How a Mathematician Solves a Problem

One of my colleagues at a previous school emailed round this problem and it caught my attention. Possibly just the bright colours, but it has a simple beauty to it, reminding me of the sort of problems that Ed Southall often shares.

I got stuck into it straight away, first putting together the triangles in a rather naive way that wasn’t any use at all. Realising that I wasn’t getting anywhere, I changed tack and went for the ‘brute force’ approach of applying trigonometry to get the answer.

Key point: if one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.

Getting an answer of 180 degrees suggested to me that there would be a neater approach., probably involving putting all three of these triangles into one large triangle. I started trying to do so.

Key point: Get as many ideas on paper as possible, don’t give up if your first couple of attempts don’t work out.

This is a bit of a mess, but in retrospect it’s my favourite page. Trying out a load of options quickly can give you a feel for what might work. I eventually got the idea that two of the angles needed to be on the ‘base’ of the triangle, and it was then a matter of varying the two angles and the distance between them until the third angle appeared at the top.

Key Point: Sometimes drawing a geometry problem more neatly will help you more than you might expect.

Then I decided to start trying to draw this out a bit more neatly; this helped me notice that I could create a key isosceles triangle.

Finally, of course you should write your work up neatly in order to present it to your readers (in my case, a triumphal email to my colleague!)

Key point: Rough working is great, but make sure you write up your solutions neatly at the end. Of course, your teacher will be even happier if they can see your rough thought process too.

I looked at my work at the end of the process.
a) I felt that sense of satisfaction that reminded me why I like maths.
b) I thought it would make a good lesson for my pupils.

In summary, here are generalised versions of my strategies:

  • If one approach isn’t working, don’t get stuck in a rut. Ask yourself what other mathematical concepts you know that may be related to the problem, and try those instead.
  • When you’re confident you’re going in the right direction, get as many ideas on paper as possible, don’t give up quickly.
  • Sometimes drawing a geometry problem more neatly will help you more than you might expect.

I know that the idea of general problem-solving skills (even within the domain of maths) is controversial. Of course, I wouldn’t have been able to solve this problem if I didn’t have a secure understanding of properties of triangles (obviously), proportional reasoning (to scale up and down the triangles) and trigonometry (to give me faith that it was worth continuing!)

However, I do feel that some of the general strategies suggested in  Thinking Mathematically have improved my ability to solve this kind of problem. It was interesting to listen to Dylan William’s thoughts on this topic: Craig Barton suggested that he was moving away from the idea that general problem-solving skills are useful and Dylan was less convinced that he learned general skills from George Polya’s classic book. He did admit, however, that ‘trying a simpler case’ (in the terms of Thinking Mathematically: ‘specialising’) is a pretty useful general principle. This, alongside writing down what you know and want and drawing a bigger, clearer diagram, are the main general strategies which I’ve seen maths students use effectively.

Personally, I’m fairly confident that I wouldn’t have been so effective at solving this problem 10 years ago, when I’d just graduated with a first class maths degree. If nothing else, teaching makes you a better mathematician.

Want to teach your pupils general problem-solving skills? Take a look at my resources here: http://lukepearce.eu/problem-solving/

#mathsresourcechat

I love collaboration, which is why I am very happy to have discovered the educational twitter community. I enjoy the general ideas and big-picture teaching discussions that take place, but I particularly love the nitty gritty: discussions of the practical side of teaching.

I find that I get the most ideas that I can apply straight into my lessons when people share resources and others makes suggestions for improvement.

I’ve done it a few times myself recently, and had really useful feedback from other twitter users. A few other people are doing it, but I’d love it if it was even more common. I find some such posts on #mathschat and #mathscpdchat, but there is also a lot of more general patter.

So to focus in on the detail, I’m going to start using #mathsresourcechat and encouraging others to do the same.

As the examples show, posting a picture of the resources makes it particularly easy for others to see and comment.

Go forth and start resource discussions!

Leaving Highgate

I spent my first seven years in teaching as part of an amazing department at Highgate School. For most of that time, my HoD was Dan Abramson and Robert Wilne (past Head of Secondary at NCTM) was Assistant Head. Robert had revolutionised the department a few years earlier with some radical changes: he introduced a scheme of work with great ideas, loads of links between topics and very specific approaches to teaching. Dan continued this tradition, bringing in many ideas from AfL and relentless energy. In the words of one of my colleagues Peter Davison, Dan could be prime minister if he wanted to.

When you teach maths at Highgate, you feel like part of something special. The whole team, now over 20 teachers, is strongly encouraged to use the same approach to teaching. I remember that Craig Barton sounded pretty shocked when Greg Ashman described a very similar style of department; I suspect that it’s pretty rare and perhaps not for everyone.

The advantages: It makes for an incredibly coherent experience for the students. When you take on a new class, you know what models and vocabulary they’ve seen before and how they’ve been taught to think about every topic. You can be sure that the highest attaining will not have been pushed through more material than the scheme dictates and you can reliably call upon the standard models to helped lower-attainers.

Outside of the classroom, the number of conversations about approaches to teaching was probably 20 times what I got at the (academically very similar) school in which I subsequently taught. I think that this is a great advantage of shared teaching methodology: if you wanted to change how you taught, you had to convince the rest of the department that your ideas should go in the scheme. Many teachers stayed in the office to work until 6 or 7pm; it helped that many were young and family-free but I like to think that it was partly because the sharing of ideas made it an inspirational place to work.

If I loved it so much then why did I leave, you ask. Mostly because my wife and I wanted to buy a house, which is not very achievable for a teacher in London.

What followed was two fairly depressing years. My new colleagues were passionate, highly knowledgeable and interested in the success of their students. But there was no scheme of work beyond the chapter list from the textbooks and discussions about how to teach maths were few and far between.

Then I discovered Twitter.

I remember very distinctively one mid-winter run, I was listening to Jo Morgan on Craig Barton’s podcast, talking about how she found twitter and all the great ideas out there. I’d just been through a similar process and it made me feel quite emotional to be part of a community again: I think I almost cried.

Despite my improved mood, I still childishly felt a bit sorry for the people on twitter because I suspected that the reason they were online was because they lacked departments like Highgate. Over time, I’ve come round the view that twitter actually has some advantages over Highgate: I can draw from a much wider range of experiences and ideas, and it has exposed me to many more ideas from the world of educational research. There are negatives too: there are still times on Twitter when I ask for ideas or opinions and don’t get any. In the Highgate maths office it was harder for people to ignore me!

I can’t actually remember if I discovered edu-twitter or Craig’s podcast first, but if I hadn’t found either of these, then I think it’s quite possible that I would have left teaching. Two years later and I’ve just started a job in which I’m the only maths teacher in the school. That doesn’t phase me because I’m safe in the knowledge that I have my online community of teachers.

Three Weeks In

I started teaching at a new school 3 weeks ago. Overall, I’m absolutely loving it. My job is more varied than anything I’ve done before and for the first time in my life, I actually look forward to going into work on a Monday.

I’ve been writing the timetable. It has been really interesting to learn how our part time staff prefer to work and try to balance this with providing a balanced week of lessons, alongside management discussions about what proportions staff should work. I’m also excited to lead outdoor education: my climbers seem to be really enjoying our weekly indoor club and I can’t wait to get them out into the mountains proper. It has been great to be involved in discussions about the curriculum: how many periods per week should we give to each subject is not a question I’ve ever considered before.

More mixed has been my work as assessment lead. Inspired by Tom Sherrington, I’ve started with the theme of feedback as actions, sharing some of my own attempts to put this into practice, but I have yet to garner much response from my colleagues. Similarly, initiating the process of collecting pupil data is taking some time.

Despite the fact that I have very few lessons and much more experience in this area, probably the hardest part of my job has been teaching maths! Small class sizes mean that it’s been possible to personalise my teaching more than ever before, and I’m enjoying the opportunity to implement some of the ideas I’ve read about during my nine month break. However, a few issues are challenging me.

1. I’ve never taught mixed-attainment classes before and I’m struggling to find a lot of concrete advice on how to best deal with it.. Do I split the class by task or try to keep them all together? Let the children choose their own tasks or assign them myself? Do I provide extra explicit instruction to some? Should this be within, or in addition to lessons?

This flow chart from @mathsmrgordon has provided some inspiration.

2 . How much to use technology? All my pupils now have a Macbook and iPad. This gives great opportunities, I’ve got them using Desmos, Geogebra and Quizlet, but am I going too far? It’s hard to tell when it’s genuinely educational and when it’s just more fun than pen-on-paper maths. And on that note…

3. I’m following in the footsteps of a teacher who sounds like he was much more fun than me! I’ve heard that he was a great teacher, very inspirational and played lots of games. My insistence on copying down worked examples and setting of written practice and extension tasks sounds pretty boring in comparison. To their credit, the pupils have generally been working very well, but I get a sense that we haven’t fully bonded yet.

Any advice? I’d love to hear it.

Bansho

I read earlier this year that Japanese teachers spend years perfecting their “Bansho.”  This refers to a specific practice of recording the thought process of the whole class solving problems together.  I just like the idea of a special word for “boardwork” as mine has historically been pretty awful so I’ve decided to improve it.

My one major change: Make notes during lessons on Microsoft Word rather than a traditional whiteboard or equivalent software. I took this idea from a Spanish course I took at Oxford University, where my teacher always made notes on word. I think it has revolutionised my bansho!

Here is a case study on the topic of solving problems by forming quadratic equations, both lessons taught to year 10 classes aiming for A*-A grades.

This is what my boardwork looked like previously:

I’m actually pretty embarrassed about sharing this in public! In fact, this is probably the neatest my handwriting gets. At least I’ve kept my equals in line, and the algebra is fairly legible, but as notes to look back on, I’m dubious about its use to my students.

This is what my boardwork looks like now:

Thanks to equation editor shortcuts, I have learned to type maths pretty much as quickly as I can write it on a board.  You’ll also notice that I can still add hand written steps in, which I usually do by copying and pasting from word to my board software, then back again: slightly clunky, but I can do it fast enough that the students don’t complain!

Of course, part of the difference between the two sets of boardwork is the detail I’ve included has increased in the second example. Typing my notes has encouraged me to write more explanations as well as just the algebra or geometry involved (although some people may think I’ve included too much commentary?). This is probably because I don’t enjoy hand-writing on the board and so I try to avoid it.

What do my students say? They almost universally prefer the new approach. Do you have terrible handwriting? Why not give mathtype a try.

Helen Hindle @ Mr Barton’s Podcast

Having taught in two selective schools in the UK, I have just started teaching in a non-selective school with mixed-attainment classes. So, my first foray back into Craig Barton’s podcast had to be listening to his interview of Helen Hindle.

To start, here are some of the advantages of mixed attainment teaching that Helen mentioned during the podcast:

  • It removes the danger of lower expectations for pupils in lower sets and gives all pupils the opportunity to access the most challenging material.
  • As mixed attainment teaching tends to improve the performance of lower-attaining pupils and setting tends to place pupils from lower socioeconomic groups in lower sets, Helen sees it as more socially just. I agree, although when Helen said, ‘we don’t segregate in the workplace’, I’m not so sure about this: you need a degree to become a teacher…
  • Helen claimed that higher attaining students are more likely to seek out challenge and push themselves, not just be happy to find it easy.
  • Pupils are less worried about asking silly questions, as they’re used to hearing comments with a wider range of sophistication.

I know that the research on setting is inconclusive, but most people seem to agree that bottom sets are bad for the pupils in them, so I definitely think that mixed-attainment classes are worth considering.

One of Helen’s key points is that a different approach is needed for mixed-attainment classes than that used with sets. She talked through three key parts of her sequence of lessons:
Showing the students a ‘learning journey’ with relevant questions helps pupils to choose material appropriate for them, and to see their progress clearly.
Inquiries help to build the teacher’s picture of pupils prior attainment, as well as giving a sense of the whole class staying together, even when many pupils are working on different tasks.
The tasks Helen uses are either self-selected from a variety of options or multiple entry point / low threshold, high ceiling.

Having never taught mixed attainment classes, I think it’s fair to say that Craig was a little sceptical of this approach and was playing devil’s advocate even more than usual! Here are some of his questions, and Helen’s responses.

What if pupils select inappropriate work?
Part of the teacher’s role is to check and redirect if necessary. Choosing appropriate tasks is a life skill that pupils need to gain.

What about a single top set / streaming / exam years / bottom set?
Top set: are you removing the extra challenge for the rest of the pupils? What about the fact that different pupils have different start points in different topics?

Isn’t it better for teacher to focus all their effort on explaining one idea?
Whole class explanations aren’t necessarily better. I would add that in a small group, the instruction can be better tailored to the individual pupils. Once the teacher has helped some higher-attaining pupils, they can disseminate this knowledge throughout the class and they will benefit from this. Lower attaining students could gain from listening to explanation of more advanced material (I’m a little dubious about this), or could be doing something different.

Is it harder work for the teacher?
It would be harder if you just used the same approach as you did for sets classes, but if you change your approach as suggested, it isn’t necessarily more difficult.

Don’t the highest and lowest attaining students get more support in sets?
Perhaps the top set, but pupils in bottom sets experience lower expectations and worse behaviour. In a class of twenty students all struggling, there is still only one teacher.

What about non-specialist maths teachers?
In Helen’s department, resources are planned collaboratively, which makes it easier for non-specialist maths teachers. This also sounds like something I would really enjoy being part of, though I also know plenty of teachers who would rather work independently.

Craig also asked Helen about her promotion of a growth mindset. I really liked a couple of things Helen said here:
Referring to how students are sometimes asked to use Red, Amber, Green to describe how they feel about the work they’re doing: “Green is the target, but amber is when you’re learning.”
And secondly, that she aims to: “change pupils perception of what constitutes success”. It made me think back to a discussion I had with Greg Ashman, in which he pointed me to research showing that learners are most motivated by success. If this is the case, then what constitutes success is very important.

I have two questions of my own:

In his podcast conversation with Craig, Dylan William encouraged keeping the class together, rather than spreading them out. This also seems to be the driving idea behind the ‘mastery’ approach and the focus of Craig’s research section on differentiation. Helen did make reference to this, through class discussion/inquiry and pupils explaining ideas to each other. However, it sounds like pupils are often working on different material, counter to this advice. I’d like to know of any studies / personal experience in which differentiation was successful and what made it effective?

Similarly to most schools that I’ve visited, it sounds like Helen’s scheme of work spends several weeks on one topic before moving on. I feel that this fails to allow for a sufficiently spiral curriculum, where each topic is revisited (and added to gradually) at least every year and mostly once or twice per term. So, will I have time within my short (3-4 hour) sequences to apply some of Helen’s ideas?

On my to do list: Look into the mixed attainment maths conference and spend some more time reading the mixed-attainment website.

Thanks Craig and Helen. As ever, it was enlightening.

Jamie Frost @ Mr Barton’s Podcast

I chose to listen to Jamie’s appearance on the Podcast as a friend and ex-colleague Peter told me that his resources were good. I wasn’t disappointed: there were lots of good ideas in on the podcast. I’ve since investigated the resources and I agree with Craig – they are consistently very high quality and contain some great ideas.  Here are some of the points he made on the podcast

Teach general ideas, not specific methods
I think this is an excellent point: rather than getting pupils to follow a specific method like “finding the zeroth term”, encourage them to understand how they can find a constant term by thinking about making the sequence ‘fit’. Crucially, general ideas like this extend much more effectively to more difficult problems (in this case, non-linear sequences).

Traffic Lights
A classic idea from AfL. I bought a class set of red/amber/green cups and trialled them for about 6 months but they proved unpopular. I teach in a school where the students don’t generally lack confidence and are very mature. They said that they would rather just put their hand up and they found it a bit childish. But I’d love to make better use of them in future. Tips, anyone?

A lesson doesn’t necessarily need a plenary
I totally agree: when I’m circulating and checking individual work constantly, I agree that sometimes I’d rather just maximise the lesson time with pupils challenging themselves more individually than bring the whole class back together.

Prime Factor Buckets
A really nice visual idea – imagine the prime factorisation like a bucket of prime factors, which you draw from to find LCM or HCF as required. I wish people would share more of these kind of ideas in teaching. There are a lot of people blogging about general teaching principles, which I do find useful and interesting, but I’d love to read more blog posts sharing ideas of how to teach specific topics.

Work-Life balance
Jamie’s sounds quite bad – mine is “better” – the inverted commas are necessary because Jamie appears very happy with his. If you’re keener to have more time for life than work – I’ll write about how I achieve this soon.

The M-word
I’m glad to hear that Jamie avoids using the word minus. In my younger classes, it’s a banned word and pupils can earn / lose reward points for using subtract and negative / misusing the m word!

Subject Knowledge for Teaching and Learning
Jamie talked about how he has gained this in his first few years of teaching. This is true of all teachers and I’ve heard a lot of Craig’s guests say something similar, but I think we should promote better sharing of this, so that you don’t have to pick it up through experience but can learn it from more experienced teachers. My previous school offered new teachers 2-3 hours a week of one on one meetings with a more experienced mentor within the department. I found this invaluable in my first two years and then enjoyed giving back to the process as I became a mentor myself. I suspect that this is very unusual but I wish it were more common.

Grammar-School Specialist
I’m currently also specialise in teaching high attaining, but unlike Jamie I’m not sure I want to remain like this for my whole life. It’s a difficult point: I like teaching further maths every year and plenty of A-level, but I went to a comprehensive school myself and I’ve seen the statistics: they seem to be better for social mobility.

Homework platform
I had a bit of a go on the trial section of this and it looked really impressive. I’m excited to see  how it develops.

Overall, it was another cracker of a podcast. Thanks Craig and Jamie.

Explicit Instruction

Three months into my foray into the world of blogging / twitter, I realised that my choice of twitter handle (@discoverymaths – now changed!) is more controversial than I thought! Whilst I champion guided discovery, explicit instruction still takes up more of my lesson time than discovery.

One time when the Head of my previous school observed my class, he saw a pretty standard lesson involving only explicit instruction. He said that he was impressed by my questioning and pushed me to think about how to pass on ideas to the new teachers I was mentoring. Here are my thoughts:

Everything is a question
Only add something to the class board notes once a student has said it, so all the ideas have to come from the class. Constant questioning means that you’re less likely to overestimate understanding and that pupils have to remain alert as they may be asked at any time. And on that matter…

What did Sarah just say?
This is one of my favourite questions, deployed in almost every lesson with larger classes. If you ever sense that someone isn’t listening to their peers, ask them to repeat what was just said. They’re usually pretty embarrassed if they can’t. Even if they can, it helps to reinforce important points and encourage listening skills.

Personalised
Target questions according to understanding to challenge all students appropriately. Break down the problem into many more steps than an experienced mathematician would.  This enables you to ask lower-attaining students to make small logical steps and higher attaining to come up with bigger ideas.

No opt-out
One of my mentors used to say “You have to have an idea. It doesn’t have to be a good idea, but you have to have an idea”. If you reach a total block, ask a simpler question which will help, either to the that or another student, before returning to the first student with the original question.

Bounce back
Hopefully your students will be inspired by all the questions you’re asking to ask their own. How to respond… Can you use your expertise to ask them an easier question which will help them come to the answer themselves? If not, pass it on to another student to keep all involved.

Pause / Think-Pair-Share
Pausing to get all students to individually think about or have a rough go at a question you have posed (and then discussing in pairs if you wish) during a period of explicit instruction is a good way to break up the time and give pupils a chance to refocus.

Subject Knowledge for Teaching and Learning
What really helps you ask good questions is knowing the misunderstandings that students often have and how to break your questions down into smaller steps or re-frame them in a different context to overcome these. As Jamie Frost emphasised on Mr Barton’s Podcast, this is not just about knowing the subject, it needs to be learned.

Aspirin for a Headache?

I really like Dan Meyer’s metaphor of presenting maths as the solution (aspirin) to a problem (headache), simply because it reminds me that I should start a lesson by making clear in some way what the problem is, before diving in to introducing new ideas.

I like it so much I was talking to some (mostly non-teaching) friends about it the other day. It went a little like this:

Luke: …isn’t that a great metaphor?

Alice: It sounds a bit negative.

Becky: It sounds as if you’re saying that maths is a headache.

Luke: no, no, no…

Alex: Yes, but I don’t want to have to take Aspirin.

Alice: I never take Aspirin anyway, even if I do have a headache. I think that’s more of an American thing?

Luke: Have you got a better version?

Alex: You want to undertake a journey but there is a river in the way. Maths is the bridge you need to cross the river.

All: *wretch*

So, I’m sticking with Headaches and Aspirin for now… anyone have a better suggestion?!