What impact, if any, do interesting lessons have on the types of questions students ask? To explore this question, we used lesson observations of six teachers from three high schools in the Northeast who were part of a larger study. Lessons come from a range of courses, spanning Algebra through Calculus. After each lesson, students reported interest via lesson experience surveys (Author, 2019). These interest measures were then used to identify each teachers’ highest and lowest interest lessons. The two lessons per teacher allows us to compare across the same set of students per teacher.
We compiled 145 student questions and identified whether questions were asked within a group work setting or part of a whole class discussion. Two coders coded 10% of data to improve the rubric for type of students’ questions (what, why, how, and if) and perceived intent (factual, procedural, reasoning, and exploratory). Factual questions asked for definitions or explicit answers. Procedural questions were raised when students looked for algorithms or a solving process. Reasoning questions asked about why procedures worked, or facts were true. Exploratory questions expanded beyond the topic of focus, such as asking about changing the parameters to make sense of a problem. The remaining 90% of data were coded independently to determine interrater reliability (see Landis & Koch, 1977). A Cohen’s Kappa statistic (K=0.87, p<0.001) indicates excellent reliability. Furthermore, both coders reconciled codes before continuing with data analysis.
Initial results showed differences between high- and low-interest lessons. Although students raised fewer mathematical questions in high-interest lessons (59) when compared with low-interest lessons (86), high-interest lessons contained more “exploratory” questions (10 versus 6). A chi-square test of independence shows a significant difference, χ2 (3, N = 145) = 12.99, p = .005 for types of students’ questions asked in high- and low-interest lessons. The high-interest lessons had more student questions arise during whole class discussions, whereas low-interest lessons had more student questions during group work. By partitioning each lesson into acts at points where the mathematical content shifted, we were able to examine through how many acts questions remained open. The average number of acts the students’ questions remained unanswered for high-interest lessons (2.66) was higher than that of low-interest lessons (1.68). Paired samples t-tests suggest that this difference is significant t(5)=2.58, p = 0.049.
Therefore, student interest in the lesson did appear to impact the type of questions students ask. One possible reason for the differences in student questions is the nature of the lessons students found interesting, which may allow for student freedom to wonder and chase their mathematical ideas. There may be more overall student questions in low-interest lessons because of confusion, but more research is needed to unpack the reasoning behind student questions. more »« less

Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education

Singh, R.; Nieves, H. I.; Barno, E.; & Dietiker, L.(
, Proceedings of the Psychology of Mathematics Education - North American Chapter)

Olanoff, D.; Johnson, K.; & Spitzer, S.
(Ed.)

How does the design of lessons impact the types of questions teachers and students ask during enacted high school mathematics lessons? In this study, we present data that suggests that lessons designed with the mathematical story framework to elicit a specific aesthetic response (“MCLEs”) having a positive influence on the types of teacher and student questions they ask during the lesson. Our findings suggest that when teachers plan and enact lessons with the mathematical story framework, teachers and students are more likely to ask questions that explore mathematical relationships and focus on meaning making. In addition, teachers are less likely to ask short recall or procedural questions in MCLEs. These findings point to the role of lesson design in the quality of questions asked by teachers and students.

Dietiker, Leslie; Singh, Rashmi; Riling, Meghan; Nieves, Hector I.(
, Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)

Sacristán, A. I.; Cortés-Zavala, J. C.; Ruiz-Arias, P. M.
(Ed.)

How can we design mathematical lessons that spark student interest? To answer this, we analyzed teacher-designed and enacted lessons that students described as interesting for how the content unfolded. When compared to those the same students describe as uninteresting, multiple distinguishing characteristics are evident, such as the presence of misdirection, mathematical questions that remain unanswered for extended time, and a greater number of questions that are unanswered at each point of the lesson. Low-interest lessons did not contain many special narrative features and mostly had questions that were answered immediately. Our findings offer guidance for the design of lessons that can shift student mathematical dispositions.

Riling, M.(
, Proceedings of the 40th annual meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education)

As computer-focused policies and trends become more popular in schools, more students access math curriculum online. While computer-based programs may be responsive to some student input, their algorithmic basis can make it more difficult for them to be prepared for divergent student thinking, especially in comparison to a teacher. Consider programs that assess student work by judging how well it matches pre-set answers. Unless designed and enacted in classrooms with care, computer-based curriculum materials might encourage students to think about mathematics in pre-determined ways. How do students approach the process of mathematics while using online materials, especially in terms of engaging in original thought?
Drawing on Pickering’s (1995) dance of agency and Sinclair’s (2001) conception of students as path-finders or track-takers, I define two modes of mathematical behavior: trail-taking and bushwhacking. While trail-taking, students follow an established approach, often relying on Pickering’s (1995) disciplinary agency, wherein the mathematics “leads [them] through a series of manipulations” (p. 115). The series of manipulations can be seen as a trail that a student may choose to follow. Bushwhacking, on the other hand, refers to actions a student takes of their own invention. It is possible that, unknown to the student, these actions have been taken before by others. In bushwhacking, the student possesses agency, which Pickering (1995) describes as active (rather than passive) and as hallmarked by “choice and discretion” (p. 117).
In this study, students worked in several dynamic geometric environments (DGEs) during a geometry lesson about the midline theorem. The lesson was originally recorded as part of a larger study designing mathematically captivating lessons. Students accessed both problems and online addresses for corresponding DGEs via a printed packet. Students interacted with the DGEs on individual laptops, but were seated in groups of three or four. Passages of group conversations in which students transitioned between trail-taking and bushwhacking were selected for closer analysis, which involved identifying evidence of each mode and highlighting the curricular or social forces that may have contributed to shifts between modes.
Of particular interest were episodes in which students asked one another to share results, which led to students reconsidering previously set approaches, and episodes in which students interacted with DGEs containing a relatively high proportion of drag-able components, which corresponded to some students working in bushwhacking mode, spontaneously suggesting and revising approaches for manipulating the DGE (e.g., “unless you make this parallel to the bottom, but I don’t think you... yes you can.”). Both types of episodes were found in multiple groups’ conversations. Further analysis of student interactions with tasks, especially with varying levels of student control and sharing, could serve to inform future computer-based task design aimed to encourage students to productively engage in bushwhacking while problem-solving.

Weingarden, M.; Buchbinder, O.(
, Proceedings of the Annual Conference on Research in Undergraduate Mathematics Education)

Karunakaran, S. S.; Higgins, A.
(Ed.)

The critical role of teachers in supporting student engagement with reasoning and proving has long been recognized (Nardi & Knuth, 2017; NCTM, 2014). While some studies examined how prospective secondary teachers (PSTs) develop dispositions and teaching practices that promote student engagement with reasoning and proving (e.g., Buchbinder & McCrone, 2020; Conner, 2007), very little is known about long-term development of proof-related practices of beginning teachers and what factors affect this development (Stylianides et al., 2017). During the supervised teaching experiences, interns often encounter tensions between balancing their commitments to the university and cooperating teacher, while also developing their own teaching styles (Bieda et al., 2015; Smagorinsky et al., 2004; Wang et al., 2008).
Our study examines how sociocultural contexts of the teacher preparation program and of the internship school, supported or inhibited proof-related teaching practices of beginning secondary mathematics teachers. In particular, this study aims to understand the observed gap between proof-related teaching practices of one such teacher, Olive, in two settings: as a PST in a capstone course Mathematical Reasoning and Proving for Secondary Teachers (Buchbinder & McCrone, 2020) and as an intern in a high-school classroom. We utilize activity theory (Leont’ev, 1979) and Engeström’s (1987) model of an activity system to examine how the various components of the system: teacher (subject), teaching (object), the tasks (tools), the curriculum and the expected teaching style (rules), the cooperating teacher (community) and their involvement during the teaching (division of labor) interact with each other and affect the opportunities provided to students to engage with reasoning and proving (outcome).
The analysis of four lessons from each setting, lesson plans, reflections and interviews, showed that as a PST, Olive engaged students with reasoning and proving through productive proof-related teaching practices and rich tasks that involved conjecturing, justifying, proving and evaluating arguments. In a sharp contrast, as an intern, Olive had to follow her school’s rigid curriculum and expectations, and to adhere to her cooperating teacher’s teaching style. As a result, in her lessons as an intern students received limited opportunities for reasoning and proving. Olive expressed dissatisfaction with this type of teaching and her desire to enact more proof-oriented practices. Our results show that the sociocultural components of the activity system (rules, community and division of labor), which were backgrounded in Olive’s teaching experience as a PST but prominent in her internship experience, influenced the outcome of engaging students with reasoning and proving. We discuss the importance of these sociocultural aspects as we examine how Olive navigated the tensions between the proof-related teaching practices she adopted in the capstone course and her teaching style during the internship. We highlight the importance of teacher educators considering the sociocultural aspects of teaching in supporting beginning teachers developing proof-related teaching practices.

Self, B. P.; Koretsky, M.; Nolen, S. B.; Dal Bello, D. J.; Widmann, J. M.; Prince, M. J.; Papadopoulos, C.(
, 2023 ASEE Annual Conference & Exposition)

Several consensus reports cite a critical need to dramatically increase the number and diversity of STEM graduates over the next decade. They conclude that a change to evidence-based instructional practices, such as concept-based active learning, is needed. Concept-based active learning involves the use of activity-based pedagogies whose primary objectives are to make students value deep conceptual understanding (instead of only factual knowledge) and then to facilitate their development of that understanding. Concept-based active learning has been shown to increase academic engagement and student achievement, to significantly improve student retention in academic programs, and to reduce the performance gap of underrepresented students. Fostering students' mastery of fundamental concepts is central to real world problem solving, including several elements of engineering practice. Unfortunately, simply proving that these instructional practices are more effective than traditional methods for promoting student learning, for increasing retention in academic programs, and for improving ability in professional practice is not enough to ensure widespread pedagogical change. In fact, the biggest challenge to improving STEM education is not the need to develop more effective instructional practices, but to find ways to get faculty to adopt the evidence-based pedagogies that already exist. In this project we seek to propagate the Concept Warehouse, a technological innovation designed to foster concept-based active learning, into Mechanical Engineering (ME) and to study student learning with this tool in five diverse institutional settings. The Concept Warehouse (CW) is a web-based instructional tool that we developed for Chemical Engineering (ChE) faculty. It houses over 3,500 ConcepTests, which are short questions that can rapidly be deployed to engage students in concept-oriented thinking and/or to assess students’ conceptual knowledge, along with more extensive concept-based active learning tools. The CW has grown rapidly during this project and now has over 1,600 faculty accounts and over 37,000 student users. New ConcepTests were created during the current reporting period; the current numbers of questions for Statics, Dynamics, and Mechanics of Materials are 342, 410, and 41, respectively. A detailed review process is in progress, and will continue through the no-cost extension year, to refine question clarity and to identify types of new questions to fill gaps in content coverage. There have been 497 new faculty accounts created after June 30, 2018, and 3,035 unique students have answered these mechanics questions in the CW. We continue to analyze instructor interviews, focusing on 11 cases, all of whom participated in the CW Community of Practice (CoP). For six participants, we were able to compare use of the CW both before and after participating in professional development activities (workshops and/or a community or practice). Interview results have been coded and are currently being analyzed. To examine student learning, we recruited faculty to participate in deploying four common questions in both statics and dynamics. In statics, each instructor agreed to deploy the same four questions (one each for Rigid Body Equilibrium, Trusses, Frames, and Friction) among their overall deployments of the CW. In addition to answering the question, students were also asked to provide a written explanation to explain their reasoning, to rate the confidence of their answers, and to rate the degree to which the questions were clear and promoted deep thinking. The analysis to date has resulted in a Work-In-Progress paper presented at ASEE 2022, reporting a cross-case comparison of two instructors and a Work-In-Progress paper to be presented at ASEE 2023 analyzing students’ metacognitive reflections of concept questions.

Nieves, Hector I., Singh, Rashmi, and Dietiker, Leslie. Student inquiry in interesting lessons. Retrieved from https://par.nsf.gov/biblio/10211151. Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education .

Nieves, Hector I., Singh, Rashmi, & Dietiker, Leslie. Student inquiry in interesting lessons. Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education, (). Retrieved from https://par.nsf.gov/biblio/10211151.

Nieves, Hector I., Singh, Rashmi, and Dietiker, Leslie.
"Student inquiry in interesting lessons". Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education (). Country unknown/Code not available. https://par.nsf.gov/biblio/10211151.

@article{osti_10211151,
place = {Country unknown/Code not available},
title = {Student inquiry in interesting lessons},
url = {https://par.nsf.gov/biblio/10211151},
abstractNote = {What impact, if any, do interesting lessons have on the types of questions students ask? To explore this question, we used lesson observations of six teachers from three high schools in the Northeast who were part of a larger study. Lessons come from a range of courses, spanning Algebra through Calculus. After each lesson, students reported interest via lesson experience surveys (Author, 2019). These interest measures were then used to identify each teachers’ highest and lowest interest lessons. The two lessons per teacher allows us to compare across the same set of students per teacher. We compiled 145 student questions and identified whether questions were asked within a group work setting or part of a whole class discussion. Two coders coded 10% of data to improve the rubric for type of students’ questions (what, why, how, and if) and perceived intent (factual, procedural, reasoning, and exploratory). Factual questions asked for definitions or explicit answers. Procedural questions were raised when students looked for algorithms or a solving process. Reasoning questions asked about why procedures worked, or facts were true. Exploratory questions expanded beyond the topic of focus, such as asking about changing the parameters to make sense of a problem. The remaining 90% of data were coded independently to determine interrater reliability (see Landis & Koch, 1977). A Cohen’s Kappa statistic (K=0.87, p<0.001) indicates excellent reliability. Furthermore, both coders reconciled codes before continuing with data analysis. Initial results showed differences between high- and low-interest lessons. Although students raised fewer mathematical questions in high-interest lessons (59) when compared with low-interest lessons (86), high-interest lessons contained more “exploratory” questions (10 versus 6). A chi-square test of independence shows a significant difference, χ2 (3, N = 145) = 12.99, p = .005 for types of students’ questions asked in high- and low-interest lessons. The high-interest lessons had more student questions arise during whole class discussions, whereas low-interest lessons had more student questions during group work. By partitioning each lesson into acts at points where the mathematical content shifted, we were able to examine through how many acts questions remained open. The average number of acts the students’ questions remained unanswered for high-interest lessons (2.66) was higher than that of low-interest lessons (1.68). Paired samples t-tests suggest that this difference is significant t(5)=2.58, p = 0.049. Therefore, student interest in the lesson did appear to impact the type of questions students ask. One possible reason for the differences in student questions is the nature of the lessons students found interesting, which may allow for student freedom to wonder and chase their mathematical ideas. There may be more overall student questions in low-interest lessons because of confusion, but more research is needed to unpack the reasoning behind student questions.},
journal = {Mathematics Education Across Cultures: Proceedings of the 42nd Meeting of the North American Chapter of the International Group for the Psychology of Mathematics Education},
author = {Nieves, Hector I. and Singh, Rashmi and Dietiker, Leslie},
editor = {Sacristán, A. I. and Cortés-Zavala, J. C. and Ruiz-Arias, P. M.}
}

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