Figure 2 - uploaded by Natasha Artemeva
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A page from an experienced professor's lecture notes Note: The professor's L1 is Russian; L2 is English. Language of instruction is Hebrew (L3).
Source publication
This article reports on an international study of the teaching of undergraduate mathematics in seven countries. Informed by rhetorical genre theory, activity theory, and the notion of Communities of Practice, this study explores a pedagogical genre at play in university mathematics lecture classrooms. The genre is mediational in that it is a tool e...
Contexts in source publication
Context 1
... professors teaching in languages other than their L1, lecture notes often are of a multilingual nature, reflecting professors' linguistic and educational experiences. Figure 2 depicts a page from lecture notes prepared by a professor whose L1 is Russian, L2 is English, and language of instruction in this case is Hebrew. Figure 2 illustrates the cross-linguistic (i.e., English, Russian, mathematical notation); multilayered (i.e., mathematics to be articulated in the lecture both in the equation and graphic forms, comments to himself, comments about what the students need to know and do); multicolored (emphasis); multidimensional (i.e., both the process-see crossing out in Figure 2-and the outcome) nature of this text. ...
Context 2
... 2 depicts a page from lecture notes prepared by a professor whose L1 is Russian, L2 is English, and language of instruction in this case is Hebrew. Figure 2 illustrates the cross-linguistic (i.e., English, Russian, mathematical notation); multilayered (i.e., mathematics to be articulated in the lecture both in the equation and graphic forms, comments to himself, comments about what the students need to know and do); multicolored (emphasis); multidimensional (i.e., both the process-see crossing out in Figure 2-and the outcome) nature of this text. ...
Context 3
... 2 depicts a page from lecture notes prepared by a professor whose L1 is Russian, L2 is English, and language of instruction in this case is Hebrew. Figure 2 illustrates the cross-linguistic (i.e., English, Russian, mathematical notation); multilayered (i.e., mathematics to be articulated in the lecture both in the equation and graphic forms, comments to himself, comments about what the students need to know and do); multicolored (emphasis); multidimensional (i.e., both the process-see crossing out in Figure 2-and the outcome) nature of this text. ...
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Citations
... Because lectures are the usual teaching format for mathematics at universities (e.g. Artemeva & Fox, 2011;Pritchard, 2015;Weinberg et al., 2016), many researchers attributed high dropout rates from mathematics study programmes to shortcomings of lectures (e.g. Fritze & Nordkvelle, 2003). ...
... Because German and international undergraduate mathematics courses during the study entrance phase usually consist of lectures (e.g. Artemeva & Fox, 2011;Pritchard, 2015;Weinberg et al., 2016), we take a closer look at the micro-level-quality of advanced mathematics lectures and investigate their process quality (good teaching). ...
... Advanced mathematics lectures are usually in a 'definition-theorem-proof' (Weber, 2004, p. 116) format. Moreover, they usually consist of 'chalk talk' (Artemeva & Fox, 2011): a lecturer writes mathematical content on the board or screen and makes some oral comments concerning this content. ...
This paper reports a study investigating the quality of advanced mathematics lectures following three goals. First, we present our conceptualisation and operationalisation concerning the quality of mathematics lectures. Second, we test the reliability of a structured observation tool that is based on the aforementioned conceptuali-sation. Third, we describe the quality of mathematics lectures based on data collected with the observation tool and illustrate its use. The results show that our observation tool is suitable for collecting data about characteristics of mathematics lectures observed on-site and from video recordings. The analysis of the data shows that our findings are in line with previous research: formal aspects of mathematical statements are presented in verbal and written forms mostly correctly; informal aspects of mathematical statements (such as examples, visual or mental representations) are, if they are presented , usually presented only verbally. Furthermore, students have the opportunity to participate during lectures, but there are seldom presentations of alternative solutions, examples or proofs and only a few opportunities for formative assessment. Finally, we discuss implications for future research.
... Alsina, 2001;Jaworski et al., 2017;Mesa et al., 2014). A usual organization is for the lecturer to deliver a spoken-written presentation for the students at the blackboard in a chalk-talk manner (Artemeva & Fox, 2011;. Students are normally familiar and comfortable with the lecturing style of teaching and appreciate a classroom organized for lectures (Love et al., 2014). ...
... During live lectures there are more digressions than in videos due to questions and supplementary information. Artemeva and Fox (2011) point to the possibility of talking to and with students during lectures. Videos are more straight to the point. ...
The present paper holds a twofold perspective on a new teaching format in a linear algebra course given to engineering students. The design resembles the flipped classroom, but with various adjustments. The first part highlights the basis on which the design was built, both research results and practical issues, but also the implementation of the teaching. It did not proceed quite as expected. The second part focuses on the outcome. It concentrates on students’ feedback on learning in the course. To inform this part, answers to a questionnaire is related to results from a previous linear algebra course that was taught traditionally, and to students’ interpretation of achievements. Drawing both on the analyses of data and research literature, an adjusted teaching format is suggested to better facilitate for learning. This contributes to knowledge on how research-based teaching arrangements may develop.
... To lay the groundwork for this study, we first highlight prior work on conceptual understanding and instruction in proof-based mathematics courses. In particular, instructors often model their own mathematical activity (Fukawa-Connelly, 2012;Fukawa-Connelly et al., 2017;Pinto, 2019), typically by providing running commentary of their thinking as they teach (Artemeva and Fox, 2011). This running commentary frequently consists of formal statements (reading what is written on the board) and informal ways of thinking, both of the content and meta-commentary on how they do mathematics. ...
... Some other studies have explored classroom presentations of mathematical content that also included the presentation of definitions in proof-based courses. Artemeva andFox (2011), Fukawa-Connelly et al. (2017), Fukawa-Connelly and Newton (2014), and Paoletti et al. (2018) all described definition presentation and suggest broad commonalities in presentation. Broadly, the construct of "chalk talk" includes presenting mathematical content along with narration of more informal ways of thinking and meta-mathematical claims (Artemeva and Fox, 2011). ...
... Artemeva andFox (2011), Fukawa-Connelly et al. (2017), Fukawa-Connelly and Newton (2014), and Paoletti et al. (2018) all described definition presentation and suggest broad commonalities in presentation. Broadly, the construct of "chalk talk" includes presenting mathematical content along with narration of more informal ways of thinking and meta-mathematical claims (Artemeva and Fox, 2011). The study of Fukawa-Connelly et al. (2017) of 11 proof-based lecturers suggested that these informal ways of thinking are common and include both examples (typically written) and informal and metaphorical ways of understanding content (typically said aloud and not written). ...
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.
... Another finding worthy of attention which is not present in Iannone, Czichowsky and Ruf (2020) is that students recognise the importance of the oral comments that the lecturer makes during a lecture to be successful in the oral assessment, and report focusing not only on what is written on the board during the lectures but also on what the lecturer says. This is very important for university mathematics -a subject still taught very traditionally in what Artemeva and Fox (2011) call a 'chalk-talk' format (preferred also in the Czech Republic). In a traditional university mathematics lecture, the lecturer writes formal mathematics content on the board and then gives an oral commentary to the students which will often include remarks about the methods used, links to other parts of the syllabus and so on (Weber, 2004). ...
Acknowledgments We would like to thank Kristýna Nižňanská, David Zenkl and Pavel Sovič for collecting the interview data and help with their analysis. We thank Adrian Simpson for his help with the statistical analysis. Finally, we would like to thank the students who took part in the study for giving up their time to engage with our project. Ethics statement The project was approved by the Committee for Ethics in Research of the Faculty of Education, Charles University, Prague, application number KEV2003. In line with BERA requirements participants agreed with the points in the informed consent forms which describe the use of the data. Names have been changed to protect anonymity. Abstract This paper reports on a qualitative replication study investigating the impact of the novelty effect on findings from interventions about assessment of mathematics at university. The replication study used the same data collection tools of a previous study on oral assessment of mathematics, but data were collected in a context where oral assessment is the norm. We aimed to find whether the results of the two studies were comparable and whether there was plausible evidence of an impact of novelty effect on the findings of the original study. Findings of the current study appear to be comparable to those of the original study. Students associate oral assessment to assessment of conceptual understanding and written assessment to assessment of procedures; they report being more anxious about the oral assessment, but they perceive oral assessment as a better learning experience than closed book exams. However, in a culture where oral assessment is the norm, we found students engaging with learning also following considerations of the difficulties of other modules taken in the same period of their degree. Finally, in this culture, oral communication of mathematics is also much valued.
The whole article is here: https://rdcu.be/dvTV6
... Therefore, time pressures in preparation reduce the number of appropriate teaching methods generated and the process of assessing the utility of each method is curtailed or might be excluded altogether if teachers lock onto a single strategy. The strategy that is most used in the teaching of mathematics means that the 'chalk and talk' strategy the frequently implemented [87,88]. The 'chalk and talk' strategy is not conducive to technology integration as it is a direct instruction method of teaching [89,90]. ...
... As anticipated by Artemeva and Fox [87], participants claimed that the 'chalk and talk' method is the quickest strategy to prepare lessons. However, they said that it is not conducive to digital technology use due to its reliance on non-digital technologies such as pens, paper, and whiteboards. ...
... Cameron and Daniel noted that, in senior classes, learning was more like a lecture than anything else, as "there is just so much to cover" (Daniel). This issue is commonly faced by teachers of mathematics, where the burden of heavy content restricts the possible pedagogical options to lecture style lessons [87], because there is "too much to cover with too little time" ( [59]: 361). ...
Over the past 30 years, teachers have been urged to increase their use of digital technology in the classroom. However, mathematics teachers have been slow to integrate ICT, even though mathematics is naturally aligned with technology. While researchers have documented a variety of time and other related factors that contribute to this resistance, there has been little in-depth analysis of teacher reasoning that inhibits technology integration in mathematics. This article presents four case studies of secondary mathematics teachers employed in Australian schools that investigates the adverse effects of time pressures in not only inhibiting a teacher’s desire to use technology but removing as an option altogether. Data was collected in the form of interviews, lesson planning documentation and notes from observation lessons. Thematic analysis was used to determine how time pressures inhibited participants ability to use technology in their pedagogy. Three time-related obstacles were identified. The first was a lack of time to prepare lessons, the second was content-laden syllabuses and finally, the need to prepare students for traditional assessments. Participants claimed that these obstacles often proved too great to overcome, causing them to abandon any use of technology. This article argues that when the obstacles to technology integration are perceived as too difficult to overcome, it is not enough to provide poorly targeted professional learning or encouragement to work harder to integrate technology. Rather, existing time pressures must be alleviated in terms of workload and syllabus demands if we want to remove the inhibitors to technology integration in mathematics.
... Paoletti et al. (2018, p. 2) state that mathematics lectures "have been unsuccessful in promoting student learning". Nevertheless, lectures are still the most common teaching mode at many universities (e.g., Artemeva & Fox, 2011). Despite the critical responses to lectures, Fritze and Nordkvelle (2003, p. 328) say: "[T]he lecture survives, probably because it serves many functions not so well observed in the present research". ...
... Weber (2004, p. 116) describes mathematics lectures as a "definition-theorem-proof" format. Moreover, mathematics lectures usually consist of "chalk talk" (Artemeva & Fox, 2011): a lecturer writes mathematical content on the board or a screen and makes some oral comments concerning this content. In this paper, we present results of an observation study concerning the presentation of theorems and proofs in mathematics lectures -as a possible quality criterion -supported by a structured observation protocol. ...
... Kiendl-Wendner (2016) states that university quality can be characterized according to the following features: forms including macro level (whole university) and micro level (single courses), and dimensions including process quality (clearly defined processes with standardized procedures), result quality (achievement of the goals set) and structural quality (adequacy of resource allocation). Because German and international undergraduate mathematics courses usually consist of lectures (e.g., Pritchard, 2015;Artemeva & Fox, 2011), we take a closer look at the micro-level-quality of mathematics lectures and investigate their process quality. ...
This paper reports a study investigating the quality of advanced mathematics lectures, in particular regarding the presentation of theorems and proofs. We compare the presentation of theorems and proofs given in two real analysis courses by two different lecturers using a structured observation tool. The results show, that informal parts of theorems and proofs were underrepresented in both lecture courses. Nevertheless, we could identify differences in the ways theorems and related proofs were presented with regard to different phases of the processes of proof construction. Thus, students might have developed different proving skills as well as different understanding of proof construction and methods of proving by both lecturers. Finally, we discuss implications for future investigations based on our results.
... According to students, the long presentation with high technology caused less communication between a lecturer and students. The 'Chalk and Talk' approach, that is writing out a mathematical narrative on the board while talking aloud, (Artemeva & Fox, 2011) was recommended. ...
This article is a case study that examines how the different teaching and learning strategies influence mathematical knowledge acquisition at university. The research hinged on how mathematics lecturers at universities teach and how students acquire the disseminated knowledge. Research stresses that most lecturers have teacher-centred teaching approaches and mathematics teaching is with minimal student participation. Research further asserts that teaching approaches that emphasise student participation is critical in enhancing effective classroom interaction. This means that, as students’ learning strategy is dominantly a participatory and collaborative one, lecturers are challenged to create a learning-as-participation environment for effective mathematics classroom interaction. The article reports a study that was conducted among university students and lecturers in a mathematics course at a South African university. The findings were that most lecturers at the university use traditional non-interactive teaching approaches that create passive environments in contrast with the predominant participative learning strategies of most students.
... Lecturing is the most researched mode of instruction in proof-based courses (e.g., Artemeva & Fox, 2011;Fukawa-Connelly et al., 2017;Johnson et al., 2018). In the context of real analysis, Weber (2004) characterized different styles of proof presentation of a single teacher-mathematician 1 (specifically, logico-structural, procedural, and semantic). ...
... The imperative of lecturing as a means for modelling has not been confined to proof or to advanced mathematics courses (e.g., Artemeva & Fox, 2011;Pritchard, 2010). For example, in the context of functions, Viirman (2021) explored how seven teacher-mathematicians introduced new mathematical objects and explained what counts as a valid mathematical argument. ...
Coming from a social perspective, we introduce a classroom organizational frame, where students’ proofs progress from collaborative construction in small groups, through whole-class presentation at the board by one of the constructors, to a posteriori reflection. This design is informed by a view on proofs as successive social processes in the mathematics community. To illustrate opportunities for mathematics learning of proof progressions, we present a commognitive analysis of a single proof from a small course in topology. The analysis illuminates the processes through which students’ proof was restructured, developed previously unarticulated elements, and became more formal and elaborate. Within this progression, the provers developed their mathematical discourses and the course teacher seized valuable teachable moments. The findings are discussed in relation to key themes within the social perspective on proof.
... Traditional 'chalk talk' still prevails in tertiary mathematics instruction (Fukawa-Connelly et al., 2016;Woods & Weber, 2020;Artemeva & Fox, 2011) report an international study on teaching tertiary mathematics interviewing 45 lecturers. All interviewees used 'chalk talk' and accounted for it as a way to expose students to the process of 'doing mathematics' and introduce them to disciplinary thinking. ...
... These 'lists' of pedagogical considerations accumulate, complete each other, and reflect a consistency that may stem from a stable and shared basis of practices, values, and norms -a pedagogical culture. In fact, as Artemeva and Fox (2011) demonstrated, mathematics lecturers in many countries share many pedagogical similarities, for example the prevailing pedagogy of 'chalk talk' in the undergraduate mathematics lecture where lecturers "verbalize everything they write on the board (running commentary) and talk about what they write on the board (metacommentary)" (Artemeva & Fox, 2011, p. 355). Our methodological approach was based on intentionally triggering the general discussion by exploring a concrete problem (TPP) and comparing two proofs; the comparison prompted the lecturers to attribute relative value to different features and revealed three pedagogical issues that we wish to highlight. ...
We propose the ‘flow of a proof’ as a construct that relates to the balance lecturers achieve in their presentation between the proof as a whole and its different parts, taking into account various aspects (informal, contextual) of proof classroom presentation. We conducted expert interviews in which we presented two different proofs of the same theorem to five experienced tertiary level mathematics lecturers and asked them to reflect on features of these proofs and of proofs in general, and on their considerations and practices when teaching proofs. The lecturers related to a rich collection of proof features and pedagogical considerations. Our findings demonstrate the subjectivity of lecturers’ considerations as well as tensions they experience between striving towards mathematical accuracy and adjusting the proof presentation to their students; these tensions influence the flow of the proof, which is an outcome of the choices that the lecturers make. We discuss our findings in light of existing literature about mathematicians’ pedagogical considerations and the existence of a shared pedagogical culture.
... In traditional mathematics lectures, the instructor usually writes the definitions, theorems, and proofs on the board, and provides additional oral explanations (Artemeva & Fox, 2011). In such lectures, students are often busy copying everything correctly, and have trouble paying attention to the instructor's explanations (Freitag, 2020). ...
One approach to address the problem that in mathematics lectures students are often busy writing, and cannot pay attention to the instructor's oral explanations might be the use of guided notes. These are a modified version of the instructor's notes that contain blanks the students have to fill in during the lecture. However, the extent to which the use of guided notes actually supports students during their note-taking probably depends on the positions of the blanks. We investigated students' perspectives on where to put the blanks by exploring in which elements of a mathematics lecture they appreciate blanks and why. The results yield some suggestions on where to put blanks and preprinted parts in a guided notes script, so that many students might benefit from guided notes for their note-taking in mathematics lectures.
