> ... numbers in general, not particular numbers. And the human brain is not naturally suited to think at that level of abstraction.
This is so wrong (at least as a stereotype).
All throughout my early education I HATED arithmetic, and found almost everything about it mind-numbingly boring and repulsively repetitive. At that point in my life, I hated math. The moment I encountered algebra though, it was "love at first sight", and ever since I've absolutely been fascinated and engaged with every type of high-level math I encounter (the more abstract, the better). And not just "fascinated" in the "I like it" sense -- math, CS, etc. is more easy/natural to me than most humanities subjects, by far.
So although I can only speak for myself, I quite disagree with any claim that the brain isn't naturally suited to abstract thinking. While I know not all people think the way I do, certainly quite a few do.
His area of expertise is mathematical teaching/learning. He was undoubtedly talking about the average person, I doubt it was supposed to be an absolute neurological-level statement. And as a general statement/stereotype, there are expected to be exceptions. I doubt he got to be a Stanford math professor without seeing some gifted math students himself. His statement can still stand, despite yourself as a counterexample.
A little more about him: His CourseEra course is "Introduction to Mathematical Thinking". It isn't about math, it's about how to think mathematically. He commonly talks about the pitfalls people make with basic mathematical approaches. He works with helping them understand approaches to math and how to deal with thinking abstractly and purely logically. Some people pick up all that stuff implicitly with little effort, some people never really master it. Given his position, I think he sees a pretty raw view of the average person's approach toward math.
Watching my kids play DragonBox, methinks a major problem with teaching algebra is the insistence on forcing steps from arithmetic to algebra, bogging down in numeric & non-numeric symbols which students have little or no cognitive relationship with at that age. Starting with pure algebraic concepts, devoid of explicit numeric meaning, may be much easier to absorb then transition into meaning-laden symbols.
Is that learning mathematics though or is it just learning symbolic manipulations by rote.
Yes some rote learning is necessary - and I'd warrant very useful in maths. However, generally in order to build on what you're learning you need to understand why you should perform certain actions.
There seems little point in learning to simply mechanically do the actions necessary to solve an equation. It is the meaning that is the reason for doing the learning and I worry that the last transition will be missed and make the entire prologue void of worth.
generally in order to build on what you're learning you need to understand why you should perform certain actions.
Before knowing why you should perform certain actions, it helps to know that you can perform certain actions. Methinks getting these basic concepts ("combine something with its inverse and it disappears", "thing over same thing is 1", "1 'times' something is that something", ...) into a kid's head very early is a good thing - may not yet understand why, but it's a mental tool that can be applied. Don't underestimate the value of having tools even if you don't know why/how they work; give a kid a hammer and he'll figure out it's for pounding nails.
>give a kid a hammer and he'll figure out it's for pounding nails. //
How old is the kid? I think you'll just end up with everything broken, unless you present it with the nails then the combination of the two is unlikely to happen naturally I feel until the child is quite old (if then).
I don't think the basics of algebra will have such results, but both are indeed tools that a kid will figure out a good use for if such tools are on hand and have been played with enough.
I could credit a lot of my creative skill to spending inordinate time as a kid just fiddling with a broad range of tools in the basement, regardless of whether I understood them at the time.
What you are saying makes a lot of sense. However, I often used that philosophy in my five years of teaching high school math/physics and found that most students had a very hard time transitioning into the symbolic versions of things. Even with exact parallels it was still almost impossible.
The difference between categorical and analytical modes of thought is not always clear to categorical thinkers, but is usually well-known to the analytical. I'm heavily in the analytical camp, went straight from high-school level algebra to the theory of formal systems and Godel numbering, and didn't 'click' with my scholastic maths education until linear algebra.
Linear algebra was exactly and precisely when I realized that "algebra" was a special case of algebras, which was what I needed to contextualize it. Before that it was a bunch of wasted rote effort.
Yep, I agree. I think that mathematics curriculum in High School should be divergent. Setting up kids, who after being introduced to Mathematical topics still hold disdain for it, to be allowed to check out statistical-based mathematics (with applications to personal finance, etc.)with introductions to basic probability theory and applications as well possibly (I have seen John Baez mentioning this before as well).
Although, in today's age there is no such thing as "practical mathematics" if you are going into a non-mathematical-touching field. Technology basically has you covered.
BUT for those who have an obvious love of Mathematics they really do need to be introduced into abstract thought which basically equates to questioning, generalizing, and enhancing notions/ideas the student has already come across.
This is so wrong (at least as a stereotype).
All throughout my early education I HATED arithmetic, and found almost everything about it mind-numbingly boring and repulsively repetitive. At that point in my life, I hated math. The moment I encountered algebra though, it was "love at first sight", and ever since I've absolutely been fascinated and engaged with every type of high-level math I encounter (the more abstract, the better). And not just "fascinated" in the "I like it" sense -- math, CS, etc. is more easy/natural to me than most humanities subjects, by far.
So although I can only speak for myself, I quite disagree with any claim that the brain isn't naturally suited to abstract thinking. While I know not all people think the way I do, certainly quite a few do.