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Why do some mathematics lessons captivate high school students and others not? This study explores this question by comparing how the content unfolds in the lessons that students rated highest with respect to their aesthetic affordances (e.g., using terms like “intriguing,” “surprising”) with those the same students rated lowest with respect to their aesthetic affordances (e.g., “just ok,” “dull”). Using a framework that interprets the unfolding content across a lesson as a mathematical story, we examine how some lessons can provoke curiosity or enable surprise. We identify eight characteristics that distinguish captivating lessons and show how some, such as the average number of questions under consideration at any point in the lesson, are strongly related to student aesthetic experiences. In addition, the lessons that students described as more interesting included more instances of misdirection, such as when students’ false assumptions provide opportunities for surprising results. These findings point to the characteristics of future lesson designs that could enable more students to experience curiosity and wonder in secondary mathematics classrooms. 

This paper explores how a professional learning community (PLC) redesigns high school mathematics lessons towards a shared commitment. We describe the nature of a PLC’s collective curricular vision to illuminate how teachers can come to new understandings as a group in order to shift the ways students experience mathematics. Using the curricular noticing framework (attending, interpreting, and responding), we analyzed the meetings of a PLC with six teachers as they individually presented lessons to be redesigned with a focus on the group’s shared commitment. Findings indicate three ways ideas were introduced that led to expansive responses, which suggests this analytic approach could identify ways in which a PLC can work towards new curricular decisions. 

When mathematics educators work towards making mathematics more relevant, they often think about including more real-world applications into mathematics lessons. But what happens when a lesson is devoid of real-world contexts? In what ways can students find it relevant? This study explores how high school students perceived relevance when they were asked to describe their experiences during decontextualized mathematics lessons. Students highlighted how they found certain characteristics of the lessons to be useful in their learning and how they perceived relevance through different feelings experienced in the lessons. 

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. 

In this study, we explore the relationships between the types of student exclamations in an enacted lesson (e.g., “Wow!”) and the varying dramatic tensions created by the unfolding content. By analyzing student exclamations in six specially-designed high school mathematics lessons, we explore how the dynamic tension between revelations of mathematical ideas at the moment and what is yet to be known connects with the aesthetic pull to react by the student. As students work through novel problems with limited information, their joys and frustrations are expressed in the form of exclamations. 

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. 

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. 

The importance of curricular coherence has been emphasized by leaders in mathematics education, who explain that coherence enhances deeper understanding by enabling students to see connections between mathematical ideas. Although there are different forms of curricular coherence in teaching and learning mathematics, the coherence within a lesson has received considerably less attention. In particular, little is known about what constitutes coherent lessons or how to measure the degree of coherence. Using lesson data from a larger study in which lessons are intentionally designed for coherence, we propose a tool for examining lesson coherence and describe characteristics of the lessons with different levels of coherence. 

Secondary students do not often have positive experiences with mathematics. To address this challenge, this paper shares findings of a design-based research project in which a mathematical story framework was used to design mathematically captivating lesson experiences (“MCLEs”). We provide evidence that designing lessons as mathematical stories shows promise. That is, students reported improved experiences in MCLEs when compared to randomly-selected lessons. The MCLEs also impacted the students’ descriptions of their experience. 

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.