Personal Excursions: Investigating the Dynamics of Student Engagement

We investigate the dynamics of student engagement as it is manifest in self-directed, self-motivated, relatively long-term, computer-based scientific image processing activities. The raw data for the study are video records of 19 students, grades 7 to 11, who participated in intensive 6-week, extension summer courses. From this raw data we select episodes in which students appear to be highly engaged with the subject matter. We then attend to the fine-grained texture of students’ actions, identifying a core set of phenomena that cut across engagement episodes. Analyzed as a whole, these phenomena suggest that when working in self-directed, self-motivated mode, students pursue proposed activities but sporadically and spontaneously venture into self-initiated activities. Students’ recurring self-initiated activities – which we call personal excursions – are detours from proposed activities, but which align to a greater or lesser extent with the goals of such activities. Because of the deeply personal nature of excursions, they often result in students collecting resources that feed back into both subsequent excursions and framed activities. Having developed an understanding of students’ patterns of self-directed, self-motivated engagement, we then identify four factors that seem to bear most strongly on such patterns: (1) students’ competence (broadly construed); (2) features of the software-based activities, and how such features allowed students to express their competence; (3) the time allotted for students to pursue proposed activities, as well as self-initiated ones; and (4) the flexibility of the computational environment within which the activities were implemented.     

Investigating undergraduate students’ proof schemes and perspectives about combinatorial proof

Combinatorics is an area of mathematics with accessible, rich problems and applications in a variety of fields. Combinatorial proof is an important topic within combinatorics that has received relatively little attention within the mathematics education community, and there is much to investigate about how students reason about and engage with combinatorial proof. In this paper, we use Harel and Sowder’s (1998) proof schemes to investigate ways that students may characterize combinatorial proofs as different from other types of proof. We gave five upper-division mathematics students combinatorial-proof tasks and asked them to reflect on their activity and combinatorial proof more generally. We found that the students used several of Harel and Sowder’s proof schemes to characterize combinatorial proof, and we discuss whether and how other proof schemes may emerge for students engaging in combinatorial proof. We conclude by discussing implications and avenues for future research.     

Proof schemes combined: mapping secondary students’ multi-faceted and evolving first encounters with mathematical proof

In this paper, we propose an enriched and extended application of Harel and Sowder’s proof schemes taxonomy that can be used as a diagnostic tool for characterizing secondary students’ emergent learning of proof and proving. We illustrate this application in the analysis of data collected from 85 Year 9 (age 14–15) secondary students. We capture these students’ first encounters with proof and proving in an educational context (mixed ability, state schools in Greece) where mathematical proof is explicitly present in algebra and geometry lessons and where proving skills are typically expected, and rewarded, in key national examinations. We analyze student written responses to six questions, soon after the students had been introduced to proof and we identify evidence of six of the seven proof schemes proposed by Harel and Sowder as well as a further eight combinations of the six. We observed these combinations often within the response of the same student and to the same item. Here, we illustrate the eight combinations and we claim that a dynamic use of the proof schemes taxonomy that encompasses sole and combined proof schemes is a potent theoretical and pedagogical tool for mapping students’ multi-faceted and evolving competence in, and appreciation for, proof and proving.     

Capturing student mathematical engagement through differently enacted classroom practices: applying a modification of Watson's analytical tool

This study examined student mathematical engagement through the intended and enacted lessons taught by two teachers in two different middle schools in Indonesia. The intended lesson was developed using the ELPSA learning design to promote mathematical engagement. Based on the premise that students will react to the mathematical tasks in the forms of words and actions, the analysis focused on identifying the types of mathematical engagement promoted through the intended lesson and performed by students during the lesson. Using modified Watson's analytical tool (2007), students’ engagement was captured from what the participants’ did or said mathematically. We found that teachers’ enacted practices had an influence on student mathematical engagement. The teacher who demonstrated content in explicit ways tended to limit the richness of the engagement; whereas the teacher who presented activities in an open-ended manner fostered engagement.     

Are self-constructed and student-generated arguments acceptable proofs? Pre-service secondary mathematics teachers’ evaluations

This study examined 14 pre-service secondary mathematics teachers’ productions and their evaluations of self-constructed and student-generated arguments in the domains of algebra, geometry, and number theory. Pre-service secondary mathematics teachers’ (PSMTs) evaluations of their own arguments indicate if they considered self-productions as proofs from a learner perspective. Similarly, PSMTs’ evaluations of student-generated arguments indicate if they decided given students’ productions could be counted as proofs from a teacher perspective. Our results show that the majority of PSMTs suspected that their invalid productions did not qualify as proofs. Furthermore, the PSMTs who were confident with their work and claimed that they had constructed a proof were more likely to make a correct judgment on four of the six student-generated arguments. We discuss implications of these findings for supporting PSMTs’ learning of proof and future research on the construction-evaluation activity.     

Schülerprobleme beim Lösen von geometrischen Beweisaufgaben —eine Interviewstudie—

In this article we report on an interview study involving ten grade 8 students. These interviews served as a qualitative supplement for a large-scale quantitative study on proof and argumentation (N=659). During videotaped interviews the students were asked to solve geometrical proof problems. The results indicate that students’ difficulties with proof and logical argumentation may be explained by insufficient knowledge of facts, deficits in their methodological knoledge about mathematical proofs, and a lack of knowledge with respect to developing and implementing a proof strategy. Low-achieving students show difficulties with respect to all these three aspects, whereas high-achieving students’ difficulties are mainly based on deficits of developing an adequate and correct proof strategy.     

Understanding authority in small-group co-constructions of mathematical proof

Authority becomes shared in mathematics classrooms when perceived sources of valid mathematical knowledge extend beyond the teacher/textbook and allow both students and disciplinary modes of reasoning to hold authority. The goal of this research is to better understand classroom situations that are intended to facilitate shared authority over proof, namely small-group episodes where students are granted authority (Gerson & Bateman, 2010) to co-construct mathematical proofs. We sought to better understand the content of undergraduate students’ attention during group proving and the sources of legitimacy for students. Using Stylianides’ (2007) definition of proof as an analytical frame, we found that student discourse focused primarily upon the mode of argumentation, followed by the mode of argument representation, and then the set of accepted statements. We identified four themes with respect to the sources of authority students relied upon in their group proving: (1) the course rubric, (2) peers’ confidence, (3) form and symbols, and (4) logical structure. Implications for research and practice are presented.     

Instructional identities of geometry students

We inspect the hypothesis that geometry students may be oriented toward how they expect that the teacher will evaluate them as students or otherwise oriented to how they expect that their work will give them opportunities to do mathematics. The results reported here are based on a mixed-methods analysis of twenty-two interviews with high school geometry students. In these interviews students respond to three different tasks that presented students with an opportunity to do a proof. Students’ responses are coded according to a scheme based on the hypothesis above. Interviews are also coded using a quantitative linguistic ratio that gauges how prominent the teacher was in the students’ opinions about the viability of these proof tasks. These scores were used in a cluster analysis that yielded three student profiles that we characterize using composite profiles. These profiles highlight the different ways that students can experience proof in the geometry classroom.     

A survey of mathematics undergraduates' interaction with proof: some implications for mathematics education

Proving is an essential activity in mathematics but there are serious difficulties encountered by mathematics undergraduates in engaging with proof in the intended way. This article presents an initial analysis of (i) a quantitative study of a large sample of UK mathematics undergraduates which describes their declared perceptions about proof, and (ii) a qualitative study of a subsample of these students which analyses their actual proof perceptions as well as their actual proof practices. A comparison is also made between their publicly declared perceptions of proof and their personal proclivities in proving.     

Constructs of engagement emerging in an ethnomathematically-based teacher education course

This paper presents a case study, in which we apply and develop theoretical constructs to analyze motivating desires observed in an unconventional, culturally contextualized teacher education course. Participants, Israeli students from several different cultures and backgrounds (pre-service and in-service teachers, Arabs and Jews, religious and secular) together studied geometry through inquiry into geometric ornaments drawn from diverse cultures, and acquired knowledge and skills in multicultural education. To analyze affective behaviors in the course we applied the methodology of engagement structures recently proposed by Goldin and his colleagues. Our study showed that engagement structures were a powerful tool for examining motivating desires of the students. We found that the constructivist ethnomathematical approach highlighted the structures that matched our instructional goals and diminished those related to students’ feelings of dissatisfaction and inequity. We propose a new engagement structure “Acknowledge my culture” to embody motivating desires, arising from multicultural interactions, that foster mathematical learning.     

Divisibility tests for low-valued primes and their use as generators of simple proofs

This article considers a family of divisibility tests for low-valued primes and their place as activities which develop the use of proof both within school and in teacher training. The way in which these tests can be seen as developing one from another is seen as of value in encouraging students to produce their own proofs.     

Supporting disciplinary and interdisciplinary knowledge development and design thinking in an informal,pre‐engineering program: A Workplace Simulation Project

In this paper, we examined students’ engagement in an implementation of a Workplace Simulation Project (WSP). The WSP was designed to actively engage students in learning disciplinary content by inviting engineers from industry to have a physical presence within the school building to collaborate with teachers and students to complete projects which simulate the tasks authentic to their work. We focus on the first year implementation of the program that partnered a high school in the rural Midwest with an engineering unit of a government organization. Using a multiple methods study design, we analyzed disciplinary and interdisciplinary pre and posts test along with students’ interviews to determine learning gains as well as students’ interpretations of creative and critical thinking as experienced in the project and their knowledge of the engineering design process. Effect sizes showed that students in the WSP group had notable gains over the control group participants. Additionally, students’ knowledge of core elements of the design process were identified in inductive analyses of the interviews. Findings from this study will provide usable knowledge about effective ways to support systems and design thinking and ways to support expert‐novice collaboration to ensure success.     

Cognitive trajectory of proof by contradiction for transition-to-proof students

History and research on proof by contradiction suggests proof by contradiction is difficult for students in a number of ways. Students’ comprehension of already-written proofs by contradiction is one such aspect that has received relatively little attention. Applying the cognitive lens of Action-Process-Object-Schema (APOS) Theory to proof by contradiction, we constructed and tested a cognitive model that describes how a student might construct the concept ‘proof by contradiction’ in an introduction to proof course. Data for this study was collected from students in a series of five teaching interventions focused on proof by contradiction. This paper will report on two participants as case studies to illustrate that our cognitive trajectory for proof by contradiction is a useful model for describing how students may come to understand the proof method.     

Two proving strategies of highly successful mathematics majors

We examined the proof-writing behaviors of six highly successful mathematics majors on novel proving tasks in calculus. We found two approaches that these students used to write proofs, which we termed the targeted strategy and the shotgun strategy. When using a targeted strategy students would develop a strong understanding of the statement they were proving, choose a plan based on this understanding, develop a graphical argument for why the statement is true, and formalize this graphical argument into a proof. When using a shotgun strategy, students would begin trying different proof plans immediately after reading the statement and would abandon a plan at the first sign of difficulty. The identification of these two strategies adds to the literature on proving by informing how elements of existing problem-solving models interrelate.     

How some research mathematicians and statisticians use proof in undergraduate mathematics

The paper examines the roles and purposes of proof mentioned by university research faculty when reflecting on their own teaching and teaching at their institutions. Interview responses from 14 research mathematicians and statisticians who also teach are reported. The results suggest there is a great deal of variation in the role and purpose of proof in and among mathematics courses and that factors such as the course title, audience, and instructor influence this variation. The results also suggest that, for this diverse group, learning how to prove theorems is the most prominent role of proof in upper division undergraduate mathematics courses and that this training is considered preparation for graduate mathematics studies. Absent were responses discussing proof's role in preparing K-12 mathematics teachers. Implications for a proof and proving landscape for school mathematics are discussed.     

Using Culturally Responsive Stories in Mathematics: Responses From the Target Audience

This study examined how Black students responded to the utilization of culturally responsive stories in their mathematics class. All students in the two classes participated in mathematics lessons that began with an African American story (culturally responsive to this population), followed by mathematical discussion and concluded with solving problems that correlated to the story. The researcher observed and recorded responses by students during each part of these lessons with protocols. Students independently reflected weekly by answering five questions to share their perspective on the African American stories. The teacher reflected on each lesson as well, describing thoughts on how these students responded to the story in each lesson. This paper examines the analyzed data from the target audience: Black students. Results revealed that Black students responded to the use of African American stories with high self‐rated levels of engagement and enjoyment and that the stories helped them think about mathematics to varying degrees. Since students who are engaged and are thinking about mathematics are more likely to achieve mathematical understanding, the researcher concludes that this strategy should continue to be tested in diverse classrooms with an emphasis on student reflection to determine if the outcomes are transferable and generalizable.     

Impacts of inquiry pedagogy on undergraduate students conceptions of the function of proof

Mathematicians and mathematics educators agree that proof is an important tool in mathematics, yet too often undergraduate students see proof as a superficial part of the discipline. While proof is often used by mathematicians to justify that a theorem is true, many times proof is used for another purpose entirely such as to explain why a particular statement is true or to show mathematics students a particular proof technique. This paper reports on a study that used a form of inquiry-based learning (IBL) in an introduction to proof course and measured the beliefs of students in this course about the different functions of proof in mathematics as compared to students in a non-IBL course. It was found that undergraduate students in an introduction to proof course had a more robust understanding of the functions of proof than previous studies would suggest. Additionally, students in the course taught using inquiry pedagogy were more likely to appreciate the communication, intellectual challenge, and providing autonomy functions of proof. It is hypothesized that these results are a response to the pedagogy of the course and the types of student activity that were emphasized.     

It's not the done thing: social norms governing students’ passive behaviour in undergraduate mathematics lectures

Students often play a passive role in large-scale lectures in undergraduate mathematics courses: they observe the lecturer demonstrate mathematical procedures, but they rarely engage in authentic mathematical activity themselves. This study uses semi-structured interviews of undergraduate students to investigate the implicit and explicit social norms and expectations that influence students to maintain their passive roles during lectures. Students were aware that their passivity was influenced by social norms, but perceived these norms as necessary for allowing the lecturer to get through the content in the allotted lecture time, while enabling students to avoid being publicly embarrassed in the lecture. However, the students appreciated opportunities to work on examples in small groups during lectures. We argue that the success of small group interactions during large-scale lectures depends on students and lecturers establishing supportive social norms, and adjusting their lecture goals from ‘covering the content’ to ‘developing mathematical understanding’.