Especially word problems -- most of them felt like they were written for students wearing intellectual blinders... if you had any modicum of relevant knowledge outside of the lesson oftentimes word problems were impossible to solve
Don't the countries that rank better use the rote method even more?
>Though lesson study is pervasive in elementary and middle school, it is less so in high school, where the emphasis is on cramming for college entrance exams
I think the implication is that kids are supposed to be taught to play with math in addition to the rote memorization, not that rote memorization is evil.
Many of the more "fun" math techniques rely on knowledge acquired through rote memorization, just like dynamic programming uses the results of inefficient calculations to speed up subsequent calculations.
> Many of the more "fun" math techniques rely on knowledge acquired through rote memorization
I wasn't aware of this. I have a nine-year-old. Are there any examples of this?
For example, take a look at this lesson: http://www.homeschoolmath.net/teaching/md/distributive.php
Ignore the stupid rectangles, and ignore the part where it says "They actually use the distributive property, but we do not need to explain that to 4th grade students." (seriously, wtf)
A child who practices lots of sums, such that she knows how to add 420 and 56 without counting with their fingers or writing things down, will be able to learn this kind of multiplication with ease. Once they learn that and practice a lot, they will be able to generalize the method to multiply any two two-digit numbers.
Memorizing things is important because if you have to count with your fingers to calculate 3x3, you will never be able to calculate, for example, 23 squared in your head. But if you know the times tables your thought process might look like this:
20x20 = 400
23x20 = 400 + 60
23x23 = 460 + 23x3 = 460 + 69 = 529
(23 times 3 is "the hard part" where most kids who know all the theory (distributive property) but are out of practice (not enough rote memorization) will lose track)
A more advanced example is what engineers used to do a lot before they had calculators: They memorized log tables, so when they wanted to multiply big numbers they just added their logs together, because log(a x b) = log(a) + log(b).
An unrelated example is converting miles to kilometers. The official relationship is that a mile equals 1.609 kilometers, an ugly number that doesn't work well with mental arithmetic. But that number is kinda close to the golden ratio, don't you think? So if you are the kind of weirdo who memorizes the Fibonacci sequence, you can quickly calculate that 13 miles should be about 21 kilometers (and 21 miles is about 34km, and so on), because the ratio between a Fibonacci number and the next approaches phi.
It's not about "fun", it's about experimentation. Free-form intellectual play.
But students are never told why they should care about abstractions in the first place, which is unfortunate. Many high schools simply refuse to speak the students' language.
I suspect this will change in this century. IMO, if a high school really wanted to be progressive, they would totally reform their math curriculum to include more exposure to applied math and computer sciences. Young folks should be using math to build their own Instagram or Minecraft clones that they can deploy to their devices that VERY DAY - using the concepts they've been introduced to in mathematics.