Where: bedroom
When: after lights out
Without any particular prompting, this conversation started, probably as an excuse to stay up a couple more minutes.
J2: 23 isn't prime because it is divisible by .1
J0: Interesting. What number multiplied by .1 is 23?
J2: 13
J0: Hmm. 13 x 0.1 is thirteen 0.1s, so I think that would be 1.3
<J1 and J2 proceed to guess what values for 23/0.1>
J2: it has to be a multiple of 10
<a couple more guesses
J1: 230
J0: So, if you let yourself have fractions or decimals, then 23 isn't a prime. Another way to say that is 23 isn't a prime in the rationals.
J2 (initially copying): 23 isn't a prime in the rationals. Hey, nothing is prime in the rationals!
J0: Usually when we say "prime" by itself, we are just talking about the integers, like -2, -1, 0, 1, 2. if we stick to that, what do you think about 23?
J1 and J2 together: its prime!
J1: what about e, is it prime?
<brief continuation where I say that I know some other primes and extensions of the definition of prime, but I don't know one in which e is prime>
Math lies
This reminded me of a math blogosphere exchange recently where teachers were talking about the "little white lies in math." What do I mean:- You can't subtract 7 from 3 (said when introducing subtraction and using a model of taking away physical objects.
- You can't divide 5 by 2 (distinguishing evens and odds, sharing whole objects, division within integers)
- You can't take a square root of -5
- You can't divide by 0
- You can't sum +1 - 1+1 -1 +1 -1 +....
- etc etc
I don't like it.
Ideally, we would take time to explore what it would mean if you could do those "forbidden" things. If time's not available now, offer to make time later. Maybe these are luxuries for someone who doesn't have to teach to an upcoming test, so if you can't explore with them, encourage the students to think about it on their own.
At the very least, use the right words to describe what you are (not) doing: "we can't take a square root of -5 in the real numbers." Sure, not all the students will get it, but hearing this caveat will clue them in that (a) this isn't a universal rule, so something special is happening, (b) there is more coming in the future and (c) what to do with their old understanding when they are finally shown the extension/clarification
Oh well, it seems to work for us.
What is the white lie behind divide by zero and 1+1-1+1-1...?
ReplyDeleteDivide by zero:
ReplyDelete(1) Studying limits, we recognize that there is a sense in which we can understand division by zero, with 0/0 cases being the most exciting, but 1/0 and -1/0 type limits having an important place, too. A related example is the Dirac delta-function.
(2) It also suggests you can always divide by things that are non-zero. This isn't true for rings with zero divisors, for example matrix rings or integers mod 6.
1-1+1-1+1... for this, there are two ways that I know to extend the convergence definition of summability:
(1) Cesaro Summation
(2) Analytic continuation
These ideas have been popularly presented in Numphile's -1/12 video. Frankly, I don't love their presentation because it takes something many people already find uncomfortable, convergent infinite series, and then seems to violate their intuition. That leaves them with the unsettling seeming confirmation that they never really understood what was going on.
Admittedly, Numberphile also made this about my specific example: Grandi's series video. In that video, they do a better job of being more helpfully explicit. I will resist the temptation to make the observation that one presenter is a physicist and the other is a mathematician.
I found the Chris Lusto post that was talking about pretty little lies.
ReplyDeleteJust to be clear: Chris is awesome. I don't mean to criticize him specifically and some of the examples he cites are tricky. Perhaps it is a subtle case-by-case call on which approach will lead to greater student understanding.
A related discussion focusing on the old quip about how angle-side-side doesn't prove congruence, Fibs, and an extensive follow-up analysis of S-S-A.
ReplyDeleteI have another math lie to add to this list, inspired by TJ Zager straight-but-wiggled. My comment:
ReplyDeleteI'm now adding the "math teacher" definition of straight-line to my list of unintentional half-truths that we tell students. This one is a little complex, so probably most students will never learn the variety of full, technical definitions of a straight line. Here's one version in a metric space. I'm not totally sure, but have the feeling that all of the (visually) wiggly lines in this post could actually be straight (under suitably defined metric for the plane).
Vector spaces have another definition. Fun exercise: will the two notions be the same in a normed vector space?
A crazy exercise (or fun game for the teacher to play at home) would be to use the students' (apparent) misconception and explore the following:
- what properties do we want a [straight line] to have? For example, shortest distance between two points.
- what if we assumed that these [lines are straight] and these other [lines] are not? This means we are magically giving the property we just agreed to these examples and saying these other ones don't have that property.
What else would be true in our mathematical system? Do we get any results that can't work together (a contradiction)? If not, what other cool things can we discover about the new world we have just created? What's the same/different compared with our "usual" world?
It just happens that this particular exploration playing with the definition of "straight line" is really rich and includes: non-euclidean geometry, differential geometry, general relativity.