Although students in traditional introductory college statistics courses see a frequentist defini... more Although students in traditional introductory college statistics courses see a frequentist definition of probability, they are rarely asked to use this definition, instead relying on technology to conduct parametric tests. In contrast, in a simulationand randomization-based statistics course, the relative frequency of simulated outcomes becomes the central focus of the process of drawing inferences. In this study, eight college students who were enrolled in an introductory simulationbased statistics course were interviewed and asked to describe sampling distributions and make inferences; the results and analysis describes the ways they used and appeared to think about empirical probabilities. Although the students appeared to be able to make connections between various aspects of inferential reasoning, they also encountered difficulties that may be related to their focus on empirical probabilities.
Textbooks play an important role in undergraduate mathematics courses and have the potential to i... more Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed to describe how they used their textbook. The results indicate that students tend to use examples instead of the expository text to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself as well as students' beliefs about reading and the nature of mathematics. Although many instructors may not clearly convey how they want their students to use the textbook, students do report using it more productively when they believe they are asked to do so. This suggests that instructors should carefully choose text materials to promote mathematical reasoning and actively encourage their students to read the text in a way that supports the development of this reasoning.
International Group for the Psychology of Mathematics Education, 2016
Mathematics textbooks are a significant pedagogical tool, particularly in light of the growing in... more Mathematics textbooks are a significant pedagogical tool, particularly in light of the growing interest in “flipped” classrooms. However, there has been little research on how mathematics students read and comprehend textbooks. This article uses the ideas of the implied reader, sensemaking frames, and sense-making gaps to analyze students’ reading of a section of a calculus textbook. In order distinguish potential weaknesses in students’ content knowledge from their reading abilities, the article also compares students’ reading strategies to the reading strategies of “expert” readers: professors in technical fields other than mathematics.
Algebra is a focal point of reform efforts in mathematics education, with many mathematics educat... more Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6-8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed. Kur...
We construct a noncomplete excellent regular local ring A with maximal ideal M such that the gene... more We construct a noncomplete excellent regular local ring A with maximal ideal M such that the generic formal $ber ring, ˆ A ⊗A K, (where ˆ A is the M -adic completion of A and K is the
This article describes the design, implementation, and assessment of four hands-on activities in ... more This article describes the design, implementation, and assessment of four hands-on activities in an introductory college statistics course. In the activities, students investigated the ideas of the central limit theorem, confidence intervals, and hypothesis testing. Five assessments were administered to the students, one at the beginning and end of the course, and three in between the activities. We found that, despite our attempts to engage our students in active reflection, their performance on the assessments generally did not improve. These results raise important issues about the design of pedagogical tools and activities as well as the need to gather data to assess their effectiveness. The central limit theorem Hypothesis testing and p-values The meaning of confidence intervals The role of variability We believe that interactive data collection activities may increase student understanding of statistical concepts. Our belief is based on the idea that students do not learn passively, but instead learn by making new connections with their previous understanding to develop new knowledge structures (Von Glaserfeld 1987). In the learning process, the student's mind responds to cognitive conflict by actively arranging and rearranging mental structures. Our goal as educators is to create and structure this cognitive conflict in order to facilitate our students' active prediction and reflection that will generate new knowledge and understanding. Our theory is borne out by other researchers. Snee (1993) makes an argument for experiential learning, and Gnanadesikan, Scheaffer, Watkins and Witmer (1997) claim: "Activity-based courses and use of small groups appear to help students overcome some misconceptions of probability and enhance student learning of statistics concepts." Some researchers have suggested the importance of experiencing data collection (e.g. Hunter 1977; Hogg 1991; Mackisack 1994) and many have recommended laboratory-based courses, in-class activities, and class projects (e.g.
Student learning and sense-making from video lectures
International Journal of Mathematical Education in Science and Technology
ABSTRACT While ‘flipped’ classrooms have been heralded as a valuable tool for student learning, l... more ABSTRACT While ‘flipped’ classrooms have been heralded as a valuable tool for student learning, little is known about the process by which students learn from the out-of-class resources. This study investigates 12 undergraduate students’ sense-making activities while watching videos about topics in introductory statistics. Analysis of interview data indicated that the students made sense of the videos in very different ways and that the challenges students faced varied widely from person to person. These results highlight the need to take into account the wide variety in students’ background knowledge and sense-making practices when creating videos. These results also suggest the importance of structuring students’ interactions with videos so that video lectures may become more useful instructional tools for a larger number of students.
This article describes the design, implementation, and assessment of four hands-on activities in ... more This article describes the design, implementation, and assessment of four hands-on activities in an introductory college statistics course. In the activities, students investigated the ideas of the central limit theorem, confidence intervals, and hypothesis testing. Five assessments were administered to the students, one at the beginning and end of the course, and three in between the activities. We found that, despite our attempts to engage our students in active reflection, their performance on the assessments generally did not improve. These results raise important issues about the design of pedagogical tools and activities as well as the need to gather data to assess their effectiveness. The central limit theorem Hypothesis testing and p-values The meaning of confidence intervals The role of variability We believe that interactive data collection activities may increase student understanding of statistical concepts. Our belief is based on the idea that students do not learn passively, but instead learn by making new connections with their previous understanding to develop new knowledge structures (Von Glaserfeld 1987). In the learning process, the student's mind responds to cognitive conflict by actively arranging and rearranging mental structures. Our goal as educators is to create and structure this cognitive conflict in order to facilitate our students' active prediction and reflection that will generate new knowledge and understanding. Our theory is borne out by other researchers. Snee (1993) makes an argument for experiential learning, and Gnanadesikan, Scheaffer, Watkins and Witmer (1997) claim: "Activity-based courses and use of small groups appear to help students overcome some misconceptions of probability and enhance student learning of statistics concepts." Some researchers have suggested the importance of experiencing data collection (e.g. Hunter 1977; Hogg 1991; Mackisack 1994) and many have recommended laboratory-based courses, in-class activities, and class projects (e.g.
Inferential reasoning is a central component of statistics. Researchers have suggested that stude... more Inferential reasoning is a central component of statistics. Researchers have suggested that students should develop an informal understanding of the ideas that underlie inference before learning the concepts formally. This paper presents a hands-on activity that is de-signed to help students in an introductory statistics course draw informal inferences about a bag of bingo chips and connect these ideas to the formal T -test and confidence interval. This activity is analyzed using a framework and recommendations drawn from the research literature.
The concept of a sampling distribution plays a central role in the process of making statistical ... more The concept of a sampling distribution plays a central role in the process of making statistical inferences. However, students typically struggle to understand and reason about sampling distributions. This study seeks to characterize the ways undergraduate students think about sampling distributions in scenarios involving repeated sampling and making statistical inferences. Eight students in an introductory statistics class worked on problems involving sampling distributions during a semi-structured interview. A framework was developed based on their responses to describe the ways they discussed and coordinated various aspects of the population and sampling distributions by focusing on the processes of sampling and repeated sampling; these descriptions suggest that explicitly coordinating particular aspects of these processes may correspond to the robustness of students’ conceptions of sampling distributions.
Although lecture is the traditional method of university mathematics instruction, there has been ... more Although lecture is the traditional method of university mathematics instruction, there has been little empirical research that describes the general structure of lectures. In this paper, we adapt ideas from narrative analysis and apply them to an upper-level mathematics lecture. We develop a framework that enables us to conceptualize the lecture as consisting of collections of narratives, to identify connections between the narratives, and to use the narrative structure to identify key features of the lecture. In particular, we use the idea of framing and embedded narratives to identify central ideas in the lecture and to understand how examples, diagrams, and smaller claims contribute to the development of those ideas. Additionally, we create a graph structure for the lecture using the repetition of mathematical ideas across embedded narratives; and we employ graph-theoretic analysis to support our identification of framing narratives and to highlight particular roles of embedded narratives.
Page 1. Students' Initial and Developing Conceptions of Variable Aaron D. Weinberg, Ana ... more Page 1. Students' Initial and Developing Conceptions of Variable Aaron D. Weinberg, Ana C. Stephens, Nicole M. McNeil, Daniel E. Krill, Eric J. Knuth, & Martha W. Alibali University of Wisconsin - Madison A primary goal of the ...
Students’ sense-making frames in mathematics lectures
The Journal of Mathematical Behavior, 2014
ABSTRACT The goal of this study is to describe the various ways students make sense of mathematic... more ABSTRACT The goal of this study is to describe the various ways students make sense of mathematics lectures. Here, sense-making refers to a process by which people construct personal meanings for phenomena they experience. This study introduces the idea of a sense-making frame and describes three different types of frames: content-, communication-, and situating-oriented. We found that students in an abstract algebra class regularly engaged in sense-making during lectures on equivalence relations, and this sense-making influenced their note-taking practices. We discuss the relationship between the choice of frame, the students’ sense-making practices, and the potential missed opportunities for learning from the lecture. These results show the importance of understanding the ways students make sense of aspects of mathematics lectures and how their sense-making practices influence what they might learn from the lecture.
Textbooks play an important role in undergraduate mathematics courses and have the potential to i... more Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed to describe how they used their textbook. The results indicate that students tend to use examples instead of the expository text to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself as well as students' beliefs about reading and the nature of mathematics. Although many instructors may not clearly convey how they want their students to use the textbook, students do report using it more productively when they believe they are asked to do so. This suggests that instructors should carefully choose text materials to promote mathematical reasoning and actively encourage their students to read the text in a way that supports the development of this reasoning.
We construct a noncomplete excellent regular local ring A with maximal ideal M such that the gene... more We construct a noncomplete excellent regular local ring A with maximal ideal M such that the generic formal ÿber ring, ⊗A K, (where is the M -adic completion of A and K is the quotient ÿeld of A) is local. In addition, given a small set L of prime ideals ofÂ[[X1; : : : ; Xn]] (where X1; : : : ; Xn are indeterminates) satisfying some necessary conditions, every element of L is in the generic formal ÿber of A[X1; : : : ; Xn] (M; X 1 ;:::; Xn) . In other words, Q ∩ A[X1; : : : ; Xn] = (0) for every Q ∈ L.
Although students in traditional introductory college statistics courses see a frequentist defini... more Although students in traditional introductory college statistics courses see a frequentist definition of probability, they are rarely asked to use this definition, instead relying on technology to conduct parametric tests. In contrast, in a simulationand randomization-based statistics course, the relative frequency of simulated outcomes becomes the central focus of the process of drawing inferences. In this study, eight college students who were enrolled in an introductory simulationbased statistics course were interviewed and asked to describe sampling distributions and make inferences; the results and analysis describes the ways they used and appeared to think about empirical probabilities. Although the students appeared to be able to make connections between various aspects of inferential reasoning, they also encountered difficulties that may be related to their focus on empirical probabilities.
Textbooks play an important role in undergraduate mathematics courses and have the potential to i... more Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed to describe how they used their textbook. The results indicate that students tend to use examples instead of the expository text to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself as well as students' beliefs about reading and the nature of mathematics. Although many instructors may not clearly convey how they want their students to use the textbook, students do report using it more productively when they believe they are asked to do so. This suggests that instructors should carefully choose text materials to promote mathematical reasoning and actively encourage their students to read the text in a way that supports the development of this reasoning.
International Group for the Psychology of Mathematics Education, 2016
Mathematics textbooks are a significant pedagogical tool, particularly in light of the growing in... more Mathematics textbooks are a significant pedagogical tool, particularly in light of the growing interest in “flipped” classrooms. However, there has been little research on how mathematics students read and comprehend textbooks. This article uses the ideas of the implied reader, sensemaking frames, and sense-making gaps to analyze students’ reading of a section of a calculus textbook. In order distinguish potential weaknesses in students’ content knowledge from their reading abilities, the article also compares students’ reading strategies to the reading strategies of “expert” readers: professors in technical fields other than mathematics.
Algebra is a focal point of reform efforts in mathematics education, with many mathematics educat... more Algebra is a focal point of reform efforts in mathematics education, with many mathematics educators advocating that algebraic reasoning should be integrated at all grade levels K-12. Recent research has begun to investigate algebra reform in the context of elementary school (grades K-5) mathematics, focusing in particular on the development of algebraic reasoning. Yet, to date, little research has focused on the development of algebraic reasoning in middle school (grades 6-8). This article focuses on middle school students’ understanding of two core algebraic ideas—equivalence and variable—and the relationship of their understanding to performance on problems that require use of these two ideas. The data suggest that students’ understanding of these core ideas influences their success in solving problems, the strategies they use in their solution processes, and the justifications they provide for their solutions. Implications for instruction and curricular design are discussed. Kur...
We construct a noncomplete excellent regular local ring A with maximal ideal M such that the gene... more We construct a noncomplete excellent regular local ring A with maximal ideal M such that the generic formal $ber ring, ˆ A ⊗A K, (where ˆ A is the M -adic completion of A and K is the
This article describes the design, implementation, and assessment of four hands-on activities in ... more This article describes the design, implementation, and assessment of four hands-on activities in an introductory college statistics course. In the activities, students investigated the ideas of the central limit theorem, confidence intervals, and hypothesis testing. Five assessments were administered to the students, one at the beginning and end of the course, and three in between the activities. We found that, despite our attempts to engage our students in active reflection, their performance on the assessments generally did not improve. These results raise important issues about the design of pedagogical tools and activities as well as the need to gather data to assess their effectiveness. The central limit theorem Hypothesis testing and p-values The meaning of confidence intervals The role of variability We believe that interactive data collection activities may increase student understanding of statistical concepts. Our belief is based on the idea that students do not learn passively, but instead learn by making new connections with their previous understanding to develop new knowledge structures (Von Glaserfeld 1987). In the learning process, the student's mind responds to cognitive conflict by actively arranging and rearranging mental structures. Our goal as educators is to create and structure this cognitive conflict in order to facilitate our students' active prediction and reflection that will generate new knowledge and understanding. Our theory is borne out by other researchers. Snee (1993) makes an argument for experiential learning, and Gnanadesikan, Scheaffer, Watkins and Witmer (1997) claim: "Activity-based courses and use of small groups appear to help students overcome some misconceptions of probability and enhance student learning of statistics concepts." Some researchers have suggested the importance of experiencing data collection (e.g. Hunter 1977; Hogg 1991; Mackisack 1994) and many have recommended laboratory-based courses, in-class activities, and class projects (e.g.
Student learning and sense-making from video lectures
International Journal of Mathematical Education in Science and Technology
ABSTRACT While ‘flipped’ classrooms have been heralded as a valuable tool for student learning, l... more ABSTRACT While ‘flipped’ classrooms have been heralded as a valuable tool for student learning, little is known about the process by which students learn from the out-of-class resources. This study investigates 12 undergraduate students’ sense-making activities while watching videos about topics in introductory statistics. Analysis of interview data indicated that the students made sense of the videos in very different ways and that the challenges students faced varied widely from person to person. These results highlight the need to take into account the wide variety in students’ background knowledge and sense-making practices when creating videos. These results also suggest the importance of structuring students’ interactions with videos so that video lectures may become more useful instructional tools for a larger number of students.
This article describes the design, implementation, and assessment of four hands-on activities in ... more This article describes the design, implementation, and assessment of four hands-on activities in an introductory college statistics course. In the activities, students investigated the ideas of the central limit theorem, confidence intervals, and hypothesis testing. Five assessments were administered to the students, one at the beginning and end of the course, and three in between the activities. We found that, despite our attempts to engage our students in active reflection, their performance on the assessments generally did not improve. These results raise important issues about the design of pedagogical tools and activities as well as the need to gather data to assess their effectiveness. The central limit theorem Hypothesis testing and p-values The meaning of confidence intervals The role of variability We believe that interactive data collection activities may increase student understanding of statistical concepts. Our belief is based on the idea that students do not learn passively, but instead learn by making new connections with their previous understanding to develop new knowledge structures (Von Glaserfeld 1987). In the learning process, the student's mind responds to cognitive conflict by actively arranging and rearranging mental structures. Our goal as educators is to create and structure this cognitive conflict in order to facilitate our students' active prediction and reflection that will generate new knowledge and understanding. Our theory is borne out by other researchers. Snee (1993) makes an argument for experiential learning, and Gnanadesikan, Scheaffer, Watkins and Witmer (1997) claim: "Activity-based courses and use of small groups appear to help students overcome some misconceptions of probability and enhance student learning of statistics concepts." Some researchers have suggested the importance of experiencing data collection (e.g. Hunter 1977; Hogg 1991; Mackisack 1994) and many have recommended laboratory-based courses, in-class activities, and class projects (e.g.
Inferential reasoning is a central component of statistics. Researchers have suggested that stude... more Inferential reasoning is a central component of statistics. Researchers have suggested that students should develop an informal understanding of the ideas that underlie inference before learning the concepts formally. This paper presents a hands-on activity that is de-signed to help students in an introductory statistics course draw informal inferences about a bag of bingo chips and connect these ideas to the formal T -test and confidence interval. This activity is analyzed using a framework and recommendations drawn from the research literature.
The concept of a sampling distribution plays a central role in the process of making statistical ... more The concept of a sampling distribution plays a central role in the process of making statistical inferences. However, students typically struggle to understand and reason about sampling distributions. This study seeks to characterize the ways undergraduate students think about sampling distributions in scenarios involving repeated sampling and making statistical inferences. Eight students in an introductory statistics class worked on problems involving sampling distributions during a semi-structured interview. A framework was developed based on their responses to describe the ways they discussed and coordinated various aspects of the population and sampling distributions by focusing on the processes of sampling and repeated sampling; these descriptions suggest that explicitly coordinating particular aspects of these processes may correspond to the robustness of students’ conceptions of sampling distributions.
Although lecture is the traditional method of university mathematics instruction, there has been ... more Although lecture is the traditional method of university mathematics instruction, there has been little empirical research that describes the general structure of lectures. In this paper, we adapt ideas from narrative analysis and apply them to an upper-level mathematics lecture. We develop a framework that enables us to conceptualize the lecture as consisting of collections of narratives, to identify connections between the narratives, and to use the narrative structure to identify key features of the lecture. In particular, we use the idea of framing and embedded narratives to identify central ideas in the lecture and to understand how examples, diagrams, and smaller claims contribute to the development of those ideas. Additionally, we create a graph structure for the lecture using the repetition of mathematical ideas across embedded narratives; and we employ graph-theoretic analysis to support our identification of framing narratives and to highlight particular roles of embedded narratives.
Page 1. Students' Initial and Developing Conceptions of Variable Aaron D. Weinberg, Ana ... more Page 1. Students' Initial and Developing Conceptions of Variable Aaron D. Weinberg, Ana C. Stephens, Nicole M. McNeil, Daniel E. Krill, Eric J. Knuth, & Martha W. Alibali University of Wisconsin - Madison A primary goal of the ...
Students’ sense-making frames in mathematics lectures
The Journal of Mathematical Behavior, 2014
ABSTRACT The goal of this study is to describe the various ways students make sense of mathematic... more ABSTRACT The goal of this study is to describe the various ways students make sense of mathematics lectures. Here, sense-making refers to a process by which people construct personal meanings for phenomena they experience. This study introduces the idea of a sense-making frame and describes three different types of frames: content-, communication-, and situating-oriented. We found that students in an abstract algebra class regularly engaged in sense-making during lectures on equivalence relations, and this sense-making influenced their note-taking practices. We discuss the relationship between the choice of frame, the students’ sense-making practices, and the potential missed opportunities for learning from the lecture. These results show the importance of understanding the ways students make sense of aspects of mathematics lectures and how their sense-making practices influence what they might learn from the lecture.
Textbooks play an important role in undergraduate mathematics courses and have the potential to i... more Textbooks play an important role in undergraduate mathematics courses and have the potential to impact student learning. However, there have been few studies that describe students' textbook use in detail. In this study, 1156 undergraduate students in introductory mathematics classes were surveyed to describe how they used their textbook. The results indicate that students tend to use examples instead of the expository text to build their mathematical understanding, which instructors may view as problematic. This way of using the textbook may be the result of the textbook structure itself as well as students' beliefs about reading and the nature of mathematics. Although many instructors may not clearly convey how they want their students to use the textbook, students do report using it more productively when they believe they are asked to do so. This suggests that instructors should carefully choose text materials to promote mathematical reasoning and actively encourage their students to read the text in a way that supports the development of this reasoning.
We construct a noncomplete excellent regular local ring A with maximal ideal M such that the gene... more We construct a noncomplete excellent regular local ring A with maximal ideal M such that the generic formal ÿber ring, ⊗A K, (where is the M -adic completion of A and K is the quotient ÿeld of A) is local. In addition, given a small set L of prime ideals ofÂ[[X1; : : : ; Xn]] (where X1; : : : ; Xn are indeterminates) satisfying some necessary conditions, every element of L is in the generic formal ÿber of A[X1; : : : ; Xn] (M; X 1 ;:::; Xn) . In other words, Q ∩ A[X1; : : : ; Xn] = (0) for every Q ∈ L.
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