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[quote=Anonymous][quote=Anonymous][quote=Anonymous]

This. These strategies are mental math tricks. Highly effective in understanding numbers. Countries who do well on PISA tend to emphasize these a lot more. I grew up in China and did all sorts of calculations like this. [/quote]

I grew up in Russia.  We had a very strong math curriculum and math culture when I was growing up. http://www.wsj.com/articles/SB10001424052748703740004574513870490836470

We did not do any of these tricks.  We had a very strict, very rigorous system.  We were not allowed to use calculators, at all.  No multiple-choice answers EVER. [b]We had to learn to prove all theorems ourselves.[/b]  We had to memorize a lot of things.  All the concepts were drilled, drilled, drilled to ensure we had a good foundation before moving on.  I bet most of my Russian friends, after thirty years of not touching algebra or geometry will still be able to recipe Pythagorean theorem and solve a system of linear inequalities.

And no there was no way you would get to 7 by doing 4+4-1.  [b]My math teacher's favorite saying was "You shouldn't scratch a left year with a right hand", meaning that a math problem should be solved the simplest way.[/b]

I agree with PP. I don't see the link to critical thinking.

[/quote]

Well that is the point right there. You may not have a full recollection of your early childhood math foundations as much as you remember algebra. But, for your teacher to have used that quote, it would appear that, in addition to the drills (which we still do here too) you must have been shown that there were other ways and your math skills were refined to make you figure out on your own which was the best and fastest way. You may not remember the methods used in early childhood to give you this understanding that then enabled you to in upper level math to attempt to prove the theorems yourself. Here, by contrast, having only ever seen one way to do it and math facts drilled with no other skills or understanding added, our teachers would have had no reason to use that colorful quote. The process was expedited and the simplest method taught in a vacuum.

So your system, which you remember as drill drill drill (because we tend to remember the least pleasant parts), actually might not have been solely drill and kill, but more like the system that is being introduced now, which is aimed at giving kids mental flexibility with numbers so that they can learn to prove the theorems themselves later on instead of just trying to memorize proofs without understanding them. It seems that your contemporaries might remember the theorems better than mine because you figured them out on your own making mistakes along the way (correcting mistakes begets understanding), then memorized them; we just memorized them in a vacuum, and most of us without understanding them.

I was very good at math, and very nearly majored in math, but I hit a wall. Now that I am re-learning along side my children, that wall is coming down, and I am reading math books for fun and understanding everything on a whole new level. I see the value in my children not only memorizing that 3 + 4 =7, but also understanding that 3 + (2+2) = (4 + 4) - 1. Learning and understanding that at age 8 will make algebra at 12 a breeze.[/quote]

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