## How can we teach problem solving?

It isn’t that they can’t see the solution. It is that they can’t see the problem.

G. K. Chesterton

In an uncertain future, the ability to solve novel problems will become increasingly in demand. Now, some folk might, correctly, point out that we’re hard-wired to problem solve. We can’t *not* attempt to solve problems when we have an idea of what the answer might be. However, we tend to be much worse at solving problems when we have no idea what the solution might look like. Everyone, regardless of their ideological beliefs about education, will surely agree that problem-solving is a vital skill, so, obviously, we should put some effort into teaching students how to do it. The question is, how?

Last week I observed a maths lesson in which Year 10 students were given problems from the MEI’s Senior Team Mathematics Challenge, aimed at sixth formers, to solve. They were incredible! They spent almost the entire hour working diligently on what, to me, looked like utterly incomprehensible maths problems. And what’s more, they left the lesson proud and happy. Clearly this must be the future of maths teaching, right?

Some context: these students are seriously able mathematicians; they all took their GCSE in Year 9 and got an A*. This year they are spending some of their maths lessons on a bespoke problem-solving course. You might well expect that this would be a relatively straightforward endeavour – how could these students not excel at problem-solving in maths? Apparently though it’s been a hard slog. At the beginning of the year, the teacher warned her class, “This is going to make your heads hurt and you are going to hate me.” Sure enough, it did, and they did.

The fact that they were talented mathematicians who knew a lot about maths, didn’t seem to help. When confronted with problems like “*In how many different ways can 2015 be written as the difference of the squares of two positive integers? “*they took one look and said, *I can’t do that. *

Certainly when I look at that problem, my mind boggles. I have literally no idea where to start. There’s nothing for me to cling on to and I slip right off. All I learn from being given this sort of question is that I can’t do it. Failure is encoded, making sure that the next time I see such a problem I already* know* I can’t do it.

These students had an advantage because they knew they were good at maths, but they had little experience of solving this kind of problem. Being able to solve problems is any domain of knowledge is about recognising the deep structure of the problem. There are, apparently, a finite number of structures for most maths problems and, with time and experience, you can learn to recognise them. *If* you recognise the structure of the problem *and* you know how to do the maths, solving the problems is trivially simple.

These Year 10 mathematicians have been learning a series of steps, distilled from the wisdom of George Polya’s 1960s classic, How To Solve It:

**Step 1: Understand the problem**

- What is the unknown?
- What is the condition you are trying to meet?
- What information is given?
- Is any information redundant?
- Do we expect one answer, or many, or none?

Strategies:

- Draw a diagram
- Introduce suitable notation
- Separate, and write down, the various bits of the condition

**Step 2: Creating a plan**

- Have you seen it before?
- Have you seen a similar problem, maybe in a different form?
- Do you know a related problem?
- Do you know a theorem that could help?
- Can you solve it directly, or do you need a ‘stepping stone’?

Strategies:

- Look familiar? Think of a similar problem having the same or a similar unknown.
- Could you use the result?
- Could you use the method?
- Can you imagine a more accessible related problem?
- Can you imagine a more general or a more specialised problem?

- Breaking it up: Keep only part of the condition and drop another part.
- How far does that determine the unknown?
- How can it vary?
- Can you derive anything useful from this data?

- Stepping stones: If I could find
*this*then I could solve the problem- Can you find anything useful from the given data?
- What data would you need to find to determine the unknown?
- Could you change the unknown, or the data, or both so that the new unknown and the new data are nearer each other?

- Oops I forgot!
- Did you use all the data?
- Did you use the whole condition?
- Have you taken into account all the definitions?

**Step 3: Carrying out the plan**

- Have you checked each step for accuracy?
- Can you see clearly that each step is the correct thing to do?
- Can you prove that it is correct?

**Step 4: Looking back**

- Can you check the result?
- Can you check the argument? (Push the limits?)
- Can you derive the results another way?
- Can you derive the result more elegantly? (At a glance)
- Can you use the result, or the method, for some other problem?

It seems to me that while these steps are tailored to fit mathematical problems they could be easily adapted to fit the demands of problem-solving in other subject areas. Experts will have internalised these four steps and will often be able to ‘just do’ whatever needs to be done to solve a thorny issue in their field of expertise. Novices are novices precisely because they have yet to go through this process.

Back to the maths lesson. Most of the students spent the vast majority of the hour working in silence on their chosen problem. Their body language was relaxed and confident. They had learned how to apply what they knew about maths to a wide range of problem types and what to do when they encountered an unfamiliar problem.

The teacher had predicted that one student would struggle more than the others. Despite being one of the most able mathematicians in the group she still often feels helpless in the face of difficulty. Her body language was markedly different: she seemed almost to squirm as she grappled with the problems. I was told that in most lessons, she gives up quickly and then gets another student to explain the answer. Then, when she’s told the answer, she feels stupid for not having seen the solution herself. Accordingly, we planned a strategy to prevent her from relying on others for help and luckily it paid off. The teacher made one or two brief interventions to assure her that she was heading in the right direction and she was independently able to solve two of the problems. As the lesson progressed, her posture changed. By the end of the class she was smiling and joking. The hope is that the this experience of having encoded success will transfer to future lessons.

So, can we teach problem-solving? Yes, undoubtedly we can. These students demonstrated just how much they’d learned in the few weeks they’d been taking this class. But, the ability to solve problems is dependent on knowing a lot about the field you’re attempting to solve problems in. The steps above would be insufficient to get *me *(and most students) to solve most maths problems as I just don’t know enough mathematics. Yet.

*NB this is a blog about problem-solving, not a blog about mathematics.*

**Coda**

After the lesson, the teacher gave me this problem to solve:

I have 10 green socks and 4 blue socks. It is pitch black and I want to get a matching pair of socks. What is the fewest number of socks I must take in order to guarantee that I have a matching pair?*

I’m pleased to say I managed to work out the answer, and with my (admittedly small) success came a sense of confidence that I might be able to work out ever so slightly more difficult problems.

*Do please give your answer below and *don’t forget to show your working out!* 🙂

Surely it’s 3… One green, one blue and then one of either…

Yep! Well done. Satisfying, isn’t it?

With 3 socks chosen the possible combinations are BBB, GGG, GBB, BGG. Either way, you’ve got a pair.

Same sock problem, one red sock, twelve blue socks, seven green socks and 10000 yellow socks, how many to guarantee a pair now?

5

Glad to see two English teachers working hard on their maths….2 house points each!

Do you think the 4-steps approach works equally well for students who find writing essays hard? As a mathematician, essay writing is sometimes as daunting as you describe the maths problems.

I think so. The key is, I think, removing scaffolding after success has been encoded.

It is 3

3.

Fist sock not a pair, second sock may be a pair or may be different, but third sock guarantees a match to one of the first two if they were not already a pair.

Indeed.

3 is correct.

Possible outcomes when selecting 3 socks:

BGG, BGB, BBG, BBB

GBB, GBG, GGB, GGG

All result is in a pair of one colour.

I believe this will hold true provided there is at least a pair of each colour.

One of my main problems with this is that as a mathematician I don’t really do this. I just ‘throw maths’ at a problem until something seems worthwhile. Except in specific cases where I have a very similar problem in mind.

I’m sure problem solving would improve with a greater range of knowledge of different types of problem. As opposed to trying to train ‘problem solving’ as a discrete idea.

I just don’t know enough about maths to comment on what you might be doing when you ‘throw maths’ at a problem, but it does seem to me that even if brute force might result in a workable solution, there may be more efficient and elegant ways to arrive at an answer.

The point of my post is that a great range of knowledge is necessary but possible not sufficient. Teaching problem-solving as a discrete idea is definitely insufficient and is possibly unnecessary, but it could be more efficient.

2 points –

they already had a good grasp of basic maths. Too often kids (who don’t even know their tables) have been expected to spot a pattern or formula.

next, I find it a little sad that they had had exam success and yet found these puzzles difficult to start. This is NOT a criticism on the teacher you observed. I have recently experienced sixth formers who got an A* at GCSE yet couldn’t understand that there may be more than one way to solve an A level problem. Some just wanted the answer or a method for dealing with every possible exam question of its type.This a reflection on the system – teaching to the test, teacher accountability, etc

David, I want to jump up and down and shout Halleluiah! One more who who is beginning to see the light. 🙂

I have always seen learning as a problem solving process, admittedly I was trained to teach D&T (or CDT as it was then). I bet many a D&T teacher is also saying something along the lines of “At last, the rest of the school subjects are beginning to wake up to the power of problem solving in learning” if they read this that is. If we teach a man to fish… etc. If we show learners how to approach learning as a problem then we make them life long learners surely.

My concept of Learning Intelligence (I see intelligence as a problem solving measure) is based on overcoming the problems of a learning environment. LQ is the ability of the learner to manage their learning environment to meet their learning needs. The set of skills, attitudes, attributes and behaviours I have identified as part of LQ include many problem solving strategies and actions.

Have a read of this article “Learning Quotient and the Design Process” http://wp.me/p2LphS-40 It is a simplified model of a design process (please note problem solving or design is not a linear process, nor is there a set start and finish point).

I also have a model of the Hero’s Journey adapted for learning using the problem solving approach if anyone is interested.

Where I started on the 2015 problem:

1. The difference of squares should bring to mind the conjugate formula, i.e., x^2 – y^2 = (x – y)(x + y). We’re looking for factor pairs of 2015 that satisfy the condition that there are two integers, x and y, where their difference is one of the factors and their sum is the other.

2. The difference of consecutive integers is an odd number, with every odd number represented. Specifically, if n is a positive integer, then n^2 – (n-1)^2 = n^2 – n^2 + 2n – 1 = 2n – 1. Since 2n – 1 = 2015, n = 1008 and m = 1007.

I kept going…

3. This yields the factor pair {1, 2015}. How could I get from {1007, 1008} to {1, 2015}? By noticing that 1008 – 1007 = 1 and 1008 + 1007 = 2015.

4. How can I generalize this strategy to find all solutions? First, list all the factor pairs of 2015. These are: {{1, 2015}, {5, 403}, {13, 155}, {31, 65}}. If I want two numbers with a difference of 1 and a sum of 2015, I can take 2015, divide by 2, and subtract half the difference to get the lower number.

5. Writing that as a formula gives: m = 2015/2 – 1/2, which can be rewritten as m = (2015 – 1)/2. I generalized that to m = (b – a)/2, then applied it to each factor pair.

6. This gave the solutions: {{1007, 1008}, {199, 204}, {71, 84}, {17, 48}}.

7. I believe there’s a proof that this list is exhaustive, but just to double check my work, I wrote a quick Python program to list the solutions by brute force. Here’s the code I ran on Trinket:

for i in range(1, 2016):

lowsq = i**2

hisq = lowsq + 2015

j = int((hisq)**(0.5))

if (j**2 == hisq):

print i, j

It listed those solutions and no others.

To your points: If you DON’T know the conjugate, then this problem indeed has no clear entry point. While I’m a proponent of teaching strategies more than facts, this particular problem is a case where if you know a fact (“difference of squares” suggests doing something with the conjugate), the problem is fairly straightforward. If you don’t know this fact, you’ll likely stare at the problem hoping it will start some sort of magic.

The second fact I used (that the differences of squares of consecutive integers, taken together, represent all odd numbers) isn’t nearly as important to this problem. What it did for me was tell me there had to be at least one solution, since 2015 is odd. So it enabled me to quickly find one solution that I could then pick apart with my knowledge of the conjugate to see how to find it algebraically, leading to my generalization. But if I didn’t have that entry point, I could have worked it out using the first fact.

Point being, the questions you list under Step 2 are indeed crucial: Use prior knowledge and experience with similar problems to see what you can use in this case.

Oh, and for the socks question, the answer is 3, as others have said. 😉 The general rule is, take the number of colors of socks and add one.

since 2015 = 5 x 13 x 31 (prime product form) the only four possible pairings (a+b)(a-b) are:

2015 x 1 a=1008,b=1007

(5 X 13) x 31 which 65 x 31 a=48,b=17

(5 x 31) x 13 which is 155 x 13 a=84,b=71

(31 x 13) x 5 which is 403 x 5 a=204, b=199

in each case a & b are found (& unique) by noting e.g.

a + b = 65

a – b = 31

adding: 2a=96, a=48, b=17.

Just reread your solution and figured out what you were doing at the end of step 5 through step 6. Don’t think my solution process is substantially different than yours. Just didn’t generate as explicit an algorithm and justification for how I found (a+b)(a-b).

I think the methods are all substantially the same. My point in commenting was not just to give an answer, but rather to show my thought processes about how I get there, in the spirit of the original post.

Phil’s comment does illustrate that the problem can be solved without an explicit appeal to the conjugate.

I though that the use of the difference of two squares was ‘obvious’ but I went down a couple of blind alleys until using prime factorisation. One aspect of problem solving that I think is addressed poorly by examinations is the willingness to try different approaches. I often fumble my way through questions and would rarely complete them if constrained by time in a way that exams are.

@phil m: I agree. There have been problems I’ve worked on for hours, only to realize that I’ve been missing something really obvious. We’re socially trained to feel “stupid” when that happens, but I usually wind up noticing a bunch of patterns and relationships that are new-to-me, so hey, it wasn’t time wasted. Searching in the weeds is only wasted time if you don’t learn anything about the weeds while you’re out there.

Alternative method that gets the same result. Fully factor 2015 to 1,5,13,31. Generate factor pairs {(1,2016), (5,403),(13,155),(31,65)} rewrite factor pair as (ave-dev)(ave+dev). Ave^2-dev^2=24. Generates same 4 pairs mentioned by Paul. Verified no other pairs can be generated. Sorry so brief and poorly edited. Worked it out over lunch and replied on iPhone.

If you want to run the Python, WordPress messed up the indentations. Here’s the code again. Replace the > with spaces to satisfy Python’s formatting:

for i in range(1, 2016):

> lowsq = i**2

> hisq = lowsq + 2015

> j = int((hisq)**(0.5))

> if (j**2 == hisq):

> > print i, j

I love that all this is happening on this blog’s thread. I have no idea what it all means, but I love it.

Isn’t behaviour crucial for this sort of teaching? I’ve just moved from a grammar school (where I routinely taught this sort of material to this sort of class) to a middling comprehensive (in a grammar school area). The first change is that my teaching has had to become much more punchy and episodic – the kids do not (yet) have the self-discipline to tackle this sort of work.

In particular, they don’t understand the primacy of the journey over the destination. They don’t feel comfortable with trying and failing. Most of all, they are not convinced that the entire business is worthwhile. Clearly, I would like to inculcate this sort of behaviour so that we could tackle these sorts of extended problems, but it seems like it will be a tough process to convince them.

I agree with your observations. My grammar school kids were not only academically bright, but had staying power and responded with relish rather than despair when they realised a problem was harder than they thought.

The expertise and depth of knowledge you mentioned is, as we mathematicians say, necessary but not sufficient. There is an entire culture to change.

Every mathematics teacher in the country is currently grappling with this as it is likely to be a key element of the new GCSE. I agree with you that it can be taught. I think that the process should begin in primary school.

Well done on the socks problem. It’s a classic. Not bad for an English teacher…:)

Yes, but then good behaviour is essential for any kind of meaningful teaching

I recall having this sock problem, along with many others, when doing maths at teacher training college. It is a typical example of a pseudo real-life problem, where the logical solution does not require sock counting in the dark. I would either 1. pair up the socks after washing and before putting away (which I do, in “real life”) or 2. use the torch on my phone to try to avoid waking my partner. It was a salutary lesson which I learned at college, albeit not the one my tutor intended. I always know if I’ve come up with a meaningless maths problem that no sane person would use maths to solve, if there is a perfectly logical common sense solution. The children are great at calling teacher’s bluff if we do come up with silly problems and I do come up with pointless problems sometimes! There used to be a brilliant series hosted by Dara O’Briain and Marcus De Sautoy on TV which involved problems which genuinely required maths, although the contexts were at times bizarre. My 10 year old (maths loving) daughter and I thoroughly enjoyed tackling these problems. I would rather have a proper maths problem with a strange or theoretical context than one which had a real life context but which is better solved without maths.

So-called ‘real-life’ problems are the enemy of education. I couldn’t give a stuff whether a maths word problem accurately reflects real life. The reason it’s worth teaching this stuff (if there is a reason) is because it’s worth knowing.

I agree with the above and any help is welcomed regardless of subjects.

I think problem solving can be hampered by the text book style questions we use in lessons. They can remove the problem solving elements by providing you with all the information you need and then the steps needed to solve it.

When I think of problem solving I use this as my starting point..if you got 11mins 36 secs its worth it and funny to (from a Maths teachers point of view).

https://www.ted.com/talks/dan_meyer_math_curriculum_makeover?language=en

This is very similar to the 4 problem solving process I learnt as a Physics undergraduate (Identify, Set-Up, Execute, Evaluate – I SEE) and one that I try to get my Physics IGCSE students to use when solving problems too. Lots of good examples of strategies and questions that I can add to this now. Thanks David

You’re welcome but it’s Polya who deserves your thanks 🙂

Great article, thanks for sharing. Problem solving is a good way of learning key maths skills that students can use in real-life scenarios.

Really interesting blog within a Maths context. You have raised some excellent points! Although I do not have your eloquent way of saying things I have written blogs on the important skill of problem solving over the last year.

For my take on Problem Solving within the context of a qualification please see http://www.theorb.org.uk/blog/57/activity-planning-solving-the-problem-of-problem-solving

Another blog https://thiseducationblog.wordpress.com/2015/06/14/real-world-problem-not-at-all/ also discusses how problem solving needs to be based in the real world scenarios. This may or may not (depending on your view point) add to what you have said in your blog.

I agree with you that we all have to solve problems, all the time and for me the problem is never the destination, the journey you choose to go on and find your destination is what the problem actually is. This could be even more relevant in other subjects because as you have pointed out, Maths MAY have only a finite number of ways to solve problems.

Thanks very much for the read. Gareth.

reviewing the replies to the original question I think it highlights that there is a difference between “solving problems” and problem solving” ! Agree?

Interesting take on problem solving. As a maths teacher in a curriculum where it is to the core this is something I battle with daily, yearly, whenever really.

As a department we have discussed a common problem solving flow in a wide range of ages and came up with something that was not so widely different to the above, but I found difficulties in explicitly teaching the steps to our mixed ability classes.

In looking for some easy to grasp mantra which epitomises these, I realised that while problem solving I was generally asking myself the same questions: “What is the question asking?” and “What do I know?” – pretty much on a continuous loop, slowly make connections between facts, concepts and the question. Decoding the question, if you like. When done, I ask myself “Does my answer make sense?”.

In particular, I realised that writing a plan was not something that you always need as often you just doing something. Making any connection makes the problem clearer. Although clearly if you can do a plan, it is preferable.

I guess that my questions for myself is whether I should have tried to persevere with explicitly teaching all the steps or continue with with my easier to remember format, while highlight steps/options during the explanation/ discussion?

On different note, I know that there must be a way of incorporating mind maps or the like in the creative process of problem solving. (my homework if you like)

With these issues I always go back to Willingham’s classic article on why teaching critical thinking is so hard. He argues that even with all these prompts people don’t make really good headway with critical thinking as they can’t ‘see’ what can be applied without expertise. It sounds like he’d approve of this problem solving class because it is giving some (hopefully systematic) experience of solving various types of problems. Despite it sounding like these are high quality prompts to push this approach too hard is surely perpetuating the teaching of ‘problem solving skills’ which is a cul de sac?

[…] It isn’t that they can’t see the solution. It is that they can’t see the problem. G. K. Chesterton In an uncertain future, the ability to solve novel problems will become increasingly in demand. Now, some folk might, correctly, point out that we’re hard-wired to problem solve. We can’t not attempt to solve problems when […]

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what is the difference between teaching probem solving and teching with problem solving

Problem solving is a domain-specific skill. The ability to solve problems within a domain is the result of mastering foundational concepts. Teaching with problem solving is only effective with experts; novices need explicit instruction on the domain. So, the difference is that one is an end and the other an (ineffective) means.