God is a real entity, theology is an ancient discussion about God. Teachers who dispense theological factoids to their students will end up (if they're lucky) with students who dispense theological factoids. Teachers who facilitate partication in the theological dialogue will end up with conversation partners. Math is the same way; it is dialectical, it is about something real that is out their that you have to get your mind around. If I spit mathematical factoids at my students they will spit mathematical factoids back at me. If I journey with my students to the world of mathematical thought as it is and has been, from there we may visit the objects of mathematics themselves, and come back thoughtful lovers of beauty and wisdom, or even mathematicians.

Arithmetic for Real

I did some of this (below) in a lecture on the nature of mathematical thought at a camp this summer, as a way of illustrating the distinction between thinking about symbols that represent mathematical objects versus thinking about the objects themselves. There is nothing like division by a fraction that so neatly captures the failure of the american educational establishment to faciliate authentic mathematical dialogue. I gave the lecture to over a 100 bright students over the course of the summer, and not a single one had any notion of why the reciprical of the divisor should be multipled by the dividend. In fact, although the purpose of the example was to illustrate the difference between symbol thoughts and number thoughts, I literally had everyone's jaws on the floor 2 out of the 3 times I gave the lecture. I got tons of buy in just from giving the kids (and the staff) a little taste of authentic arithmetic thinking.

So why did they tell you to flip that blasted fraction and multiply? Well, when you do a problem like 6 divided by 2, you have a rule for figuring out what the answer is. Usually you ask, "how many times does 2 fit into 6?" This works for dividing by fractions but it is not very easy. For instance, suppose I want to divide 10 by 2/3. How many times does 2/3 fit into 10? Well, it fits in at least once for every whole in 10, so that is 10 times so far. Once a 2/3 has been fit into each of the 10 wholes in 10, there should be 1/3 left over in each of the 10 wholes for a total of 10/3. 2/3 fits into 10/3 five times, so with the 10 2/3 that we already crammed in, that makes 15. This checks out with the old "flip it an multiply" operation if you have 2 seconds to check.

But this method is too much work. There is another rule that can be used to figure out any division problem that works better with division by fractions. Instead of looking at 10 divided by 2/3 and asking "how many times does 2/3 fit into 10," say, "10 is 2/3 of some number, what is the number." If 1o is 2/3 of some number then what is 1/3 of the number? Since 1/3 is 1/2 of 2/3, 1/3 of the number must be 5 if 2/3 of it is 10. If 1/3 of the number is 5, then the whole number must be 15, as before. Note that the first step was to cut 10 into as many parts as there were thirds, namely, 2. This is why the numerator of the divisor has to be relocated to the denominator. Similarly, once a unit fraction of the whole has been obtained the resulting quantity must be multiplied by the denominator of the divisor in order to obtain the whole. This is why the denominator of the divisor must be relocated to the numerator. Notice that the rule works with 6 divided by 2, but not very well. 6 is 2 wholes of some number, what is the number?

It seems like the various rules for determining the outcomes of a single binary operation ought to be somehow equivalent, if division, for instance, is really one thing. How do the two rules that I gave relate to one another? I have a partial answer. Another rule for doing a division problems like 6 divided by 2 is, "if 6 is divided into 2 parts, how much is in each part?" But what does it mean to divide something into 2 parts? If it means something like 6 is to be distributed evenly over two parts or wholes, then the same rule can be used for 6 divided by 2/3. In the case of 6 divided by 2, 6 is distributed evenly over two wholes and the outcome of the operation is the quantity that ends up distributed over a single whole part. So if 6 is distributed evenly over 2/3 of a whole the answer should be again how much fits into a single whole part, so that the determination of the outcome is identical in all cases. This is the case since if 6 is distributed evenly over 2/3 of a whole then quantity in the single whole part would be 9. So if division means "even distribution over a specified quantity," then "let 6 be divided into 2 parts" and "if 6 is 2 wholes of a number, what is the number" are really the same rule.

Now the challenge is to figure out how "how many times does 2 fit into 6" is really the same question as the other two.

An Exercise

These notes refer to Exercise 1 on page 12 of Godel's Incompleteness Theorems by Raymond M. Smullyan. They are not really intended to be read by anyone, it is just kind of cathartic to write them out and post them. They are, however, an example of what I think any rational person can do if they have the guts to try a hard thing. I have put about 15 hours of slow, contempletive reading and processing into 8 or so pages of material. It took me 2 hours just wrap get my mind around the set (P complement)*. This was about as hard for me as my freshman classical geometry course with Peter Woo and almost as gut wrenching.

There is a function phi into which one can plug any expression and any positive integer and get another expression. A subset of the expressions are called predicates and another subset are called sentences. The only thing that is known about phi other than its domain and range is that whenever the input expression is a predicate, the output expression is a sentence. Every expression in the language L has a unique positve integer associated with it called its godel number. En is the expression with godel number n. En can be plugged into phi with any positve integer, but the expression to which phi maps En and its godel number n is called the diagonalization of n.

By hypothesis, the P* is expressible in L. P is the set of all the godel numbers of all provable sentences in L. P* is the set of all positive integers n such that the godel number of the diagonalization of n is in P. In other words P* contains all positive integers that have provable diagonalizations. P* is expressible in L means the there exists a predicate H such that phi maps H and a positive integer n to a true sentence if and only if n is and element of P*. So Phi maps H and n to a true sentence if and only if the diagonalization of n i provable. Suppose n is the the godel number of H. H would be a better name for this number. Then the expression to which phi maps H and h is true if and only if the diagonalization of H is provable. But the expression to which phi maps H and h is the diagonalization of h so the expression to which phi maps H and h is true if and only if it is provable. Not very helpful.

Incidentally, if the hypothesis had stated (as the abstract form of Godel's theorem proven in this chapter does) that (P complement)* were expressible, things would have been different. Again, P contains the godel numbers of all the provable sentences in L. The complement of P contains all positive integers that are not the godel numbers of provable sentences in L. (P complement)* contains all the positive integers n that have the godel number of their diagonalizations in P complement. In other words, (P complement)* contains all positive integers n whose diagonalizations are not provable. So if (P complement)* is expressible there exists a predicate H such that the sentence to which phi maps H and any n is true if and only if n is an element of (P complement)*. In other words, The sentence to which phi maps H and n is true if and only if the diagonalization of n is not provable. Again taking the godel number h of H, it follows that the expression to which Phi maps H and h is true if and only if the diagonalization of h is not provable. But the expression to which phi maps H and h is the diagonalization of h, so the expression to which phi maps H and h is true if and only if it is not provable. So L is incomplete. The reason why exercise 1 does not work like this is becuase it follows from "P* is expressible" rather than "(P complement)* is expressible" that the expression to which phi maps H and h is true if and only if it is provable rather than not provable, which seems trivial.

I clarified the above in my head for about an hour before it occured to me that it would be much more helpful if it were true that the expression to which phi mpas H and h were false if and only it were provable. Since L is assumed to be correct, this would means that the expression is true but not provable so the system would be incomplete. So I wondered whether there were any way to turn the result on its head.

The second part of the hypothesis states that for every predicate H there exists another predicate H prime such that the expression to which phi maps H prime and n is provable if and only if the expression to which phi maps H and n is refutable.

The trouble in the above thought process is that I was looking at the Godel number of H instead of at the godel number of H prime. Let's call it h prime. Since P* is expressible there exists a predicate H such that the sentence to which phi maps H and n is true if and only if the diagonalization of n is provable. So the sentence to which phi maps H and h prime is true if and only if the sentence to which phi maps H prime and h prime is provable. But, by hypothesis, this sentence is provable if and only if the sentence to which phi maps H and h prime is refutable. So the sentence to which phi maps H and h prime is true if and only if it is refutable. Since L is correct this sentence must be false and not refutable. Since the sentence is false and it is already given that it is true if and only if it is provable, then it isn't provable either. So the sentence is neither provable nor refutable and the language L specified here is incomplete.

Another Dialogue

Meanwhile I had the following dialogue with another 7th grader who was working on a test prep book. Her mother had indicated by circling the problem that 768 is not the sum of 3/8 and 11/20.

Instructor: Here. Give it another try. (Writes 3/8 plus 11/20 on a piece of paper, student multiplies the fractions and obtains 33/160.)
Instructor: Hmm... (writes 3/8 times 11/20 on the paper, student multiplies and obtains 33/160) uh oh... you got the same answer to both those problems, which one is wrong?
Student: This one (indicates addition problem).
Instructor: You want to try it again? (student adds the numerators and then the denominators obtaining 14/28 and then reduces this to 1/2)
Instructor: hmm... Is 11/20 bigger than 1/2 or smaller?
Student: Smaller.
Instructor: What is half of 20?
Student: 10.
Instructor: Is 11 bigger than 10 or smaller?
Student: Bigger.
Instructor: So is 11/20 bigger than 1/2 or smaller?
Student: Bigger.
Instructor: Good. So if you add 3/8 to 11/20 can the answer be 1/2?
Student: No.
Instructor: So what did you do wrong then?
Student: I dunno.
Instructor: Try this problem. (Writes 1/2 plus 1/3 on the paper. Student changes 1/2 to 3/6 and 1/3 to 2/6 and adds obtaining 5/6) Excellent. So what did you do wrong before?
Student: (perplexed look) oh. (Student changes 3/8 to 15/40 and 11/20 to 22/40 and adds obtaining 37/40.
Instructor: Excellent. Let's look at the next problem.

Allow me to interpret this phenomenon. This student is good at doing mechanical processes. She just has trouble knowing when to execute the right program. She looks at the symbols on the page and then executes whatever mechanical process is loosely connected in her mind with symbols of that type. If she were thinking about what the symbols meant she would at least say she didn't know how to do it. If she works enough problems over a long enough period of time she will connect the right process with the right type of symbols most of the time. Not even then will she have number thoughts. "Success" will be looking at a "+" and thinking, "oh, ya do this," and getting it right. But she will never look at "+" and think about addition. Or maybe after twenty or thirty years of balancing her checkbook she will be ready to carry her idea of addition to fractions when she tries to help her kids with her homework.

A Transcript

Below is the transcript of a 7th grader trying to calculate the total simple interest charged on a $500 loan over two years given that the interest rate is 18%. The section on interest is that last section of a chapter on percents.

Instructor: When a bank loans you money you have to pay back what you borrowed to begin with plus some extra money called interest. There are lots of way to calculate interest but the easiest way is called "simple interest." Interest is called simple when a fixed percetage of the original amount is owed for every year that borrower has the money. In this case the original amount is $500, the fixed rate is 18%, and the borrower has the money for 2 years. In order to calculate the simple interest for one year we need to calculuate 18% of 500. So what is 18% of 500?
Student: I dunno.
Instructor: Well, you multiply .18 by 500. (The student finds the product using a calculator, the result is 90.) So the interest for the first year is $90. If you owe $90 interest each year for two years how much interest do you owe in all?
Student: 1800.
Instuctor: Can you tell me how you got 1800?
Student: I dunno.
Instructor: Suppose you own a bookstore and I come in on Monday and buy a $5 book and then on Friday and buy a $15 book. After that I do not come back. How much did I spend at your store in all?
Student: $20
Instructor: How did you get $20?
Student: 5 plus 15.
Instructor: Excellent. So if pay $90 interest this year and $90 in interest next year how much interest do you pay in all?
Student: 1800.
Instructor: How did you get 1800?
Student: I dunno.
Instructor: What is 90 plus 90?
Student: 1800
Instructor: Can you write down the problem? 90 plus 90?
Student: uhh...
Instructor: Here you go... (Instructor writes the problem on a piece of paper, student obtains 180.)
Student: oh.
Instructor: Good. Let's try the next problem.

Example

Elementary students learn that when a number is divided by a fraction, the answer can be obtained by flipping the second one and multipying is by the first. For instance, twenty one divided by three sevenths is the same as twenty one times seven thirds. If you remeber how to multipy fractions (top times the top, bottom times the bottom), you'll see that the result is 49.

This is a simple process that is easy to teach even to people who do not pass a numeracy check like the one below. But why does it work? I would like to discuss why this process works as a way of examining in the simplest possible context the kind of mathematical knowledge that makes mathematicians tick.

Numeracy Check

How many thirds are there in two thirds? The number two thirds is impossible to understand without first knowing what a "third" is. If you cannot answer this question, your perception of the objects to which the phrases "one third" and "two thirds" refer is weak to non-existent.

Before coming to Biola for my undergraduate work, I read a book called Calculus, by Michael Spivak. I had read books about calculus before, and was expecting more of the same. I was truly shocked by this book. The book presented elementary calculus from an advanced standpoint; the calculus of mathematicians. When considered from an advanced standpoint, the calculus (and mathematics in general) offers the brave soul an epistemic-aesthetic experience that is both sublime and mind-altering.

This was a natural continuation of an experience I’d had while abroad, two years before. While sitting in a dark corner of a hostel in Liverpool, I observed a group of German travelers chatting amongst themselves in their mother tongue. I grasped while sitting there what I could not grasp in high school. The reason one would learn German is to speak with the Germans. I was so affected by this experience that I cut short my time in Europe and spent the remainder of my year abroad in Latin America, where I studied Spanish intensively.

A passion for pure knowledge, coupled with a passion for pure communication, has engendered in me a desire to see the souls of young people open to a larger world of knowledge and human experience. In the mathematics classroom, I will seek to facilitate an epistemic-aesthetic experience of the kind I described above.

This experience accomplishes several things. In the first place, the objects of mathematics stamp the mind with an image of their rational structure. The mind needs to be formed in the image of God, and mathematics provides this formation in a powerful way. In the second place, the experience of higher mathematics is simply the sort of thing that our minds were designed to do. It is as essential and as enriching to the human soul as listening to great music and reading great poetry. If you never learn to read, you are seriously hampering your ability to express the image of God. The same is true for never having moved your mind through mathematics, or great poetry. They enrich the soul, and this is part of what it means for the soul to become good. Finally, the Genesis injunction to subdue the earth is powerfully fulfilled through science, as it gives us the ability to understand and manipulate the physical world. All this is made possible by mathematics—not the object-less machinations of small souls, but by the direct apprehension of the mathematical objects that underlie the structure of the universe. I want my students to be great-souled, and mathematics is one of the keys to this result.

In my classroom, I will try to recover the epistemic-aesthetic experience of higher mathematics in the elementary/secondary curriculum. The approach will be dialectical, because the nature of true mathematical thought is dialectical. Students will take their time, developing ideas over the course of several years, and only be exposed to modern mechanizations of these ideas once they have achieved a high-level understanding of the underlying concepts. As a part of recovering the dialectical nature of the material, I will incorporate original mathematical works, which in their undigested state reflect how mathematics occurs at the frontier of knowledge. Students whose minds develop in this context will find themselves much more capable of functioning at the frontier of knowledge, where there are no textbooks. Students who are exposed only to predigested material will find that they are capable only of learning what others have already learned

Mathematics and Culturally Sensitive Education

Educators these days are excited about integrating the student’s culture into curriculum. American indian students, for instance, are likely to value their heritage. If the indigenous peoples of america have an ancient tradition of numeracy, this tradition should be integrated into the curriculum. This will increase the likelyhood that numeracy will become an immanent value for these students; a value that will lead them into numeracy and perhaps into higher mathematics.

The chapter on culturally sensitive education in my foundations of ed book did not talk about efforts to integrate the mainstream culture into the classroom. Nevertheless, much is being done in mathematics education, at least, toward this end. For instance, in the Algebra II book that I am using the section on polynomial functions begins with a Calvin and Hobbes comic in which Calvin proposes going to Africa to migrate with the wildebeasts instead of going to school that morning. The text then informs the student that the wildebreast population over the last several decades can be modeled using a third degree polynomial function (to what end I know not). My best explaination for what is going on here is that the author is trying to connect what the student really values, i.e., humor, with the topic of the day. This is culturaly sensitive education: integrating what the student cares about into the curriculum.

The problem is that what makes mathematics worth doing and what serious mathematics has been done for is not valued by mainstream culture.

Mainstream culture values what is useful to the individual. Unfortunately for this route, few students will use school mathematics (basic arithmetic aside) to produce results that they could not have produced without it. Lectures on how solutions of linear equations are basic to modern engineering fall on the deaf ears of students who are certain that they will never go into engineering. The question is specifically, “When will I ever use this?” The answer is "Probably never, and that's why you are a boring person."

Mainstream culture is also entertainment driven. Many classroom walls are graced by posters informing students that “Math can be fun!” This is like entering a cheetah into a salt water speed race featuring a dolphine, a blue whale, and a PT109. Numbers do not flash brightly colored lights at students, tell jokes, or go to commercial break every five minutes. Numbers do not have breasts or wear tight pants. While numbers do register some entertainment value, they simply cannot compete with the mainstays of entertainment culture. Placing mathematics in the same category as baseball and casual sex is a serious category error that will not convince anyone that it is worthwhile. It would be much more accurate to put up a poster that said "Can't find another vien? Try Galois Theory!"

So what are we going to tell these kids?