Mathematics as a Constructive Activity

Learners Generating Examples

Anne Watson, John Mason

• Publisher: Routledge
• Year: 2006
• ISBN:
  9781135630010
• Edition: 1
• Topic: Education
• Subtopic: Education General

This book explains and demonstrates the teaching strategy of asking learners to construct their own examples of mathematical objects. The authors show that the creation of examples can involve transforming and reorganizing knowledge and that, although this is usually done by authors and teachers, if the responsibility for making examples is transferred to learners, their knowledge structures can be developed and extended. A multitude of examples to illustrate this is provided, spanning primary, secondary, and college levels. Readers are invited to learn from their own past experience augmented by tasks provided in the book, and are given direct experience of constructing examples through a collection of many tasks at many levels. Classroom stories show the practicalities of introducing such shifts in mathematics education. The authors examine how their approach relates to improving the learning of mathematics and raise future research questions.*Based on the authors' and others' theoretical and practical experience, the book includes a combination of exercises for the reader, practical applications for teaching, and solid scholarly grounding.
*The ideas presented are generic in nature and thus applicable across every phase of mathematics teaching and learning.
*Although the teaching methods offered are ones that engage learners imaginatively, these are also applied to traditional approaches to mathematics education; all tasks offered in the book are within conventional mathematics curriculum content. Mathematics as a Constructive Activity: Learners Generating Examples is intended for mathematics teacher educators, mathematics teachers, curriculum developers, task and test designers, and classroom researchers, and for use as a text in graduate-level mathematics education courses.

1
Introduction to Exemplification in Mathematics

We start with a task that concerns factors of whole numbers:

Task 1a: Two Factors

Think of some integers that have only two factors.

Nothing amazing will have happened yet! You might have to decide what you mean by factors. Do you include or exclude the number itself, and what about 1? However, because you are reading this without us you do not have to worry about our definitions; just use your own definitions, and stick to them. Whatever you decide, it gets more interesting when you develop this task:

Task 1b: More Factors

Think of some integers that have only three factors.

Only four factors.

Only five factors.

As we said in the preface, reading a task and then moving directly on to our commentary may be tempting, but it may not give you an experience of full participation in or an appreciation of the ideas of this book.¹ You may want to read no further until you have considered the numbers you have just produced. What do they have in common? How do they differ? Are these classes of objects useful or interesting in any way?

A central theme of the book is that mathematics is learned by becoming familiar with examples that manifest and illustrate mathematical ideas and by constructing generalizations from examples. The more of this we can do for ourselves, the more we can make the territories of mathematics our own.

A higher level question that we expect teachers to ask themselves is “would I use this task with students? If so, with whom, why, and how? If not, what adjustments would be needed?’’ It is very tempting to reject tasks found in books as inappropriate for your situation. The tasks that we offer are not just specific tasks but rather illustrate possible task structures that have proved at least interesting and productive. So rejecting a task immediately may block access to potential value.

Task 2 addresses the issue of helping learners distinguish among several technical terms by working explicitly on what distinguishes them.

Task 2: Mode, Median, and Mean

Construct a data set of seven numbers for which the mode is 5, the median is 6, and the mean is 7.

Alter it to make the mode 10, the median 12, and the mean 14; alter it to make the mode 8, the median 9, and the mean 10.

Is it possible to preassign any value to each mode, median, and mean independently and restrict the data set to just five data points?

How small a data set can achieve any preassigned mode, median, and mean? How much choice of data set is there then?

A common approach is to start with the mode because it has to be repeated, tinker with adding extra data values so as to make the mean work, and then adjust those values to achieve the median. One pedagogical aim of the task is to promote awareness of the range of possibilities and to raise questions about what it is about a data set that is captured by each of the three statistical summarizers.

But what was your experience of doing this task? How did you choose to start, and why? If you started with particular numbers, why those? If you started with some general relationships, what was the pathway by which you moved toward particular data sets if, indeed, you bothered to do that? You may have chosen to stay with the generalities. You may have become aware of generalities as you did the task; if so, what did you do with this awareness? What choices were you making, and how were they linked to your knowledge of underlying relationships?

Similar tasks could be used to emphasize the difference between products, sums, and differences (e.g., for pairs of numbers) as well as between commutativity and associativity in groups and rings.

Having promoted your reflection, perhaps it is already time to address a fundamental question.

WHAT IS AN EXAMPLE?

In this book we focus on the learner’s experience rather than on definitions. Because most of what is offered to learners in schools and colleges is intended to indicate some kind of generality (a concept, a class, a technique, a principle, etc.), we use the word example in a very broad way to stand for anything from which a learner might generalize. Thus, example refers to the following:

• Illustrations of concepts and principles, such as a specific equation that illustrates linear equations or two fractions that demonstrate the equivalence of fractions.
• Placeholders used instead of general definitions and theorems, such as using a dynamic image of an angle whose vertex is moving around the circumference of a circle to indicate that angles in the same segment are equal.
• Questions worked through in textbooks or by teachers as a means of demonstrating the use of specific techniques, which are commonly called worked examples.
• Questions to be worked on by students as a means of learning to use, apply, and gain fluency with specific techniques, which are usually called exercises.
• Representatives of classes used as raw material for inductive mathematical reasoning, such as numbers generated by special cases of a situation and then examined for patterns.
• Specific contextual situations that can be treated as cases to motivate mathematics.

There are deep and significant questions concerning just how examples actually illustrate or exemplify, and these will be addressed later in the book. We use exemplification to describe any situation in which something specific is offered to represent a general class with which the learner is to become familiar—a particular case of a generality. For instance:

• A trial examination question may be offered to represent the kind of questions learners will have to answer.
• A specific object may be used to indicate what is included and what is excluded by a condition in a definition or theorem; for example, a drawing of a particular rectangle to accompany a definition of rectangles given in a textbook.
• A specific object may be used to indicate the significance of a particular condition in a definition or theorem by highlighting its role in a proof or by showing how the proof fails in the absence of that condition; for example, using f(x)=|x| to indicate how continuity on its own is not enough to ensure that every value of x has a derivative associated with it.
• A specific object may be offered to indicate a dimension of variation implied by a generalization; for example, in Task 1b: More Factors we could have offered or been offered 25 and 225 as numbers that both have odd numbers of factors, are both square, but do not have the same number of factors; thus, they offer some ideas about a possible generalization and also put limitations on what can be claimed.
• An object may be chosen to illustrate a complex structure but is made up of simpler objects familiar to the learner; for example, the expression 3(4+5) to introduce distributivity.
• A generic diagram may be used to indicate something that remains invariant while some other features change; for example, as suggested earlier, a drawing of one generic triangle (that does not look equilateral, isosceles, or right-angled) can be used to illustrate the coincidence of medians.

All of these situations require the learner to see the general through the particular, to generalize, to experience the particular as exemplary to appreciate a technical term, theorem, proof, or proof structure, and so on.

Possibly you already find yourself disagreeing with us about some of these matters. For instance, textbooks typically give examples of rectangles whose sides are parallel to the edges of the paper, often in the ratio 2:1 and with the longest side horizontal. How, then, can such a diagram represent all possible rectangles? One diagram of a triangle does not convince learners about all possible triangles, so why not use dynamic geometry software instead of a static drawing? The first numbers you find with 3, 5, and 7 factors are all squares, but will this always be the case?

These are precisely the kinds of questions that we have been asking ourselves for some time. Clearly, one special example may not enough to give learners an idea of the full extent of what is possible, and it may indeed be misleading in its details.

In this book we develop the idea of example spaces: collections of examples that fulfill the kinds of functions just listed and suggest that these collections might be seen as central in the teaching and learning of mathematics. We have found ourselves using and extending language introduced by Ference Marton to describe the structure of example spaces in terms of dimensions of possible variation (our adaptation from Marton & Booth, 1997; see also Leung, 2003), which constitutes a generality that can be read into or through examples, and the associated notion of range of permissible change in each of the dimensions of variation. These terms appear through the book and are elaborated in chapter 5. A different way to describe the structure of example spaces is in terms of detecting invariance of some features while others are changing, which is powerfully accessed through use of the prompt “what is the same and what is different’’ about a collection of objects (see, e.g., L.Brown & Coles, 2000). Again, this notion is developed through the book.

At the start of this chapter we offered a task with variations that was intended to trigger an exploration of an example space emerging in your response to the task itself. You may have made use of decomposition of multiples into their prime factorization and stated that only prime numbers have precisely two factors—themselves and unity. You may have been on familiar territory. Then we asked for numbers with three factors. You may have continued to look at prime factorizations and tried to construct numbers using three factors including unity; but then you noted that the number created also has to be a factor as well, so every time you multiply two of the factors you get a new factor! How can you have two factors that, when multiplied together to make the final number, do not produce a new factor as well? Or, you may have used a decomposition approach and listed the factors of various numbers systematically, noticing that they occur in pairs. Aha! If they occur in pairs, how can you ever have an odd number of them? But as you decompose several numbers you may begin to group them according to the number of factors they have and thus create classes that you may not previously have recognized as classes. What do 15 and 21 have in common? What do 12 and 18 have in common? Eventually, you may have extended your search to some numbers other than small integers.

If counting factors is familiar territory, you could have asked yourself whether a prime number of factors produces a recognizable class or whether you can characterize cubes or higher powers in terms of num-bers of factors. Any task can be opened up to further exploration by altering constraints.

Notice how working on this task invites an interplay between what is familiar and what is unfamiliar. Every now and then you may have experienced a rush of recognition as a new idea took shape and, maybe, was recorded in some way. The numbers you used as examples are all old friends, but you looked at some of them in new ways perhaps. You may even have chosen to use particular numbers because of properties you already knew, like choosing 24 because it has several factors, 81 because it is a square of an odd square, 1,760 because it is a larger nonprime number, or 2²×3⁴×5² because you can “see” how to count factors (Campbell & Zazkis, 2002). But 1,760 may not have been big or complex enough, so you may have had to construct particular numbers that have the desired properties. You have explored your example space, looked at what you know in new ways, and constructed new objects to occupy new or extended example spaces. At the heart of this approach to teaching lie two important pedagogical principles:

• Learning mathematics consists of exploring, rearranging, and extending example spaces and the relationships between and within them. Through developing familiarity with those spaces, learners can gain fluency and facility in associated techniques and discourse.
• Experiencing extensions of your example spaces (if sensitively guided) contributes to flexibility of thinking not just within mathematics but perhaps even more generally, and it empowers the appreciation and adoption of new concepts.

These principles are illustrated in what follows through tasks and commentaries. You can examine your own experience throughout this book, either by trying the tasks yourself or with learners or by imagining what would happen in your usual teaching situation to get a sense of the effect of such prompts.

ON WHOSE SHOULDERS ARE WE STANDING?

We ground our work in a well-established tradition of questioning the relations between experiences, ideas, and knowledge as well as in the pedagogical wisdom of friends, colleagues, and others. Authors’ expressions of this wisdom can be found in every period of recorded history.

Plato wondered how we could come to understand color, shape, or indeed any abstract noun. If we see examples of white and hear them being called white, we will eventually construct our own understanding of what it means to be white and will be able to use the word in ways that others understand. How, he asked, can color be explained to a blind person who has to rely on definition and abstraction? Thus, he highlighted an apparent paradox that must be resolved by any constructivist view of learning: How can we construct what we do not already know? This is only a paradox if you assume there is a fixed meaning for white that is independent of the knower, and a true meaning the learner has to strive to understand. Similarly, learners of mathematics strive to make sense of the examples they are offered, use the terms their teachers use to describe generalities, and ultimately are expected to construct new objects and understandings that match those of their teachers.

George Polya (1981) was prolific both as a mathematician and as a reflective and articulate educator. He made use of the terms extreme, leading, and representative as types of examples (p. 10), which other authors also use. Extreme examples involve going to the edge of what usually happens within the particular mathematical context and seeing what unusually happens. For instance, young children might believe that multiplication makes things bigger; but even if we restrict multiples to positive integers, we find that multiplying by one does not make things bigger. Furthermore, multiplication by zero obliterates everything! Extreme examples, therefore, confound our expectations, encourage us to question beyond our present experience, and prepare us for new conceptual understandings. Multiplication by zero prepares us for new understandings of multiplication just as treating a circle as an n-sided polygon when n grows very large prepares us for work with limits.

Drawing on Polya’s ideas, Edwina Michener (1978) experimented with the explicit use of several different types of examples of mathematical concepts in her teaching of undergraduates. We draw particularly on the notion of a reference example in chapter 5. A reference example is one that becomes extremely familiar and is used to test out conjectur...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Preface
• 1: Introduction to Exemplification in Mathematics
• 2: Learner-Generated Examples in Classrooms
• 3: From Examples to Example Spaces
• 4: The Development of Learners’ Example Spaces
• 5: Pedagogical Tools for Developing Example Spaces
• 6: Strategies for Prompting and Using Learner- Generated Examples
• 7: Mathematics as a Constructive Activity
• Appendix A: Some Historical Remarks on Teaching by Examples
• Appendix B: Suggestions About Some of the Tasks
• References
• Notes