Math Relearning/Progressions/6-8 The Number System and High School Number
These are my comments on the Common Core Math Progressions document The Number System: Grades 6-8 & Number and Quantity Standards (number part): High School (PDF).
"mysterious acceptance that 'of' must mean multiplication" - Yes, I've never quite understood that translation. I suppose this explanation with the commutative property makes sense. The scaling model also seems to make sense of it.
"measurement interpretation of division" - This makes me think of remainders and fractions as results of division and ask how they're related. What does it mean that 9 / 4 = 4 * 2 + 1 = 2 1/4? The larger point is that there are all kinds of angles on these operations to relate to each other. I may want to try to identify what can be related in case the curriculum doesn't do it all and I want to explore them. It's probably not necessary, but it could help with the flexible thinking needed for problem solving. Noting examples like this one I happen to think of will help me think about what to look for.
"8/3 ÷ 2/3 = 4, because 4 is how many" - If you're muddle headed like me, you might be confused about what to do with which parts of this problem, and remembering something about dividing lengths of the number line into smaller parts from some earlier problem, you might try to solve this by dividing each third into thirds and then count off two of them at a time, which would give you 12. If you were on a better track, you might remember that third is a unit here, so you're just dividing 8 by 2. But if you're still a bit muddled, you might not know why one is a more correct approach than the other, so it would help to explore how the grammar of such an equation translates into a procedure.
"linguistically different ... mathematically the same" - In this case it's because the answer to "how many" is "a fraction of one," which leads to the second question, "how much of one?"
"find a common unit" - If you're working only with the visual model and not applying rules like "multiply the denominators," how would you decide on the common unit?
"2/3 of a cup fills 3/4 of the container" - When there are multiple fractions in a situation, it starts to get hard to think about. It would help to have frameworks or guidelines for thinking about such problems, things to notice and ways to relate them. For example, you could make sure to pay attention to the units (cup, container) and what number is "of" another number (the liquid is filling an amount of the container).
"The shaded area is 3/4 of the entire strip." - Maybe this will be clear with practice, but at first glance I'm confused by how the diagram relates to the equations.
"leads us directly to the invert-and-multiply" - Ah, another procedure I've been wondering about.
"denominator equal to a power of 10" - So are the denominators of irrational numbers infinity? What would that mean? I still need to find out how long division works in terms of place value and fractions and such.
"prime factorization ... can be time-consuming and distract" - What do you have to say to that, James Milgram? They skip it on purpose.
"In some cases 0 has an essential meaning" - Yes, though it seems relatively rare that 0 means what people normally think of, which is nothing, so that negative numbers are somehow less than nothing. It just means some chosen reference point.
"line segments acquire direction" - That relates to a conclusion I came to when initially thinking about numbers in this project, that numbers, at least signed ones, are vectors, since they have both magnitude and direction. So even a simple number has some level of complexity.
"larger in magnitude" - So there is a sense in which -7 is larger than -5.
"the number located a distance |q|" - Interesting bringing absolute value into addition.
"one-dimensional vector addition" - Aha!
"integer chips are not suited" - I think Chapin and Johnson bring up another problem with using chips.
"how you get from" - This way of expressing subtraction is very clear to me.
"rely increasingly on the properties of operations" - I both welcome and dread this. Welcome because it's moving along the development of mathematical thinking that Tall describes toward abstract reasoning. Dread because I feel like it'll be harder for me to get a sense of what the math means without visual models.
"a choice we make" - Are there really no real world situations where it applies?
"you want to be able to say that" - What would happen if the distributive property didn't apply to negative numbers? And how would we know it didn't? I think the demonstration of multiplying p and q shows why the distributive property applies.
"can extend division" - We are definitely getting into territory that ties my brain in knots, at least until I get more used to these ways of thinking about fractions. Right now my mind automatically applies the rules I learned, but I know I don't really understand what they mean.
"an extension of the fraction notation" - Very interesting. This gets to C&J's list of interpretations of fractions. It's like some kind of trick. Somehow we apply a meaning of fractional notation that makes sense for one kind of number to another kind of number, where it still makes sense in one way but is nonsensical in another. I must ponder this when I get back to it in the curriculum.
It's interesting that the physical world is limited in the ways it can directly express math, yet somehow we can still use other parts of math to reason about the world. I've read that some of these uses of math are like shortcuts for making real-world calculations, such as the use of imaginary numbers for the time dimension in equations related to the Big Bang. I'm sure math plays other indirect roles. I'll try to explore those as I go.
"a sort of address system" - Is this not true for fractions with denominators that aren't powers of ten? Maybe this analogy is just meant to illustrate the progressive refinement of increasing the denominator or something.
"1/3 is always sitting one third of the way" - How do we know this? Apparently the answer ("a rigorous treatment of this mysterious infinite expansion") will have to wait till after middle school.
High School, Number
The Real Number System
"rational exponents" - Are there irrational exponents?
"good practice for mathematical reasoning habits" - Is there really no practical relevance for irrational numbers? For example, knowing a number is irrational means you expect to be able to calculate it more precisely if needed. Is that too trivial?
How did we conclude complex numbers exist, that the square root of -1 has an actual solution? How do we know when some operation doesn't have a solution, such as dividing by zero?