Talking about nothing

                                

 My Mojojo likes to talk to me while she plays video games.  She really likes for me to give her my undivided attention, but honestly, watching someone play video games is not challenging enough to keep me interested on a Friday night.  So tonight, when she wanted to fight some salmonids in Splatoon 3, I asked if we could talk numbers.  She was on-board.

Now Mojojo is only 11, and I usually tutor and teach kids who are older than she is, but she has a pretty advanced conceptual understanding of math, so sometimes I talk to her about how to teach things older kids have trouble with.  Tonight, because of one of my tutees, I asked her about negative numbers.  She has a basic understanding of what they are, because I started her on the idea a few years ago when she asked what the last number was and what the first number was.  (Infinity is a fascinating topic, and I'm sure we'll get into that more later.)

For now, I started by asking her what she thought 5 + -2 was.  She thought for a moment and said 3.  I tried a few more with her: -4 + 1, 13 + -6, and so on.  And then I asked her if she thought she could come up with a rule for adding positive and negative numbers.  Right away, she said, "well, you subtract."  So I asked how she knew the sign of the answer.  At first, she told me that the sign of the first number gave you the sign of the answer.  So I asked her, since -5 + 1 is -4, what's 1 + -5.  She knew that answer had to be the same, so she eventually came up with taking the sign of the larger number.  She said a few things about the number line here and how that led her to know she needed to subtract, because the numbers went in opposite directions.

I figured this was a good start, and I asked about adding two negatives, like -7 + -2.  Right away, she told me that because they go the same way on the number line the answer would be "more negative".  (I really liked that phrasing, because some of my older kids have trouble with the idea that negative numbers are ordered the opposite direction.  ie -8 is bigger than -10.)

From there we got into subtraction.  She had no problem at all with subtracting positives and negatives.  When I asked how she knew, she told me, "well it's like a sock, and subtracting a negative flips it around".  That doesn't make tons of sense to me, but it worked really well for her.  And that analogy carried her into a negative subtracting a negative as well.

So then I asked if we could kick it up a notch and try multiplication.  She knew that -2 * 1 had to be -2 and that -1 * 2 had to be -2, but she faltered a little on -2*-1.  She said, "It's either 2 or -2," but she couldn't quite decide which.  After going back and forth, I asked if she wanted to try playing with some patterns.  She said "sure, I'm good at patterns!"

So we did this:

4 * 3,  4 * 2, 4 * 1, 4 * 0, 4 * -1, 4 * -2 and then once we were sure that a positive times a negative is negative, we said, ok how about -4*2, -4 * 1, -4, * 0, -4 * -1.  From there, we were pretty convinced.  Mojojo says "ok, so with multiplication it's a double sock flip,"  She says "because if you go to the edge of the sock, you have to go up the inside, so you have to flip it around, and if you get to the edge again you have to flip it around again."  We didn't do as much with division, but we're in agreement that since division and multiplication undo each other, the rules must work the same way.

And even though it doesn't make tons of sense, I like her sock analogy, because she's clearly thinking of numbers as continuous (infinitely divisible) instead of discrete (countable).  Because each number connects to the ones next to it, like a sock!  In hearing her reason through it, it was clear that she was not thinking about counting numbers, and that's interesting to me because I hadn't realized that negative integers make much less intuitive sense than negative measurements.  I had viewed discrete and countable numbers the same way, but Mojojo straightened me out.

Counting in Different Bases

                                                 

A suggestion that came up when I spoke to my friend D----- today was that she introduce her 5 year old to the idea of counting in different bases.  Her son likes to count and add and subtract, and she was looking for something else he might enjoy.  So I remembered when my little Mojojo was about that age, and she loved learning to count in different bases.  There's lots I like about this activity, but one thing is that it's a perfect example of a low floor, high ceiling activity.  Kids don't need much to start working with different bases, but they can use it to tie together lots of math later on.

Essentially, I explained it to Mojojo like this:

We say we count in Base 10, because there are 10 different numbers that we use: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.  When we count, after we reach 9, we have to start re-using numbers: 1-0, 1-1, 1-2, etc.  But what if we only had 4 different numbers? Then we would count 0, 1, 2 3 and run out of numbers!  So we'd have to count 10 next!  When Mojojo was 4-5, she would laugh hysterically when I skipped from 3 to 10, and then we'd count together: 11, 12, 13, 20!  And she would burst out in giggles again.  By the time we went from 33 to 100, she could barely speak from laughing at all the numbers we were "skipping".

Doing this with young children makes it easy for them to understand place value when it comes up at school.  Putting a 1 in the tens place means you've run through all ten numbers once -- or whatever your base is.  If you're counting in base 4, instead of a tens and hundreds place, you'd have a fours and sixteens place.  This is pretty easy to prove to kids because you can count with them and they'll find that 10 is the fourth number they reach and 100 is the 16th.  Later, when they learn more about exponents, they can tie this understanding to squares and cubes.  (Another cool activity for older kids is to see whether they can express fractions and decimals in different bases -- full disclosure, this bends my brain around a little.  To write 1/4 in base four is 0.1, and 1/16 is 0.01.) 

Operations in bases are fun, and they can be a big help to a deeper understanding of algebra later on.  But getting kids started with them can happen at 4 or 5!

One last thought on these -- if you have a kid who wants to know if we ever use this, YES.  I don't necessarily think practical applications make something more fun to learn, but computers work with binary, which is base 2.  And if you have a cable box that you reset, you can watch it count in hexadecimal (base 16!) which includes the letters A, B, C, D, E & F.  Mojojo points out that colors on the computer have hex values which are written in hexadecimal, which is a different way of writing a specific color than the RGB values.

Why this blog?

 

 

I moved to Brooklyn almost exactly 21 years ago to get my Masters in Education paid for and start teaching high school mathematics in New York City.  Since then, everything about my life has changed.  For the purposes of this blog, however, there are two main important changes: 1) I helped raise my stepson, had my daughter, and have taught and tutored -- so I've been talking with kids and 2) By working with schools as a teacher, a coach of teachers, a tutor, a parent, and the head of a tutoring organization, I've learned a whole lot about how schools teach math.

Today, I was in a little coffee shop in Red Hook, tutoring a student in geometry.  After I finished, the student left and I went to throw out my coffee cup.  I heard a voice behind me and discovered an old friend with whom I taught about a decade earlier.  We got into a conversation about our kids and she asked me a question: "My son is really into math, but his dad and I aren't very mathy -- how do I figure out what to do next with him?"  I have gotten versions of this question from friends for years: "What's a good book for a kid who really likes math?"  "How do I make sure my child stays excited about learning math?" and "My child is bored by math in school but likes playing with numbers outside of school -- how do I build on that?"

My dear friend D----- got me thinking.  I'm headed back into the classroom in the Fall, and I'm over the moon about getting to focus on kids learning math again.  This is the first time, however, that I've been a classroom teacher every day alongside the daily pressures of raising a child.  I'm really interested in how parents contribute to their kids' education.  Specifically, I'm interested in ways we get kids motivated and engaged and learning.  Sometimes this leads to formal academic success, but I don't care as much about that part.  My daughter and I talk math frequently, and we both walk away having learned something almost every time (AND IT'S FUN).

So I'd like for this blog to be a place where I keep track of ideas for how to build on what our children are learning in school.  Parenting is hard, and none of us have all the answers -- so I'd like to make this blog a place where we can ask questions / share ideas about how to keep kids learning interesting things and being excited about stretching their brains.  I'm going to try to post ideas/questions/experiences at least once a week.  And if you're following along and have specific questions or ideas, I'm happy to address those.