How to Make Your Kids Love Maths

I once wrote a post called How to help your child fall in love with maths (even if they hate it) in which I talked about how my children learn maths without a curriculum.

That was three years ago now. I’m pleased to say our approach is still going strong and that Cordie (12) and Jasper (11) love maths more than ever.

Today I thought I’d reflect on what aspects of our maths approach have been most successful, then in my next post I’ll share in more detail what each of my kids are doing for maths right now and what our maths plans are for their senior school years.

When I learned maths at school my goal was only ever to get the right answers. I would watch the teacher do an example, memorise the procedure and obediently do my sums. It never once occurred to me to ask why a particular maths method worked. (Or perhaps I just learned to suppress that curiosity very early on.)

When it came to exams, I crammed a bunch of procedures into my head, passed with A grades, then promptly forgot everything.  For the next 20 years I had nightmares about going into a maths exam unprepared and not being able to answer a single question.

In contrast, my kids have always been allowed – encouraged – to ask questions. To be honest, by this point there’s no stopping them. In maths, as in life, they don’t accept anything unless they know why. Yes, sometimes regret this. 😉

Over the years I’ve learned to anticipate their questions by encouraging them to work things out for themselves in the first place.

A few examples

So when Cordie wanted to know how the long division algorithm worked, we went in search of answers (thank you, Denise’s Cookie Factory Guide to Long Division).

In geometry, most textbooks just present formulae for the area of shapes. But I knew that my children would want to know why these worked, so we figured them out for ourselves (see below for more details).

And when we recently came upon trigonometry, Cordie not only wanted to know what this branch of maths was used for but also what all those strange words actually meant. Which was possibly the first time it had occurred to me that sin, cos and tan were anything other than magic buttons on a calculator that when pressed while reciting the appropriate incantation – ‘SohCahToa!’ – spewed out the correct answer.

Of course when your kids ask questions, you need to . . .

2.  Be willing to go off on tangents

The best thing about not following a curriculum is that you’re never tempted steamroll over your child’s curiosity in an effort to finish a bunch of material by the end of term.

So there’s always time for games…

And you have time to investigate questions, like ‘What’s the area of a non right-angled triangle?’ Which leads to several weeks of playing with shapes as you figure out the relationships between rectangles, parallelograms, triangles, trapeziums and even circles. Which means your kids never panic about forgetting a formula, because they know everything follows from cutting up a rectangle.

Having time to follow rabbit trails means you have time to explore questions like, ‘What’s trigonometry used for?’ Which leads you to research the history of trigonometry, from right back when ancient astronomers used it to calculate the positions of the stars, to how triangulation is used today in everything from MRI scanners to animation software.

And when your daughter who’s passionate about linguistics asks where the words sine, cosine and tangent come from, you can spend a pleasant half hour discovering how ‘sine’, like the word used to describe our facial cavities (sinuses) comes from the Sanskrit word for ‘bowstring’.

Of course, your child may not be as interested as mine in the etymology of maths terms, but by following whatever it is they are interested in, you’ll deepen their understanding of what they’re learning and make it more memorable.

3. Do buddy maths

Since the early days of homeschooling right up until now when they’re 11 and 12, I’ve done maths alongside my children.

In this way I’ve been able to share my passion for maths, clear up any confusion as soon as it occurs, and head off boredom by moving on as soon as I can see a concept’s been mastered. (Plus of course I’m there to help navigate rabbit trails and answer ‘why’ questions.)

4. Don’t drill them on maths facts (unless they ask you to)

This is a controversial one, and I won’t pretend I’ve never casually suggested to my kids how useful it might be for them to rote-learn their multiplication tables, but they were having none of it.

Jasper couldn’t see the point of memorising something he can quickly work out every time, and (like a lot of bright people) Cordie gets stressed by time pressure.

So I decided I may as well trust mathematicians like Jo Boaler who says that not drilling kids on maths facts is a sure way to increase both their maths confidence and their number sense.

“Drilling without understanding is harmful … I’m not saying that math facts aren’t important. I’m saying that math facts are best learned when we understand them and use them in different situations.”

Jo Boaler

Guess what? It worked! After years of having fun with numbers, neither of my children has a problem with doing rapid mental calculations – for numbers both below and above 12.

5. Do maths anywhere that works for your child

In an ideal world I’d teach my kids sitting down nicely at the table as they write neatly using pencil and paper. I’m easily distracted and repetitive movement in my peripheral vision drives me crazy.

However… as the adult I’ve had to adapt myself to accommodate how my kids learn best – on any given day.

So if my son wakes up with the wiggles and wants to do maths while he leaps  around on a giant ball, I take deep breaths, read problems aloud, and hold up a whiteboard for him.

I used to worry that Jasper would never be able to be still enough to write out his answers, but I’ve noticed that when his mind is sufficiently engaged he’s quick to grab a whiteboard to draw a diagram or scribble some notes to help him figure out a problem. As maths gets more complex (and interesting) I anticipate him naturally doing this more and more.

* * *

How do you do maths in your house?

What approach to maths works best for your children?

I’d love to hear from you. 🙂

* * *

PS   Bonus maths story – For a behind-the-scenes story about what maths is really like in our house, hop over to my blog about life in a family that embraces its quirkiness, Laugh, Love, Learn.

PPS   Remember my maths nightmares? I haven’t had a single one since I started learning maths this way alongside my children!

Resources

Books

Let’s Play Math  Denise Gaskins. A wonderful book all homeschoolers should read. See my review here.

What’s Math Got to Do with It?  (UK title The Elephant in the Classroom) Jo Boaler

Living maths activities we’ve done04

How to make a multiplication tower

Fun with tessellations

Pythagoras for kids

5 Days of maths playtime

How to teach maths without a curriculum

Let’s Play Math – My review of the book that has acted as our guidebook through our years of learning maths without a curriculum

When every day is maths playtime – Our living maths approach when my children were 8 and 9 years old

Living maths curriculum 2013-14

How my autodidactic 9 year old is learning maths without a curriculum

How we do maths without a curriculum

Why we love Edward Zaccaro more than Khan Academy – About my 9 and 10 year olds’ favourite maths books

Maths – Why Faster Isn’t Smarter

Teaching trigonometry

Teaching trigonometry Resourceaholic. When you introduce your child to trig, I highly recommend printing off a set of logarithmic ratios in table form before you reach for the calculator, and that you start out with the kind of  approach outlined in this article.

Applications of trigonometry Dave’s short trig course

Origin of the terms Sine, Cosine, Tangent The Math Forum

Geometry

Animated explanations for area of basic shapes I can’t find the one we used, but this is very similar. We cut the various shapes out of coloured card.

Times tables

Fluency Without Fear Jo Boaler, Fluency Without Fear

How my autodidactic 9 year old is learning maths without a curriculum

In my last post I described J(9) as fiercely autodidactic, which makes me giggle because it’s so literally true. J(9) is bright, funny, creative – and very independent. When you add in the emotional regulation challenges that come with Sensory Processing Disorder, you have a child who keeps homeschooling life very interesting.

Like his sister, J(9) didn’t get on with any maths curriculum. We stopped looking for one that worked a long time ago. He happily joins C(10) and me for maths stories and hands-on activities, but until recently it was impossible to do one-to-one maths with J(9).

I’ve heard enough stories about unschooled kids and maths to know that he’ll get there in the end. J(9) has a natural aptitude for numbers – he knows most of the multiplication tables without ever having consciously learned them, for example. So I didn’t worry about his long-term future. But maths is fun, and I didn’t want him to miss out.

An obvious solution for someone who doesn’t like to be taught is to use a self-teaching curriculum. Unfortunately, J(9) finds these boring. I sympathise. It’s difficult to bring out the joy of real-world maths in a self-teaching curriculum aimed at 9-year-olds.

I thought, briefly, that Khan Academy might be an exception. I liked how its maths curriculum is laid out, and  the sophisticated way coaches can monitor pupils’  work. Unfortunately, Khan Academy didn’t work out for J(9). I’ll share more about that in my next post.

What to try next?

One of my favourite homeschool mum roles is detective. I love quietly observing my children, gathering clues about how I can support their learning.

I considered what I knew about J(9) and maths. He has strong spatial skills and likes playing with numbers. He’s easily bored, and to focus his mind he often needs to move his body. He loves puzzles and games – but if there’s one thing even more likely to trigger a meltdown than making a mistake, it’s losing at a game. We’re working on these challenges. I know about the importance of a growth mindset, and one day I hope that J(9) will see the value of mistakes, too.

In the meantime, I relied on my own growth mindset. I took everything I’d learned from each of our maths “failures” and just kept on trying new ways to work with J(9). It only occurred to me recently, looking back over the last few months, that we seem finally to have found our groove.

The solution (for now)

What has evolved for us is an extremely relaxed version of the buddy maths I do with C(10). Maybe “relaxed” isn’t right word. “Mindful” might be a better description of my role in the process. Here’s how it looks in detail.

The book we use – J(9) chooses a book to work from (e.g. a maths story, or a source of problems). Every day for the last month he’s chosen Becoming a Problem Solving Genius: A Handbook of Math Strategies. (I’ll say more about why we love this book in my next post.)

Where we do maths – We take our book, together with whiteboards and markers, to the sofa.

Topic of the day – J(9) picks a chapter. We rarely follow books sequentially, though we often continue with the next level of problems in a topic we left off last time.

Time of think – One of us reads out a problem. Then I stay quiet and give J(9) time to think. I only offer hints  when he asks for them (I’ve learned this the hard way). Instead, I take deep breaths and remind myself that crawling under the sofa being a snake, or jumping on top of it like a monkey, helps him concentrate.

Writing things down – If I don’t instantly know the answer to a problem, I use a whiteboard to figure it out. J(9), ever independent, doesn’t look at my workings. His brain works differently from mine and he often mentally calculates things I can’t.

I don’t force him to write anything down, but he sees me doing so, and recently he’s started to make his own notes and diagrams when he solves more complex problems. I do my secret happy-dance when he does this, because representing problems in different ways is an important mathematical strategy. It also allows him to retrace his steps when he goes wrong. (And – less importantly – one day he’ll need to show his workings in exams.)

Dealing with mistakes – J(9) tells me his answer when he’s done. Whether I agree or disagree, I set my voice to neutral and ask, “How did you get that?” If he’s made an error, he often spots it as he explains his process. He can then change his answer, so he doesn’t feel like he’s got it “wrong”.

Occasionally, when we get different answers, I realise I’ve made a mistake. J(9) likes it when that happens.

If I’ve got the same answer via a different process, I ask J(9) if he’d like to hear how I did it. Then  I try to respect his answer! He’s gradually learning that one tends to make fewer mistakes using simpler processes, but if I’ve learned not try to foist a method on him.

And, I admit, there are still times when J(9) can’t see where he went wrong, doesn’t want to talk about it, and he ends the session early, frustrated. I’m learning not to get upset when that happens – it doesn’t negate the learning that’s gone before. We’ll come back to the topic another time, when he’s ready.

When are we done? – There’s no minimum time for our sessions or number of problems we do. We might do one question or thirty. J(9) is in control of his learning.

The results (so far)

One-to-one maths with J(9) has transformed from something we both dreaded into an absolute pleasure (mostly).

I’m hopeful that our buddy maths routine will continue after we’ve exhausted the questions in Becoming a Problem-Solving Genius. Perhaps we’ll move on to Murderous Maths or try out one of the many great sources of maths videos.  I’ll let you know.

Perhaps the best outcome of our new way of doing maths is that J(9) is beginning to trust me as his learning mentor. I know he will always want to learn as independently as possible. As he gets older that’s not necessarily a bad thing. But I want him to know he can always come to me for help and support.

***

Weekly Wrap-Up at Weird Unsocialized Homeschoolers

The Hip Homeschool Hop at Hip Homeschool Moms

The Home Ed Link Up week 4 at Adventures in Home Schooling

How to make a Multiplication Tower

Do you have projects pinned that you’ve been meaning to do for years? Multiplication towers was one of mine. When I first saw them on The Map is Not the Territory I knew we’d all learn something from playing with this 3D multiplication model.

The catalyst for getting around to it was a Christmas gift. My lovely parents-in-law sent us some fine meat which arrived from Scotland packaged in polystyrene and dry ice. The steaks were delicious – but you know you’re a homeschooler (or a toddler) when you get even more excited about the packaging than the gift itself. Yes, that giant slab of polystyrene was the perfect base for our multiplication tower.

What you need

Here’s where my learning curve began. (Skip this bit if you are an instructions-reader, you won’t get it.) You know how you sometimes start gathering materials for a project without consciously thinking it through?

So I had my polystyrene base, a pack of beads, and now I needed some skewers. How many? I thought of the multiplication grids we’ve been working on recently. Should our multiplication tower go up to 10 x 10, or all the way to 12 x 12? “Oh, why not go for it?” I thought, throwing caution to the wind and putting two 100 packs of bamboo skewers in my basket.

Back home, I used a sharpie to mark out a grid of 144 dots on the polystyrene. It wasn’t until I began inserting the skewers that I saw the flaw in my plan, and realised that (a) no skewer would be tall enough to hold 144 beads, (b) the 12 times table alone would use up almost all our 1000 beads and (c) as enthusiastic as my kids are about hands-on maths, I may not be able to persuade them to spend a week threading beads.

At this point it occurred to me to refer back to the project, where I discovered that Malke made a 5×5 multiplication tower, requiring 225 beads. Good idea.

I thought about making the model myself first, to make sure I understood how it worked. But after a few beads I remembered that sometimes a student learns more if the teacher doesn’t know all the answers.

So I called C(10) and we figured it out as we went along. (See the labelled photo of our Lego multiplication tower below if you’re not sure of the reasoning behind the colour-coding.)

Of course this isn’t difficult maths – many people make multiplication towers with slightly younger kids as an introduction to the concept – but it does get you thinking logically about number patterns and the commutative property.

Lego multiplication tower

I showed J(8) the Lego multiplication tower Frugal Fun for Boys made.  He enthusiastically set about gathering 2×2 Lego bricks to make his own. He only had enough bricks to make up to 4×4, but it was enough to get the idea.

When we compared C(10)’s bead tower with J(8)’s Lego one, we noticed they were different.  In our bead tower we had used the same colour beads for the “ones” on both the x and y axes, as they did at The Map is not the Territory. In the Lego tower, we had used different colours for each “one” on the y axis (as Frugal Fun for Boys did).

So in the photo above, on the x axis we have a blue Lego for “1 one”, two blues for “1 two”, three blues for “1 three” and four blues for “1 four”.

Meanwhile, on the y axis, we have one blue and one white Lego representing “2 ones”, a blue, a white and a red for “3 ones” and a blue, a white, a red and an orange for “4 ones”.

The children and I discussed the two models, and decided the Lego model made more sense to us. C(10) changed her bead tower.  More great learning.

Minecraft multiplication tower

Next, C(10) decided to make a multiplication tower on Minecraft. You can’t write numbers on Minecraft so she decided to use colours to represent the numbers (the small dark blue blocks along the axes in the picture below).

She spent some time viewing the tower from different angles in a way that made me slightly dizzy. At one point she exclaimed, “Hey, look at this pattern here, on the diagonal – 1, 4, 16, 25!” She’d discovered the squares.

C(10)’s Minecraft tower reminded me to show her the computer-generated multiplication towers at Moebius Noodles. Click over to see quite how high 12×12 multiplication towers get.

C(10) decided to save her Minecraft multiplication tower in a new “Maths World” and went on to demonstrate some maths puzzles she could play with there. {I see more fun ahead.}

If you didn’t get polystyrene for Christmas

If you don’t have a polystyrene or foam block, you could perhaps use sand to support the skewers, or follow the example of Highhill Homeschool and use cotton. Or make a Lego tower.

Have you ever made a multiplication tower? Did you learn anything surprising?

***

The Hip Homeschool Hop

Wonderful Wednesdays

Entertaining and Educational

Collage Friday

Weekly Wrap-up

In England, maths is the second most hated subject in schools, second only to science.

Most homeschooled kids I know love science. Why? Because they learn science in a fun, hands-on way that bears little resemblance to the dry textbook science of most schools.

But I’ve heard of more than a few homeschooled children who dislike maths. Which begs the question – why don’t homeschooling parents share the joys of maths with their children in the same way they do science?

The answer, sadly, is because of how most adults were taught maths. Whether they hated maths or excelled at it, most people have no idea what real mathematics is.

I know I didn’t. I was one of those kids who enjoyed maths at school because I was lucky enough to have a good memory. That, combined with a competitive streak, got me A’s through to age eighteen. At that point I quit maths and ran for the hills, amazed I’d managed to make it through to the end without anyone finding out that I hadn’t a clue what all those symbols actually meant.

It wasn’t until three years into homeschooling my kids that I began to get a glimpse of what maths really is.  I’m still at the start of this journey of discovery so I’m certainly no expert, but I’d love to share with you some of what I’ve learned so far.

I can’t imagine any child who knows what real maths is finding it boring.

Maths is the art of making patterns

Maths is not about memorising a bunch of dry facts and procedures. Memorisation may have its place, just as learning vocabulary does. But it bears no more relation to real mathematics than a list of French verbs has to the lifetime works of Victor Hugo.

Just like music and painting, maths is an art – the art of making patterns with ideas.

Real maths is a fascinating process of creative discovery.

And what child doesn’t enjoy engaging their curiosity as they play with ideas?

To fall in love with maths, your child has to know what maths really is. And you can’t show them unless you know what maths really is.

The good news is, you can start to think like a mathematician in just twenty minutes. After that you can just jump in and learn alongside your kids.

Learn how to think like a mathematician in 20 minutes

A Mathematician’s Lament

Read A Mathematician’s Lament (links to a free 25 page PDF).  The twenty minutes it will take you to read might be the best homeschool investment you ever make.

After you’ve read A Mathematician’s Lament, read at least one of these books:

Let’s Play Math

Let’s Play Math: How Homeschooling Families Can Learn Math Together, and Enjoy It! Read this first if you’re keen to get started doing real maths with your kids, especially if they are elementary age.

What’s Math Got to Do With It?

Jo Boaler, a maths professor at Stanford University, has conducted extensive, long-term research into how children learn maths. The approach she outlines in this book teaches all children to think and problem-solve – even those who think they’re maths failures who could never enjoy maths.

After you’ve read these, I guarantee you’ll be excited to share what you’ve learned with your children. You may be a little overwhelmed about where to start, but don’t worry, there are plenty of options.

And really, it doesn’t matter what part of maths your child falls in love with first. Just jump in – or if you’re an organised type, make a one month plan of fun maths activities, and get stuck in.

Many parents integrate maths play-days into their homeschool schedule alongside their regular curriculum. If your curriculum is working, this can be a good approach. All children will benefit from the opportunity to play with real mathematical ideas, even if they love their curriculum.

But if your child hates maths, don’t be afraid to ditch the curriculum – at least for a while – and jump into mathematical fun. You don’t even have to call it maths.

Our living maths experiment

My own kids have always been mathematically able, but neither got on well with traditional curricula. We tried Singapore Math and Math Mammoth. Each worked for a while but didn’t last.   It was frustrating at the time, but I’m grateful now for my kids’ honesty – without it, I might never have discovered the joy of real maths.

We’d always read maths stories and done a few hands-on activities, but too often these got pushed aside as “extras” when we were trying to get through the curriculum.

Then, six months ago, we began a one-term living maths experiment, which worked so well we’ve made the change permanent.

“This is all very well,” you might be saying, “but my child won’t be able to get a job or into college if she can’t pass maths. How is all this playing with patterns going to help her pass her exams?”

In answer to that question, consider this story of two girls, Lilly and Katy.

1. Lilly

Lilly is twelve years old. Her mother knows how much pleasure playing a musical instrument can bring, so she surrounds Lilly with beautiful music and encourages her to take up an instrument.

Lilly chooses violin, and soon enjoys playing simple tunes. Her enjoyment inspires her to play more often. As her love of music grows, Lilly decides she wants to learn about time signatures and note values, key signatures and scales –  all the while learning to play more and more complex works.

Lilly enjoys her music so much that she even begins to compose her own little pieces, transcribing the notes onto staff paper and transposing the songs into different keys to share with her musician friends.

2. Katy

Katy is eight years old. Her mother, Tracey, has been told how good it is for children to learn music. And she knows that it will be useful for Katy’s future to pass her musical theory exams at age sixteen.

So Tracey has Katy get out her staff paper each day and copy notes from a book, making sure she gets her clefs and key signatures right. Tracey reminds Katy how important it is to neatly fill in her quarter-notes and get all her stems pointing the right way.

Katy is told that once she has a thorough grounding in music notation and theory and has passed her music theory exams, she will be allowed to listen to and play music – perhaps when she is at college.

Katy grows to hate this boring subject called music, and to Tracey’s frustration begins to dawdle longer and longer over her work, preferring instead to stare out of the window and hum tunes to herself.

What if Katy’s mother were to ask you for advice. Tracey is in despair, saying that Katy hates music, cries through her lessons and begs to be allowed to stop them. “It’s obvious she just doesn’t have a music brain,” says Tracey. “But how will she ever pass her music exams if she doesn’t keep working at it?”

How would you respond?

Which girl will pass the exam?

Which girl do you think will do best in her musical theory exam at age sixteen? Katy, who has spent eight years learning dry musical theory – which she has come to hate – but has never heard a melody?

Or Lilly, who spent four years making and enjoying real music, and who was inspired to learn its technical language along the way to enhance her enjoyment even more?

If children are allowed to experience the creative art that real maths is, everything else will fall into place much more easily.

Resources

Once you’re in on the secret of what maths really is, you’ll begin to notice opportunities for maths play everywhere.

Let’s Play Math is a great place to start finding concrete activities to do with your children.

Before long you’ll be solving puzzles, playing games, crafting intricate geometric shapes, reading biographies of great mathematicians, cracking codes, learning from videos, getting acquainted with the maths section of the library and generally having lots of fun while you develop your mathematical muscles.

Here are just a couple of ideas to get you started.

Blogs

Let’s Play Math – the blog by Denise Gaskins, the author of the Let’s Play Math book

My Living Maths posts here at Navigating By Joy

Moebius Noodles

Stories from an Unschooling Family lists a stack of fun resources in her recent post Bite-Sized Pieces of Unschooled Maths

Math Monday Blog Hop at Love2Learn2Day – a themed blog hop, featuring posts on a different maths topics each week

Talking Math With Your Kids – how to talk with your children about numbers, shapes and other mathematical ideas in daily life

AngelicScalliwags‘ recent Helping a Struggling Maths Student series has several great activities for bringing maths to life

All these blogs contain links to other real maths activities. Enjoy the rabbit trail!

Books

There are so many to choose from, but here is a diverse selection of my favourites that will appeal to all ages:

The Great Number Rumble: A Story of Math in Surprising Places

Murderous Maths

Mathematicians are People Too: Stories from the Lives of Great Mathematicians

The Sir Cumference series (not just for younger kids – middle-school children will appreciate these too)

The Number Devil

For more views on homeschooling maths, check out:

Highhill Education – Math Curriculum Not Required

Hammock Tracks – Math, Tears and Frustration – Not the Perfect Arithmetic Trinity

One Magnificent Obsession – When Math Brings Tears

Barefoot Hippie Girl – Learning Flexibility Via Math

Every Bed of Roses – Math is a Problem – Now What?

***

Hip Homeschool Hop 9/24/13

Collage Friday

Weekly Wrap-Up

TGIF Math Games for Preschool & Homeschool

The Homeschool Mother’s Journal

Living Maths Curriculum 2013-14

We started last year using a combination of workbooks and Life of Fred, and we ended it with a full-time living maths experiment inspired by Denise Gaskins’ Let’s Play Math. I’m pleased to say that the experiment has been a huge success and we plan to continue with it next year.

Why I judged our living maths experiment a success

* both C(9) and J(8) eagerly agree to do maths

* I’ve noticed big improvements in their problem-solving abilities

* they’re more confident tackling challenging maths problems

* because our maths sessions include a lot of conversation, they’re more articulate in using mathematical language and talking through problems logically

* this has extended to their spontaneous use of mathematical charts and diagrams to help solve problems

Our living maths routine

I prefer routines to structured schedules so our plans are loose. Some days J(8) likes more structure – on these days he asks to use Life of Fred which we read together.

I try to balance the kind of activities we do over a week, and tailor the day’s activity to our mood. If we get caught up in a long project like discovering pi I don’t worry about fitting in anything else.

I usually do maths with each child separately, though often the other will join in when they see us playing a game or swapping story-problems.

Problems and Puzzles

We grab a few puzzles or problems, settle ourselves comfortably on the sofa with a whiteboard and dry-wipe marker each (and usually the dog. He likes living maths) and get to work (play).

Sometimes we make up the problems, other times we get them from books or websites. Recently we’ve been enjoying puzzles from Mindbenders and Brainteasers and Primary Grade Challenge Math.

Next year I’m planning to add in the Murderous Maths series and a few other Rob Eastaway books, and I’m sure many more will make their way onto our shelves.

Stories

This term we’ve learned about circles and measuring angles with the Sir Cumference series. I have several more of these on our shelves, which we’ll use as a springboard for more geometry play next year.

We’ll no doubt review and extend our investigation of Fibonacci, perhaps using Wild Fibonacci.

And I’m very excited about doing a project using The Librarian Who Measured the Earth, which tells the story of how Ancient Greek mathematician Erastosthenes measured the circumference of the Earth. (Modern scientific estimates differ by less than 2%!)

Games

At least once a week we’ll play maths or logic games.  Some of our favourites this year have been

Blokkus

Mastermind

Yahtzee

We’ll also continue to try out games we find online, like Contig Jr and make up our own games using a hundred chart.

Manipulatives

Maths is very hands-on round here. Some days we get out our tangrams, pattern blocks, Lego, metre ruler, compasses, measuring cups or weighing scales and just play.

Preparation

This summer I’m preparing by reading books like Knowing and Teaching Elementary Mathematics and taking Jo Boaler’s free Stanford University How To Learn Math course. (I highly recommend Boaler’s highly readable and eye-opening The Elephant In The Classroom – titled What’s Math Got To Do With It? in the US.)

I’m looking forward to sharing more of our living maths adventures over the next year.  What maths fun do you have planned?

Discovering Pi – the living maths of circles

Do you remember learning to calculate the circumference, diameter and area of a circle? If your experience was anything like mine, you were given this random 3.14 number (which for some reason the teacher kept referring to as pie), and told to plug some other numbers into a formula which you were then expected to memorise for the test.

Not exactly magical.

I wanted my daughter’s introduction to the mysterious properties of circles to be different. So last week I set up an activity to help her discover the magic for herself.

What we did

I gathered a variety of flat circular objects (mostly lids), a ruler and some string, and was beginning to take some measurements in preparation for introducing the activity to C(9), when she came over and asked what I was doing (I love it when an activity starts that way).

I told C(9) that I was wondering whether, if I knew how much a big wheel measured across the middle, I could (without measuring) figure out the distance around its edge. I said I was starting out by measuring some smaller circles to see if I could find any pattern.

“Can I help?” she asked, grabbing a ruler and a lid.

Making a chart

As she worked, C(9) recorded her measurements in a chart.

Once she had four pairs of measurements, we were ready for the next stage.  I asked C(9) what our goal was, and wrote down her words: “To find out what the length across the middle is in relation to the length around the edge.”

Finding a relationship

C(9)’s first suggestion was that we calculate the difference between the two numbers in each pair of measurements.

But no pattern emerged, and she wasn’t sure where to go next.

Session 2 – Functions

I began our next session by reminding C(9) of our goal: “To find out what the length across the middle is in relation to the length around the edge,” and asked whether she’d had any thoughts about what to do next (no).

Then I asked her if she remembered learning about functions in Life of Fred. I wrote out some pairs of numbers and asked her to guess the function and apply it to a new case (I used “add 2”). Then she did the same for me (she chose doubling), and we did a few more.

Session 3 – Algebra

After our play with functions, C(9) returned to trying to find a relationship between the numbers in her chart.  She suspected multiplication was the key: “15.5 times something is 50.  But how do we know what the something is?”

By chance,  the previous week C(9) had learned how to balance simple algebraic equations (we’d picked a random chapter from Primary Grade Challenge Math), so I suggested that we try using algebra to calculate the missing something.

This may seem an unnecessarily complicated step, but given what we’d been doing recently, balancing equations was the easiest way for C(9) to see that if we want to know what we multiply x by to get y, we have to divide y by x.

She then solved the equation, first by estimating and then with a calculator. The first missing number came out as 3.2.  (At this point we had a little recap of decimal place value and rounding, and I reminded C(9) that our initial measurements had been rough and ready, using string.)

C(9) did the same for each pair of measurements, and got h=3.2 for all but one set, which came out as 3.4. (I’m not sure why the numbers were so consistent, when pi rounds to 3.1. But I guess it’s not too big a margin of error.)

I was pretty excited at this point, but managed to keep quiet because although C(9) could see the pattern, she wasn’t exactly jumping up and down yet.

We decided to test our newly-found relationship between the lid’s “middle” and “edge” on a new circle. C(9) drew a circle with compasses and wrote out the formula, “middle x 3.2 = edge”.

She measured the circle’s diameter, multiplied it by 3.2 and wrote down the answer, 23. Then we measured around the circle using string and a ruler – 23cm! C(9) was genuinely gobsmacked, like we’d just performed a magic trick – what a great learning state!

Session 4 – Introducing “pi”

In our next session, C(9) and I read Sir Cumference and the Dragon of Pi, in which a young knight has to solve a puzzle involving the relationship between a circle’s diameter and circumference, in order to save his father’s life. After the knight solves the puzzle, the book talks about how the Greek letter pi is used to represent the mathematical constant which is the key to that relationship.

Many paths to pi

I’m sharing this story not because I think what we did is the only way to teach a child about pi. I’m a near-beginner at this living maths business, and I’m sure a better mathematician could have guided C(9) through the process of discovering pi much more efficiently.

I’m sharing what we did because we both learned so much as we happily worked together, and I’d love for others to experience that joy.

If I’d thought in advance about teaching C(9) everything she learned as we did this puzzle, I’d probably never have got round to starting, and who knows how she would have reacted if I’d listed all the concepts  she was going to use beforehand. But as we worked though our puzzle, C(9) had a reason to learn each new skill, and I had real examples to work with to teach her.

When the time is right to teach J(8) about pi, I have no doubt our path will be different, with tangential learning that fits his needs – I’m looking forward to it.

Has your student “discovered” pi? I’d love to hear of the learning route you took.

A Living Maths Approach to Angles

For our living maths story this week I’d planned to build on what we learned recently about right angles by reading  Sir Cumference and the Great Knight of Angleland. As usual, I think I learned as much as the children – and not just about maths.

Sir Cumference and the Great Knight of Angleland

The book tells the story of Radius, son of Sir Cumference and Lady Di of Ameter, who sets off on a quest to earn his knighthood. He takes with him a family heirloom – a circular medallion with mysterious numbers around its edge (the book comes with a cardboard copy of the medallion).

During Radius’s quest we discover with him how to use the numbers on the medallion to measure right, acute and obtuse angles. With the medallion’s help, Radius succeeds in navigating a path through the perilous maze to complete his quest.

Maths Playtime

C(9) actually jumped ahead and read the book while I was working on something else with J(8)*. I came over to find her playing with the “medallion” (protractor). She’d drawn around it and marked 0, 90 and 180 degrees.

We talked about acute and obtuse angles, and I asked her what we might also call the 0 point (“360”.)

Then I pointed to the six o’clock position on the circle and asked how many degrees round that would be, counting clockwise from zero. I left the room to transfer some washing to the dryer – our “maths lesson” wasn’t meant to have started at this point. 😉

A few minutes later she came and found me with the answer – “270 degrees”. I asked how she’d worked it out. Finding out how a student’s minds works is such a valuable part of the mentoring process.

She told me she’d measured a degree with a ruler and found that “this amount” on the protractor [ten degrees] was the same as “that amount” on the ruler [a centimetre]. By working around the circle she’d found the answer. “Then,” she continued, “I realised there was a pattern – you add 90 each time you go round a quarter of the circle.”

I congratulated her on thinking like a mathematician!

If we used a formal curriculum, I’m sure my third grader would have “learned” about acute and obtuse angles by now and maybe even used a protractor to measure them. But what I love about this approach is seeing her sheer joy at figuring it all out for herself.

Hearing her animated explanation of how she’d solved the puzzle showed me without doubt that she really understood the concept. It also gave me valuable insight into her learning process, which is quite different from her brother’s.

If I’d asked J(8) the same question, I’m pretty sure he would have come straight out with the answer “270” – but he wouldn’t have been able to explain how he found it. Not having to “show your workings” when your brain doesn’t consciously do workings is one of the joys of homeschooling for the right-brained visual-spatial learner. Teaching J(8) to “backwards-engineer” and thus extend his thinking process (as well as pass exams, later on) is one of my long-term goals.

A final indication that C(9) took ownership of what she learned was that she decided to make a notebook page about what she’d learned for her maths journal.

*Incidentally, while C(9) was teaching herself about angles, I was helping J(8) understand the steps of long division using Life of Fred (Honey) – at his request.  I love how a living maths mentoring approach means I can help each of my children learn in the way that’s right for them. (Which might be a different way next week – there’s never a dull moment!)

For more hands-on maths ideas, visit the Math Teachers at Play Carnival #63.

Fun With Tessellations

After we read about tessellations in The Great Number Rumble: A Story of Math in Surprising Places we decided to make our own artistic versions. I got the directions from Big Ideas for Small Mathematicians.

Tessellation is about regular patterns that split the plane up into lots of little tiles which fit together perfectly, without overlapping or leaving any gaps. Tessellation is fundamental to maths, because it’s all about symmetry.

We started with a cardboard square each (ours were about 5x5cm).  We talked about how we could cover a page with squares without leaving any gaps.

First we cut a piece from the bottom of our square. We were careful not to cut the corners off, and we found it easiest to cut from corner to corner (to avoid having to measure where to reattach the cut piece on the other side). We slid the cut-off piece upwards, and attached it with tape to the top edge of the square.

Then we did the same on the left side of our square. We cut a piece out, slid it along to the right side, then reattached it.

I asked the children if we had added any cardboard to our shapes, or taken any away (no). We agreed, then, that our shapes should take up the same total amount of space as our original squares.

We traced around our shape on a blank piece of paper, then carefully moved it along and traced around it again.  And again, and again until we’d covered the page.

Our tessellations looked so pretty, we decided to paint them.

J(8)’s didn’t cover his paper without gaps – he was adamant he wanted to create his art his way – but he understood the idea!

The artist M.C.Escher used tessellation to create amazing art.  This BBC video clip is excellent!

Mathematicians know that their subject is beautiful.  Escher shows us that it’s beautiful.

Prof. Ian Stewart, University of Warwick

For more maths ideas, visit the inspiring monthly carnival Math Teachers At Play over at my favourite maths blog, Let’s Play Math.

Join me at:

Collage Friday at Homegrown Learners

Weekly Wrap-Up at Weird Unsocialized Homeschoolers

Look What We Did at Hammock Tracks

Hobbies and Handicrafts at Highhill Homeschool

When Every Day Is Maths Playtime

After I linked up our Pythagoras and the Knotted Rope activity at the Hammock Tracks Homeschool Review,  Savannah offered to interview me about our full-time living maths approach for this week’s Homeschool Review.  I jumped at the chance – I love talking about maths!

My children (aged eight and nine) don’t use any formal maths curriculum. Instead, we have a living maths routine.  The move away from curriculum was gradual. I’d always liked living maths – the fun my children have with it, and how it gives them a sense of maths in the real world – but in my head “real maths” was the curriculum, and living maths was an extra.  And we all know what tends to happen to “extras” in a busy homeschooling household!

Then I read Denise Gaskins’ book Let’s Play Math, which gave me the confidence to flip the balance. Let’s Play Math is one of those precious books which is both inspiring and practical – it makes you want to change, and tells you how to do it.

Here’s how our routine looks:

Monday – maths games like KenKen, Shut The Box or Yahtzee to practise arithmetic and maths facts.

Tuesday – oral story problems.  We grab a whiteboard and take turns making up problems for each other. They learn from watching me solve their (usually very convoluted!) problems, and I learn how their minds work from seeing how they approach each problem.  It’s a great opportunity for me to model, and the kids to practise, how to use notes and diagrams to solve real maths puzzles.

Wednesday – maths literature. We read aloud from a living maths book – maybe a mathematician’s biography or a maths picture book.  I allow time afterwards to play with the ideas we’ve heard about. For example, when we read What’s Your Angle, Pythagoras we knotted ropes to make our own right-triangles and proved Pythagoras’ Theorem using Lego.

Thursday – manipulatives and hands-on geometry. Recently we’ve played with pattern blocks and tangrams, made geometric shapes with toothpicks and mini-marshmallows, and used isometric graph paper to make Maori taniko designs when we were studying the history of New Zealand.

Friday – children’s choice of any of the above.

As I was writing this, I put the question to my nine year old daughter:”Tell us more about this full-time living math approach.” Her reply: “We do more real life maths and story problems, which are really funny because you can make up extremely crazy things.  And often we find maths in real life.”

What do you see as the benefits to this learning style?

Seeing my kids enjoy maths is very important to me, but in itself that wouldn’t be enough to satisfy me that a full-time living maths approach is right for our family. What does convince me is noticing my children beginning to think like mathematicians… Read the rest of the interview at Hammock Tracks

Pythagoras and the Knotted Rope

Now we’ve switched to a full-time living maths approach, we’re actually making time to play with some of the wonderful resources we’ve had on our shelves for years.

On Friday we read What’s Your Angle, Pythagoras, a picture book which tells the story of how the young Pythagoras learns how to make a right-angled triangle using knotted rope, and discovers how to calculate the length of its hypotenuse using square tiles.

Obviously the book is mostly fictional, and it takes some historical liberties – the boy Pythagoras visits Alexandria, for instance, several hundred years before the city was built! – but these are discussed at the back of the book in a way that made my kids laugh and was a handy review of Ancient Greece and Alexander the Great.

How to make a right-angled triangle using rope

In the book, the young Pythagoras notices what happens when buildings are constructed with less-than-accurate right-angles. On a trip to Alexandria with his father, he learns how the Egyptians use knotted rope to overcome this problem.

We tried it out for ourselves. We tied eleven knots at equal distance along our rope before joining the ends in a final knot, so that we ended up with twelve short lengths of rope between each knot.

Then we used our rope to make different shaped triangles. We counted how many lengths of rope were on each side of each triangle.

To make a right-angled triangle, we found that we needed the sides to be 3 lengths, 4 lengths and 5 lengths of rope respectively.

(Top Tip: Take care to make the knots evenly spaced. C(9)’s rope worked perfectly for making right-angled triangles, whereas the one I helped J(8) make didn’t, oops!)

Using Lego to demonstrate the Pythagoras Theorem

While playing with floor tiles, the young Pythagoras in the story discovers that if he makes a square along each side of a right-angled triangle, the square on the longest side uses the same number of tiles as the other two sides’ squares put together.

We tried this for ourselves with 2×2 Lego bricks.

Pythagoras uses what he has learned to work out how long a ladder is needed to reach the top of a wall. He also helps his father calculate the sailing distance to Rhodes.   Both excellent demonstrations of the usefulness of maths!

I would never have thought to teach my kids the Pythagoras Theorem at the ages they are (8 and 9) – all we did was read a picture book. But that living book inspired us to play, and before we knew it we were formulating mathematical proofs.  Another living maths success!