Tuesday, May 12, 2009

rational numbers revisited

Rational Numbers Revisited


Numbers like 1/3 seem to be on the edge. It is a fraction that is also an infinite decimal. If I am considering eating one of three apples, a third of the apples, then 1/3 would seem to be a real world faction. But if I want a third of a pie, I have to use decimals and
1/3 = 0.333..., which is in the Ideal World. A third of a pie is an Ideal World piece of pie.

Should these more or less controversial distinctions be made to students? I think, “Yes.” I think that it is good for students to think about the meaning of numbers. Whether they agree with me or not, they have thought about the meaning of numbers and hence made them less fearsome.

Every computation is made with rational numbers, in particular decimals. We see √2 and π in formulae so often that it is easy to think that they are numbers that exist in the real world but they do not. I know people that disagree with me on this but they are wrong. Before √2 was invented, it didn’t exist. It was invented as a convenience in constructing a model of a line. A model is not the reality of that which it models. Lines do not exist. Lines are part of the model that was made as an aid in thinking about the mark made on a paper with a pencil and straight edge. √2 was added to the Ideal World to fill in a hole in the Ideal World line.

When geometry appeared on the scene, the eye was the best measuring instrument available. This makes it easy to think that real world lines didn’t have holes. The proof that the square root of two isn’t rational just proves that √2 doesn’t represent something in the Real World but is a symbol for something in the Ideal World.

Evidently I believe that if I construct an eight-legged, three-headed, seven-horned character to fill in a gap in the plot of my Sci-Fi book, I have not added to the population of the Real World.

These last remarks may appear to be a digression but they aren’t really. I think that the classroom discussion of the philosophy of mathematics would be helpful. The mystery of mathematics isn’t in using the rules and symbols, it’s in wondering what the existential meanings of the rules and symbols are. And if the student sees where the real mystery lies, using the rules and symbols may lose its mystery. Anyway...

If a person wants to do something in their life that involves computation, they going to have to deal with rational numbers. I think it was Feynman who said that to understand physics, you had to know calculus. In the same way, if you want to compute you have to know decimals.

I have seen old arithmetic books that introduced decimals before fractions. Decimals were more useful and the author felt that they were a simpler approach to rational numbers than fractions. I think he had a point. It’s surprising how hard it is to get a kid facile with a carpenter’s tape measure.

I have given thought to how to teach fractions and sketched a possible approach but even as I sketched I could see problems teaching it. I kept asking myself, “What if a kid just doesn’t care about fractions?” I was saying things that would only be heard by students who cared.

I think that consideration should be given to making fractions an elective. It seems to me that the early years in school should be devoted to teaching things that the kids want to learn. I would make the point of the early schooling to give the student the experience of learning something. I found that university students didn’t know what it felt like to learn something. They didn’t know what it felt like to ‘know’ something. Giving the student this experience gives them the ability judge for themselves if they know something or not.

A student came in for help on implicit differentiation. I often tried to find the first place where the student got stuck. This particular student felt that he had everything under control up to implicit differentiation. But it turned out that he didn’t know what ‘differentiation’ meant. And when we looked at ‘differentiation’ it turned out he didn’t know what a function was, much less what an implicit function was. He really thought that he knew these concepts; he wasn’t trying to con me. He didn’t know what it felt like to ‘know’ something.

I have lived long enough to see kids grow up, mine and other people’s, and then see their kids grow up. Kids seem to like 1st grade and somewhere along the way their interest wanes. Not every kid, but a lot. I, myself, always liked school. I went back to school in one form or another every September from 1941 to 1995. I didn’t want to get out of graduate school; I loved graduate school. So I don’t know what it feels like firsthand what it feels like to ‘not like school’, but I believe them when students tell me they don’t like school.

When I got involved in home schooling I was surprised at how little time it took to cover the daily lesson plan. Since these lesson plan books were apparently covering the same material that regular school was covering, I assumed that a school room was an inefficient way to teach regular stuff.

It seems obvious to me that when you take a bunch of people and decide that they all need to know something, your success in teaching that ‘something’ depends on the size of the bunch and to what degree the bunch of students agrees that they need to know what’s being taught. The student is forced into classes that someone, often in the distant past, decided they should take.

But what about trying a different way to group students? Instead of grouping students by courses that are decided by forces external to the students, group them by common interests that they have. At the beginning of the school year, there would be a period of time when the students would form groups of common interest and then teachers would be assigned to the groups where they were competent in the interest of the group. A group can’t be an empty set of students but it can have a single element. A group could be quite large; the students may have a group that wants to be a drum and bugle corps or a symphony orchestra or a football team or a chess team...whatever.

If a person knows how to learn and knows how to recognize the difference between knowledge and ignorance, they have the world open to them. The basic elements of mathematics are not many and the different fields of mathematics just put them together in different ways and give them different names. A particular question may be difficult to answer but not hard to state.

I think that if you know how to learn one subject, you can learn any subject should the spirit so move you.

Now in my utopian school, when it comes to teaching fractions, the teacher would start with students who more or less want to be there. There is some ground floor to stand on.

I also think that the educational process is hindered not only by the counter-productive grouping of students into fixed courses but by the age restriction. When a kid is ready to learn, teach him or her. The mixture of ages in a classroom would lend stability. The younger kids would see an older person who feels that learning is worthwhile. I do not understand why there is segregation by age in schools.

When I was 30 years old I took up the motorcycle. I would ride on a desert area a couple of blocks from home in the company of teenagers. These boys could ride better than I and knew more about motorcycles generally. The experience of being taught by these kids was thought provoking. They were surprisingly adult in their attitude toward motorcycles and telling me what they knew. Learn is learning and these kids taught me a lot.

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