3 + 2 = 5 Let me count the ways!

That we live in a print-rich environment is undisputed. Even in country areas one cannot travel far without being bombarded by print. In addition to road signs there is a plethora of billboards advertising the best places to sleep, eat, or play that can be found just ahead.

Environmental print is the genre with which many children first engage successfully with reading.  Ask any parent who’s detoured around fast food outlets, hidden shopping catalogues, or camouflaged cheaper brand names of identical products.

That we are immersed in mathematics in our daily lives is just as evident but doesn’t always receive the same recognition. I think this may in part be because people often think of mathematics as abstract algorithms and theorems that we (they try to get us to) learn in school; and that have no apparent application to our lives beyond the walls of the classroom.

However, even the examples mentioned above are just as rich in mathematics are they are in print. They include distances, and perhaps time, to the destination, cost of items, opening hours, and number of attractions. Anyone travelling a distance with young children will have answered questions such as “Are we there yet?”, “How much further?”, and “What time is it?

As I say in my statement about mathematics on my readilearn site,

“Mathematics is all around us. We use it every day for a huge range of purposes from deciding on the sequence in which we dress ourselves, to calculating how much time we have available for an activity.”

One of the resources suggests 25 ways for parents to keep their children thinking mathematically over the school holidays. I have shared these ideas previously in Counting on the holidays.

Recently I was at the gardens with my two grandchildren (G1 and G2, aged 6 and 4), their mother, and my Hub. The children consulted a map and signposted paths to follow the Children’s Trail which had various sculptures along the way. I am undecided about the value of distracting children from the trees and plants, as if the vegetation itself would not be interesting enough. However, the children enjoyed locating the sculptures in the sequence numbered on the map, and reading the accompanying information. They were engaged in purposeful reading and mathematical thinking in context: real life learning!

Pandas on the Children’s Trail © Norah Colvin 2016

As we headed back, G2 made a comment that showed she was engaged in mathematical thinking of her own. She observed that there were two children and three adults, which made five of us all together.

“That’s right,” I confirmed. “There are five’. I thought for a little while, then added, “And do you know what? As well as two children and three adults, there are two boys and three girls.” The children looked at the group and confirmed that I was right. They laughed – a different interpretation.

This gave me an idea for a thinking game: how many other arrangements of three and two could there be?  I wondered if the children would like to play along. I had never attempted this before and had no idea if there’d be more, or if we had already exhausted all options.

I looked at the group. I noticed our shoes: three had closed shoes and two had open shoes. I thought about our names: three shared one surname, two another. Then we were on. Everyone was thinking of ways we could be arranged into groups of two and three.

Sometimes we sorted according to different characteristics, as in the previous examples. Others times we used a simple yes or no sorting, such as two have hats with brims and three don’t have hats with brims. This is the easiest sorting to do, and the first that children learn.

G2 made many suggestions of this type of sorting for one and four.  One has the characteristic, the others don’t. This was age appropriate for her, and it was great to see her joining in confidently and contributing to the discussion. G1 was able to engage in the more complex thinking required for the groupings of three and two.

I was amazed at the number of different combinations we came up with, and that each of us was combined with others in many different ways.

These are some of the ways we arranged ourselves into groups of two and three (not physically, just in our discussion).

This seemingly mundane activity has potential for developing thinking and learning by encouraging:

• thinking about things in new and different ways
• looking for similarities and differences
• observing detail
• sorting according to different characteristics – which is important to both maths and science (think animal and plant classification)
• having fun with maths
• having fun with family

But wait, there’s more: When we left for home, two went in one car and three in the other!

I think this would be a great activity to do with young children learning about number. It may be a challenge for teachers in Australia where children wear uniforms to school but I’m already thinking of how it could be done with toys or illustrations. It’s not quite the same as doing it with the children themselves, but it could be fun. What do you think?

You won’t be surprised to discover that I have prepared a readilearn resource for sorting as well!

35 thoughts on “3 + 2 = 5 Let me count the ways!”

1. Sarah Brentyn

“Mathematics is all around us. We use it every day…” Absolutely. But you’ve found a way to make it fun. 🙂 I’m beginning to think you can do that with pretty much anything.

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1. Norah Post author

I think most things should be fun – for kids anyway. And I’m still a 6-year old at heart, so I try to keep it that way for me as well. Some things just aren’t fun, but I watched a movie the other night called “What we did on our holiday” with Bill Connolly and he almost made dying seem like fun. A lot of people fear maths as much as death. I think maths should be fun!

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2. Bec

That sounds like a fun afternoon with the wee kids! I can imagine how much they would have enjoyed your encouragement for their thinking. I also immediately thought of the teddy bear resource, which is fun. It’s quite ..calming? for a “grown up”.

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1. Norah Post author

I’m not sure what you mean by your last statement. Did I make a mistake?
I love engaging in this mathematical thinking with the wee ones. It is great when they make their own discoveries. I remember when you discovered that you could count in twos (no, it wasn’t last year!). You were very proud of you discovery, but I was impressed! 🙂

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1. Bec

Hi Nor – no mistake, it just felt silly referring to myself as a “grown up”. But I do enjoy the teddy bear sorting. Maybe a sorting activity for adults would be a nice de-stressing game or app.

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3. Sherri

I love how you incorporate such wonderfully educational ideas into an outing with your grandchildren. And you make it so much fun! So yes, I think it’s a wonderful idea. And the photo of the billboards reminds me of the first time I went to America, I could not believe how many there were and how huge! Mathematics was something I struggled with in school but ended up using it in all my jobs. Something we use in ways we take for granted, as you’ve perfectly illustrated here Norah! 🙂

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1. Norah Post author

Thank you for your lovely comment, Sherri. I do have a lot of fun with my grandchildren, and I do try to make the learning incidental and not seem as if I am “teaching” them. That’s a sure way to turn children off, I’ve found. They put up with it in school because that’s the way it is, but they don’t want to be “taught” outside of that. It’s a fine, fun, line. 🙂
I’m looking forward to seeing a few of those billboards when I visit America soon. Maybe there’ll be more opportunities for maths fun with my grandchildren!
Thanks for sharing your thoughts. 🙂

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1. Sherri

Absolutely…making it fun and natural is what it’s all about! And wow…you’re going to America? Oh do tell…when and where? Have you been before? I am itching to get back…next year God willin’ and the creek don’t rise as they say. And yes, you’ll be seeing plenty of billboards there! 🙂

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1. Norah Post author

I haven’t been to America before. I’ve been to very few places. I am very much looking forward to it. I will have great fun with my grandchildren and their parents. I can’t wait! 🙂
I hope the creeks don’t rise and you get back for a visit next year.

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4. Annecdotist

I read this a couple of days ago but didn’t have time to comment, although it’s stayed on my mind since then. Firstly, I think you’re absolutely right that it’s a pity that children aren’t helped to notice everyday mathematics in the same way that they might do with the written word and your exercise sounds great.
I do wonder about the widespread aversion to mathematics and where it stands from. I haven’t looked for any evidence to back this up (although it chimes with something I heard a few years ago on the radio – BBC no less!) but I wonder if there is an overemphasis on basic number skills when most of mathematics is about concepts rather than numbers. I’m not saying we don’t need addition and subtraction skills but that perhaps knowing (perhaps intuitively, not necessarily being able to put into words) what those processes mean, but perhaps teachers’ and students’ anxieties about multiplication tables might impede them progressing to the far more interesting stuff about x and y and the absolutely beautiful i, the square root of minus one (which I’m sure I’ve mentioned before).
On that basis, I think your exercise is excellent: the numbers involved are easy but the concept that things can be divided up in different ways is exciting and creative. Looks like you all had fun on the walk.

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1. Norah Post author

Thanks for your wonderful comment, Anne. I appreciate that you have put forward an explanation for maths aversion. I agree with you. There is far too much drill and practice of abstract notions. Learning multiplication tables would be much easier if children had many concrete experiences manipulating groups and arrays and actually knew what 3×4 meant! I think what is happening too much in schooling now, is rushing children into the abstract, the drill and practice of things that can be measured, without setting a firm foundation of the concepts that underpin it all.
I don’t remember your discussing the beautiful i (assuming you don’t mean yourself!) I’d love to be educated!
We did have loads of fun, but I think I may have had the most!! 🙂

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1. Annecdotist

I’d actually go a bit further and say that multiplication tables aren’t that important at all. That might be because I still struggle with mine (although perversely I still insist on trying to do calculations in my head) beyond the fives but had no problem with mathematics secondary school to degree level. And i is an imaginary number (j in physics) equal to the square root of minus one (if you think about it any number multiplied by itself, even a negative number, comes out positive, so there’s no such thing in ordinary mathematics as the square root of a negative number) but if you allow it is a hypothetical construct it enables, and indeed simplifies, complex mathematical operations (and does something amazingly practical in physics, but I don’t really understand that part).

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1. Norah Post author

Thank you for the explanation, Anne. It sounds fascinating. That imaginary numbers have a purpose fascinates me.
I love watching my grandchildren as they make their own mathematical discoveries and explorations. Grandson has just discovered he can add 2-digit numbers in his head. He was a bit disappointed today when he challenged me to add two 2-digit numbers and I was able to tell him the answer immediately. He said that his mum had told him most adults would be slower than he was in calculating. I had to tell him that it was because he had a very clever grandmother!
Granddaughter was calculating all the way home the number of strawberries eaten and how many left. When she got home she made up a counting-back song about eating the strawberries (and drove her father crazy repeating it!). She also noticed the numbers on apartment mailboxes near her home and realised that some numbers were missing and asked why. I love seeing children in command of their own learning. It is very exciting. And I’m sorry I’ve given you a long answer, but I wanted to share your enthusiasm for maths, even though it’s at the opposite end of the spectrum. 🙂

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1. Norah Post author

Thank you, Sacha. The best thing is to be aware of opportunities when they arise, and play games with them rather than try to “teach”. I’m sure you and your boy will have much fun together. 🙂

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5. Steven

I like how many different combinations were identified and that through examination, it is possible to have a guess at which individuals fell into which category of the other arrangements.

Maybe you could further enhance your resource (or have a more advanced activity) where more involved Group problems could be investigated. For example, there are 4 green blankets and 1 blue blanket, 2 blankets have a name tag and 3 do not. How many arrangements of un-tagged blankets?

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1. Norah Post author

Thank you, Steven. I’m pleased you put a little more thought into the possibilities of the combinations. Some were obvious, but others perhaps less so.
Thank you for your suggestion for a resource. I’ll give some thought to it. Some good open ended opportunities to think creatively would be great.
As I see the answer to your problem, there are only two possible combinations: 2 green and 1 blue un-tagged blankets, or 3 green un-tagged blankets. Am I right?

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6. thecontentedcrafter

I think it was a wonderful game Norah – and just goes to show how much fun can be had by everybody with numbers and groupings. Some examples caught my eye also as activities for beginning algebra………….

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1. Norah Post author

Thank you, Pauline. I think it’s important to have fun with Maths. Otherwise they end up being turned off it, like so many I know were. Including me. Fortunately I re-found enjoyment in the basics and beauty of it! 🙂

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