Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn

Melhuish, Kathleen; Fukawa-Connelly, Timothy; Dawkins, Paul Christian; Woods, Christian; Weber, Keith

doi:10.1016/j.jmathb.2022.100986

Cited by 21 publications

(13 citation statements)

References 86 publications

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“…It is unsurprising that the predominant face-to-face lecture classification was straight lecture given its central pedagogical role in the history of the university (Friesen, 2011 ). In mathematics, the literature suggests it is the predominant mode of instruction in proof-based courses (Melhuish et al, 2022 ). Given that 42% of the straight-lecture classifications were perceived by students as being beneficial or very beneficial to learning is a reminder that it is what happens in the traditional lecture, how the students engage with it, and indeed how it sits in relation to other components of the module, that matters.…”

Section: Discussionmentioning

confidence: 99%

“…While a detailed discussion of the literature on the format of mathematics lectures and mathematicians’ beliefs about the best way to teach mathematics is beyond the scope of this paper, we conclude this section by mentioning two studies that we believe are relevant. The first is a recent survey of the literature on the teaching and learning of proof-based mathematics courses (Melhuish et al, 2022 ). The authors note that the predominant form of instruction in such courses is the traditional lecture, where mathematicians write the formal mathematics on the board while providing informal meta-commentary.…”

Section: Literature Reviewmentioning

confidence: 99%

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The university mathematics lecture: to record, or not to record, that is the question

Meehan

Howard

2023

Math Ed Res J

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While recordings of lectures proved invaluable for students’ learning during the pandemic, as our university transitioned back to in-person teaching there was a call from many lecturers to remove the requirement to provide lecture recordings due to the perceived negative impact on attendance. To examine in detail the relationship between recordings and the corresponding face-to-face lectures, we conducted a study on the formats of lectures across our undergraduate mathematics programmes pre- and post-lockdown in March 2020, and students’ perceptions of how beneficial they felt each was for their learning. In May 2020, 156 mathematics students completed a survey containing both quantitative and qualitative questions. Findings indicate that pre-pandemic almost 70% of the lecture formats classified by students were traditional in nature, with 20% classified as containing some interactions, and the remainder as including group work. While students did not perceive great differences in terms of the benefits to learning between the face-to-face and online lecture formats in the majority of modules, those who experienced interactions or group work in lectures before lockdown, reported a greater drop than those who experienced a traditional lecture. Irrespective of preferences for face-to-face or online learning, students were clear about the value of lecture recordings and interactions with peers and lecturers to their learning. Based on our findings, and those of others, we discuss implications for mathematics lecturers’ practice. Specifically, we argue for the provision of lecture recordings or short pre-recordings, especially when the lecture is traditional in nature.

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Section: Discussionmentioning

confidence: 99%

Section: Literature Reviewmentioning

confidence: 99%

The university mathematics lecture: to record, or not to record, that is the question

Meehan

Howard

2023

Math Ed Res J

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“…The second strand of literature we draw on is that of mathematicians presenting content. While there is significant literature exploring classroom presentations of proof (c.f., Melhuish et al, 2022), there is relatively little documentation about classroom teaching practices regarding definitions. The study of Pinto (2019) explored how two mathematicians presented "the same" lesson about the formal definition of derivative in a real analysis course.…”

Section: Observations Of Classroom Practices Regarding Definitionsmentioning

confidence: 99%

“…All of these might be understood as exploring how mathematicians present definitions as well as additional ideas related to mathematical concepts. There has been significant research exploring how mathematicians think about teaching, especially about the presentation of theorems, and explorations of how mathematics is presented in lecture (see Melhuish et al (2022) for a recent review). Yet these studies have only seldom explored what the mathematicians want their students to learn from these presentations of definitions and other mathematics, and have typically done so in the context of a single definition.…”

Section: Introductionmentioning

confidence: 99%

How mathematicians characterize and attempt to develop understanding of concepts and definitions in proof-based courses

Rupnow,

Fukawa-Connelly

2024

Front. Educ.

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Mathematics education research has long focused on students’ conceptual understanding, including highlighting conceptions viewed as problematic and looking for ways to develop more desirable conceptions. Nevertheless, limited research has examined how mathematicians characterize understanding of concepts and definitions or promote activities beneficial for students. Based on interviews with 13 mathematicians, we present thematic characterizations of what it means to understand a concept and definition, highlight activities mathematicians believe assist students’ learning, and examine their reasons for promoting these activities. Results include mathematically grounded descriptions of what it means to understand a concept but general descriptions of approaching and supporting learning. Implications include a need for attending to intended meanings for “understanding” in context and how this impacts appropriate activities for developing understanding, as well as a careful examination of the extant research literature’s claims about seemingly unified notions of conceptual understanding.

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“…However, under the banner of active learning, a new movement is gaining momentum in tertiary mathematics education -Inquiry-Based Mathematics Instruction [6,7]. In inquiry classrooms, students are challenged to -reinvent or create mathematics that is new to them‖ ( [6], p. 131).…”

Section: Introductionmentioning

confidence: 99%

Inquiry-based mathematics education: a call for reform in tertiary education seems unjustified

Evans

Dietrich

2022

STEME

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<p style="text-indent:20px;">In the last decade, major efforts have been made to promote inquiry-based mathematics learning at the tertiary level. The Inquiry-Based Mathematics Education (IBME) movement has gained strong momentum among some mathematicians, attracting substantial funding from US government agencies. This resulted in the successful mobilization of regional consortia in many states, uniting over 800 mathematics education practitioners working to reform undergraduate education. Inquiry-based learning is characterized by the fundamental premise that learners should be allowed to learn 'new to them' mathematics without being taught. This progressive idea is based on the assumption that it is best to advance learners to the level of experts by engaging learners in mathematical practices similar to those of practicing mathematicians: creating new definitions, conjectures and proofs - that way, learners are thought to develop 'deep mathematical understanding'.</p><p style="text-indent:20px;">However, concerted efforts to radically reform mathematics education must be systematically scrutinized in view of available evidence and theoretical advances in the learning sciences. To that end, this scoping review sought to consolidate the extant research literature from cognitive science and educational psychology, offering a critical commentary on the effectiveness of inquiry-based learning. Our analysis of research articles and books pertaining to the topic revealed that the call for a major reform by the IBME advocates is not justified. Specifically, the general claim that students would learn better (and acquire superior conceptual understanding) if they were not taught is not supported by evidence. Neither is the general claim about the merits of IBME for addressing equity issues in mathematics classrooms.</p>

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Collegiate mathematics teaching in proof-based courses: What we now know and what we have yet to learn

Cited by 21 publications

References 86 publications

The university mathematics lecture: to record, or not to record, that is the question

The university mathematics lecture: to record, or not to record, that is the question

How mathematicians characterize and attempt to develop understanding of concepts and definitions in proof-based courses

Inquiry-based mathematics education: a call for reform in tertiary education seems unjustified

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