Posted Wednesday, 28 August 2019

### What do you see? How many?

Scott and Ryan use unit chats as quick and engaging ways to warm-up a maths lesson.

By applying a process of reasoning, analysis and logic, a group of students work through abstract mathematical ideas and discover the exciting journey of discovery and reward that pure maths can offer.

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Hi, it’s Shyam again!

Today I want to talk to you about some university level thinking in mathematics…. “Wait – huh? Why would you do that to me?” I hear you say. The answer is that this was thinking done by some year 6 students – 11 to 12 years of age – in our Alcoa Champions of Maths program. And seeing students working at 7 years above their level is one of the amazing outcomes we’ve seen as a result.

So how did this happen? We’ve been working on the ReSolve Special Topic: Assessing Mathematical Reasoning. The teachers have explored what the proficiency of reasoning really means. Here we get into territory of analysing, generalising and justifying. We talk about conjectures, counterexamples and proof.

If this is all new to you (and it will be to most!) here’s a few quick definitions to help you out:

Analysing: looking at different examples and seeing what’s the same / what’s different, seeking a pattern or rule.

Generalising: making a claim about something that holds true in many instances.

Justifying: Testing the truth of statements and providing supporting examples and reasons, counterexamples or proofs.

The Teacher’s Handbook from ReSolve goes into much more detail about these.

Conjecture: a mathematical statement that is not yet proved. E.g. a statement like all even numbers are multiples of 4.

Counterexample: The statement above is wrong. To prove that it is wrong we only need one example e.g. 6 (it’s even but not a multiple of 4). The example that disproves a conjecture is a counterexample.

Proof: This is a watertight argument showing that a conjecture is true. E.g. Conjecture: There are no odd numbers that are multiples of 2. Proof: Every single multiple of 2 is even (by definition) and odds are never even, so no multiple of 2 can be odd. This might seem a bit silly because it’s obvious, but it just gives an idea of how a proof might work. This is a conditional proof and there are many types of proof. Kate Archibald’s Year 6 students at Treendale Primary School showed us a couple of these last Friday…

Students were working on the Magic V task. In this task, they must fill V’s of 5 circles as drawn here with the numbers 1 to 5.

They then sum up all the numbers along each arm and if they are equal, we call it a Magic V. If the sum of the two arms is different it is not a Magic V.

Here is an example of a Magic V:

1 + 4 + 5 = 3 + 2 + 5 = 10. The sum of each arm is the same.

In the task, students are asked to determine whether this conjecture is true or false:

**Using only the numbers 1 to 5 (once each) it is impossible to make a magic V with an even number at the bottom.**

As usual, the class was full of engaged students, great discussions and brilliant ideas. I want to talk about two particular solutions that the students came up with. You can see the students discussing these two ideas in this video:

In this class Louise is the fictitious student who proposed the conjecture. And she is right. These two groups had methods to prove it.

Luke’s group attempted a *proof by exhaustion*. In this type of proof, we test every possible example, or (usually) we test a set of examples that represents every other example. Their reasoning is correct, and the proof is complete. The really clever part of this type of proof is in finding the representative examples. In this case, the 6 examples that the group found are the smallest possible set of representative examples. The members of the group also explain why this set of 6 examples is enough. It’s the most efficient proof by exhaustion we can perform for this problem.

Jy and Jaegar attempted a *conditional proof*. This is the type of proof where you say if **** is true then **** must also be true. For example, if all rectangles are parallelograms and all squares are rectangles, then all squares are parallelograms. (This is true! Ever thought about the fact that a square is a parallelogram?). Here are the steps of the proof that they were attempting to explain (in my words):

1. 1. If you put an even number at the bottom, then the remaining four numbers cannot be grouped into pairs that have equal differences:

eg:

These have the same difference. So, the arms balance, and the V is magic, but it’s an odd number at the bottom.

These arms have different differences. So, the arms are not balanced, and the V is not magic.

2. If the numbers cannot be grouped into pairs that have equal differences, then they cannot make a magic square. Hopefully, if you look at the examples above and think about it, you can see that this is true.

This is a more difficult type of proof to attempt and it really impressed me that they came up with this. However, Luke was not convinced by the proof. I believe when he gave his reason as “unless you test them all you can’t know for sure”, he was saying that this proof wasn’t complete. So without completing the proof, or providing another one, you must test all examples.

He’s right, but Jy and Jaegar are close to providing a full proof. They know it, they just can’t quite fully explain it yet. You see in the video that the debate gets really passionate here. That is something I love to see in a maths class: students really caring about their ideas.

Sarah, at a distance from both sides of the debate, with an objective view, understands the whole picture. She begins to express her understanding when she says she thinks both groups are right.

Jy and Jaegar have not proved the statement **1. If you put an even number at the bottom, then the remaining four numbers cannot be grouped into pairs that have equal differences.**

It’s not obvious that this statement is true, and they have not given us a reason to believe that when you put an even number at the bottom, there cannot be any way to group the remaining numbers to have equal differences. Later, I encouraged the students in this class to persist and try to fill in the missing explanation and complete the proof. And they replied! Kate (their teacher) shared my email with the 4 students in the debate and gave them an hour alone to work on the problem. At the end of the hour this is part of the message they wrote back to me:

It’s a complete and elegant proof that skips out the need to talk about differences in the numbers – we just need to look at oddness and evenness in the arms. Great work! Kate also filmed the process of them working through ideas to arrive at this point. It was full of powerful messages about learning maths.

First, they debated the merits of the exhaustion method vs the conditional proof (or as they call it ‘the theory way’):

What they’re getting at here is the concept of rigour, a very important idea in mathematics.

Then, Sarah explains to the group the significance of odd and even numbers in the problem:

Next, she goes on to convince the others of the full proof and they each make sense of it in their own way:

And, finally, they delight in the satisfaction of completing a proof:

What messages do you take away from watching these videos? I think there’s a lot your students might benefit from, so I’d encourage you to share them with them. For me there are three big messages.

- Let’s forget the notion that girls can’t do maths as well as boys, it’s utter rubbish as Sarah clearly shows.
- Students can get deeply engaged in pure mathematical ideas – with no real-world context – when they are presented in the right way; pure maths is an exciting journey of discovery that has its own rewards.
- You get to see here what it’s like when you completely convince yourself of something through logic, what it means to truly know something without the need for any external validation – that’s an experience that I believe transforms a student’s view of themselves, it builds a special type of confidence that comes from their own ability to reason.

In the Australian Curriculum, the concept of proof is introduced, first in 10A, then throughout the Specialist course in Year 11 and 12. Some of the ideas here – like proof by exhaustion – aren’t encountered by many people until their first year of university.

I was truly impressed by the level of thinking that these students demonstrated and how engaged they were with the maths. I was also impressed by the respectful way they share and test each other’s ideas. These are direct outcomes of the approach we have been taking in our lessons in the Alcoa Champions of Maths. This is what can happen when you bring together a well-designed task, such as this ReSolve task, with effective pedagogies, such as Building Thinking Classrooms, 5 Practises for Orchestrating Productive Mathematics Discussions and Talk Moves. This is what we do in the program and it’s exciting to see what students are capable of in these conditions.

I hope today’s blog was inspirational and educational for you. It certainly was for me!

Keep thinking and enjoy the struggle!

Shyam

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