Keynote Part I

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Problem #3: Defining Terms and Concepts

If you have ever taught any elementary grade, you've probably heard this question. Children often wonder, and ask, "Is zero even or odd?"

Defining Concepts and Terms


So here we will examine a third kind of work that teachers do, one that I'll call defining concepts and terms.

I'm going to take us through two stages of this. One thing one might hope is that teachers, when confronted with questions like this, would appreciate that mathematics is a domain where definitions play a central role. Addressing this question about the parity of 0 requires being able to identify a definition that can help students to resolve the question.

The first thing I'm going to show you is a set of textbook definitions that teachers would likely encounter:

Examining Textbook Definitions

How well does any of these help to answer the question?

  1. An even number is a number of the form 2k, where k is an integer.
  2. An even number is a natural number that is divisible by 2.
  3. An even number is any multiple of 2.
  4. An even number is a number that has 0, 2, 4, 6, or 8 in the ones place.

The first is an example of a definition a teacher is likely to have learned in a math course for teachers. The second, third, and fourth are examples drawn directly from U.S. textbooks. What I'd like you to think about is the adequacy of each definition for helping the teacher answer the student or to help the students answer their own question. Spend a moment looking through these definitions asking yourself, "How well does any of these definitions help to answer the question about whether 0 is even or odd?" Keep in mind that the context here is, let's say, third grade. In thinking about how well the definition helps to answer the question, you're thinking about both its mathematical features and its usefulness as a definition.

What do you think about these definitions and their usefulness in resolving the question? Do you find any of them particularly helpful? Are any quite useless? What are some of their differences?

One interesting observation is that the definitions differ with respect to which numbers are included in the domain to be considered as even or odd. The second definition does not include integers and the first, obviously, explicitly does. The other two definitions leave it tacit: We don't know exactly what is being intended there.

Let's stretch just a little now. Again, in this stretching, I'm still not moving to teaching but I am moving in the direction of knowledge use. In mathematics, a definition needs to function in a way that's usable by a particular user, in a particular context, in order to help make certain kinds of distinctions. Therefore, if you have a definition that is mathematically correct but incomprehensible to the person who wants to use it, it's not a very good definition.

In many ways, the first definition is fine. It's a perfectly conventional definition and it would be useful to many of us in the audience to discern whether zero is even or odd. It's probably not a useful definition for a third grader, however. It's not useful because it contains terms and notation that wouldn't be usable yet. The kind of thinking a teacher must do here is quite complex. It includes reasoning through the question, "How does one make up a mathematically viable definition that's accurate - correct - and yet usable by students?"

It's a very subtle form of mathematical understanding to realize what mathematical definition is and needs to do. From the perspective of a mathematical user who is only eight years old, each of these definitions has various features: The first is relatively incomprehensible; the second includes the term "natural number" which may or may not be comprehensible; the third relies on knowing what "multiple" means. The fourth is a useful criterion for evenness, but it is not so natural as a definition since it is not conceptually intrinsic, relying as it does on special feature of place value notation. For example, does the number zero have the zero in the ones place? It is a description that focuses on how a number is written using place value notation instead of a concept of "evenness" and "oddness." It doesn't permit you to determine whether numbers are even or odd if they are written using Roman numerals, or in base five.

Let's move to a classroom context where students can - and are encouraged to - offer definitions. For example, the teacher may ask, "So what is a definition of an even number?" I'd like you to look at several student definitions and think about the work of the teacher in sizing up each of these student responses:

Considering Students' Definitions of Even Number

How well does any of these help to answer the question?

1. An even number is a
number you can split in
two equal parts
without having to
break anything in half.

image007.gif      image012.gif

2. An even number is a
number that when you
group it in twos, there
are none left.


3. The even numbers are
every other number on
the number line, like 2,
4, 6, and so on.


These are three definitions we often hear students using. Again, the question is still, "Is zero even or odd?" Try to think about how well any of these helps to answer the question.

How well does each of these work as a mathematical definition? How well does it help to answer the question of whether 0 is to be considered even or odd?

(Note: To engage in a discussion on these questions, click on the link above.)

Being able to think through this and how these are similar to or different from one another is important mathematical work. How do these student definitions map back to the textbook definitions? How do they map to each other? There's a lot we could do together about the mathematical adequacy and the usability of these definitions. This kind of detailed thinking about what makes a definition good enough for use is a kind of mathematical judgment that we've not often thought about as central to elementary or middle school teaching but in fact comes up frequently. Rephrased, then, the mathematical problem for a teacher is:


What is a mathematically appropriate
and usable definition of even number
for third graders?


This question requires subtle use of and knowledge about definitions. Additionally, it is not the kind of knowledge of definitions that would necessarily grow from having a mathematics course where you learned the first textbook definition presented above. Such knowledge alone wouldn't be sufficient to help a teacher work through this question.

So, what would you consider to be a mathematically appropriate and useable definition of even numbers for third graders? If you can make one that you think satisfies the two requirements -- that it be mathematically correct, and also usable by eight-year-olds -- share it with others. As we examine proposed definitions, we can talk about what makes a definition usable -- a mathematical question, to be sure. In so doing, we will see some of the mathematical reasoning and knowledge required for the task.

What is your idea of a mathematically appropriate and usable definition of even and odd numbers for third graders? What is your justification that it satisfies both of these considerations?

(Note: To engage in a discussion on these questions, click on the link above.)

We will move next into the third section of this keynote in which we will talk about the implications of thinking of content knowledge for teaching as comprising the use of ideas and reasoning needed to solve problems endemic to practice.

What have you noticed about the mathematical ideas, habits of mind, and practices that you employed as you worked on these problems that arise in teaching mathematics?

(Note: To engage in a discussion on these questions, click on the link above.)

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