This is the title of an excellent book by Paul Zeitz that I'm currently working through. You can also watch him in a video about solving problems here, or get his video course here. In his work, Zeitz aims to answer many of the questions I also have regarding how we can teach/learn problem solving.

He starts by comparing exercises with problems. He calls an

He goes on to define three 'levels' of problem solving: strategy, tactics and tools.

I like this idea of splitting problem solving into different aspects. It feels similar to playing chess, where a strategy is an overall scheme you follow to win a game, upon which all your moves are based, whereas a tactic is a short sequences of moves you use time and again, perhaps to capture a piece, such as a pin or skewer.

Zeitz discusses a (quite large) number of strategies and tactics in his work. I will talk about many of these in future posts, but for now let's look at a few of the key strategies in action: '

He starts by comparing exercises with problems. He calls an

**exercise**a mathematical question that you know how to answer immediately, such as 237 x 68. A**problem**is a mathematical question that**you**do not know how to answer immediately (although it may be an exercise for someone else).He goes on to define three 'levels' of problem solving: strategy, tactics and tools.

**Strategies**are mostly non-mathematical, often psychological, general ideas for starting and pursuing problems - more on these below.**Tactics**are mathematical methods that work for a number of different problems, such as symmetry or parity.**Tools**are less important - they are techniques and 'tricks' that work in specific situations.I like this idea of splitting problem solving into different aspects. It feels similar to playing chess, where a strategy is an overall scheme you follow to win a game, upon which all your moves are based, whereas a tactic is a short sequences of moves you use time and again, perhaps to capture a piece, such as a pin or skewer.

Zeitz discusses a (quite large) number of strategies and tactics in his work. I will talk about many of these in future posts, but for now let's look at a few of the key strategies in action: '

**wishful thinking', 'make it easier'**and '**get your hands dirty'**. Consider this problem from the non-calculator US AMC10 test, designed for KS4 students:Assuming we know what the factorial sign means, we can access this problem. Or is it just an exercise? Well, I know what factorial means, and I know what a perfect square is, so I

So how to begin? Of course, you should try it for yourself before reading further...

Here's one way of going about it.

*could*just calculate each one and find the answer - but this will be very difficult without a calculator! Thus we are in the realm of a problem (for me, anyway) as I do not immediately know how to proceed.So how to begin? Of course, you should try it for yourself before reading further...

Here's one way of going about it.

**Wishful thinking**and**make it easier**say to me that I would be able to solve this problem easily if the numbers were smaller, and**get your hands dirty**suggests I should try calculating some of these smaller numbers and see what happens. So here goes:We can see that some of these are squares and some are not. But a closer look at the expressions reveals the structure of the problem to us; the repeated numbers in the numerators make squares, so we could rewrite the expressions like this:

Now can you see what makes these expressions a square? We only have a square if the last term in the numerator makes a square when it is divided by 2. So the next square will be 7!8!/2 because 8/2 = 4 is a square. The next one is then 17!18!/2, which is D in the AMC question, and the problem is solved.

So should we end here? No, there is reflection to be done. Firstly, our strategies of wishful thinking, make it easier and get your hands dirty have served us well for this problem, so let's remember them for future use. But was there an easier way to solve this problem? And what else have we learnt? What have we learnt about factorials and square numbers?

As a teacher I like to think about how we can

So should we end here? No, there is reflection to be done. Firstly, our strategies of wishful thinking, make it easier and get your hands dirty have served us well for this problem, so let's remember them for future use. But was there an easier way to solve this problem? And what else have we learnt? What have we learnt about factorials and square numbers?

As a teacher I like to think about how we can

**extend this question further**. We could think about the nature of square numbers. For example, in this question we used the fact that a square multiplied by a square is a square - is this always true? Why? Why not investigate this idea further by posing a question to students like this:And of course, none of this was of any use if we do not put what we have learnt into action on some more problems... Can you use any of the strategies mentioned here to solve this classic problem?