Archive for: July 2008

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Photo credits: two unknown and one from monkeymucker
Edited by Brian

I recently read a post at:

dolcevitaacademy, which talked a bit about something that is close to my heart, which is: should you focus on teaching concrete skills first, or rather on concepts? (I’ve blogged about that on the post about Math Skills v. Math Concepts.)

This seems to be about what the “math wars” are about.

If you don’t know what the math wars are, here’s the quick and dirty on them:

At some point, around the early sixties, American educators decided that what we needed to improve our math education was a new way to teach math. This was probably due to our getting our butts kicked by the Soviets in the Space Race for awhile in the late fifties.

Some genius came up with “The New Math,” which was basically a somewhat new way to teach math; it had nothing to do with any kind of actually new math . I mean, two plus two still equalled four.

Since then, “traditionalist” and “constructivists” have been sniping at each other about “which way is the best way.” (I like to snipe at both.)

(more…)

Tags: math education , math wars , new math

I’ve got lots more to post about bases, but right now I’m faced with a paradox - Hintikka’s Paradox, to be precise. 

Hintikka’s Paradox comes from Deontic Logic, a form of Modal Logic. I first read about it in Raymond Smullyan’s “Alice in Puzzleland’ (a brilliant book about logic, and Alice in Wonderland, that is worth looking into.)

In the introduction to “… Puzzleland,” Smullyan describes Hintikaka’s Paradox this way:

“Is it proper to call morally wrong something a person cannot do? Hintikka has a notorious arguent designed to show it is wrong to try to do something impossible. There is now a large literature on this strange question…”

I’ve yet to encounter much of that literature, and boy, I have looked. I probably wouldn’t have understood most of it, anyway. 

But it boils down to this, Hintikka’s Paradox implies that, “What is not possible is positively forbidden.” 

It’s important not to approach this from a “common sense” frame of mind. Common sense is usually neither, and is often a disadvantage when approaching counter-intuitive material. So try to keep an open mind. 

(more…)

Tags: deontic logic , Hintikkas Paradox , logic , modal logic

Continued from the post about  ”What is a Base?”:

The same person wrote a follow up comment:

“you are not pretending i’m stupid!!!!! Okay is a base the number you can multiply by?????
“example: base two is 2,4,6,8,10,12,14,16,18 ?????????i don’t know what you mean!”

Yeah, keep trying to convince me that you’re stupid. From your grammar and your tone, you’re starting to make some headway. 

But I generally don’t believe a child can be stupid. Misguided, full of anxiety about themselves and the world, OK, but stupid is reserved for adults (where a lot of people make up for lost time). 

Maybe I wasn’t clear enough, so let me try again.  A base is a way to write a number using place value (columns). The amount of digits you decide to use in the columns determines the number of the base. If you use ten digits per column, the number will be in base 10. If you use three digits per column (the digits 0, 1 and 2), the number will be in base 3. You will understand this better as you read on. (more…)

Tags: convert base 10 to base 2 , what do we use bases for