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This chapter addresses numeration topics that are typically developed in the middle grades yet have not been addressed in earlier chapters. These include exponents, integers, absolute value, and an investigation of rational and irrational numbers as they make up the real numbers.

Large Numbers, Small Numbers, and Exponents

Exponents and order of operations are more closely allied with algebra although they are not really algebraic topics. Order of operations is strictly a convention that many teachers seem to think should be discovered by students. I think this is a good place to distinguish between a logical or mathematical concept and an arbitrary convention. Simple four-function calculators do not use algebraic logic while most calculators designed for the intermediate grades and up do use algebraic logic. Why are these different? I have made some attempt in this section to help students see why we use exponential notation to express very large or very small quantities. Our technological world has permitted us to explore measures that are very, very small as well as some that are larger than most of us can comprehend. In many fields of study, and even reading the newspaper, these extreme measures are more and more a part of life.

Exponents may be important simply because they have traditionally been given such instrumental treatment in algebra: "When multiplying numbers with a like base add the exponents," and so on and so on. With graphing calculators, there are some nice things that can be done with these topics.


The integers do get considerable attention in the middle grades and it seems that some focus on this topic is worthwhile. Be prepared that even students who have had a significant amount of college mathematics find the modeling of the integer operations to be difficult. And so, won't students in the seventh grade find them difficult also? Absolutely! And that brings up a more serious question that you will want to explore with your class: If the rules for integer computation can be learned by students relatively easily, why would I go to this trouble to try and model them with students? After all, there is almost no evidence that actual computation with signed quantities (integers, reals or even variable quantities) is dependent on thinking about the models. The answer may be in a distinction between learning mathematics and performing routine computations. No student should be permitted to accept a rule on faith without personal justification.

The integers also provide a context for the broader discussion of the use of models in learning mathematics - a review of ideas addressed in Chapter 3. As noted in Chapter 3, the two suggested models, vectors and sets, are superficially quite different. However, onto each of these models can be imposed the concepts of opposite and quantity. Some teachers tend to find one model or the other easier to understand. (I have found this to be a function of how I have introduced and worked with the models.) Why work with both or require students to understand the use of both models. My own response is that if students (your teachers or children in school) cannot "see" the same concepts in both models then perhaps they do not really have a firm understanding of the ideas involved.

Rational Numbers

It is sometime in the middle grades that children begin to think of fractions as numbers rather than in terms of pizza or other physical objects. The research discussion of this transition is very limited. It is also in the middle grades that texts begin to use fraction notation as indicated division. This also gets little or no attention in the literature. I have made some attempt to wrestle with these issues in this book.

Indicated Division. It has always been a bit of a mystery to me that somehow children in about the seventh grade are to begin to believe that 3/5 can mean 3 whole things divided into 5 parts when for all of their prior school years they have been told that it means 3 of 5 equal parts making up one whole. A review of popular textbook series offers limited assistance here. I have offered some ideas for you to consider. Chapter 16 on fraction meanings begins with a development of fractional parts based on sharing problems or what are essentially partitive division problems. Based on this development, the denominator of a unit fraction can be seen as a divisor of one whole. Then the numerator is a multiplier - indicating the number of thee parts are shared with each sharer. Activity 24.3 on page 504, suggests a partitive division context with various symbolic solutions, all of which are correct but which look very different. The discussion around a problem like this may be useful. Perhaps the whole issue of indicated division is simply one that bothers me more than it does you. I think it is important to at least discuss it.

Density of the Rationals. It has always seemed to me that the density of the rational numbers was a fairly mind-boggling idea. Imagine being able to have an infinite number of fractions exist between any two that I pick, regardless of how close they may be to begin with. And then to think that there are even more numbers in between all of those. Wow! But I must admit that you do not run into many middle school teachers who are as blown away by that idea as I am. Further, I am not convinced that I can excite many seventh or eighth grade students about these ideas. So, there is ample argument that perhaps these topics are not worth discussing. To argue on the other side, it seems really interesting that 0.999999… is exactly the same as one and the idea of an infinite decimal may be important. Note that 1/9 = 0.1111111…, 2/9 = 0.22222…, 3/9 = 0,33333…, etc. So, 9/9 = 0.99999… . The existence of the rationals and the density of these things seems to be important if we want students to believe that even though you plot only a few discrete points from a continuous function, that it makes sense to connect those points in a continuous curve. There really are a lot of connected ideas in mathematics.

Real Numbers

An idea basic in algebra is that of an expression representing a quantity. For example, 34-5 is an expression for 76 and is exactly the same as 3x52+1 and also 76. In the same manner, rather than indications that something is to be done, i.e., find the square roots. This idea seems to me to be part of a beginning understanding of the real numbers. That the reals are composed of the rationals and the irrationals and that together they are not only dense but leave no gaps is much more easily said than understood.


How much time and effort you give to this probably depends on how well you think your students understand these topics from their methods classes and perhaps how profound or significant you believe these ideas are for middle-grade students.

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