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Some General Opening Remarks
1
o
Some General Opening Remarks 'Much ojwhat OUT students have actually learned - more precisels, what they have invented JOT themselves - is a set oj "coping skills "for getting past the next assignment OT examination. liVhen their coping skills jail them, they invent new ones. We seesome oj the "best" students in the country; what makes them "best" is that their coping skills have worked better than most for getting them past the barriers we use to S01t students. We can assure you that that does not mean our students have any real advantage in terms oj understanding mathematics. ' (Smith and Moore, 1991, quoted in Anderson et al., 1998, as continuing to be accurate)
This book is directed towards people who fmd themselves teaching mathematics, either to students who have been told they need mathematics for their own discipline (such as economics, science, engineering, or management), or to those who are studying mathematics for its own sake. It assumes that there are lectures and tutorials (possibly with additional problem classes, labs, or exercise classes), some of which may be repeated to more than one group of students, and that your students hand in work on which they receive feedback. What does one actually do as a teacher? Standing up and talking at your students, displaying diagrams, setting homework, asking questions and asking your students to ask questions or discuss what has been said among themselves are all parts of teaching. The last activity is much more precise than the others, and is typical of the level of detail provided in this book. It takes only a few seconds to do it, but its consequences can be long lasting. I want to refer to these detailed acts in some generic manner. That is, I want to distinguish within 'lecturing' a collection of specific acts, such as pointing to some part of a diagram, or pausing intentionally. The term I have decided to use is tactic, because it suggests a short-term goal rather than a long-term aim. Furthermore, it sounds a bit like tact, signalling that tactics are to be carried out tactfully, and not as an imposition or a demand. For me, it also has the sense of tacking in sailing: you make progress not by heading directly towards your goal but by taking account of the prevailing conditions. Most importantly, tactics are intended to stimulate or enable students to take the initiative, to act upon the subject matter and so to learn mathematics more effectively. A major concern about this approach that is shared by many teachers is that every tactic employed takes a few moments (or more) away from covering all the topics in the syllabus. However, I have found that, by sacrificing either or both of control and time in the short term, I can achieve the long-term goal of getting my students to learn more effectively and also enjoy it more. As a result, it is possible to cover at least as much as before, sometimes in more depth. The book consists mainly of a collection of tactics, but they are all held together by and generated from one central concern: to stimulate students to take the
Some General Opening Remarks
2
initiative to act upon the mathematical ideas and make sense of them, and not just attempt to master a succession of techniques. Some colleagues have commented that the huge range of tactics suggested here makes them feel guilty that they do not spend as much time on their teaching as perhaps they could. My own experience has been that time spent working on teaching has enhanced my research. Although this may seem unlikely, the principal effect of teaching has been to sharpen my sensitivity to my own thinking processes, to where my attention is directed. When this carries over into research, it enhances research activity as well. In addition, interesting connections and problems can come to light when constructing tasks for students (Cuoco, 2000). However, you must test this conjecture, like everything else suggested here, for yourself. It is also unwise to work on more than two aspects of your teaching at anyone time. A wide range of possibilities is offered here, in the hope that there will be something to suit everyone, whatever their teaching task. There is no implicit or explicit suggestion to work on everything!
Preparing to Teach Before embarking on this journey, it may be useful to ponder your basic assumptions and beliefs about how mathematics is most effectively learned, because these determine to a large extent what sorts of things you will try and why you might want to try them. Just as with assumptions in mathematics that need to be explicitly stated so as to be taken into account in proofs, it is valuable to bring your assumptions and beliefs about teaching and learning to the surface so that they too can be examined, questioned, and perhaps modified. Like any mathematical ccajecture, as long as they remain implicit or below the surface of your awareness, they will have a strong influence but will not be open to challenge.
Task: Assumptions and Beliefs Mark the entries in the following table that, together, come closest to capturing what you feel to be the most important aspects of learning and teaching mathematics, adding your own if you wish. Students learn mathematics most effectively by Doing lots of examples for themselves
Reading through their notes or a textbook carefully in the light of their own examples
Reconstructing theorems and techniques for themselves
Being shown how ideas can be formalised or abstracted
Following a clearly laid out presentation, line by line and symbol by symbol
Following the development of definitions, lemmas, theorems, and proofs
Posing and solving problems
Working by themselves
Working with others
Discussing topics with others
3
Preparing to Teach
Comment:
In looking through this list of short statements, you probably found something positive in most of them. Try to find or formulate one or two that sum up for you how students learn best; the sort of thing you might find yourself saying to a colleague, or thinking when listening to someone discussing various forms of teaching. It is likely that you drew on your own experience, perhaps recognising elements of what seemed to work for you. However, memories are not always entirely reliable, though we often tend to base our teaching on what we think we did as students.
Task: Teaching Now consider your responsibilities as a teacher, again adding your own entries as necessary. My responsibility as a teacher is to Set out all the details as clearly and logically as possible
Stimulate my students to making sense of something for themselves
Provide motivation and applications for the material covered in the text
Cover all the techniques they will be tested on
Show my students how to wrestle with mathematics the way a mathematician does
Startle and surprise my students, generating dilemmas that they will have to resolve
Concentrate on techniques
Concentrate on meaning and understanding
Introduce new ideas and concepts
Display links between topics
Now go back and see in what way you disagree with, or place much less emphasis on, the other entries (each entry is espoused by some very good teachers!). Comment:
As before, you probably found at least something positive in most of the statements. Try to again find or formulate the one or two that represent for you the principal contribution you can make to supporting your students in learning effectively. Did you think to add something about assessment, such as that final assessment should be similar to the assessments and exercises used during the course, or that assessment needs to be both challenging and yet confidence developing?
No matter what perspective you hold, perhaps the fundamental question for any lecturer, tutor, or marker is how to stimulate their students to take the initiative. Perhaps your students do not actually know what it means to take the initiative with respect to mathematics. After all, their experience may be of a complex subject with a multiplicity of technical terms and techniques, which all seem well worked out. They may see their task as being to reproduce set behaviour under examination conditions.
4
Some General Opening Remarks Perhaps your students do not really know what it is like to be mathematical, to think mathematically, to recognise situations as opportunities to ask questions that can be worked on mathematically. Maybe what your students need most is to be in the presence of someone who is 'being mathematical': someone who asks mathematical questions mathematically (Mason et al., 1982, Mason, 2000). In almost all cases the topics being taught are in fact well rehearsed and familiar to the lecturer, so it is tempting to act like a talking textbook, to reproduce the distilled essence without treating the topic as an example of how mathematicians work and think. Yet if we want our students to be enthused by mathematics, to approach it eagerly and positively, and if we want them to appreciate what mathematics is like as a discipline rather than simply as a body of defmitions, theorems, proofs and techniques, then it behoves us to be mathematical with and in front of our students. If we want our students to encounter not just techniques, but structures, heuristics, and ways of thinking pertinent to the particular mathematical field being taught, then we need to display these explicitly. This does not mean that it is effective to walk in and solve a lot of problems, formulate definitions and prove theorems in front of them, mindless of their presence. On the other hand, neither is it effective to give a truncated and stylised presentation which supports the impression that mathematics is completely cut, dried and salted away, that it is something that one can either pick up easily or not at all. The most effective method is to display aspects of mathematical thinking, such as forming and questioning mental images supported by diagrams, constructing examples to probe as well as illustrate theorems and techniques, asking mathematical questions about situations, and making and modifying conjectures publicly. It must be noted that in our present consumer-oriented society, student comments on their experience as students are of great importance for quality assurance. Consequently, if you are going to embark on changing your practices, you must make sure you take your students with you. One useful way to think about this is in terms of the existence of an implicit contract between you and your students: they expect you to give them facts and tasks, and they expect that, through memorising the facts and doing the tasks, learning will take place. You, in tum, expect them not only to do the tasks, but to be able to reconstruct the techniques and use these in a variety of contexts. This is one of a number of issues that have been studied and debated in some depth in the mathematics education literature and which are discussed in Chapter 7.
5
Reflection
Reflection Task: Reflection What specific issues about teaching mathematics trouble you at the moment? What examples can you find from your own experience to illustrate or challenge the beliefs outlined briefly above? Comment:
Issues mentioned by current lecturers include:
o o o o
how to interest your students in really working at mathematics, especially in service courses; how to leave a lecture feeling that some students have actually got something from it; how to help students in a tutorial or problem class without just doing it for them; how to cope with a wide range of backgrounds and mathematical facilities.
o
Some General Opening Remarks 'Much ojwhat OUT students have actually learned - more precisels, what they have invented JOT themselves - is a set oj "coping skills "for getting past the next assignment OT examination. liVhen their coping skills jail them, they invent new ones. We seesome oj the "best" students in the country; what makes them "best" is that their coping skills have worked better than most for getting them past the barriers we use to S01t students. We can assure you that that does not mean our students have any real advantage in terms oj understanding mathematics. ' (Smith and Moore, 1991, quoted in Anderson et al., 1998, as continuing to be accurate)
This book is directed towards people who fmd themselves teaching mathematics, either to students who have been told they need mathematics for their own discipline (such as economics, science, engineering, or management), or to those who are studying mathematics for its own sake. It assumes that there are lectures and tutorials (possibly with additional problem classes, labs, or exercise classes), some of which may be repeated to more than one group of students, and that your students hand in work on which they receive feedback. What does one actually do as a teacher? Standing up and talking at your students, displaying diagrams, setting homework, asking questions and asking your students to ask questions or discuss what has been said among themselves are all parts of teaching. The last activity is much more precise than the others, and is typical of the level of detail provided in this book. It takes only a few seconds to do it, but its consequences can be long lasting. I want to refer to these detailed acts in some generic manner. That is, I want to distinguish within 'lecturing' a collection of specific acts, such as pointing to some part of a diagram, or pausing intentionally. The term I have decided to use is tactic, because it suggests a short-term goal rather than a long-term aim. Furthermore, it sounds a bit like tact, signalling that tactics are to be carried out tactfully, and not as an imposition or a demand. For me, it also has the sense of tacking in sailing: you make progress not by heading directly towards your goal but by taking account of the prevailing conditions. Most importantly, tactics are intended to stimulate or enable students to take the initiative, to act upon the subject matter and so to learn mathematics more effectively. A major concern about this approach that is shared by many teachers is that every tactic employed takes a few moments (or more) away from covering all the topics in the syllabus. However, I have found that, by sacrificing either or both of control and time in the short term, I can achieve the long-term goal of getting my students to learn more effectively and also enjoy it more. As a result, it is possible to cover at least as much as before, sometimes in more depth. The book consists mainly of a collection of tactics, but they are all held together by and generated from one central concern: to stimulate students to take the
Some General Opening Remarks
2
initiative to act upon the mathematical ideas and make sense of them, and not just attempt to master a succession of techniques. Some colleagues have commented that the huge range of tactics suggested here makes them feel guilty that they do not spend as much time on their teaching as perhaps they could. My own experience has been that time spent working on teaching has enhanced my research. Although this may seem unlikely, the principal effect of teaching has been to sharpen my sensitivity to my own thinking processes, to where my attention is directed. When this carries over into research, it enhances research activity as well. In addition, interesting connections and problems can come to light when constructing tasks for students (Cuoco, 2000). However, you must test this conjecture, like everything else suggested here, for yourself. It is also unwise to work on more than two aspects of your teaching at anyone time. A wide range of possibilities is offered here, in the hope that there will be something to suit everyone, whatever their teaching task. There is no implicit or explicit suggestion to work on everything!
Preparing to Teach Before embarking on this journey, it may be useful to ponder your basic assumptions and beliefs about how mathematics is most effectively learned, because these determine to a large extent what sorts of things you will try and why you might want to try them. Just as with assumptions in mathematics that need to be explicitly stated so as to be taken into account in proofs, it is valuable to bring your assumptions and beliefs about teaching and learning to the surface so that they too can be examined, questioned, and perhaps modified. Like any mathematical ccajecture, as long as they remain implicit or below the surface of your awareness, they will have a strong influence but will not be open to challenge.
Task: Assumptions and Beliefs Mark the entries in the following table that, together, come closest to capturing what you feel to be the most important aspects of learning and teaching mathematics, adding your own if you wish. Students learn mathematics most effectively by Doing lots of examples for themselves
Reading through their notes or a textbook carefully in the light of their own examples
Reconstructing theorems and techniques for themselves
Being shown how ideas can be formalised or abstracted
Following a clearly laid out presentation, line by line and symbol by symbol
Following the development of definitions, lemmas, theorems, and proofs
Posing and solving problems
Working by themselves
Working with others
Discussing topics with others
3
Preparing to Teach
Comment:
In looking through this list of short statements, you probably found something positive in most of them. Try to find or formulate one or two that sum up for you how students learn best; the sort of thing you might find yourself saying to a colleague, or thinking when listening to someone discussing various forms of teaching. It is likely that you drew on your own experience, perhaps recognising elements of what seemed to work for you. However, memories are not always entirely reliable, though we often tend to base our teaching on what we think we did as students.
Task: Teaching Now consider your responsibilities as a teacher, again adding your own entries as necessary. My responsibility as a teacher is to Set out all the details as clearly and logically as possible
Stimulate my students to making sense of something for themselves
Provide motivation and applications for the material covered in the text
Cover all the techniques they will be tested on
Show my students how to wrestle with mathematics the way a mathematician does
Startle and surprise my students, generating dilemmas that they will have to resolve
Concentrate on techniques
Concentrate on meaning and understanding
Introduce new ideas and concepts
Display links between topics
Now go back and see in what way you disagree with, or place much less emphasis on, the other entries (each entry is espoused by some very good teachers!). Comment:
As before, you probably found at least something positive in most of the statements. Try to again find or formulate the one or two that represent for you the principal contribution you can make to supporting your students in learning effectively. Did you think to add something about assessment, such as that final assessment should be similar to the assessments and exercises used during the course, or that assessment needs to be both challenging and yet confidence developing?
No matter what perspective you hold, perhaps the fundamental question for any lecturer, tutor, or marker is how to stimulate their students to take the initiative. Perhaps your students do not actually know what it means to take the initiative with respect to mathematics. After all, their experience may be of a complex subject with a multiplicity of technical terms and techniques, which all seem well worked out. They may see their task as being to reproduce set behaviour under examination conditions.
4
Some General Opening Remarks Perhaps your students do not really know what it is like to be mathematical, to think mathematically, to recognise situations as opportunities to ask questions that can be worked on mathematically. Maybe what your students need most is to be in the presence of someone who is 'being mathematical': someone who asks mathematical questions mathematically (Mason et al., 1982, Mason, 2000). In almost all cases the topics being taught are in fact well rehearsed and familiar to the lecturer, so it is tempting to act like a talking textbook, to reproduce the distilled essence without treating the topic as an example of how mathematicians work and think. Yet if we want our students to be enthused by mathematics, to approach it eagerly and positively, and if we want them to appreciate what mathematics is like as a discipline rather than simply as a body of defmitions, theorems, proofs and techniques, then it behoves us to be mathematical with and in front of our students. If we want our students to encounter not just techniques, but structures, heuristics, and ways of thinking pertinent to the particular mathematical field being taught, then we need to display these explicitly. This does not mean that it is effective to walk in and solve a lot of problems, formulate definitions and prove theorems in front of them, mindless of their presence. On the other hand, neither is it effective to give a truncated and stylised presentation which supports the impression that mathematics is completely cut, dried and salted away, that it is something that one can either pick up easily or not at all. The most effective method is to display aspects of mathematical thinking, such as forming and questioning mental images supported by diagrams, constructing examples to probe as well as illustrate theorems and techniques, asking mathematical questions about situations, and making and modifying conjectures publicly. It must be noted that in our present consumer-oriented society, student comments on their experience as students are of great importance for quality assurance. Consequently, if you are going to embark on changing your practices, you must make sure you take your students with you. One useful way to think about this is in terms of the existence of an implicit contract between you and your students: they expect you to give them facts and tasks, and they expect that, through memorising the facts and doing the tasks, learning will take place. You, in tum, expect them not only to do the tasks, but to be able to reconstruct the techniques and use these in a variety of contexts. This is one of a number of issues that have been studied and debated in some depth in the mathematics education literature and which are discussed in Chapter 7.
5
Reflection
Reflection Task: Reflection What specific issues about teaching mathematics trouble you at the moment? What examples can you find from your own experience to illustrate or challenge the beliefs outlined briefly above? Comment:
Issues mentioned by current lecturers include:
o o o o
how to interest your students in really working at mathematics, especially in service courses; how to leave a lecture feeling that some students have actually got something from it; how to help students in a tutorial or problem class without just doing it for them; how to cope with a wide range of backgrounds and mathematical facilities.