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This week I provided games and puzzles at a welcome lunch for new students in the Mathematical Sciences degree programs. I had big logic puzzles and maths toys and also a list of some of my eight most favourite puzzles on tables with paper tablecloths to write on.

One of the puzzles is the Seven Sticks puzzle, which I invented:

Seven Sticks
I have seven sticks, all different lengths, all a whole number of centimetres long. I can tell the longest one is less than 30 cm long, because it’s shorter than a piece of paper, but other than that I have no way of measuring them.
Whenever I pick three sticks from the pile, I find that I can’t ever make a triangle with them.
How long is the shortest stick?

I sat down at one of the tables and I could see the students there were working on the Seven Sticks, so I asked them to explain what they had done. (WARNING! I will need to talk about their approach to the problem, so SPOILER ALERT!)

They told me about their reasoning concerning lengths that don’t form triangles, and what that will mean if you put the sticks in a list in order of size. They had used this reasoning to write out a couple of lists of sticks on the tablecloth which showed that a certain length of shortest stick was possible but that a longer shorter stick wasn’t possible. And so they knew how long the shortest stick actually was.

Only they said to me they hadn’t done it right.

I was surprised. “What?” I said. “All of your reasoning was correct and completely logical and you explained it all very clearly. Why don’t you think you’ve done it right?”

Their response was that what they had done wasn’t maths. They pointed across the table to what some other students were doing, which had all sorts of scribbling with calculations and algebra, and said, “See? That’s maths there and we didn’t do any of that.”

I told them that actually what they did was exactly what maths is — reasoning things out using the information you have and being able to be sure of your method and your answer. Just because there’s no symbols, it doesn’t mean it’s not maths.

Only later did I realise the implication of what had happened: these students are coming into a maths degree, which means they have done the highest level of maths at school, and they perceived that something was only maths if it had symbols in it. A plain ordinary logical argument wasn’t maths to them. And a scribbling of symbols was better than a logical argument, even if it didn’t actually produce a solution.

This made me really sad.

I wonder what sort of experiences they had that led them to this belief. I wonder what sort of experiences they missed out on that led them to this belief. I wonder how many other students have the same belief and I will never know. I wonder how I can help all the new students to see more in maths than calculations and symbol manipulation, and allow themselves to be proud of their work when they do it another way.

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A lot of people introduce the derivative at a point as the slope of the tangent at that point, which to me is quite confusing. It seems to me that the reason we want the derivative is that it is a measure of the slope of our actual function at that point, not the slope of a completely different thing. To me, the thing is that the function itself is pretty much straight if we are close enough to it, so when we’re looking really close, saying it has a slope at this point is a meaningful thing to say.

For years I got at this idea by drawing a little magnifying glass on my function, and a big version nearby to show what you could see through the magnifying glass. But last year I decided to get technology to help me do this dynamically and I made a Desmos graph with a zooming view window that will show you the view on a tiny circle around a point on a curve. (Some of the bells and whistles here were provided with a little help from Andrew Knauft.)

Here’s what the graph looks like, and a link to the graph so you can play with it yourself: https://www.desmos.com/calculator/pa1cudpc07

I’ve used this to help students see that the curve itself really does look straight when you’re very close, and so treating it like it is straight there and saying it has a slope is a reasonable thing to do. Students are usually most impressed and love to play with it. It’s also interesting to put in a function with a kink in it like  |x2-1| to show that the kink never goes away no matter how closely you view the function, so having a single slope there isn’t really reasonable.

Every time I’ve had to search my own twitter account to find the tweet where I shared it, and I couldn’t keep doing that, so here it is in a quick blog post for posterity.

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I created the Number Dress-Up Party puzzle way back in 2017 and every so often I stumble across it again when searching Twitter for other stuff. When I stumbled across it today, I decided it was time to write it up in a blog post.

The puzzle goes like this:

The Number Dress-Up Party

All the numbers have come to a dress-up party in full costume. They all know themselves which costume everyone else is wearing, but you don’t know.

If you pick any two of them and ask them to combine with +, -, × or ÷, they will point out which costume is the correct answer, and they’ll happily do it as often as you want. For example, you might ask for hamburger + bear and they will point to unicorn. (If the answer isn’t at the party, they’ll tell you that too.)

How do you correctly identify the numbers 0, 1, 2 and 3? How do you do it in as few steps as possible?

(Photo from http://www.worldrecordacademy.com/mass/most_mascots_to_do_the_same_dance_Japan_breaks_Guinness_world_record_213225.html)

It is worth clarifying now that when I say “all” numbers I originally meant it to be all the real numbers. (But it is very interesting to think about how the problem is different or even possible if it means all the rational numbers or all the integers or all the natural numbers or some other set of numbers entirely.) It’s also worth pointing out that it is actually possible for you to guarantee that you actually find these numbers — you shouldn’t have the possibility of having to go through all the infinitely many costumes to be sure of finding 0, for example.

It’s also worth clarifying that the rules say you have to ask two different costumes to combine with an operation. If you can see how using two of the same costume might help you identify actual numbers, then you are thinking along some helpful lines. However, the puzzle is much harder and much more interesting if you have to use two different costumes every time.

I love this puzzle so much! The reason I love it is that it forces you to think about numbers and algebra in a completely different way to any other puzzle or problem I have seen. You really need to think about how numbers are related to each other and how they behave under operations in order to figure out a way to correctly identify these numbers from the way the costumes interact.

Also, in order to tell someone about your solution, or even figure it out in the first place, you need to find a way of talking about or writing about this, which is a lot more difficult than it might seem at first glance. I find it really interesting to see how people attack the problem of describing what they are doing in this problem.

The feel of this puzzle to me is the feel of an abstract pure maths course like abstract algebra or number theory or real analysis, where you are digging deep into how numbers work without reference to specific numbers per se. I would love to go into such a course and use this in the first lecture/tute to get students in the right sort of mindset to attack the rest of the course. I’d also love to do an extension to this as an investigation into how the Euclidean Algorithm for natural number division works. To all those people who haven’t had that sort of training I say you are doing what a pure mathematician does when you think about this problem! You never thought it would be so much fun, did you?

Anyway, there is the Number Dress-Up Party puzzle for posterity. There are several different solutions and they are all lovely. Have fun!

PS: If you feel like seeing how people have attacked this problem, and are ok with spoilers, then check out the replies to this tweet.

 

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This post was going to be part of the Virtual Conference of Mathematical Flavours, which you can see all the keynote speakers and presentations here: https://samjshah.com/mathematical-flavors-convention-center/. The prompt for all the blog posts that are part of this conference is this: “How does your class move the needle on what your kids think about the doing of math, or what counts as […]

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Since 2013, the MLC and Writing Centre have been doing a game called Letters and Numbers at Orientation Weeks and Open Days to create interaction with people. I tweeted a photo of one of our sessions during Open Day yesterday and it has attracted a lot of attention, so I thought I might record some […]

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On the 23rd of July 2008, I started my first day as coordinator of the Maths Learning Centre at the University of Adelaide. Today is the 23rd of July 2018 — the ten year anniversary of that first day. (Well, it was the 23rd of July when I started writing this post!)
So much has happened in […]

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I think this will be my last post about Twitter Math Camp (TMC), getting in just before the TMC18 officially starts (though a lot of people are already there tweeting their TMC-eve adventures even as I write).
TMC is a truly remarkable conference, as I have described before, both in 2016 when I wasn’t there, and […]

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Here is another blog post in my series of only-a-year-late posts about Twitter Math Camp 2017 (TMC17). In this one I want to talk about the Crochet Coral workshops Megan and I did, but I don’t want to actually talk about the crochet coral. Instead I want to talk about the quietness.
TMC was a wild wild […]

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A year ago, I went to Twitter Math Camp (TMC) and it was a wonderful experience. TMC is a great conference full of all sorts of opportunities for maths teachers to learn from each other in many ways. The one way I like the best out of all the possibilities is “My Favourite”.
My Favourite is […]

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Fairy bread, in case you don’t know, is an Australian children’s party food.

Here’s how to make fairy bread: take white bread, spread it with margarine, and sprinkle with hundreds and thousands. Now cut into triangles and serve.
Notes:

It has to be white bread. If you try to make fairy bread with wholemeal bread, or multigrain bread, […]

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