Category Archives: math

Learning and teaching early math

I was delighted to find a textbook called Learning and Teaching Early Math: The Learning Trajectories Approach at an EarlyON drop-in centre. The Esso Family Math program reminded me to talk to A- about math concepts beyond counting, and it was great to learn about math in early childhood education in even more detail.

Subitizing: This is about instantly recognizing small groups without counting them. The key tip was: “Use small number words in everyday interactions as often as you can.” S straight-line arrangements of homogeneous objects are the easiest, then rectangular, then scattered. Presenting different groupings can help kids learn how to add groups up to get a total.

Counting: When A- counts too quickly, she sometimes misses items or double-counts. I can encourage her to focus on accuracy by saying something like, “Slow down and try very hard to count just right.” Pointing, touching, or moving items can help. This is a good time to introduce board games.

Comparing, ordering, and estimating: Number lines are hard to work with. 10-frames might be a good starting point. Estimating can be helped by subitizing and using benchmarks. Games to play: building stairs that are missing a step, matching place settings, asking “Who is older?,” asking “Is it fair?”

Arithmetic: Predict, then count to check. “Counting up to” can lead to subtraction (5, 6, 7, 8). When A- starts doing math in school, it can be good to help her learn how to use her non-writing hand to count as a way of confirming. The textbook had a good breakdown of different types of problems and their difficulty: change-plus, part-part-whole, change minus; a + ? = b; ? + a = b. Showing dot diagrams can help with subitizing and decomposition. (6 = 0 + 6 = 1 + 5 = …) Break apart to make 10. See which numbers can be shown with the same number of fingers raised on each hand.

Spatial thinking: Feely box? Also, talking about patterns, landmarks. Taking pictures of things and their immediate surroundings, then going on a scavenger hunt. Make my picture. Shadows.

Shape: Don’t forget to show different variants instead of just typical triangles, etc. Identify squares as a special type of rectangle. Talk about attributes (points, sides, …). Show distractors. Secret sorting – guess my rule.

Composition and decomposition of shapes: Pre-composer, piece assembler, picture maker, shape composer, substitution composer, shape composite iterator, shape composer with superordinate units. Block & LEGO building: planned, systematic; verbal scaffolding. Agam program? Pattern block: outlines, vertices, matching sides, internal lines.

Geometric measurement: Standard rules are more interesting and meaningful? Teaching kids to line up endpoints. Cut pieces of string to help with indirect measurement. Subskills: iteration, zero point, alignment. Logo programming can be helpful. Talk about bigger, smaller, longer, shorter. Area is hard; try folding/cutting/moving paper. Talk about capacity/volume, angle, finding similar angles.

Patterns: Not just visual patterns (ABAB) – the search for mathematical regularities and structures. Be careful about using = – don’t use it to list objects (John = 8, Marcie = 9), numbering collections (III = 3), strings of calculation (20 + 30 = 50 + 7 = 57 + …). Provide variety (ex: 8 = 12 – 4). Contrast with > and <. All math is a search for patterns, structure, relationships.

How do you know? is a very powerful question. Ask it to get kids reflecting on how they figure things out. Challenging tasks result in better long-term memory. Promote a growth mindset instead of a fixed one. Well-designed computer manipulatives can be worthwhile.


  • Talk about bigger numbers (4-10) for sets of present, visible objects
  • Discuss math while reading
  • Keep fathers involved
  • Talk about geometry and spatial relationships
  • Do puzzles, play math games
  • Cook with kids
  • Have high to very high expectations
  • Don’t worry about base 10 blocks, etc.

The book mentioned that many early educators tend to spend just a little time on math, and may even have a bit of math anxiety themselves. I like math, so it might be good if I handle sneaking in more of it during play time. Based on this, I think I’m going to try:

  • Bringing a die around so that we can use it for subitizing practice and impromptu dice/board games
  • Looking for developmentally-appropriate spatial puzzles at the drop-in centres
  • Using more comparative language (bigger/smaller) when we’re playing with playdough
  • Making up patterns and talking about patterns I see around me (“I noticed that…” “What do you think the next one will be?”)
  • Taking advantage of A-‘s interest in fairness, comparison, etc.

Back to school, back to study groups

We started our first study group session on Friday with a quick review of multiplication. J- and V- warmed up by reciting the multiples of 6 to 9. Good retention from last year, and we’ll see how practice helps them improve. After the warm-up, we went over a shuffled deck of multiplication flashcards.

The teachers had given them a quiz in school, so we covered some of the topics they found confusing. W- and I explained the difference between convex and concave shapes using angles and lines. I drew different figures and quizzed them on the classifications. J- and V- drew their own figures, and they classified them together.

Squares and square roots were another point of confusion. We started off with a graphical review of what squaring means, and what a square root is. I used a tip from John Mighton’s “The Myth of Ability”: I tweaked my exercise to vary in scale without varying in difficulty. (What’s the square root of 31337 x 31337?) After J- and V- understood the relationship between squares and square roots, we covered approximation and factorization as ways of finding the square root. J- and V- practised finding the square root of numbers like 225 and 144.

We’ve encouraged them to take notes so that it’s easier to review lessons. The extra study group time will definitely help, too. Grade 8 will help students learn how to solve real-life problems, so we’ll be sure to show more of the calculations of everyday life. Here we go!

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).

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.

Hypercubes, happiness, and serenity

I remember reading an excerpt from Flatland in Childcraft when I was growing up, and wondering: how would a flat square understand this three-dimensional world we live in? In high school, I read a book about mathematical curiosities. Challenged by the idea of visualizing hypercubes and other higher-dimension objects, I turned to a trick I’d come across while reading: take what you see, use time as the fourth dimension, and imagine all the moments superimposed. Non-existence, birth, life, motion, death, and oblivion collapsed into a single space, further complicated by the rotation and revolution of the earth, the other motions of our galaxy and universe…

I had an existential moment: life is so short and insignificant! 

And then I thought, “Hey, this is pretty cool.” I dipped into this imagined world occasionally, thinking about the past and future of places, objects, and people. It proved to be a useful test for relationships: what would life be like with the grief of losing this person – will it have been worth it? It also helped me let go of stuff. I could see myself before I got whatever it was, and I could see myself after.

You might say it’s an odd sort of happiness that maintains an awareness of death and insignificance, but it’s the sort of calm happiness that’s confident that everything will work out. Why get upset over something that will pass?

So when I came across the ideas of unconditional serenity and emptiness in Joseph Sestito’s Write for Your Lives (an approach that draws on Buddhism), I thought, “Hmm. That’s what they call it.”

It’s still a little strange to look at someone, stretch my imagination, and see them as child and senior, idea and memory. It’s good practice, though, and it reminds me that we’re all in the middle of our own journeys.