Saturday, September 9, 2017

What is Math?

It's the first question I pose for my capstone students, and then I ask them for the five biggest discoveries in math. A good way to start their blog (or reanimate if I've had them in class before.) Here's two of the previous classes responses: Winter 14 and Fall 15. The responses often run similar courses. Total aside: one student claimed How was that availabl

There's lots of math is everywhere and everything. I've felt and said that myself. One of my favorite teaching memories is a Kindergarten class that I visited weekly, and the first moment was someone getting to challenge me: there's no math in... bridges! Are you kidding?! Bridges are all about math! Next week, there's no math in Batman!

But is it helpful? If math is everything, then maybe people are already doing all the math they need to know. Me telling them that they're doing math either makes it irrelevant, or invalidates my line of reasoning because they very well do know what math is, they had a decade or more of it, and it is not that.

On one blog, I asked is the math the thing, the mathematical description of it, or the making of the mathematical description?

Lauren said that math is tool, but it's also an opportunity.  That's new to me, but also familiar. Isn't that the spirit behind #wcydwt and #anyqs? I had a little experience this week like that. I usually ask some kind of data question (mine or a learner's) on my sign in sheets and then make a display. Usually then shared on Twitter. For me it's a part of immersion, making the classroom mathematical. In some classes it leads directly into making representations, or becomes data for an activity. Almost always a chance to notice & wonder. Wednesday I shared one without the label, and it got some fun thinking. Chance for a joke, chance for some figuring.

As for the milestones, I was struck by a few things this time.

  • Numbers - lots of mention of numbers, sometimes specific like π, i or 0, sometimes familial, fractions or negatives. I am too eager to move past these, often, but now want to embrace them. The abstraction of quantity - that is a big freakin' deal.
  • Pythagorean Theorem. Of course. But it is a big deal. I love its history, and continuing story, and think it must be one of the first examples of hey, this means this AND how can we use that? Thank goodness right angles are useful. Or are they useful because of this property?
  • Patterns. So glad they think of this as essential. But when Eugenia Cheng says that math is the logical study of logical things, I think that math might have been born when we realized that there were patterns of patterns. When we were first meta.
  • Euclid. One of the things that comes from the course is discovering the people in many cultures who took that step of writing and organizing what we know. There's something about math that makes it naturally becomes a system.
I love teaching this course, and learners who are ready to think about the meta-patterns are the main reason.


PS>  I was listening to Anne Lamott's TED interview yesterday where she was so encouraging about just write. Just write. It really made me want to blog, to sit down and write. So when Lauren's post made me think that I wanted to think, I wrote. I can't worry about my blog being a bunch of first drafts. I can't be held back by the two open tabs on my Twitter Math Camp post and my summer calculus post. I just have to write. If you're reading this, thank you. That's already too kind.

PPS> If you don't watch the whole thing, you might watch around 25 min in (-15), where she talks about good writing is getting the reader to say "Ooh, tell me..." That set my teacher senses tingling. Her next part of that is that a confused reader is an antagonistic reader. That's exactly teaching, right? Where is the line between a learner wanting to know more, and not knowing enough to be interested. They need the beginning of a pattern, and to believe it's not just noise.

Tuesday, September 5, 2017

Top Ten Favorite Numbers

What numbers are the favorites of the people who have favorite numbers? I decided to ask on a lark, expecting a few responses, and it went crazy. (For my relatively quiet corner of social media.)

The idea had been bugging me since Joseph Nebus (who has a great weekly review of #mathcomics) linked to this comic from Cavna:

NO WAY are those the greatest, nor even the most popular. I can't even remember what tweet I saw that put this mild annoyance over the edge into asking out loud, but now I have a bunch of data on math teachers' favorite numbers.

This experience has taught me that our people care about numbers. They are more than quantities, they connect to ideas and stories.

Some things I noticed:

  • 18 is the first natural number not to appear.
  • Ironically, 2 is no one's 2nd favorite number, but is some people's 3rd or 1st favorite.
  • Having a symbol or name makes you a Big Deal number.
  • For about the top 20, number of mentions correlates to the Borda count (3 points for 1st, 2 for 2nd, 1 for 3rd).
  • No one loves negative numbers. Come on people. Transfinite numbers got more love.
  • 73 was the largest prime mentioned. Nope 163. Nope, 8675309. That number!
  • 6 was the last single digit to be mentioned. 
  • 42 did not show up for the first several hours, then stormed up in popularity.

I'm going to show this list to learners and ask them to think about why some of these numbers might be on here. In particular the larger numbers...

My top three (not included in the data) would probably be 4, 0, and Φ. 4 was my first ever favorite number. I explained to several adults how it was both 2+2 and 2 x 2. As a joke I'd get people to continue the pattern 2, 4, ... and if they said 6 I'd say 8 and vice versa. Little pain in the neck I was. (Except I was never little, as the family joke went.) 0 is the competitive spot. 10 - the first number to show place value? -1 - the huge discovery or invention? Something with a slick math history, like $sqrt{2}$, e, 1729 or 163? Something exotic, like Graham's number, a googol, or τ? In the end I have to go with 0. The digit that became a number, with cool Bahmagupta connotations. Փinally, the number about which I sometimes tell students that it was invented by my great, great, great grandfather. Even if there was no name connection, even if it wasn't so marvelously algebraic, even if I hadn't seen 3rd graders discover it through the amazing Fibonacci connection, I would have to pick it for the spiral connections.

The question elicited some great stories and tweets...

I can be pretty dense.

Bob Lochel shared this perfect kickoff to the top ten, from the show that made Top Ten a thing. Also, when asked early in my career for what I wanted to be like, I often cited David Letterman. I apologize to my students then, and to their grandchildren.

So from the home office in Grand Haven, Michigan,

Math Teachers' Favorite Numbers

10.  It's the answer to Life, the Universe, and Everything...

9. Not really...

8. Moving up one space,

7.  Lucky for us, lucky for you, this prime is one better than perfect. It's been up, it's been heavenly, it's been deadly, it is in 7th place with 7 points,

6. Often considered the first number, and still the...

5.  Pythagoras may have called this number the root of all evil, and it still gets a lot of hype. It is geometric and irrational, ...

4. Move on, folks.

Nothing to see here. Except the number that makes our place value system so craaazy good; Brahmagupta made it work. Often mistaken for a vowel, sometimes seen wearing a fashionable sash. Er, slash.

3. Fee or Fie? You won't fo-fum when you contemplate this 3rd place number, unless you go to the point where you're crazy and see it every where. Favorite of the Egyptians, the Greeks and God if you believe all the hype, it's....

2. Popular choice among mathematicians, who have denoted it after the greatest ever to be called one of their number. It turns up everywhere, and has all your base.
1. As surprising as Alabama football, we find here the number with not one, but two days dedicated to it. Half the number some claim it should be, but twice what it takes to be right. A great big slice oooooof - no. I hate pie jokes. And what's with everybody focusing on irrational, when it's transcendental?

If you want to dig more deeply, Carolyn Frye recommended the great RadioLab show on favorite numbers.

If you want to math more deeply, here's the data in a Google sheet. Thanks to everyone who participated, and sorry for clogging up your twitter feed.

I think sometimes I protest too deeply the stereotype that math is all about numbers. Maybe there are times to just go with it, and geek out.