Wednesday, January 19, 2011

Mathematics, A Eulogy?

Did you know that the area of a circle is Pi times its radius squared? Did you know the area of a triangle is one half of its base times its height? Did you know for a right triangle (a triangle with one angle of 90 degrees or pi/2 radians) the sum of the squares of the two sides adjacent to the right angle is the same as the square of the side opposite the right angle, the hypotenuse (did you know that I, a mathematician, still cannot spell hypotenuse?)? More importantly, do you know why?

Earlier today I posted a Facebook blasting the mainstream method of mathematics education. This was out of various frustrations, including my frequent interactions with people in the late stages of chronic math apathy, and conversations with colleagues on the state of our students. I have colleagues, how weird is that! Much of my despair boils down to our dogmatic adherence to a system which is fairly seriously flawed. I don't claim to have THE solution, I have thoughts that I would like to see tried out, but mostly, I want to see us have the courage and vibrancy to experiment boldly with our curriculum, changing it to try to get out of this malaise.

I believe I was unduly harsh, as math is not the only subject that suffers from some deep problems endemic to our education system. One horribly harmful practice is the passing along of students who have not sufficiently mastered current materials to higher level courses. Because math is such a rich and useful subject, attempting to understand a subject without a thorough grasp of the prerequisite courses is akin to the canonical fool attempting to build a sturdy foundation upon ground made of sand. Although factoring polynomials (mostly quadratics) and negative numbers are two fine examples of things one ought master before proceeding, as they end up in fairly frequent use, by biggest concern is fractions.

I cannot understand what part of our system so scars students that they stay scared of fractions long through their math careers. If one simply wishes to work with fractions, two simple rules suffice, common denominators to add, multiply straight across. This will not convey an understanding of the nature of fractions, but it will provide you with all the "special" tools that you need to manipulate them, they won't be things of beauty, but at least they ought no longer be feared. However, as I indicate at the end of the first paragraph, the essential point of math is NOT an isolated scattering of lifeless, lonely facts, but an intricate, intertwined network of collaborating reasons why things are true.

Were I to be given an island on which to implement my social experiments, my math curriculum would, broadly, look as follows (and mothers could keep their children...). An introduction to the positive integers, addition, and the number one. These topics are deeply connected, as you can think of one, the unit, as the starting place for all mathematics, without it there is only nothing, then addition miraculously conjures the rest of the whole numbers, as one is joined by one for two, and so forth. Here we introduce the miracle of zero, which leaves numbers unaltered Should we ever experience diminishing, as happens when one departs, we become familiar with negative numbers (not subtraction, which is a misleading myth). Now, one might find it useful to repeatedly add a number to itself, such as if students arrive on buses each carrying 32 children, 32 children after the first, 32+32 after the second, 32+32+32 after the third, and thus is multiplication formed from addition.

Now things become a little tricky, but I have faith in our citizenry to persevere. Just as zero played a special role for addition, leaving things unaltered, our unit plays the same role for multiplication. And just as the addition of -n undoes addition by n, because together they add to 0 which leaves things untouched, multiplication by 1/n undoes multiplication by n, because they multiply together to zero. Thus we run into the first fractions as reciprocals to the whole numbers, and we noticed that division too is an illusion. Consider 0*n, which we know to be n added to itself zero times, when no buses have arrived, there are no students, no matter how many are on each bus, so 0*n=0. Now suppose you are discovering this for the first time, what must you suspect about 1/0? Well, one expects multiplying by 1/0 to undo multiplication by 0, because that is the role of 1/n to undo n. So, 1/0 times 0*n should be n, no matter what n is. However, no matter what n is, 0*n always arrives at 0, so 1/0*0 needs take on more than one value to satisfy our system, since it cannot, and indeed must be 0 since anything times 0 is zero, we discover that 1/0 cannot be included in our system without breaking the patterns that we have established. Because there is no division, and n/0 is instead n*(1/0), we must conclude that, should zero occupy a denominator, we have left our reasonable system of numbers, so this is to be avoided. This has gone on long enough, but we may quickly build exponentiation from multiplication the same way we build multiplication from addition, and roots now are un-exponents, the same way negatives are un-addition and fractions are un-multiplication.

Of course, this is a bit of a mess when skated through in 2 paragraphs, rather than over the course of a year or two, but the point is that it is all interrelated. This is where the beauty of mathematics comes from, it is a mosaic, and a puzzle, and a fractal laden, dew adorned spiderweb on a cool morning. Addition and multiplication are different, not separate, and subtraction and division no longer exist, much simplifying the order of operations. Recall that these operations have the counter intuitive property that they do not commute (that is, 3-2 is not 2-3, and 2/3 is not 3/2). Since these operations no longer exist, this strange fact, an artifact of a formulaic approach to math, need no longer be considered. Now 3+(-2) and (-2)+3, as well as 2*(1/3) and (1/3)*2 can be swapped as much as is desired.

The WHY is beautiful, without it math is just an exercise in memorization and rote manipulation, two things that are fairly boring and easily done by computer. With why, math becomes a rich environment for exploring, every new piece of information not a fact, but a new piece of the puzzle which must be slotted into an awaiting rough edge. The easiest method to determine the area of a circle requires calculus, but I can prove the Pythagorean theorem on a napkin in under 5 minutes (if we are in the same place, feel free to ask me to do so, it is a beautiful proof). How amazing is that, these two facts, that we casually teach side by side in an introduction to geometry, have vastly different backgrounds and stories. Personally, I think the area of a triangle is the easiest of the three, and accordingly, I leave it as an exercise for the reader, should you be in the mood to explore. It may help to take right triangles as your point of departure, but remember that not all triangles are right triangles.

As I posted on Facebook, learning math the traditional way is akin to learning to write poetry by taking lines from the poems of the greats and sticking them next to each other in varying orders. Or, to be less absurd, it is learning about history by rote memorization of dates and events, with no thought paid to the overall correlation and causality, the beautiful, interconnected, vastness of human experience. It is learning a foreign language sentence by sentence, hardly paying heed to individual words in the phrases blindly committed to memory, let alone the system of syntax and grammar. Overall, I have to say that what is a very important question to answer, but only because you need to establish the what before you can sail off into the sunset after your elusive why's...


Karen said...

I can't remember if I told you this, so I'm sorry if I'm repeating myself. We have jokingly considered putting up a chalkboard of the "how-to for fractions" since so much of what people ask us is about fractions. We'd direct students to the board first before letting them talk to us. ;)

Seriously, though. It would be cool if we could teach people math in a way that they saw how rich it is. It is difficult though because it feels like it really depends on a solid foundation in elementary school and then again in basic algebra. It also depends on the students wanting to do more than just get by. However, perhaps if they saw the richness as a young person, they would be willing to try to understand math better.

May you always have success in your teaching!

Kelsey said...

First, enlightening post! There isn't much math in midwifery, and I do dearly miss it!

Secondly, I have seen the fallout of the way math is taught. Only knowing how, never why, leads students to be unable to realize that they already know the next concept. This leads me to my final point-when will you write a textbook?

Kenny said...

Karen, you hadn't told me about that, or I forgot, so I'm glad you mentioned it, although it is both funny and sad. You are exactly right, a deep appreciation for math must begin from the fundamentals, but since they are no less rich than the proceeding subjects, I do not see that this is a barrier in theory, only in practice. Color me naive, but I think kids are curious, and if we present math in a way that engages their curiosity, rather than stifling it with rote memorization, they will want to do more than just get by. After all, its elementary school, all you need to do to get by is not get peanut butter in someone else's hair, share the blocks, and suck it up and stay quiet during naptime.

Kelsey, I often worry about my ability to communicate math, or anything really, via one directional communications such as lectures or textbooks. But, it is something to think on. Another issue is that, to my knowledge, elementary math is not taught from textbooks, but that could just be my ignorance of the methodology.

Kenny said...

Come to think on it, my sentence in the previous comment starting with 'Kelsey' works just as well if you remove everything from 'via' on.

Lea said...

yes, please write a text book! or better yet, a teacher training course..... it's true, it's not just math. The ironic thing is you use the example of history being taught only by dates....well, to some people, it is! I think learning to write is very similar to learning math, in that we learn it in isolated obscure classes that feel like busy work and have no connection [that I could see at the time] to what I would actually need to know. I grade a lot of writing assignments and it's painfully clear 95% of my students have never learned to write. Why would they? Writing classes are often boring busy work. It's hard to connect future need to present disinterest.
I personally feel like math is a language I never learned, so all I know are a few nouns and pre-constructed sentences but I have no syntax. I can't speak it. I just trust an equation does what it does, not because I understand it, but because someone told me it does. When I understand it, though, I love it...but it's overwhelming to think of starting all over and so it's easier to avoid it, much to my own disadvantage.

Lee Ann said...

My brain is racing with responses to your thoughtful post. Wish I had the time to organize and state them clearly. For now, I'll say that your ideas sound somewhat like the "new math" that was introduced in the mid-sixties. Problem was that teachers couldn't teach it that way because most had never understood or appreciated math themselves. How many mathematicians and math lovers are willing to get students off to a good start math-wise by becoming grade school teachers?

Also, I'm not convinced that students who aren't at least one standard deviation above the norm will ever appreciate or be interested in anything beyond rote memorization and application of math concepts, especially in this time of cheap calculators.

Teaching history is another story, though, because it can be taught as a story and therefore loved and remembered by everyone. But once again, you have folks teaching history at lower levels who have no love of history. So they rely on date memorization and boring textbooks when they "teach" history, thus ruining the subject for those students who don't read about history outside the classroom and know how interesting the subject really is.

We need to attract primary and secondary teachers who are excited by the subjects they teach, excited by learning in general, and remain excited about passing along this joy of learning to their students. I was lucky to have a lot of these teachers in grades 1-8. The joy of learning was palpable in classes with good teachers -- and drudgery was equally palpable in classes with bad teachers.

Well, that's a start at trying to express the whirlwind of thoughts in my head.

Kenny said...

Never let it be said that I don't follow through with requests in the comments, eventually...

Math is Beautiful BAM!

It isn't a text book, or a teacher training course, but it is a start toward me figuring out how to explain mathematics. I'll contribute understanding math, but I need others to contribute wanting to understand math.

Unfortunately, the path to understanding most equations lies not in math, but in some form of science. Math will let you understand what equations are, in general, and how they work, but most useful equations come out of experimental observations made by people like Mr. Brugger who understand both math and some stuff about the "real world."

"New Math" actually came up in the discussion that I had with my colleague that helped inspire my FB status that prompted this post (that led to another blog entirely...). He mentioned that it failed because teachers did not adapt to it, my response was that we should start implementing it with new teaches then. Math teachers unwilling or unable to make the transition can teach math the current way, but as new teachers are hired, they teach the new way. Unless we want to keep teaching in an outdated, less effective way because we cannot be bothered to work on switching, yes, that makes sense.

In regards to me teaching elementary schoolers, or even teaching them to teach, I remain suspicious. I don't like kids, they are somewhat boring conversationalists. Furthermore, I doubt I have any idea what successful teaching of kids looks like, maybe no worse than the average Education professor, but still not a standard to which to aspire. However, I would be perfectly happy to sit around with future teachers and just chat about mathematics itself, then they can their teacher mojo and translate it for their students.

I think that if we hire teachers who are excited about their subject, and train them to teach it in a manner that it is exciting, then where the mean is will shift. It could be that students who are currently in the top 16th percentile (one standard deviation above the norm in the standard normal distribution) become the typical student.

In conclusion, thank you so much to all the amazing teachers that I have had, who kept my curiosity alive! And thank you to the next generation of great teachers who are heading out into the classrooms now or soon!