curriculum and Methods
I have a rather unique approach to tutoring math
I’m far from alone in deploring the way math is taught in public schools. Every few years some new fad is adopted but the results seem to only get worse. There is a huge amount of fraud going on in our schools. I know kids who’ve managed to ‘pass’ their grade 10 math class somehow in spite of not being able to count! The system can only flunk a certain number of kids, so others ‘pass’ even though they have no idea what math is about. A year after graduation they can’t add two 4-digit numbers. Sound familiar?
The rot starts in the early grades when kids are promoted even though their skills are questionable. Every year things get worse until, as I said above, things finally fall apart entirely in grade 10. But the system has a ‘conveyor belt’ mentality which might work to some extent in Social Studies or English or Basket Weaving, but which fails in math, where competence *must* be nearly perfect before a student is moved forward. In my view the only acceptable math mark is an ’A’. What varies is how quickly you get that ’A’. Thus, the conveyor belt must be abandoned. In my view math should *only* be taught by tutors who move students forward at their best speed.
My approach is a blend of requiring absolute competence with a rather laid-back exploration of concepts, puzzles and problems
The system not only fails students who have fallen behind, it also tends to stunt kids who are bright. Public schools are particularly hard on boys who, for reasons of dogma, are not helped to excel in math even though it is one of the few subjects in which they are (on average!) naturally more gifted. But bright girls can be equally let down. Although the system is now entirely focused on girls’ results, there remains this ghost in the culture that girls don’t really like math, and many of them have soaked up that idea. We’re surrounded by failure, thus there is now a proliferation of tutoring services. Some of them are using genuinely useful new programs. My favorite is ‘Jump Math’ which is achieving spectacular results.
However Jump Math is ‘method’ oriented; that is, it quite rightly demands perfect competence in calculation skills like division. It becomes incomplete when creativity and understanding of concepts becomes more important. My approach is a blend of requiring absolute competence with a rather laid-back exploration of concepts, puzzles and problems.
That’s because the dirty little secret is that math, up to about grade 10 or 11, is really much simpler than the system makes it seem
The strange thing, though, is that the more you relax, the faster you go. People can learn math much faster than the system permits them to. I once had a friend who quit school early and who had just about zero understanding of math beyond counting on her fingers, but she was intelligent and later in life she decided to make something of herself and applied for training in a skilled trade. However she needed grade 11 math equivalency! I taught her the whole of math, from grade 1 through grade 11, in a couple of months.
Incidentally, as to genuinely progressive ideas about math (and education in general) the Waldorf and the Montessori people both understand that the grade 1 to grade 12 ‘conveyor belt’ approach to math is counter productive. Kids are ready for different things at different ages. There is really not much point in starting math education in grade one, that is the time in their lives when kids are naturally learning language. I could easily take a bright kid of about ten years old who had taken no math classes at all and bring them up to grade 5 level in one semester. The human mind is not really ready for math before puberty, however basic methods and counting and such might be taught a little before that. In short, if your kid has fallen behind, don’t worry about it too much since a competent tutor can repair the damage quite quickly. IF their psychology is not too damaged, which it usually is.
That’s because the dirty little secret is that math, up to about grade 10 or 11, is really much simpler than the system makes it seem. They drag it out to the point where kids’ minds are not awakened, they are numbed. There are only eight axioms (fundamental laws) in algebra and they can be written down on one page. More than half of everything in grades 8 and 9 boils down to cross-multiplication. The system pretends to educate whereas in fact it obfuscates.
In my view math should *only* be taught by tutors who move students forward at their best speed
We'll organize the subject of mathematics into these categories:
- Commandments (attitudes)
- Times Tables (got to know them!)
- Concepts (the laws of math, truths, ‘axioms’)
- Conventions (how we decide to do things, traditions)
- Language (we *must* know what we are talking about!)
- Methods (calculations, techniques, algorithms)
- Solving (how we get good at solving problems)
The Commandments
Most of the trouble people have with math is not actually about math itself, it is about how they approach the subject, yet almost no one addresses that whereas I believe it is primary. To learn and to do math, we must enter a state of mind that is quite different from our ‘ordinary’ state of mind just as a Buddhist monk must enter a certain state of mind before she can meditate. And there are certain disciplines and habits that are essential.
It starts with The Commandments:
- Thou shalt memorize thy times tables.
- Thou shalt be prepared.
- Thou shalt not be afraid of being afraid.
- Thou shalt be equal, confident and bold.
- Thou shalt demand clarity.
- Thou shalt not fall behind.
- Thou shalt not study for tests.
- Thou shalt be neat.
- Thou shalt not guess.
- Thou shalt not fake it.
- Thou shalt not ask if your work is right.
- Thou shalt neither race nor compare.
- Thou shalt not flog a dead horse.
- Thou shalt have fun.
Methods and Solving
- Translate from English into Math. Both are languages.
- ‘Extract’ the math from the endless exercises they give you.
- What kind of problem: division, ratio, trig, algebra, ??.
- Have you seen this problem before? Almost always ‘yes’.
- Data + Method = Answer.
- Extract the data, determine the method.
- Do you have sufficient data? Sometimes not!
- Beware ‘red herrings’ or irrelevant data.
- Can’t remember a method? Build your own method from first principles.
- Find the pattern, math *is* patterns.
- What is the simplest case? Build from there. “Reducto ad absurdum”.
- They are trying to fool you. Don’t be fooled.
- They want it to look harder than it is. See the simplicity.
- They surprise you with stuff they have not explained, don’t be flustered.
- They give you stupid questions, don’t blow your cool.
- ‘Walk around’, look for ‘loose stones’ in the castle. Plan your attack.
- Do what can be done. Chip away at it step by step.
- Write down only what needs to be written down, but show your reasoning.
- Where possible do easy calculations in your head.
- Learn your calculator.
- Get all the logic, then do all the calculations.
- Make an estimate almost every time.
- Look hard for shortcuts, they are almost always there.
- Plug in some numbers to prove out the algebra.
