Richard Lesh - Academia.edu (original) (raw)
Papers by Richard Lesh
Springer eBooks, Oct 13, 2009
Hurford’s article is a useful introduction to the topic of complex systems and their potential si... more Hurford’s article is a useful introduction to the topic of complex systems and their potential significance in mathematics education. He focuses on two categories of issues. The first concerns the possibility of treating complex systems as an important topic to be included in any mathematics curriculum that claims to be preparing students for full participation in a technology-based age of information. The second concerns the possibility of using systems theory in general, and complexity theory in particular, to develop models to explain the development of students’ mathematical thinking in future-oriented learning or problem solving situations.
English, Lyn D. <http://eprints.qut.edu.au/view/person/English,\_Lyn.html>, Jones, Graham A.... more English, Lyn D. <http://eprints.qut.edu.au/view/person/English,_Lyn.html>, Jones, Graham A. <http://eprints.qut.edu.au/view/person/Jones,_Graham.html>, Tirosh, Dina <http://eprints.qut.edu. au/view/person/Tirosh,_Dina.html>, Lesh, Richard <http://eprints.qut.edu.au/view/person/ ...
Journal of Mathematics Teacher Education, Aug 23, 2021
Routledge eBooks, Oct 12, 2012
Evaluierte Lernumgebungen zum Modellieren, 2018
Damit die Schuler durch den heutigen Unterricht auch in der Welt auserhalb der Schule erfolgreich... more Damit die Schuler durch den heutigen Unterricht auch in der Welt auserhalb der Schule erfolgreich sein konnen, muss ihre mathematische Ausbildung uber einen festgelegten Satz an Fertigkeiten und Routinen hinausgehen. Stattdessen mussen sie in der Lage sein, den in der Schule erlernten Stoff auf subtile Weise anzupassen und anzuwenden, um mit Feingefuhl und Geschicklichkeit bei der Losung von Problemen, die gekennzeichnet sind durch Einschrankungen und Unklarheiten, reagieren zu konnen. In diesem Kapitel beschreiben wir Aktivitaten, die den Lernenden diese Art der Erfahrung ermoglichen. Wir betrachten die Erfahrungen der Schuler mit diesen Aktivitaten und wir zeigen, wie die Ungewissheit uber die Art der Anwendung des erlernten Wissens einen Raum fur die Verwendung von mathematischen Denkweisen zur Interpretation der Welt eroffnet. Wir zeigen dann, wie die Prasenz eines konkreten Kunden fur die Schuler die Moglichkeit bietet, Losungen zu formulieren, die auf die menschlichen Bedurfnisse und die Sichtweise dieses Kunden sowie anderer Menschen, die von der Losung betroffen sind, eingehen.
The Mathematics Enthusiast, 2015
Realitätsbezüge im Mathematikunterricht, 2018
The purpose of this chapter is to highlight how work focusing on student-level modelling and idea... more The purpose of this chapter is to highlight how work focusing on student-level modelling and idea development can also serve as a powerful context to investigate parallel and interacting modelling processes at the teacher level. We assert that expertise in teaching is partly reflected in how teachers interpret and respond to classroom situations-in what they see and recognize, as well as in what they do. To establish a setting for extended studies of teacher-level modelling, different research groups have found it desirable to move beyond single-activity implementations in one of several ways. We begin our description of this work by identifying several values or beliefs about the nature of mathematical thinking and learning and corresponding instructional practices, which comprise key teacher competencies for the teaching and learning of modelling
In diesem Kapitel wird verdeutlicht, wie Modellierungsprozesse von Lernenden fur die Lehrerausbil... more In diesem Kapitel wird verdeutlicht, wie Modellierungsprozesse von Lernenden fur die Lehrerausbildung zum Lehren von mathematischer Modellierung als ein wirkungsvolles Instrument eingesetzt werden konnen. Wir behaupten, dass sich die Expertise des Unterrichtens zum Teil darin widerspiegelt, wie Lehrkrafte Unterrichtssituationen interpretieren und darauf reagieren – sowohl darin, was sie sehen und erkennen, als auch darin, was sie tun. Wir beginnen unsere Ausfuhrungen mit der Identifikation verschiedener Beliefs in Bezug auf mathematisches Denken und Lernen sowie den entsprechenden Unterrichtsmethoden, die die Schlusselkompetenzen der Lehrkrafte zum Lehren und Lernen von Modellieren umfassen, die bei diesem Projekt in den Vordergrund gestellt wurden. Unsere Ergebnisse legen nahe, dass – bei Studien zur Entwicklung von Modellierungskompetenz bei Lernenden uber die Dauer eines ganzen Kurs hinweg – signifikante wirkungsvolle Veranderungen auch bei den Kompetenzen auf Lehrerebene fur das...
Foundations for the Future in Mathematics Education, 2020
Mathematical practices are the cornerstone of mathematical learning, so it is critical for resear... more Mathematical practices are the cornerstone of mathematical learning, so it is critical for researchers and educators to understand their ontogenesis. In this chapter we trace the ontogenesis of three forms of mathematical practice. The first, argument, represents a discursive view of learning, originates in everyday conversation, and culminates in specialized forms of discourse, such as proof. The second, inscription, represents a view of learning as co-originating with written expression. It begins with scribbles and culminates in systems of notation, especially those that are digital and dynamic. The third, modeling, is inaugurated by metaphor and analogy, especially those involving the body. As it progresses, it encompasses argument and inscription for the purpose of creating mathematical systems for describing the world. Although each draws on native resources and dispositions, each of these forms of practice also relies on contexts that encourage their development. Accordingly,...
The central goal of this chapter is to describe six principles that have proven to be especially ... more The central goal of this chapter is to describe six principles that have proven to be especially useful to help researchers (or teachers, curriculum designers, or assessment specialists) create thought revealing activities that we refer to as model-eliciting activities. The first section describes general characteristics and purposes of model-eliciting activities. The second section gives examples of several model-eliciting activities, and it also describes typical solutions that students generate to such problems. The third section states several disclaimers about modeleliciting activities, which tend to cause confusion if they are not addressed early. The fourth section describes difficulties that modeleliciting activities were designed to address in research; and, the fifth section describes principles for designing productive model-eliciting activities.
The problem solving process in most twenty-first century contexts involves teams working on probl... more The problem solving process in most twenty-first century contexts involves teams working on problem situations. It involves partitioning a complex situation into parts that can be addressed by specialists. It involves communicating information in forms that are meaningful for other specialists and suitable for their tools. It involves planning, monitoring and assessing intermediate results. In other words, the primary activity is team-based mathematical modeling. While the use of small groups in mathematics classes has been encouraged for over 20 years, research has focused on the achivement of individuals (as measured by standardized tests) and social benefits (such as self-esteem, social acceptance) (Davidson, 1990). These areas of emphasis and research have been confined to a frame of reference of school mathematics, whereas the needs of the current work place go beyond these interests. For example, schools are interested in the achievement of individual children, whereas busines...
Note: Before reading this chapter on iterative videotape analyses, readers may find it useful to ... more Note: Before reading this chapter on iterative videotape analyses, readers may find it useful to read the transcript that is given in the Appendix to the chapter. The transcript is one of those that is referred to in the chapter; and, many of the chapter's main points make reference to it. In this regard, the following comments are especially relevant: • The students' goal for this problem solving session was to produce a description of the information that was given; and, the purpose of this description was to enable the students to rank workers whose summer jobs involved selling things in a park. • The students' initial way of thinking about the information was quite barren and distorted; that is, they neglected to notice a great deal of relevant information, and, at the same time, they made assumptions that were not valid. • To gradually arrive at a final way of thinking about the situation, the students went through a series of modeling cycles in which they interpreted givens and the goals in systematically different ways that focused on different facts, quantities, relationships, and patterns. • The researchers' goals were to produce a description of how the students' ways of thinking evolved during the problem solving session. Consequently, the researchers also arrived at a final interpretation of the session by going through a series of modeling cycles in which their early ways of thinking were remarkably barren and distorted compared with those that evolved later. The following issues are useful to think about when reading the transcript. What do you think was the researcher's perceived purpose for the description that he was generating? Is it likely that another purpose or another perspective might have emphasized different principles for selecting and organizing information, or for interpreting their significance? In what ways did the researchers test the usefulness of alternative interpretations? What methods did they use to determine whether a given interpretation needed to be refined, modified, or extended? If the researchers undertook the job of producing a movie to convey to audiences what was going on in the problem solving session, do you think that the best kind of movie would consist of nothing more than the videotape of the session? This chapter about iterative videotape analyses is a companion to two others in this book. The first is about model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, chap. 21, this volume); and, the second is about multitiered teaching experiments (Lesh & Kelly, chap. 9, this volume). In general, all three chapters focus on research whose central purpose is to investigate the nature of the constructs that students (or teachers or groups) develop to make sense of a targeted class of problem solving or decision-making situations. Also, all three chapters emphasize research designs that are potentially shareable; that is, it is possible to coordinate the work of several researchers who are working at multiple sites using diverse practical or theoretical perspectives. Whereas the chapter on model-eliciting activities (chap. 21, this volume) describes principles for designing problem solving situations in which observations are to be made, the chapter on multitiered teaching experiments (chap. 9, this volume) focuses on principles for designing the data collection stages of research, and this chapter on iterative videotape analyses focuses on the stages of research that involve data analysis. Also, the chapter on multitiered teaching experiments concentrates on studies in which conclusions based on today's sessions influence tomorrow's data collection activities; whereas, this chapter on iterative videotape analyses focuses on research settings in which the most significant stages of data collection must be completed before the most significant stages of data interpretation begin. Therefore, in this latter case, there exist no severe constraints on the turnaround times for data interpretation.
Springer eBooks, Oct 13, 2009
Hurford’s article is a useful introduction to the topic of complex systems and their potential si... more Hurford’s article is a useful introduction to the topic of complex systems and their potential significance in mathematics education. He focuses on two categories of issues. The first concerns the possibility of treating complex systems as an important topic to be included in any mathematics curriculum that claims to be preparing students for full participation in a technology-based age of information. The second concerns the possibility of using systems theory in general, and complexity theory in particular, to develop models to explain the development of students’ mathematical thinking in future-oriented learning or problem solving situations.
English, Lyn D. <http://eprints.qut.edu.au/view/person/English,\_Lyn.html>, Jones, Graham A.... more English, Lyn D. <http://eprints.qut.edu.au/view/person/English,_Lyn.html>, Jones, Graham A. <http://eprints.qut.edu.au/view/person/Jones,_Graham.html>, Tirosh, Dina <http://eprints.qut.edu. au/view/person/Tirosh,_Dina.html>, Lesh, Richard <http://eprints.qut.edu.au/view/person/ ...
Journal of Mathematics Teacher Education, Aug 23, 2021
Routledge eBooks, Oct 12, 2012
Evaluierte Lernumgebungen zum Modellieren, 2018
Damit die Schuler durch den heutigen Unterricht auch in der Welt auserhalb der Schule erfolgreich... more Damit die Schuler durch den heutigen Unterricht auch in der Welt auserhalb der Schule erfolgreich sein konnen, muss ihre mathematische Ausbildung uber einen festgelegten Satz an Fertigkeiten und Routinen hinausgehen. Stattdessen mussen sie in der Lage sein, den in der Schule erlernten Stoff auf subtile Weise anzupassen und anzuwenden, um mit Feingefuhl und Geschicklichkeit bei der Losung von Problemen, die gekennzeichnet sind durch Einschrankungen und Unklarheiten, reagieren zu konnen. In diesem Kapitel beschreiben wir Aktivitaten, die den Lernenden diese Art der Erfahrung ermoglichen. Wir betrachten die Erfahrungen der Schuler mit diesen Aktivitaten und wir zeigen, wie die Ungewissheit uber die Art der Anwendung des erlernten Wissens einen Raum fur die Verwendung von mathematischen Denkweisen zur Interpretation der Welt eroffnet. Wir zeigen dann, wie die Prasenz eines konkreten Kunden fur die Schuler die Moglichkeit bietet, Losungen zu formulieren, die auf die menschlichen Bedurfnisse und die Sichtweise dieses Kunden sowie anderer Menschen, die von der Losung betroffen sind, eingehen.
The Mathematics Enthusiast, 2015
Realitätsbezüge im Mathematikunterricht, 2018
The purpose of this chapter is to highlight how work focusing on student-level modelling and idea... more The purpose of this chapter is to highlight how work focusing on student-level modelling and idea development can also serve as a powerful context to investigate parallel and interacting modelling processes at the teacher level. We assert that expertise in teaching is partly reflected in how teachers interpret and respond to classroom situations-in what they see and recognize, as well as in what they do. To establish a setting for extended studies of teacher-level modelling, different research groups have found it desirable to move beyond single-activity implementations in one of several ways. We begin our description of this work by identifying several values or beliefs about the nature of mathematical thinking and learning and corresponding instructional practices, which comprise key teacher competencies for the teaching and learning of modelling
In diesem Kapitel wird verdeutlicht, wie Modellierungsprozesse von Lernenden fur die Lehrerausbil... more In diesem Kapitel wird verdeutlicht, wie Modellierungsprozesse von Lernenden fur die Lehrerausbildung zum Lehren von mathematischer Modellierung als ein wirkungsvolles Instrument eingesetzt werden konnen. Wir behaupten, dass sich die Expertise des Unterrichtens zum Teil darin widerspiegelt, wie Lehrkrafte Unterrichtssituationen interpretieren und darauf reagieren – sowohl darin, was sie sehen und erkennen, als auch darin, was sie tun. Wir beginnen unsere Ausfuhrungen mit der Identifikation verschiedener Beliefs in Bezug auf mathematisches Denken und Lernen sowie den entsprechenden Unterrichtsmethoden, die die Schlusselkompetenzen der Lehrkrafte zum Lehren und Lernen von Modellieren umfassen, die bei diesem Projekt in den Vordergrund gestellt wurden. Unsere Ergebnisse legen nahe, dass – bei Studien zur Entwicklung von Modellierungskompetenz bei Lernenden uber die Dauer eines ganzen Kurs hinweg – signifikante wirkungsvolle Veranderungen auch bei den Kompetenzen auf Lehrerebene fur das...
Foundations for the Future in Mathematics Education, 2020
Mathematical practices are the cornerstone of mathematical learning, so it is critical for resear... more Mathematical practices are the cornerstone of mathematical learning, so it is critical for researchers and educators to understand their ontogenesis. In this chapter we trace the ontogenesis of three forms of mathematical practice. The first, argument, represents a discursive view of learning, originates in everyday conversation, and culminates in specialized forms of discourse, such as proof. The second, inscription, represents a view of learning as co-originating with written expression. It begins with scribbles and culminates in systems of notation, especially those that are digital and dynamic. The third, modeling, is inaugurated by metaphor and analogy, especially those involving the body. As it progresses, it encompasses argument and inscription for the purpose of creating mathematical systems for describing the world. Although each draws on native resources and dispositions, each of these forms of practice also relies on contexts that encourage their development. Accordingly,...
The central goal of this chapter is to describe six principles that have proven to be especially ... more The central goal of this chapter is to describe six principles that have proven to be especially useful to help researchers (or teachers, curriculum designers, or assessment specialists) create thought revealing activities that we refer to as model-eliciting activities. The first section describes general characteristics and purposes of model-eliciting activities. The second section gives examples of several model-eliciting activities, and it also describes typical solutions that students generate to such problems. The third section states several disclaimers about modeleliciting activities, which tend to cause confusion if they are not addressed early. The fourth section describes difficulties that modeleliciting activities were designed to address in research; and, the fifth section describes principles for designing productive model-eliciting activities.
The problem solving process in most twenty-first century contexts involves teams working on probl... more The problem solving process in most twenty-first century contexts involves teams working on problem situations. It involves partitioning a complex situation into parts that can be addressed by specialists. It involves communicating information in forms that are meaningful for other specialists and suitable for their tools. It involves planning, monitoring and assessing intermediate results. In other words, the primary activity is team-based mathematical modeling. While the use of small groups in mathematics classes has been encouraged for over 20 years, research has focused on the achivement of individuals (as measured by standardized tests) and social benefits (such as self-esteem, social acceptance) (Davidson, 1990). These areas of emphasis and research have been confined to a frame of reference of school mathematics, whereas the needs of the current work place go beyond these interests. For example, schools are interested in the achievement of individual children, whereas busines...
Note: Before reading this chapter on iterative videotape analyses, readers may find it useful to ... more Note: Before reading this chapter on iterative videotape analyses, readers may find it useful to read the transcript that is given in the Appendix to the chapter. The transcript is one of those that is referred to in the chapter; and, many of the chapter's main points make reference to it. In this regard, the following comments are especially relevant: • The students' goal for this problem solving session was to produce a description of the information that was given; and, the purpose of this description was to enable the students to rank workers whose summer jobs involved selling things in a park. • The students' initial way of thinking about the information was quite barren and distorted; that is, they neglected to notice a great deal of relevant information, and, at the same time, they made assumptions that were not valid. • To gradually arrive at a final way of thinking about the situation, the students went through a series of modeling cycles in which they interpreted givens and the goals in systematically different ways that focused on different facts, quantities, relationships, and patterns. • The researchers' goals were to produce a description of how the students' ways of thinking evolved during the problem solving session. Consequently, the researchers also arrived at a final interpretation of the session by going through a series of modeling cycles in which their early ways of thinking were remarkably barren and distorted compared with those that evolved later. The following issues are useful to think about when reading the transcript. What do you think was the researcher's perceived purpose for the description that he was generating? Is it likely that another purpose or another perspective might have emphasized different principles for selecting and organizing information, or for interpreting their significance? In what ways did the researchers test the usefulness of alternative interpretations? What methods did they use to determine whether a given interpretation needed to be refined, modified, or extended? If the researchers undertook the job of producing a movie to convey to audiences what was going on in the problem solving session, do you think that the best kind of movie would consist of nothing more than the videotape of the session? This chapter about iterative videotape analyses is a companion to two others in this book. The first is about model-eliciting activities (Lesh, Hoover, Hole, Kelly, & Post, chap. 21, this volume); and, the second is about multitiered teaching experiments (Lesh & Kelly, chap. 9, this volume). In general, all three chapters focus on research whose central purpose is to investigate the nature of the constructs that students (or teachers or groups) develop to make sense of a targeted class of problem solving or decision-making situations. Also, all three chapters emphasize research designs that are potentially shareable; that is, it is possible to coordinate the work of several researchers who are working at multiple sites using diverse practical or theoretical perspectives. Whereas the chapter on model-eliciting activities (chap. 21, this volume) describes principles for designing problem solving situations in which observations are to be made, the chapter on multitiered teaching experiments (chap. 9, this volume) focuses on principles for designing the data collection stages of research, and this chapter on iterative videotape analyses focuses on the stages of research that involve data analysis. Also, the chapter on multitiered teaching experiments concentrates on studies in which conclusions based on today's sessions influence tomorrow's data collection activities; whereas, this chapter on iterative videotape analyses focuses on research settings in which the most significant stages of data collection must be completed before the most significant stages of data interpretation begin. Therefore, in this latter case, there exist no severe constraints on the turnaround times for data interpretation.