This morning over the breakfast table my family had a great conversation about integers It started when my youngest, L (5th grade), talked about his math test tomorrow. He said the whole chapter was easy and that he wasn’t worried about it. I asked what kind of questions would be on the test, and he said that it would be things like “identify the integer in the following statement: it is -20 degrees C outside.” I’m sure the test will have more than that on it, by the way, but that launched us into some fun conversation about integers.

I asked him if he thought there was an infinite number of cells in the human body. That launched us into talking about all of these:

- Air molecules on earth
- houses
- homes
- books
- gallons of milk
- hairs on your head
- cups in the world
- heaps of sand

Some were easy: houses, gallons of milk, cups. Some got us really talking, especially “books” as we started to interpret those as fiction books.

Here are some of the thoughts that occurred to us as we argued around the table:

- If you don’t know where the end is, you can’t say you’re halfway done.
- Once it’s done, there’s a halfway point if you count pages or words, but half a story or half a plot is harder.
- We talked a lot about how the Harry Pottter books cram a lot in the last 100 pages or so, for example

- If you have a heap of sand and take a grain out, it’s still a heap. If you repeat, at some point it’s no longer a heap, but it’s never a fractional heap.
- So maybe integers are used for things that can’t be split up? If you can split them up, you should use reals or decimals or rationals or something.

- My partner is a writer and she talks about how many of her writing friends are heavy outliners. They know where the half-way point of their story is.
- Houses are measured with real numbers, but homes are like heaps: they’re a home until they’re not. Half a home doesn’t make sense.
- Human cells are interesting. They “divide” to reproduce, but my argument was that right up until it actually splits, it’s one cell, and once it splits, it’s two.

I’ve been thinking about this all day. I’m coming around to the notion that we often say something is integral (or is counted by integers) when really we should use real numbers and admit that it just works out that they’re often things like 2.00000 . . . etc (like houses, or gallons of milk, or cups, but not homes, books (maybe?), and air molecules). I think we use rational things (fractions) when maybe we shouldn’t. Maybe when someone says they’re halfway done with a story they’re really saying they are still at zero stories but will soon be at 1 story. They might be measuring time, or words, or pages, but that’s a proxy, using things that can’t be measured with integers.

One interesting thing was the different approaches of my kids. L was interested but admitted he was confused at times (now we’re a little nervous about tomorrow’s test – I joked that I should send this post to his teacher). C (10th grade) really felt that if you couldn’t clearly see the end of something, figuring out fractions didn’t make sense. A half gallon makes sense because we know what a full gallon looks like, but a half story is tough to make sense of. B (12th grade) felt that you can convince yourself that you have less than 1 of lots of things (like books), but even if you can’t figure out what the fraction actually is, if there’s a way to think about it being less than one, you can’t say it’s described by integers. Mostly that argument was on the book side, not air molecules or hairs on your head.

Overall it was a fun conversation. I love seeing the #tmwyk hashtag on twitter (talk math with your kids) but it’s often hit or miss with my own kids. This was fun mostly (I think) because I was really trying to wrap my own brain around it, and not just trying to teach them something.

So what do you think? Here are some starters for you:

- This is great. I think another great thing to talk about to see if it’s integral is . . .
- Why don’t you use the word “quantized” for this? What, are you scared of physics or something?
- This is dumb, everything is countable and split-able. I can’t believe I even read half of this post.
- #tmwyk can work great even if you’re “just” teaching them something, here’s 7.5 examples . . .
- What did you have for breakfast?
- I’m a fiction author and I’m really bothered by what you say. I often take 3/4 of one book and put it together with 1/4 of another to get a new book I can publish.

Did you also discuss the difference between discreet and continuous?

Not really, but I think that’s a cool road to go down. One question I ask people is whether mass is discrete or continuous, considering that to gain mass means gaining a discrete number of atoms.

This is awesome. I often teach number systems to elementary education majors, and C’s comment really stood out to me:

“C (10th grade) really felt that if you couldn’t clearly see the end of something, figuring out fractions didn’t make sense.”

One huge barrier I see to my elementary education students’ understanding is not thinking about “the whole” (i.e. “the end of something”), and not being comfortable changing the whole. For instance, you can view 2 cans of soda as 2 “cans of soda” or 1/3 “six-packs of soda.” They describe the same amount of liquid by they have different wholes/ends.

This is huge in understanding how our number system works. When we do the standard addition algorithm, we first add the ones-place, with our whole being “ones.” Then we change the whole we care about and start counting “tens.” So in 23+41, we first do 3 “ones”+1 “one”=4 “ones,” and then we do 2 “tens” + 4 “tens”=6 “tens” (we do not actually think of it as 20 “ones” plus 40 “ones”=60 “ones,” typically).

Tell C that he basically described exactly what I tell my education majors is the more important thing for them to learn in my class.

will do! I hadn’t even thought about “changing the whole.” That’s really useful.