Teaching and order

Students know what their interests are, not teachers

I recently looked at “Rank Your Teacher”, a webpage I found through Google. By my name there were 7 or 8 entries. There was one that essentially gave me a zero and had dropped my class in the first few days. The consensus of the rest seemed to be that I was a good teacher except when I was teaching and then I was boring.
The classes represented were trigonometry and (not quite) college algebra. In retrospect I agree with their assessment. I couldn’t see how to talk about trigonometry in a way that might be interesting. As they are presently conceived I still don’t see how to.
My solution is to not teach them as separate courses. Teach algebra and trigonometry as they arise in the context of other instruction.
It seems to me that the present philosophical underpinning of mathematics education is the there is an ideal, linear ordering of mathematics and it should be taught in that order. This “order” is the “philosopher’s stone” of mathematics education; if it could just be found we could turn base metal of mathematical deprivation into the gold of mathematical literacy.
Unfortunately I don’t think this order exists.
A hundred years ago 90% of the population was rural. High school was enough education to qualify for a job that would put a person in the financial middle class. College algebra was taught in college.
The mathematics taught in the first twelve grades was not to be learned now and used at some indeterminate later date. Weights and measures (how much milk you took to the creamery), geometry (how much land your dad put into wheat), how to make change (helping out at the store, taking eggs into town to sell) could be used everyday on the farm.
It is less clear to today’s student where they are going factor polynomials or solve trigonometric identities in their everyday life after high school. It isn’t clear to me either.
Word problems are called applied problems. When I was in K-12 I knew that this was bullshit thought up by adults that they expected me to believe. The problems are obviously contrived. The mathematics text books of today proudly say that they have applications but this is just hype for the authors to convince a publisher to publish their book and for the publishers to convince a school to buy it. What are called “applications” aren’t put in for students, they are put in for adults.
It isn’t as though word problems have no value. Part of mathematics is about the collection of data and its organization to solve problems. This is what word problems are about, not real world applications. They should be presented as such. The education establishment uses the real world application lie as a daily staple. The students know they are being lied to. I hope.
It should be kept in mind that the important thing being taught is how to develop mathematics to model a problem. The problems don’t have to be linearly ordered.

Instead of adults deciding on the student’s path to knowledge, the teacher helps the student follow their own path to knowledge. I think that all early learning should be done this way. I remember grinding through Vanity Fair as a junior in high school. It wasn’t until I was older that I had the interest and life experience to appreciate Becky Sharp.

Consider some examples:

I remember starting piano lessons when I was ten. First there were scales and then little melodies written by “Schaum” in Book I. I graduated to “The Happy Farmer.” My mother 26 years earlier had started piano the same way and learned “The Happy Farmer”.
Both my mother and I were conscientious students and could eventually play some fairly sophisticated tunes. My mother could always sight read the copy of “Poet and Peasant” that she had learned as a teenager.
But neither of us could really play the piano. We gained some appreciation of piano music but didn’t know how to play the piano.
All the people I know who can play a musical instrument started by playing tunes they liked and wanted to learn. My music teachers had me learn tunes they thought I should learn, many of which didn’t really want to learn.
The more tunes a person learns that they are interested in the broader their interest becomes and they eventually become interested in tunes they, when first learning, thought were uninteresting.

When I was six (1941) my dad told me about the Bohr model of the atom and the infinitude of natural numbers. He told me that a rectangle was an unstable structure because it can change shape without changing the length of its sides. On the other hand a triangle can’t change shape and keep its sides in tact.
So I found an interest in triangles, infinity and atoms at an early age and six years later that early interest was expanded by books on geometry, mathematics and physics.

The problem with mathematics courses as they are now constructed is that they are “learn this now because you will need it later” courses. These courses are in the ordered development of mathematics.
In point of fact very little of it is used later and the syllabus is padded with topics that won’t be used at all. There are topics in algebra courses that are used only by teachers teaching algebra courses.
There are topics in early mathematics that teach some thinking and problem solving skills but memorizing algebra stuff or trigonometry stuff is not one of them.
Suppose a Junior High School student likes to go to carnivals. They are aware of carnival rides and the types of forces their body is subjected to. By riding in a car or bus or subway or wagon or bike or tricycle very young children have felt the effects of acceleration and deceleration. They have felt the reality of objects needing a force to change direction when the vehicle they’re in makes a turn.
Young people are aware of what faster and slower mean, aware that things change and that some things change faster than others.
The ideas of differential calculus can be discussed without numbers. The idea of rates and rates of change can be introduced using only elementary arithmetic.
When more mathematics is needed, that is the time to present it. It is my belief that this approach will have positive results if one is interested in students having some facility with mathematics.

I can speak only for myself but in my 50’s, while trying to learn guitar, I realized that I didn’t listen to myself play. I had taped myself playing a tune and I thought I had done it pretty well but when I played it back it sounded terrible, really bad. I was so occupied with which fret and which string that I didn’t listen.
So I consciously tried to listen to myself play. The first thing that became evident was that I couldn’t do two things at once, namely put my fingers in the right spot and listen at the same time.
In a flash of insight I realized why the one vinyl record of my piano playing sounded so much worse than I thought it should. It had no feeling or dynamics, just the correct notes.
What comes to mind is the conscientious student of mathematics who memorizes and then is surprised when they don’t know any mathematics. I had a student in Calculus II who claimed that he had become a max-min maven in Calculus I. I let him pick a calculus book and pick a max-min problem from the book to work on the blackboard. An hour later he was completely lost; he had no idea how to work the problem.
He wasn’t trying to con me; he really thought that he understood max-min problems until he looked at what he had put on the blackboard. I really thought that I could play “Bumble Boogie” on the piano until I listened.
Learning mathematics linearly entices the student to memorize. You just have to memorize the order. If I was stopped in the middle of a tune, guitar, banjo or piano, I would have to start at the beginning. I had memorized the order and when the order was broken I had to start over.
Needless to say I have observed this when students would try to put a theorem that they had memorized on the blackboard.
This happened to me in music but why not in mathematics? I could be asked a question in the middle of a problem or theorem and pick it up without dropping a stitch. I would sometimes get off on a tangent and forget where I was in the demonstration but as soon as I was reminded, I was back on track.
My music teachers could start anyplace in a tune be it “The Happy Farmer”, Beethoven’s Sonata Pathetique or “Classical Gas”. Rock guitarists jump around and sing while they are going crazy on their instrument. I watch country singers play effortlessly as they sing. I watch banjoists nod to a friend in the crowd while in the middle of “Foggy Mountain Breakdown”.
I can start in the middle of the Heine-Borel Theorem.
Learning mathematics linearly is like trying to understand Oklahoma by driving through it on I-40. I hear that “Oklahoma City is mighty pretty”.
You learn about Oklahoma by going to Ponca City, Hugo and Enid. You learn about it on the two lane roads and by stopping at the only café in a small town for a bowl of chili.
The musicians that I have talked to who had an easy relationship with their instrument learned this and that, learning tunes that they liked. As they played more they became more musically sophisticated and their musical interest broadened to rock, jazz and Bach.
But people who know things look back at how they learned them and think that they did it in the wrong order. If they had just learned the 43 fundamentals of drumming first. This was told me by an accomplished drummer who had started out just fooling around and was teaching his girlfriend the 43 fundamentals to start with. She gave it up after the first few fundamentals.
Mathematics books are monuments to the order principle.

Topics should be chosen as interest dictates and every so often the separate pieces can be consolidated into a single instance of a concept.
The choices of topics look random from the outside but they aren’t. They follow the path of the learner’s interest.