## Math study group: Positive and negative numbers

It was Friday, so J- and her friends were singing the Friday song as they hung up their coats and got ready for our math study group. It turned out that they had been so excited about coming home (to a math study group!) that they’d forgotten to arrange things with their parents, and V-‘s dad had been waiting for her at school. Once everyone had called around and sorted things out with their parents, and everyone was well-fed, we got back to math.

One of the benefits of hosting multiple kids in a study group is that you get more information about what people are learning in school. V- said she needed help with positive and negative numbers, so that’s what we started off reviewing.

A quick review: 2 – (-3) = ? . Boggles all around.

Okay. A step down: -2 – 4 = ?. Still boggles and some guesses.

I drew a number line and labelled it with the numbers. “Imagine a cat standing on -2. Which direction does the cat go if you’re subtracting 4?”

“Left!” chorused the kids. “-6!”

I drew the cat ending up on -6. We did a couple of other exercises along those lines. Nods all around. Okay.

“What about -2 + 3?” I drew another numberline. “Right! +1.”

“What about 2 – (-3)?” I drew the cat on the numberline. “Okay, we’re starting on 2. And we’re subtracting, so we would normally move to the left, but we’re moving -3 steps… so the cat walks backward three steps.”

“5!” said the kids. One of them asked, “Do your cats really walk backwards?”

“They do more of this hopping backward thing, yes, but cats can walk backwards if they want to.”

So we did a few more of those exercises, including things like -4 – (-5) and -(-(-2)). We also reviewed multiplying and dividing positive and negative numbers. The kids seemed comfortable with that, and answered our exercises with little prompting.

As we wrapped up our review of positive and negative numbers, A- arrived. She’s in grade 6, a grade behind the other kids, so we modified our exercises. She said she was taking up decimals in class. I asked her how she felt about the multiplication table. “Bad,” she confessed, at which the other kids begged (begged!) to do multiplication practice.

“But first, we’re going to talk about algebra very quickly,” W- said. He briefly reviewed what an algebraic equation really means, and the different parts of the equation: the constants, the variables, the operators, the assertion, and so on. We hope this will help them remember to keep their equations balanced, always doing operations on both sides of the equals sign.

“All right, multiplication,” I said, and we headed outside to practise multiplication. The way we do it is good for building confidence and a sense of numbers: we go through sets of five multiples until the kids can rattle them off smoothly. For example: 6, 12, 18, 24, 30. 6, 12, 18, 24, 30. And so on, around the circle. It’s really more of an audio recall task than a calculation task, and it gets them used to what the numbers feel like. They catch themselves now, when they make a mistake. And they’re enthusiastic and run ahead of themselves, doing sets of ten instead of sets of five, or challenging themselves further by doing jumping jacks while saying the numbers.

After multiplication practice, one of the kids piped up and asked, “Can we solve the equation in the breadbox?” Ah. Yes. Those. I’d spent some time the night before writing up simple equations and hiding them around the first floor of the house – possible exercises for J- or the study group, depending on how things went. So we agreed that they could look for the five Post-It notes I’d hidden IF they solved the equations as well. I settled in to review decimal multiplication and division with A- to help her catch up, and W- reviewed the other kids’ work on the algebraic equations.

Our Friday afternoon math study groups are a great ritual. Glad we stumbled into organizing them! I hope other parents can host study groups as well – it would be good for all the kids to see active involvement – but it’s probably easiest for us, logistically speaking, because we can often work from home and we both enjoy teaching. If you can, try it!

2011-04-10 Sun 12:05

• jenny lisk

ok so THIS would make a great book…seriously…how exactly you manage to get middle school kids loving a math study group and begging to do multiplication tables!! I’ve been reading this series of posts with interest and for future reference (my kids are only 6 and 4). Never would have thought of going outside, hiding post-it notes, etc. A great book would include some of the creative lessons / approaches / exercises you’ve done…tips for starting a group…feedback from kids, parents, teachers…practical issues that make the group work, like feeding them and what not…how do you structure the sessions and how do you decide what to cover…etc!

You could even consider expanding it by reaching out to others who do creative study groups on other subjects and including chapters/ideas from them?

Let me know when you write this book and I’ll buy it! :)

• W- says the secret isn’t that the kids enjoy math, it’s that the kids enjoy spending time with each other. It’s true! The kids show a lot more energy and interest during our group study sessions than, say, when we’re coaching J- on her own. It might also help that both W- and I- create a math-positive environment and contribute a lot of energy and creativity ourselves.

I’m gathering notes for a possible book, as I haven’t found lots of really good tips on the topic. =)

• If anyone is interested in online resources, these wisweb applets from the Freudenthal Institute are rather nice on Negative Numbers.

http://www.fi.uu.nl/dwo/wismaat/mod1/en/frameset.html

There are some nrich games too that help – see Connect Three here for example
http://colleenyoung.wordpress.com/2010/05/28/nrich/

On negative numbers I find different students like the various different explanations, money analogies appeal to many – remove a bill and you are better off, patterns of numbers too are helpful.

• Aankhen

This really sounds like a fun way to learn. When I was a kid, I was in a school that didn’t follow the conventional methods of teaching for seven years, so I didn’t go through the normal grind—we learnt maths in a fairly interesting manner. Even so, it couldn’t hold a candle to what you’re describing!

As an aside, you have a curious mixture of en dashes (–) and hyphen-minuses (-) in your post. Is that the result of some sort of postprocessing?