In effect, Dr. F. said that most people--schools, teachers, homeschool parents, etc.--mistake teaching arithmetic or showing kids how to "do" math for teaching mathematics, and that that's wrong. Doing math and getting math are two very different things.
Instead of using canned problems out of a book, we should teach kids math only through natural methods, through science experiments like pendulums and bouncing balls, graphing changes in history, the weather, election results, etc. on Excel, and so on. Ask them how many oranges can fit in a box. Have them estimate the value of pi as closely as they can using only geometrical shapes and a ruler.
Rather than teaching math facts or requiring memorization, we should encourage kids to derive their math facts every time they do a problem until they have internalized them. No memorization ever. If it takes longer to do the problems, then so be it; just do fewer, deeper, harder problems. Memorization kills intuition, and should be banned.
Start with the big picture. Teach calculus to the littlest kids, but don't call it that and don't expect them to understand it all in one bite. Give it to them until you lose them and then move on to the next topic. It's the spiral method of teaching at its best: every 2 or 3 years, come back to calculus (or stats or trig or geometry or whatever), only with the next layer of complexity, picking up wherever the child stopped during the previous rotation of the spiral (if that makes sense, as I'm explaining it badly).
Above all else, teach them that math is beautiful and encourage them to use their intuition.
To me the above is either a new age mumbo-jumbo or a very complicated way of describing problem solving.
What is math, or why do we learn math? To spit out answers or to be able to problem solve?
Rusczyk writes
"true mathematics is not a process of memorizing formulas and applying them to problems tailor-made for those formulas. Instead, the successful mathematician possesses fewer tools, but knows how to apply them to a much broader range of problems. We use the term �problem solving� to distinguish this approach to mathematics from the �memorize-use-forget� approach."So why memorization without thinking (memorize-use-forget) is bad, you can't say that memorization per say is bad. It is a tool!
Memorization of certain things in math is crucial. You have to understand "why", but once you understand, cetain things should just stick with you and you should be able to recall them right away, otherwise you will be lost in more complex problem solving.
Kriston's son is 6 years old, so she is looking at a different math than I am looking at with 13 year old.
Time tables are generally the first thing that kids are asked to memorize. Thay have to understand WHY, but if they don't memorize them, how are you going to do division? Intuitively? Then you will have ton's of mistakes, even though you do understand the principle, or it will take you forever...
I would like to see an example of how do you
teach multiplication problem using a swinging pendulum? Shouldn't you make math as simple as possible?
I see memorization as a tool, not as a goal in math. Once you "have" multiplication, further persue of math should require you to memorize exponents, which will lead you to memorization of some logaritms. Having your factorials memorized is a huge advantage to problem solving - you are eliminating some brainless steps (once you know that is is a factorial you have to use, of course - and memorization won't take you there).
So....I disagree with the statement that memorization in math is an enemy - blind memorization is an enemy, not memorization per say. Smart memorization is an excellent tool!
Kriston is worried that her son will lose interest in math. Do word problems - no way you can get bored with those. Don't star calc yet :-), unless you want to lose your son completely.
