https://link.springer.com/epdf/10.1007/s12129-017-9659-z?author_access_token=stOqFhx08wuEn9MIBRNfBve4RwlQNchNByi7wbcMAY4pF6Km7OthXxcqYwy7bIQTTkzFiPQPrBZEUI-oVBSjQ25rb8GOWARgw6unbmR54uU_8n7rNst8xQ80FQ1Aji0iZRqjhulYOhgG6lyoW0x7Pw%3D%3D
Read it and be afraid. Be very afraid. The barbarians are already inside the gates.
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I recollect one time when as a graduate assistant at Marquette, I had an office with two other TAs on the second floor of what was then the Math building and is now not there at all. Three other TAs had the office next to us. The rest of the math department was on the third floor, where they needn't associate with us peasants. The remainder of the 2nd floor was given over to something called the Education Department.
One day, we math grads fell into discussion with one of the Education professors regarding the New Math, in which little kids were being taught abstract set theory. At one point we objected that not even the teachers understood such abstractions. "The teacher does not need to understand the material," the professor of education insisted. "He only needs to understand how to teach the material."
All hail Common Core, where a kid can be marked wrong for showing that 5x3=15 because 5+5+5=15 rather than 3+3+3+3+3=15 as the syllabus says!
As I explained to someone who was defending the Common Core, I delve into the mathematics of other civilizations (such as the Maya), and math in other numerical bases (did you know 1/4 and 1/3 are .3 and .4, in base-12?) for fun—but even I find the methods used by the Common Core to be confusing. They also misappropriate symbols that are already used in math, albeit not in school math; I forget what, exactly, they were using "^" for, but it took me several minutes of wondering if I'd somehow contracted numerical aphasia to realize it wasn't exponents, which is what that symbol does in computer math.
ReplyDeleteRegarding your finishing remark about the Common Core; as a non-American I'm not familiar with it, but in my country - at least in primary education - there is a deliberate effort to make pupils think about the relationship of maths with the real world. The difference between "five times three" and "three times five" is then very relevant, despite the resulting number being the same.
ReplyDeleteI suppose it can be helpful to be able to see that there is a difference between three regular basketball teams and five teams for 3-on-3.
DeleteBut that is the difference of 3@5 and 5@3, not 5X3 and 3X5, which is commutative for real numbers. Of course, by now everyone has forgotten what the "at...apiece" symbol @ meant. The three teams at five members apiece, the two numbers are not the same kind of things and so not commutative with five teams at three members apiece.
DeleteWell yes, I was riffing on the possible usefulness of the nameless outlander's approach to the commutative property. Still fifteen basketball players, though.
DeleteIs the multiplication symbol being taught without commutativity soon after? I know that addition definitely is. I don't remember when I learned that, but I think I figured it out soon after I stayed to learn to multiply. Hence if asked 9x2 I wasted no effort in adding 9 twice rather than adding 2 nine times.
ReplyDeleteAll I know is that if asked to multiply 9x2, I simply said 18 and did not agonize over it.
DeleteI teach at a community college and I had a student who, when encountering 8*7, started drawing 8 rows of 7 boxes each and counting them. I told her that she needed to memorize her multiplication table if she wanted to advance in mathematics. She proceeded to tell me about her "learning disability" which required that, as a visual learner, she had to multiply by drawing boxes. I suppose if someone was a kinesthetic learner I'd have to let them add by counting on their fingers.
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