I had explained to him that what I had planned so far for the book is only undergoing development, not a final plan, and in fact will almost certainly need to change anyway and that my reasons for thinking this were motivated by what Alain had said. Actually, I was wondering about that development even before Alain had said something, but he had encouraged me to think about other issues as well in deciding what goes into the book.
Robert Hansen had replied to me last night, and I just wrote him a reply. Because the comments I had said are of interest on this list, I will include my reply below (since the post has not appeared yet, there is no link I can give):
On 9/17/2010 at 9:31 pm, Robert Hansen wrote:
> Jonathan wrote... > > "Are fractions really necessary--other than code for > division? If so, for what?" > > How can you not include fractions? They are the basis > of ratios, percentages and all other rational > expressions. Can you be more specific by what you > mean when you say "not include fractions"? Is this > primarily a book for remediation?
First, that was a question Alain Schremmer asked, and I think it is a good question to ask and to consider. He does at least make a good point in that perhaps we beat fraction arithmetic to death too much, that many support a heavy emphasis of fractions in math courses simply because of conventional wisdom or mere tradition. If there is indeed a good reason for fraction arithmetic receiving the emphasis it does, I don't see that reason. And here I'm talking about the general student, not those interested in STEM careers or other specialized careers where a strong knowledge of how to perform fraction arithmetic is essential. I don't see that many careers where a strong knowledge of fraction arithmetic is essential.
Adding and subtracting fractional measurements seems necessary to a lot of people but only because some people need that and only because the customary system of measurement uses fractions. The metric system does not, and if America goes completely metric someday--and I'm sure it will happen, but when that will happen is anyone's guess--then all our measurements will be expressed in decimal form. I really do wonder how many people will truly need to know how to add or subtract measurements expressed as fractions, and much of that need will disappear when we go metric. I also wonder how many people can even read rulers for themselves well enough to express lengths as fractions. I would guess that relatively few know how to do that.
Yes, I agree that fractions are the basis of ratios and percentages, but not any significant knowledge of how to perform fraction arithmetic is required. We will need, of course, to know how to take a fractional or percentage amount of a quantity and need to know that nx to ny is the same ratio as x to y, but even that last statement can be proven by noting that multiplying the dividend and divisor of a division by a common number does not change the quotient.
I would not argue that fractions be omitted entirely because they do appear in the real world. But an understanding of what fractions mean in terms of partial units and in terms of division (which is actually a more general interpretation of the "parts of a whole" idea) and what it means to take a fractional amount of a quantity are the most important ideas. Many students don't get even that. And if they don't, then all this fraction arithmetic makes no sense to them. And I have observed huge numbers of students who are greatly confused enough about fraction arithmetic that they can't learn even to do fraction arithmetic correctly. Many others who do learn it quickly forget it, especially if they had struggled to learn it or that their work clearly shows that they "get" it somewhat but that their understanding of how to do the fraction arithmetic is still shakey.
Of course, education is not really career training or at least it shouldn't be reduced to nothing but career training, so we can rightfully justify teaching some ideas in mathematics if they at least help students learn to reason mathematically, even if those ideas are ideas they will never use later on. So if fraction arithmetic is indeed not going to be very useful for students later on in terms of the mathematics they will use, then can we really justify placing the emphasis on fraction arithmetic that we do in terms of what I just said at the beginning of this paragraph? Perhaps so, but there are probably better choices we could make to accomplish such goals. That is, we can help students learn to make sense of mathematics and learn to reason mathematically via other ways.
Even if we can truly justify the emphasis with fraction arithmetic in these courses, we cannot justify the current ways that we teach it because it ends up being a huge waste of time for everyone and ends up becoming a major headache for the teacher to see huge numbers of students not getting it.
I do plan to keep the material on fraction arithmetic in my book, regardless of the answers to these questions. Those who will need it later on and can learn to reason mathematically can learn the fraction arithmetic they need from the book. And I will include it also because many teachers will feel the need to teach it and because some students will be interested in learning fraction arithmetic, especially if I motivate it by the fact that nonzero terminating decimal numbers are not closed under division and also by the fact that fractions are generalizations of terminating decimal numbers in that fractions allow us to give the sizes of partial units where the unit is split into any number of equal-sized parts rather than a number of equal-sized parts that is a power of 10.
In short, Alain Schremmer has raised a good question that is worth considering, and that has motivated me to ponder it further. It has reminded me that we should not justify that this idea here or that idea there is essential to a math curriculum or should receive the amount of emphasis it typically does just because lots of people say so.