So this is exactly the same text as before, except that at the request of my wife, partner, trusted advisor, and employer, I have put in a few illustrations which are NOT me looking drunk. The things we do for love.
So, first, seven short pieces, seven little things to have in mind. Then, some thoughts about Singapore Math and number sense, kicking off my "almost a week of math-y posts." Those of you who don't find this stuff interesting, who read the blog for some other reason, well, there will probably be something for you, too, sooner or later. But this week we're back to Singapore Math.
|The general idea only continuous, made out of dimes, and going all the way to Mars. If this image makes you curious, it's part of something really cool at this piece by Ross Andersen in Aeon.|
Well, then, still going one digit all the way, when Mars is in conjunction (in a line with the sun and Earth, with the sun between) it's 2.5 times as far away as the sun, which would be 180+45, 225 million miles.
- Did math in head, so figures were not exact. Needed to ask to borrow a calculator.
- It sounded like an enclosed tube, which is a tunnel, not a bridge.
- Did not consider cost of so many dimes
- Very uncreative answer, one of the most uncreative we have ever had. He just calculated. Creative answers we have had for this included
- "I would pay everyone on earth a dime every time they thought about a bridge to Mars, and when there were enough positive thoughts, someone would build the bridge."
- "I would just need one dime, that we would spin out into a silver wire and stretch between Earth and Mars. Then I would ride there on my unicycle."
- "I would ask NASA how much it would cost, multiply by ten, and ask the March of Dimes how many dimes they get per year, and divide."
"From there we made a final assault on the concept that elementary algebra depends on: that an unknown number will behave exactly like a known number. (Willard, at first, did not see how we could know that 2x+3x=5x if we didn't know what x was, and could also clearly see that we couldn't possibly perform the experiment of trying all the infinite possible values of x to make sure; nor did he see that we wouldn't have to do that for every possible equation)."
- All I need is just to review basic math.
- Give me a rule that's not so complicated.
- I can do any problem if it's money (or cups and pints, or wood, or cars).
- If I always know the right answer anyway, why do I have to do the math?
- Just tell me what to do and I'll do it. Don't tell me nothing else.
- "Seventeen!" (or any other number) (Often shouted out before anyone else can begin working on the problem, especially when the answer should be an equation, expression, or interpretation.)
- So how come it's a percentage if you just said it's a fraction?
- We're never going to use this.
- We've been here for t weeks and you still haven't told us what x is. (where t=number of weeks they've been here. Often off by 1 or 2).
- You can't add letters.
|Nobody looks drunk, either. But at least they didn't waste any time on that algebra stuff. You can't add letters anyway.|
- when you put two bars end to end, you add;
- when you put a bunch of identical bars end to end, you multiply;
- when you back up along one bar by the length of another bar, you subtract; and
- when you slice a bar lengthwise, you divide or write a fraction (because fractions and division are the same thing).
- the student is flexible in using numbers, meaning that the student knows what a number, a relationship, an operation, or an algorithm is (not just how to manipulate the symbol for it in one narrowly specified context).
- the essence of that flexibility is that a student knows what to do with a number, a relationship, an operation, or an algorithm in an unfamiliar context
- flexibility is achieved by knowing ideas conceptually, meaning that above and beyond memories of the use of the number, relationship, operation, or algorithm in similar problems before, the student grasps the underlying ideas clearly enough to see them in unfamiliar situations and handle them accordingly.
A kid with good number sense will learn math, even from indifferent teachers using poor materials; a kid with poor number sense will never really understand math, even in a well-taught excellent curriculum.
|Mary Boole. Looking very not drunk.|
set out the basic idea. She was pushing beyond the familiar basic question of what it was about memorizing algorithms as purely symbolic operations that led so many students straight into The Wall? In doing that , she posed what turned out to be a more productive question: what did the students who avoided The Wall, or encountered it without being defeated by it, do differently?