Category Archives: teaching

The enemy of your enemy is your friend: mnemonics and negative integers

From April 26, Tuesday: J-‘s studying for Thursday’s “in-class performance assessment” on integers. (In-class performance assessment? What happened to the good old word “quiz?” Too much anxiety?) We’re spreading the review out over the next two evenings.

The test will cover adding and subtracting positive and negative numbers. J- and her study group are already off multiplying and dividing (which apparently don’t turn up until grade 8 – really?). W- made up a quick worksheet for J- to practise adding, subtracting, multiplying, and dividing integers.

“The enemy of your enemy is your friend,” I heard her say as she solved the exercises, writing down the correct signs for all the products and quotients. I grinned. I’d taught them that mnemonic two weeks ago. It’s a way to remember the results of multiplying or dividing numbers.

As I explained to the kids: you don’t have to stick to this in real life. Pou can certainly be friends with the friends of your enemy. But this might help you remember the signs for multiplication and division:

  • The friend of your friend is your friend. Positive times positive is positive.
  • The friend of your enemy is your enemy. Positive times negative is negative.
  • The enemy of your friend is your enemy. Negative times positive is negative.
  • The enemy of your enemy is your friend. Negative times negative is positive.
A B Result
Friend + Friend + Friend +
Friend + Enemy – Enemy –
Enemy – Friend + Enemy –
Enemy – Enemy – Friend +

2011-04-26 Tue 20:05

Glad to see it stuck in her head! She answered all the exercises correctly (and quickly, too).

Study group update: negative numbers, exponents, and awesomeness

W- started the kids on a review of positive and negative numbers. They got the hang of those quickly, so they worked on fractions, exponents, scientific notation, and engineering notation. They multiplied numbers with exponents, divided numbers with exponents, dealt with negative exponents, figured out the two answers to x2 = 1… Whee!

J- really wanted to review the Greek alphabet. We introduced it so that they can easily work with θ, α, β, and other characters when they encounter the letters in science and math. J- picked them up really quickly thanks to the flashcards we made. She used the same techniques to teach the other kids more of the letters, repeatedly cycling over small sets of letters, sharing original mnemonics (λ reminds her of “Mary had a little lambda” and a hill).

Watching the kids teach themselves Greek letters – and have fun doing so! – I wondered what on earth we were doing correctly, and if we could help other people do it too. Maybe it’s really just providing a space where the kids can get together and learn, and some guidance and exercises to help them grow.

J- says she learns more – and enjoys learning more – in our study groups than she does in school, because the study group is more fun, more focused, and easier to understand. It’s a happy middle between the intense focus and isolation of a one-on-one tutoring session, and the anonymity of a large class. I’m glad we’re doing it, and I’m amazed at how the kids are doing.

And they begged for more brainteasers! So now I get to dust off my collection of logic puzzles and go through them. Turnabout’s fair play, though, so they have free license to stump me with whatever they can throw at me. =)

2011-04-15 Fri 18:43

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

Why we use more than math textbooks and general-purpose resources

For last Sunday’s study group, we focused on algebraic expressions. The kids were a little out of sorts at the beginning. “Math is boring,” one said.

“The way it’s taught in school, maybe. But math is really useful in life, so it’s good to learn it,” I said. I shared a few examples of saving money with math, enjoying life with math.

The group warmed up using a matching exercise, matching the word problems on the left side with the algebraic expressions on the right. Then we worked through some of the problems I’d prepared. In one afternoon, we talked about:

  • cats and how much food they eat (1/4 cup, twice a day, 365 days, n cats…)
  • T-shirts, sleeping cat toys, and chopsticks that look like lightsabers
  • how much it might cost to eat onigiri for every meal, every day, for a year
  • how long you might be able to eat onigiri given a particular budget
  • Scott Pilgrim, Wallace, and Knives Chau
  • more cats, including Neko on my head

There are several types of exercises. Completely abstract ones (here’s an equation, solve for n) get lots of confusion and little engagement. Practical exercises (how much would this cost after tax?) get some interest. Outlandish exercises drawing on the kids’ interests get lots of laughs – and solutions. So we mix practical exercises and outlandish ones, one to show math in real life and the other to get the kids involved. It’s like improv comedy, but for education.

This is where parents and tutors really need to step in and mix things up. Textbooks are written for everyone. They can’t take individual interests into account, and they can’t be revised each month to take advantage of pop culture references. When you make up your own exercises, though, you can do whatever you want.

I know J- likes Scott Pilgrim, Fruits Basket, and cats, so they turn up in math exercises. It’s not hard to pick up some standard forms of exercises from textbooks and translate them into more interesting situations.

Helping someone learn? Make up exercises based on their interests and see what happens.

Helping kids learn algebra

In the math study groups we organize at home, we’ve moved past fractions and percentages into the wild and wonderful world of algebra. Translating a problem into an algebraic equations is somewhat familiar to J-, but the process of solving algebraic equations confuses all the kids. I have a feeling that we’re either taking up the topic before the teachers have had a chance to adequately explain things, or the real-life situation (“Town”) leaves the students little time to focus on other lessons. Well, it is what it is (this is becoming one of my mantras these days), so we just have to do our best.

The small-group format is still working well. We’re going to try it with four kids to see if pairing them up to help each other will help the kids learn more effectively and build more confidence. W- has also checked out an armful of books from the library. I’ve been paging through “Real-World Algebra” and similar books to find some ideas for exercises the kids can relate to.

We try to liven things up with energy and amusing examples. They have to eventually become comfortable with abstract exercises such as 5n + 30 = 180, and it’s difficult to make that more interesting. I don’t want to just repeat the fake word problems of standard algebra textbooks, so I’m keeping an eye out for real-life situations in which I’ve used algebra myself. It can be hard to notice when you take math for granted, but math is everywhere, so I should be able to collect examples.

In the meantime, there are small things we can do to help them keep their attention on math or to remember the concepts more vividly. I tried this example for distribution:

2 * (number of lions + number of tigers + number of bears) = 2 * number of lions + 2 * number of tigers + 2 * number of bears.

I drew a lion, a tiger, and a bear instead of writing the corresponding phrases. =) Then J- said, “Oh my!” and everyone laughed.

The kids often forget that whatever they do to one side of the equation, they need to do to the other. As a result, J- once ended up with the interesting equation 2 = 4. Looks like we need to review how to use the equals sign. ;) We might try the see-saw metaphor. If you have a balanced see-saw, you can keep it balanced by adding or removing the same amount from both sides. You can keep it balanced by multiplying and dividing from both sides. If you add, subtract, multiply, or divide one side without doing the same to the other, you end up with an imbalanced seesaw. We’ll see if that helps them remember.

Because we’re discussing new material for them, we have to walk through the exercises together before they can try things on their own. When they try things out, progress can be slow and frustrating. We’re seeing if we can take advantage of group dynamics by posing a question and encouraging the kids to talk out loud about the strategies they might use. They help each other out, too. The group format definitely pays off – seeing other kids struggle or succeed helps a great deal.

Do you have any favourite middle school group study resources or tips? =)

Helping kids learn about automation

J- shuffled in and out of the living room, listless and bored. As part of a 9-week simulation of real life in school, she and her classmates had been assigned jobs. Her job was to be an accountant, and the tedium of checking dozens of pretend tax returns had long sunk in. W- had encouraged her to use a calculator, so at least she didn’t have to multiple all those figures by hand, but there were still so many numbers to verify.

My geek sense tingled, as it does whenever there’s an opportunity for a quick win through automation. I coaxed her back to her homework. “Come on, let’s set up a spreadsheet,” I said. “That way, you don’t have to redo each of the calculations or worry about getting things wrong.”

We brought up OpenOffice.org Calc. She was still lackluster, so I took the lead in creating the spreadsheet. I asked her which tax return we could use as a model, and she picked hers. We started filling in the formulas, checking her work along the way. (We found and fixed an error in her tax return, too!) Then we tested the spreadsheet on a few other tax returns she had manually done, and she used it to check the rest.

Result: Not only could she verify a correct tax return in less than a minute, but she perked up and started having fun with it. She made a pile of correct tax returns and a pile of incorrect ones, with sticky notes pointing out the deficiencies. She still doesn’t want to be an accountant again, but at least she knows that tedious tasks might be automated away.

The next time J- finds herself doing tedious calculations or verifications, I hope she thinks about how much faster, more reliable, and more enjoyable the spreadsheet was compared to calculating things step by step, and perhaps invest time into learning how to automate whatever she needs to do.

How do people learn how to automate? It’s such a time-saving skill, but it doesn’t seem all that common. Maybe people are intimidated by spreadsheets and programming languages, and that fear of losing more time keeps them from gradually building the knowledge they need to save lots of time. If we can show J- and other kids the benefits of automating, maybe that light at the end of the tunnel will encourage them to learn. If we expose them to the methods for automating tasks, such as putting calculations into a spreadsheet, creating keyboard macros, or writing short programs, maybe they’ll realize it’s not scary – and maybe they’ll start modifying or creating new tools.

In my experience, working with new automating frameworks is always slow and somewhat frustrating in the beginning. It helps that I don’t usually need or want to automate everything right away. I break things down into small things, small wins. I might start by figuring out the most time-consuming parts and automating that 10%, or automating the most common operations. As I become more familiar with the tools and the process, I automate a little bit more, and more, and more. Eventually I might even create a tool that other people can use, like the way my Community Toolkit for Lotus Connections is off and running.

The hardest thing, I guess, is knowing where to start. I run into that problem a lot, because I work with lots of different technologies and frameworks. It’s like looking for the end of a tangled piece of string. That can be hard to find in the confusion, but once you do, you can start unknotting the mess. I want J- to be able to think: ah, this has to do with calculations, maybe I can get a handle on it by using a spreadsheet, putting in manual steps if needed.

How do you use teachable moments to encourage people to automate?

2011-03-29 Tue 21:09