experience difficulties with it; a prevailing explanation is that students rely too much on transformations. And yet, transformational activity is absolutely essential: it is the primary means by which one generates more tractable representations that are better suited to the situation at hand. Strikingly, we found no studies that directly examine students’ productive uses of transformational activity. To this end, we conducted a series of task-based interviews with undergraduate students in order to illustrate and account for productive instances of transformational activity across undergraduate mathematics. Our findings affirm a hypothesis from the literature that supplementing one’s transformational activity with notions of equivalence can support productive reasoning. Additionally, we extend this idea by providing detailed analyses of what these supplementary notions of equivalence entail.
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Cook, J. P.*, Richardson, A., Reed, Z., & Lockwood, E. (2024) Analyzing students’ reasoning about equivalence: The importance of transformational activity. In Proceedings of the 15th International Congress on Mathematical Education. Sydney, Australia. https://icme15-c10000.eorganiser.com.au/data/clients/1/773/submissions/173146/abstract.pdf.
In Defense of Transformational Activity: Analyzing Students’ Productive Reasoning about Equivalence (2024)
April P. Richardson, John Paul Cook, Zackery Reed, O. Hudson Payne, Cory Wilson, & Elise Lockwood
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Equivalence is a foundational idea in mathematics and a key fixture in the K-16 curriculum. There is considerable evidence, however, that students at all levels experience difficulties with it. A prevailing explanation is that students rely too much on transformations; and yet, transformational activity is absolutely essential: it is the primary means by which one generates more tractable representations that are better suited to the situation at hand. Strikingly, we found no studies that directly examine students’ productive uses of transformational activity. To this end, we conducted a series of task-based interviews with undergraduate students in order to illustrate and account for productive instances of transformational activity across undergraduate mathematics. Our findings affirm a hypothesis from the literature that supplementing one’s transformational activity with notions of equivalence can support productive reasoning. Additionally, we extend this idea by providing detailed analyses of what these supplementary notions of equivalence entail.
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Richardson, A., Cook, J. P., Reed, Z., Payne, O. H., Wilson, C., & Lockwood, E. (2024). In Defense of Transformational Activity: Analyzing Students’ Productive Reasoning about Equivalence. In Cook, S., Katz, B., & Moore-Russo, D. (Eds.) Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education. Omaha, NE. http://sigmaa.maa.org/rume/crume2024/papers/044.pdf.
From an Inclination to Subtract to a Need to Divide: Exploring Student Understanding and Use of Division in Combinatorics (2024)
Zackery Reed, Elise Lockwood, & John S. Caughman IV
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In this report, we provide an initial exploration into a key but under-studied phenomenon in enumerative combinatorics — the use of division in solving counting problems. We present a case of one undergraduate student solving a combinatorics problem; this case is representative of a broader phenomenon in which students may intuitively desire to account for an overcount using subtraction, when division is a productive and useful approach. We highlight the conceptions a student demonstrated as she progressed from using subtraction to using division successfully. We frame our analysis in terms of a set-oriented perspective (Lockwood, 2014).
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Reed, Z., Lockwood, E., & Caughman, J. S. (2024). From an inclination to subtract to a need to divide: Exploring student understanding and use of division in combinatorics. In Proceedings of the 26th Annual Conference on Research in Undergraduate Mathematics Education. Omaha, NE. http://sigmaa.maa.org/rume/crume2024/papers/087.pdf.
How Do We Disentangle Equality from Equivalence? Well, It Depends (2023)
Zackery Reed, John Paul Cook, Elise Lockwood, & April Richardson
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In this paper, we report on interviews with mathematicians exploring the ways in which they think about the relationship between equality and equivalence. Given sometimes unclear and conflicting presentations of equality and equivalence in the literature, we are motivated to understand subtleties about how these constructs interact; doing so can have pedagogical implications, which we explore in this paper. We present three major themes that emerged from our analysis of the mathematicians' discussions of equality: 1) that equality represents a well-specified equivalence relation, 2) that audience matters when specifying equivalence and equality, and 3) that context is imperative when discussing equivalence and equality.
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Reed, Z., Cook, J. P., Lockwood, E., & Richardson, A. (2023). How do we disentangle equality from equivalence? Well, it depends. In Cook, S., Katz, B., & Moore-Russo, D. (Eds.) Proceedings of the 25th Annual Conference on Research in Undergraduate Mathematics Education. pp. 315-323. Omaha, NE. http://sigmaa.maa.org/rume/RUME25v2.pdf#page=339.
A framework for analyzing students’ reasoning about equivalence across undergraduate mathematics
Zackery Reed, John Paul Cook, Elise Lockwood, & April P. Richardson
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Establishing and leveraging equivalence is a central practice in mathematics. Though there have been many studies of students’ uses of equivalence, much of the research thus far has been domain-specific, and the literature generally lacks coherence within and across mathematical domains. In this theoretical paper, we propose an initial unifying framework for capturing the different ways that students might establish equivalence. Using constructs born out of the K-12 literature, we discuss how this framework can be applied to student reasoning in undergraduate settings. We do so by presenting the results of conceptual analyses of students’ possible uses of equivalence when thinking about vectors, isomorphisms and homeomorphisms, and single- variable limits. We then conclude with a detailed analysis of student data from combinatorics that identifies productive aspects of their uses of equivalence when constructing permutations.
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Reed, Z., Cook, J. P., Lockwood, E., & Richardson, A. (2022). A framework for analyzing students’ reasoning about equivalence across undergraduate mathematics. Karunakaran, S. S., & Higgins, A. (Eds.) Proceedings of the 24th Annual Conference on Research in Undergraduate Mathematics Education. pp. 857-866. Boston, MA. http://sigmaa.maa.org/rume/RUME24.pdf#page=875.