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This is really how to fix this: http://carymillsap.blogspot.com/2011/01/axiomatic-approach-to-algebra-and-other.html http://carymillsap.blogspot.com/2011/01/ax

This is really how to fix this: http://carymillsap.blogspot.com/2011/01/axiomatic-approach-to-algebra-and-other.html http://carymillsap.blogspot.com/2011/01/axiomatic-approach-t... I was an undergrad with Cary, and he got A's, I got B's with the odd C and this is exactly why. Until I took the Discrete Math course ( w/ the Eggers book - which is fantastic ) , I was floundering. I knew what the rules were, roughly, but I didn't KNOW know them. I'm not name-dropping Cary. It's just a coincidence, but I googled him and found the essay. I bring up the "we were undergrads" thing because it's a nearly perfect natural experiment. Use one of the Algebra sections in junior high or high school to teach rigor. There are no moves in a solution unless you name the principle by which the move was made. I know it seems impossible to implement, but there can't be a better time than 8th or 9th grade to do this. You get exposure to rigor for the kids that aren't pointed at STEM, the kids that are will actually be prepared - win-win. If we have to hire ringers as math teachers - people outside the education industrial complex - so be it. Whatever it takes.

Gravityloss 13y ago
That was excellent. One big part was that the kids agreed on the rules together, before they started using them.
revelation 13y ago
That seems like a positively annoying and pointless thing (see the line where they eventually just start referencing them by number; it's become just another 'thing to do'). Not the value of teaching math through axioms and deriving everything you do from them! That's very useful and proper preparation for higher math. But just because I can express everything in propositional logic with just NAND, I don't do that. Just as I would prove the mean-value theorem once and then simply invoke it when appropriate, I define conjunction, disjunction, negation etc. once and move on with the world. This is the lecture you are trying to teach, after all: you can deconstruct the universe into very simple blocks.
- ArkyBeagle 13y ago
  Just to reinforce things: we are talking about kids very early in their math career. And math is drill; Khan academy, all that. It's an investment. You can do drill in 8th grade, or you can do it in your freshman year of college. The point is: 1) I thought it annoying and pointless in the 8th grade. Oh clever me, I knew so much better. 2) People who sucked it up and did it, with whom I shared classes, blew my doors off in college math courses for a couple of years. A teacher made this the central fact of the algebra class they took. 3) Then I took the discrete math course, and stopped having trouble. There is something to this. Rigor is not natural. Learning rigor is essential to being able to operate your life well.
wwweston 13y ago
What's the Eggers book? It doesn't seem to be referenced in the linked blog post, and I can't seem to find anything with a Google or Amazon search on "Discrete Mathematics Eggers".
- ArkyBeagle 13y ago
  http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025 http://www.amazon.com/Transition-Advanced-Mathematics-Dougla... My bad - I mispelled "Eggen" as "Eggers"
  - wwweston 13y ago
    Thanks!

Gravityloss 13y ago

That was excellent. One big part was that the kids agreed on the rules together, before they started using them.

revelation 13y ago

That seems like a positively annoying and pointless thing (see the line where they eventually just start referencing them by number; it's become just another 'thing to do'). Not the value of teaching math through axioms and deriving everything you do from them! That's very useful and proper preparation for higher math. But just because I can express everything in propositional logic with just NAND, I don't do that. Just as I would prove the mean-value theorem once and then simply invoke it when appropriate, I define conjunction, disjunction, negation etc. once and move on with the world. This is the lecture you are trying to teach, after all: you can deconstruct the universe into very simple blocks.

ArkyBeagle 13y ago
Just to reinforce things: we are talking about kids very early in their math career. And math is drill; Khan academy, all that. It's an investment. You can do drill in 8th grade, or you can do it in your freshman year of college. The point is: 1) I thought it annoying and pointless in the 8th grade. Oh clever me, I knew so much better. 2) People who sucked it up and did it, with whom I shared classes, blew my doors off in college math courses for a couple of years. A teacher made this the central fact of the algebra class they took. 3) Then I took the discrete math course, and stopped having trouble. There is something to this. Rigor is not natural. Learning rigor is essential to being able to operate your life well.

wwweston 13y ago

What's the Eggers book? It doesn't seem to be referenced in the linked blog post, and I can't seem to find anything with a Google or Amazon search on "Discrete Mathematics Eggers".

ArkyBeagle 13y ago
http://www.amazon.com/Transition-Advanced-Mathematics-Douglas-Smith/dp/0495562025 http://www.amazon.com/Transition-Advanced-Mathematics-Dougla... My bad - I mispelled "Eggen" as "Eggers"
- wwweston 13y ago
  Thanks!