Teachers need to be well informed about current initiatives in mathematics and see how these can ... more Teachers need to be well informed about current initiatives in mathematics and see how these can be translated into rich and stimulating classroom strategies. This series provides a practical illustration of the skills, knowledge and understanding required to teach in the secondary classroom.
By treating collections of questions as mathematical objects, that is ordered sets containing ind... more By treating collections of questions as mathematical objects, that is ordered sets containing individual questions as elements, we gain insight into the potential role of exercises in learning mathematics. We use the notion of 'dimensions of possible variation', derived from Ference Marton, to discuss some exercises. There are implications for the design of question sets, for pedagogical decisions in the use of question sets, and for reflective questioning by learners.
University Science and Mathematics Education in Transition
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection w... more , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
We report a study which investigated how mathematics graduates and engineering undergraduates stu... more We report a study which investigated how mathematics graduates and engineering undergraduates studied sequential tables of values with the aim of deriving a function. By recording the eye-movements of participants as they tackled this task, we found that there were substantial individual differences in the strategies adopted by participants. However, these strategy choices appeared to be unrelated to both the mathematical background of participants, and their success rates.
International Journal of Mathematical Education in Science and Technology, 2000
In order to work with the inevitable didactic tension (the more clearly the teacher indicates the... more In order to work with the inevitable didactic tension (the more clearly the teacher indicates the behaviour sought, the easier it is for students to display that behaviour without generating it from understanding) which is part of the contrat didactique described by Brousseau, it is useful to work with all six modes of interaction between teacher, student, and content: expounding, explaining, exploring, examining, expressing, and exercising. Posing students questions with the purposes or effects of focusing, testing or enquiring forms the basis for many of these modes, but like most interactions, questioning can become habitual and limited in scope. I conjecture that we are moved to pose questions as tasks, or to ask questions of students, when we ourselves experience a shift in the structure of our own attention, and this is why we often 'catch students out' with our tasks, and why we often get caught in a funnelling sequence of questions (in the sense of Bauersfeld). I offer the conjecture that questions arise, whether in planning or in the classroom, as a result of us experiencing a shift in our own attention; the purpose of our question is often, perhaps usually, to try to provoke a corresponding shift in our students. My claim is that by paying attention to the types of questions we ask, we can influence the development of students' awareness of mathematical thinking. I will look in detail at Probing Boundaries, Story Telling, and Probing Understanding. La tension didactique (Plus le professeur indique clairement la conduite recherchée, plus les étudiants trouvent facile de la suivre sans la comprendre) est une partie du contrat didiactique décrit par Brousseau. Pour travailler avec la tension didactique inévitable, il est utile de travailler avec tous les six modes d'interaction entre le professeur, l'étudiant et le contenu: exposer, expliquer, explorer, examiner, exprimer, exercer. L'habileté de poser des questions aux étudiants afin qu'on puisse concentrer, examiner, ou s'informer, elle constitue la basse à beaucoup de ces modes; mais comme la plupart des interactions, l'habileté de questioner peut devient habituelle et limitée. Je prétend que si on fait attention aux genres des questions posées, on peut influencer le development de la conscience des opinions mathématiques des étudiants. Je propose la conjecture qu'un question nous se présente avant de ou pendant une classe, parce que notre attention est transformé soudain; le but de la question est souvent, peut être toujours, essayer provoquer un trnsformation correspondent pour nos étudiants. J'éxaminerai en détail à les frontières serrés, la narration des histoires mathématiques et la construction des problèmes fondamentales qui examine la comprehension.
1. BACKGROUND There is evidence from earliest historical records that examples play a central rol... more 1. BACKGROUND There is evidence from earliest historical records that examples play a central role in both the development of mathematics as a discipline and in the teaching of mathematics. It is not surprising therefore that examples have found a place in many ...
Foram propostas duas tarefas a alunos portugueses do ensino básico para os encorajar a raciocinar... more Foram propostas duas tarefas a alunos portugueses do ensino básico para os encorajar a raciocinar matematicamente sem o recurso a cálculos, com o objetivo de procurar evidências sobre diferentes formas de atenção (Mason, 2003) e de usar essas evidências para refletir sobre as tarefas e sobre a capacidade dos alunos raciocinarem "com razoabilidade" em matemática. A primeira tarefa, centrada na localização de um lugar secreto usando uma applet, foi resolvida por dois pares de alunos do 4.º ano de escolaridade. A segunda, envolvendo a estrutura de quadrados mágicos, foi proposta a duas turmas do 7.º ano de escolaridade. Em qualquer dos casos, procedeu-se à gravação e transcrição das interações entre alunos e professor e na segunda tarefa recolheram-se, ainda, as resoluções escritas dos alunos. Os dados obtidos foram analisados usando um modelo composto por cinco formas ou microestruturas de atenção. No caso da primeira tarefa, evidencia-se que as crianças são capazes de racio...
This paper elaborates the notion of a personal example space as the set of mathematical objects a... more This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to three studies spanning the age range of learners, from school-age learners to pre-service teachers learning maths and professional mathematicians. Their constructions of examples are used as evidence of their personal example spaces. From these, we identify characteristics of such spaces that provide insight into learning mathematics. This perspective informs teaching by giving access to how personal knowledge is structured and what might enhance that structure.
This paper elaborates the notion of a personal example space as the set of mathematical objects a... more This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to three studies spanning the age range of learners, from school-age learners to pre-service teachers learning maths and professional mathematicians. Their constructions of examples are used as evidence of their personal example spaces. From these, we identify characteristics of such spaces that provide insight into learning mathematics. This perspective informs teaching by giving access to how personal knowledge is structured and what might enhance that structure.
The word rational is often associated with reasoning and ‘being reasonable’. Henri Poin- caré (19... more The word rational is often associated with reasoning and ‘being reasonable’. Henri Poin- caré (1956) expressed surprise that people find mathematics difficult to learn, because from his perspective it is entirely rational, and humans are rational beings. Jonathon Swift (1726) had however already challenged this notion, proposing that human beings are at best ‘animals capable of reason’.
Teaching students to reason mathematically seems therefore to belong to a disputed domain between ‘they do it already so it is only a matter of evoking and provoking it and drawing it to attention’ and ‘it requires specific and explicit teaching in order to come to students’ minds when required’. Indeed many teachers would claim from their expe- rience that it is difficult to persuade students to use various forms of reasoning such as reasoning by contradiction or by contrapositive. On the one hand very young children reason empirically (they pick up patterns in human and non-human behaviour and act as if they have internalised these); on the other hand they don’t seem to know what to do when asked to justify something mathematically. The stance taken here is that there are delicate shifts of attention involved in reasoning ‘reasonably’1 in mathematics, and that teachers can be of considerable support if they are attuned to these different ways of attending.
In this paper we briefly describe two tasks that we conjecture can provoke young stu- dents to display mathematical reasoning, and report the responses to these tasks of some students in grades 4 and 7. We use the theoretical framework of structures of attention (Mason 2003) to highlight different aspects of student’s reasoning.
Teachers need to be well informed about current initiatives in mathematics and see how these can ... more Teachers need to be well informed about current initiatives in mathematics and see how these can be translated into rich and stimulating classroom strategies. This series provides a practical illustration of the skills, knowledge and understanding required to teach in the secondary classroom.
By treating collections of questions as mathematical objects, that is ordered sets containing ind... more By treating collections of questions as mathematical objects, that is ordered sets containing individual questions as elements, we gain insight into the potential role of exercises in learning mathematics. We use the notion of 'dimensions of possible variation', derived from Ference Marton, to discuss some exercises. There are implications for the design of question sets, for pedagogical decisions in the use of question sets, and for reflective questioning by learners.
University Science and Mathematics Education in Transition
, except for brief excerpts in connection with reviews or scholarly analysis. Use in connection w... more , except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
We report a study which investigated how mathematics graduates and engineering undergraduates stu... more We report a study which investigated how mathematics graduates and engineering undergraduates studied sequential tables of values with the aim of deriving a function. By recording the eye-movements of participants as they tackled this task, we found that there were substantial individual differences in the strategies adopted by participants. However, these strategy choices appeared to be unrelated to both the mathematical background of participants, and their success rates.
International Journal of Mathematical Education in Science and Technology, 2000
In order to work with the inevitable didactic tension (the more clearly the teacher indicates the... more In order to work with the inevitable didactic tension (the more clearly the teacher indicates the behaviour sought, the easier it is for students to display that behaviour without generating it from understanding) which is part of the contrat didactique described by Brousseau, it is useful to work with all six modes of interaction between teacher, student, and content: expounding, explaining, exploring, examining, expressing, and exercising. Posing students questions with the purposes or effects of focusing, testing or enquiring forms the basis for many of these modes, but like most interactions, questioning can become habitual and limited in scope. I conjecture that we are moved to pose questions as tasks, or to ask questions of students, when we ourselves experience a shift in the structure of our own attention, and this is why we often 'catch students out' with our tasks, and why we often get caught in a funnelling sequence of questions (in the sense of Bauersfeld). I offer the conjecture that questions arise, whether in planning or in the classroom, as a result of us experiencing a shift in our own attention; the purpose of our question is often, perhaps usually, to try to provoke a corresponding shift in our students. My claim is that by paying attention to the types of questions we ask, we can influence the development of students' awareness of mathematical thinking. I will look in detail at Probing Boundaries, Story Telling, and Probing Understanding. La tension didactique (Plus le professeur indique clairement la conduite recherchée, plus les étudiants trouvent facile de la suivre sans la comprendre) est une partie du contrat didiactique décrit par Brousseau. Pour travailler avec la tension didactique inévitable, il est utile de travailler avec tous les six modes d'interaction entre le professeur, l'étudiant et le contenu: exposer, expliquer, explorer, examiner, exprimer, exercer. L'habileté de poser des questions aux étudiants afin qu'on puisse concentrer, examiner, ou s'informer, elle constitue la basse à beaucoup de ces modes; mais comme la plupart des interactions, l'habileté de questioner peut devient habituelle et limitée. Je prétend que si on fait attention aux genres des questions posées, on peut influencer le development de la conscience des opinions mathématiques des étudiants. Je propose la conjecture qu'un question nous se présente avant de ou pendant une classe, parce que notre attention est transformé soudain; le but de la question est souvent, peut être toujours, essayer provoquer un trnsformation correspondent pour nos étudiants. J'éxaminerai en détail à les frontières serrés, la narration des histoires mathématiques et la construction des problèmes fondamentales qui examine la comprehension.
1. BACKGROUND There is evidence from earliest historical records that examples play a central rol... more 1. BACKGROUND There is evidence from earliest historical records that examples play a central role in both the development of mathematics as a discipline and in the teaching of mathematics. It is not surprising therefore that examples have found a place in many ...
Foram propostas duas tarefas a alunos portugueses do ensino básico para os encorajar a raciocinar... more Foram propostas duas tarefas a alunos portugueses do ensino básico para os encorajar a raciocinar matematicamente sem o recurso a cálculos, com o objetivo de procurar evidências sobre diferentes formas de atenção (Mason, 2003) e de usar essas evidências para refletir sobre as tarefas e sobre a capacidade dos alunos raciocinarem "com razoabilidade" em matemática. A primeira tarefa, centrada na localização de um lugar secreto usando uma applet, foi resolvida por dois pares de alunos do 4.º ano de escolaridade. A segunda, envolvendo a estrutura de quadrados mágicos, foi proposta a duas turmas do 7.º ano de escolaridade. Em qualquer dos casos, procedeu-se à gravação e transcrição das interações entre alunos e professor e na segunda tarefa recolheram-se, ainda, as resoluções escritas dos alunos. Os dados obtidos foram analisados usando um modelo composto por cinco formas ou microestruturas de atenção. No caso da primeira tarefa, evidencia-se que as crianças são capazes de racio...
This paper elaborates the notion of a personal example space as the set of mathematical objects a... more This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to three studies spanning the age range of learners, from school-age learners to pre-service teachers learning maths and professional mathematicians. Their constructions of examples are used as evidence of their personal example spaces. From these, we identify characteristics of such spaces that provide insight into learning mathematics. This perspective informs teaching by giving access to how personal knowledge is structured and what might enhance that structure.
This paper elaborates the notion of a personal example space as the set of mathematical objects a... more This paper elaborates the notion of a personal example space as the set of mathematical objects and construction techniques that a learner has access to as examples of a concept while working on a given task. This is different from the conventional space of examples that is represented by the worked examples and exercises in textbooks. We refer to three studies spanning the age range of learners, from school-age learners to pre-service teachers learning maths and professional mathematicians. Their constructions of examples are used as evidence of their personal example spaces. From these, we identify characteristics of such spaces that provide insight into learning mathematics. This perspective informs teaching by giving access to how personal knowledge is structured and what might enhance that structure.
The word rational is often associated with reasoning and ‘being reasonable’. Henri Poin- caré (19... more The word rational is often associated with reasoning and ‘being reasonable’. Henri Poin- caré (1956) expressed surprise that people find mathematics difficult to learn, because from his perspective it is entirely rational, and humans are rational beings. Jonathon Swift (1726) had however already challenged this notion, proposing that human beings are at best ‘animals capable of reason’.
Teaching students to reason mathematically seems therefore to belong to a disputed domain between ‘they do it already so it is only a matter of evoking and provoking it and drawing it to attention’ and ‘it requires specific and explicit teaching in order to come to students’ minds when required’. Indeed many teachers would claim from their expe- rience that it is difficult to persuade students to use various forms of reasoning such as reasoning by contradiction or by contrapositive. On the one hand very young children reason empirically (they pick up patterns in human and non-human behaviour and act as if they have internalised these); on the other hand they don’t seem to know what to do when asked to justify something mathematically. The stance taken here is that there are delicate shifts of attention involved in reasoning ‘reasonably’1 in mathematics, and that teachers can be of considerable support if they are attuned to these different ways of attending.
In this paper we briefly describe two tasks that we conjecture can provoke young stu- dents to display mathematical reasoning, and report the responses to these tasks of some students in grades 4 and 7. We use the theoretical framework of structures of attention (Mason 2003) to highlight different aspects of student’s reasoning.
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Papers by John Mason
Teaching students to reason mathematically seems therefore to belong to a disputed domain between ‘they do it already so it is only a matter of evoking and provoking it and drawing it to attention’ and ‘it requires specific and explicit teaching in order to come to students’ minds when required’. Indeed many teachers would claim from their expe- rience that it is difficult to persuade students to use various forms of reasoning such as reasoning by contradiction or by contrapositive. On the one hand very young children reason empirically (they pick up patterns in human and non-human behaviour and act as if they have internalised these); on the other hand they don’t seem to know what to do when asked to justify something mathematically. The stance taken here is that there are delicate shifts of attention involved in reasoning ‘reasonably’1 in mathematics, and that teachers can be of considerable support if they are attuned to these different ways of attending.
In this paper we briefly describe two tasks that we conjecture can provoke young stu- dents to display mathematical reasoning, and report the responses to these tasks of some students in grades 4 and 7. We use the theoretical framework of structures of attention (Mason 2003) to highlight different aspects of student’s reasoning.
Teaching students to reason mathematically seems therefore to belong to a disputed domain between ‘they do it already so it is only a matter of evoking and provoking it and drawing it to attention’ and ‘it requires specific and explicit teaching in order to come to students’ minds when required’. Indeed many teachers would claim from their expe- rience that it is difficult to persuade students to use various forms of reasoning such as reasoning by contradiction or by contrapositive. On the one hand very young children reason empirically (they pick up patterns in human and non-human behaviour and act as if they have internalised these); on the other hand they don’t seem to know what to do when asked to justify something mathematically. The stance taken here is that there are delicate shifts of attention involved in reasoning ‘reasonably’1 in mathematics, and that teachers can be of considerable support if they are attuned to these different ways of attending.
In this paper we briefly describe two tasks that we conjecture can provoke young stu- dents to display mathematical reasoning, and report the responses to these tasks of some students in grades 4 and 7. We use the theoretical framework of structures of attention (Mason 2003) to highlight different aspects of student’s reasoning.