Classic Blog

Nuggets

Freddy: “Mr Pearce, we found your blog.”

Me: “Yes?”

Freddy: “It’s all about teaching.”

Me: “What did you expect?”

Freddy: “Some mathematical nuggets?”

Me: “I’m a teacher, not an entertainer.”

OK, so I’m still quite happy with my last line, just because I like the ring of any line of the form “I’m not a …”, which I stole from a friend Sam Bartlett who employs it comedically better than I ever could. But I was being rash – a teacher is, to some extent, an entertainer.

I clarified by saying that I think there is rather a glut of mathematical nuggets out there and so I’m looking to provide something different.  However, I do like nuggets, so I’ll allow myself one (and only one) blog post to share some of my favourites:

The Movie Maths Quiz

movie-maths

Don’t work hard…

dont-work-hard-work-intelligent

The Venn Diagram of Bollocks

venn-diagram-of-bollocks

George Ford on Countdown

Report Cards for Mathematicians

maths-reports

The Three-Switches Problem

three-switches

An interesting problem

Causes of Death

 

DIY Homework

Virtually all of my teaching ideas are stolen, but I think I came up with this one myself last summer.  Maybe that’s a bad sign.  I’d love to hear from others who have tried something similar…

It’s not exactly complicated: students choose their own homework.  Obviously they require some guidance. Click here for the document I show the students outlining my advice.  The basic structure:

  1. Students must choose questions which they can check the answers for (so I’m expecting that they’ll mostly work from the textbook, which has answers at the back – and this has turned out to be the case)
  2. I suggest a minimum of 90 minutes per week for most of my classes, rather than a specific number of questions.
  3. Students should try to challenge themselves, and find questions with which they struggle.  If they’re doing the final few (most difficult) questions from each exercise fully correctly, that’s fine, but otherwise I expect to see mistakes and (attempted) corrections, with questions for me if they can’t get to the correct answer.

I should point out that this is not very different to my usual expectations for homework.  I have always asked students to check their own answers, so that I can focus my time on helping with the problems with which they’ve struggled.

What Went Well

The quantity of work done has impressed me – normally more than I would have set. I obviously can’t tell if a student has actually spent 90 minutes but in general, I think they have done so and many have done much more.

There has been much better differentiation in the difficulty of problems tackled: some students have tackled drill problems from the starts of exercises, others more varied problems from the harder end, and some have extended themselves with Olympiad or university entrance exams.

There has also been differentiation of the topics covered: students have spent different amounts of time on different topics, according to their needs. Some students have also varied the topic they’re working on, and worked on topics covered some time ago, giving them a chance to constantly revise.  I continue to encourage all to do this.

It has encouraged greater reflection and independence: I’ve seen more evidence than before of students learning from their mistakes, and many have engaged in very useful written dialogue with me, asking specific questions about how to tackle a particular problem and letting me know exactly what they don’t understand.

Even Better If

As I discovered in my end of term report, some students have found it difficult to choose questions and hence haven’t enjoyed the freedom as a result. I feel that this issue will diminish as they get more used to the approach. I do suggest a set of questions on each topic during classwork, so if they really don’t want to use initiative, they don’t have to!

A couple of students wanted to be more frequently in a situation where they weren’t able to look up the answers, so that they are forced to check their work more carefully. This is a fair criticism and so I’ll balance DIY homework with occasional ‘assessment homework’ next term.

It is taking me a little longer than previously to give written feedback, as I have to look through a variety of different topics and often have to find the relevant questions for my reference. I don’t mind this too much as I have enjoyed the process of engaging with my students more than writing repetitive comments on a set of near-identical work.

I’ve been keeping a record of which questions each pupil has struggled with, and so far not many have been good at returning to these questions in future homework. I need to continue to encourage them to do so, and this term I’m going to set up a shared spreadsheet to check their progress on them.

Overall

It was probably a bit much to go for this approach with all my sixth form homework this term: when I have a new idea, I do tend to go for it in quite a big way!

However, I do feel that the WWW’s outweigh the EBI’s and so I will continue to use it for at least half of sixth form homework time, probably more.

I’d love to hear from anyone else who has tried something similar and has suggestions on how to make it more effective.

New Year’s Resolutions

Many of these ideas come from a survey of my pupils at the end of last term. So this starts with a brief summary of that rather long post.

  1. Ensure that I’m providing enough guidance and support when asking pupils to investigate unfamiliar problems, by creating more guided resources, and preparing a back up text-book option for certain pupils.
  2. Refer more frequently to the skills from Thinking Mathematically to encourage pupils to know what to do when stuck on a problem and other strategies from Helen to help them develop a growth mindset.
  3. Make use of mini-tests: Mathsbot looks like it will be a good source for these in KS3+4, I need to source something similar for KS5.
  4. Set one or two summative homeworks per class per term, in addition to supporting pupils in choosing their own questions by using Google sheets to share questions which pupils have found difficult and track their progress in re-attempting them.
  5. Take opportunities for whole-class interactivity, particularly with year 10, making use of Dan Meyer’s 3 act tasks.

And others which don’t come from the survey.

  1. Ensure I always make clear the Headache before providing the Aspirin.
  2. Write (type where possible) board notes more clearly and slowly; learning Spanish and trying furiously to copy quickly-disappearing notes down from a board has taught me this!
  3. Continue trialling Inquiry Maths lessons, in particular bringing them to sixth form as well as younger pupils.
  4. Use shared Google doc with each pupil to track their general progress and targets, alongside my target setting form.
  5. Build a website to share my approach to providing summative and formative feedback, both directly to pupils and in written reports.

Trigonometry

The Resource

This is an approach to introducing trigonometry I took from my previous department at Highgate.  It focuses on just the tangent scale factor in this initial intro – we would then return to introduce sine and cosine a month or so later, following a spiral curriculum.

I would start with an introduction on the aim of the lesson: we want to find a link between the sides of a triangle and the angles within it. This could make use of the 3-act math format alongside a real life problem such as finding the height of a mountain: personally I often do this with one of my favourite tracks playing the background: Everest by Public Service Broadcasting.

The worksheet guides pupils to discover that if you divide opposite by adjacent for similar right angled triangles, you always get the same number.  This gives us the link we are looking for. It can be followed up (or proceeded) with a note on why you would expect triangles with the same internal angles to have the same scale factor.

Then we can split up the job of working out these scale factors: give each pupil an angle. I then tabulate the pupil’s results and encourage the class to look for the pattern: any incorrect/wildly inaccurate values for tan will stand out and can be quickly corrected.

I’d then demonstrate how we can use this table (for now, avoiding the calculator tan button) to estimate sides / angles.  Using the pupils’ own numbers gives them a sense of ownership over the method, which adds a lot of value to the approach. I have heard that at Highgate, from where I got this approach in the first place, they now also graph the tan values at this stage, to make the pattern even clearer and then use the graph to find the required tan values, but I haven’t tried that myself.

Personally, I then apply the same approach to introduce the sine and cosine scale factors, only returning to show how the calculator knows all the values later in the year.

Suggestions for alternative approaches welcome as ever.

Pythagoras

The resource

A cutting and sticking exercise which leads pupils towards the discovery of pythagoras’ theorem. Beyond introducing the aim of the lesson, this requires very little whole class input from the teacher.

I’ve used this with higher-attaining pupils and it’s worked very well. I’m not sure whether more structure in the questions would be needed for others – I don’t have too many ideas how to break it down further: each individual question is relatively simple so at least the first two numerical examples should be achievable by most.

At Highgate, we then spent a lesson or two in which pupils found missing sides in a triangle by drawing both diagrams out each time – quite a substantial use of time, but it meant that they were very comfortable with the ‘proof’ when we came to generalise it.

I now tend to dive straight into the algebraic generalisation (either individually, in small groups or as a class, depending on how the class are finding it) before proceeding with more traditional applications of Pythagoras.

A follow up to show the pupils which demonstrates the idea in a subtlety different way was tweeted by @solvemymaths. This would make a nice follow up to show pupils.

Suggested improvements please!

My End of Term Report

I’ve used surveys before. This is the first time I’ve written up the results publicly; I’m hoping that it will help me remember the key points. I’ve decided to present the data, a selection of comments and my responses for each questions I asked.

You can see the original survey here

Inevitably when you ask formative questions, it can feel as if the feedback is quite negative, and I have had to remind myself that most of the data is positive!

Mr Pearce’s lessons have been _________ other lessons.

much more interesting than 5%
more interesting than 41%
similar to 36%
less interesting than 16%
much less interesting than 2%
More field trips please! The difficult extension questions are really interesting – could we go through the answers more often.
More practice of exam style questions. Less time spent proving and understanding why things work. – This sentiment was repeated by 2 others.
Maybe more interaction like going up to the board and doing this interactive
More math-robics please! It makes me concentrate on the task at hand.
More really generic examples we can look back to for if we are stuck

I phrased the question as being ‘relative to other lessons’ because it was the best way I could think to make it meaningful.  I teach in a very good school with outstanding colleagues so I’m surprised that it came out positive on average.  There is probably a ‘being nice to the teacher’ effect – I imagine that all lessons would be more interesting than average.

My y10 class suggested more interactivity – my recent inquiry lessons would have been an excellent opportunity to do this: I could have asked pupils to present their contexts for the time-graphs on the board, rather than writing them up myself.  I’ll try to be more alert for such opportunities.  I’m glad they have enjoyed my math-robics, it’s a nice way to break up a double lesson in the afternoon, but I don’t think I’ll be using it any more regularly!

Three sixth form students would like more exam practice in place of “understanding why things work” – I’m dubious about whether this will really make lessons more interesting?! I think they answered the substitute question: “what do you want me to do differently?” which I’ll come back to later.  A suggestion of ‘more generic examples’ is similar.

The request for more time spent on ‘difficult extension questions’ comes from a mixed class with some incredibly high achieving pupils – as much as I’d love to spend more time discussing the STEP problems I’ve been offering as extensions, I don’t think that this would be a good thing to do with the whole class, as these questions are not suitable for others.

I feel ___________ comfortable to work hard and make mistakes in Mr Pearce’s classes than in other lessons

much more 8%
more 38%
similarly 49%
less 5%
much less 0%
you react in a friendly manner and i do not feel pressured to get it right and therefore i try to work without the teachers aid and only call him over to check my answers or when i am totally stuck and my partner is too
Mr Pearce allows us to be open in class & is a much better teacher than my previous maths teacher!
Good class atmosphere – feels like more of a discussion.
Should be allowed to ask for help more
Again, the fact that the lessons generally are aimed towards the top 3-5 in the class. Fear I may seem stupid asking some questions

Generally quite positive, which I’m sort of surprised about, as my friends often joke that I’m far too harshly critical to be a teacher – I’m glad to see this doesn’t come through too strongly!

Even so, two people are unhappy: One of my y12 pupils thinks that the lessons are aimed towards the top end of the class, so I must make sure I’m differentiating more carefully.  The other doesn’t feel that they can ask for help, which worries me slightly: my classes are generally small so I always have lots of time to help.  I guess I should keep repeating pleas for pupils to ask and targeting particularly quiet ones.

The balance between independently working things out for myself and Mr Pearce telling me how to solve problems has been…

much too teacher-led 0%
too teacher-led 0%
about right 71%
too independent 27%
much too independent 2%
Often the maths is not fully explained, so I get stuck. Making notes on a topic with explanations of how to solve different kinds of problems would also be very useful
Too much time focusing on derivation of equations etc not enough making sure they are securely in our head and that we have an understanding of their applications
I don’t think the methods are sufficiently discussed in class, as generally we cover them too quickly (or not enough questions) and then I am not completely confident in the method.
Sometimes I feel like it takes me too long to work things out for myself and by the time I do we have completely moved on to a different topic.
When I am stuck, a lot of the time I don’t know how to continue

As the author of a blog called ‘discovery maths’ it’s not surprising to see that no pupils think that my lessons are too teacher-led. I also teach in a relatively traditional school, which makes my teaching stand out more than in my previous job, and so I feel that 71% of pupils saying that my balance is ‘about-right’ is quite surprising.

There are clearly a significant minority of pupils who are not comfortable with my approach, so I need to tone it down a little, with certain classes in particular. I agree that there are definitely certain topics where I need to provide more/better guidance. Similarly, sometimes I should move the class more quickly to generalise, allowing more time for application of rules. Finding the right balance in these two areas in one of my major aims as a teacher and I’m constantly trying to improve my decision making and target it more appropriately for individuals and classes. Next term I’m going to try to provide more back-up guidance, such as ‘gap fill’ worksheets for pupils who are stuck when trying to generalise.

Additional, I will try to sometimes offer alternative options to pupils who are particularly set against discovery, such as studying examples in their textbook. I also need to make sure I always conclude the process of discovery with a summary and example: I think I currently do this 95% of the time but apparently not often enough.  I’m also trying to make my notes clearer by typing them (now that I’m quite fast on equation editor!) and ensuring I include discussion notes/annotation as well as just the mathematical process.

Finally, not knowing how to continue when stuck is a common problem and one I should definitely try to address. Several years I had a big push towards instilling some of the problem solving strategies from Thinking Mathematically, but I need to constantly remind myself to bring these ideas into classes every time I teach a new class.

Mr Pearce’s written feedback is helpful.

Strongly agree 18%
Agree 77%
Disagree 5%
Strongly disagree 0%
Sometimes set your own questions so there is no way of checking that the answer is correct before submitting it, so it enforces rigorous checking of the work beforehand.
A much more formal prep structure would be helpful to gain an accurate assessment of how we’re doing periodically.
I find it hard to identify which questions to do so would appreciate if more specific ones were recommended
Please could you send us the solutions to the extension problems set the week before.
It would help if you gave a worked solution for a question that i struggled on.

I had a new idea in August, so as usual went with it wholeheartedly: allowing sixth form pupils to choose their own homework.

Generally I think it has been effective. Most pupils have targeted their work carefully to tackle their weaknesses. There has been much better differentiation in the difficulty of problems tackled. It has encouraged greater reflection and independence: I’ve seen more evidence than before of pupils learning from their mistakes, and many pupils have engaged in very useful written dialogue with me. The quantity of work done has impressed me – normally more than I would have set. In fact, the only negative from my point of view is that it is taking me longer to give written feedback!

However, some pupils clearly aren’t enjoying it! One pupil doesn’t feel that they can choose appropriate questions so I need to spend longer explaining how to do so. (I’ve already dedicated a fair amount of time to this and I do give a suggested list of questions for each topic so I’m surprised, though it’s useful to know).

A more legitimate concern is that a couple of pupils don’t feel they are getting a sense for their progress. Personally, I’ve never found homework to be a particularly useful summative assessment tool: the variety of effort exerted and assistance sought means it rarely gives a fair impression of a pupil’s understanding. Instead, I use it formatively and set fairly substantial tests in class to get an idea of progress. Next term, I’m going to set more regular mini-tests (inspired by Colleen Young) which should hopefully address this.

Another good point was that a pupil wanted to be in a position where they couldn’t look up the answers, in order to enforce them checking their work more carefully.  I can see the advantages of this, especially as we get closer to exams, so I’ll set one or two assessment homeworks later this term.

Two other comments request written solutions to problems. I’d rather not give these out directly, as educational research and personal experience tells me that they won’t help pupils to remember ideas in the long term. However, I must remind my pupils that if my hints / solution starters are not helpful enough, they must ask me in class (or on the next homework) for further assistance.

Wow – this ended up being quite long, so I summarised the main points here.

Inquiry 1

Here you can see everything that went on the board during this inquiry: https://t.co/s7Ac1r7Mfl.

I decided to try my first inquiry with a year 7 extension group I teach once a week. This consists of just 8 pupils: a luxuriously small number to work with, and great for a trial run of a new approach.

What Went Well: 

The pupils really got into it. They are generally very motivated but until now, they had been quite reserved and less willing to share their ideas. The investigative nature and the chance to write up their ideas for the class brought them out of their shells. One unforseen advantage of this was that I felt that I learned much more about their strengths than I had in the preceding 4 lessons.

It encouraged excellent sharing of skills: several of the class were quick to generalise, but others were algebraically stronger so helped them to prove their generalisations. Others focussed on finding examples which then showed the generalisers that their theories were incomplete.

The process gave them the experience of being real mathematicians, something which is far too rarely the case in schools. They loved it.

Even Better If:

The regulatory cards are quite general and so needed more explanation than I gave. At first, pupils chose a regulatory card like “practice a procedure” but when asked what procedure, couldn’t answer.  Similarly, they chose “change the prompt” but had no suggestions for how to change it!

The process of pupils sharing their ideas was pretty chaotic, and that’s with a class of 8! Moderating an inquiry so that all pupils get to contribute as they wish will be tough with a more normal class size. This will be somewhat offset when I am ready to open up inquiries to several branches.

Summary:

The inquiry only lasted 2 x 35 minute lessons and could easily have gone on for longer but for the Christmas holidays! I suggested that they might continue working on it over the break so I’m interested to see if they have done so in January.

Overall, I think it’s going to be a great approach with this group which I’ll use very regularly. Next, to a full sized class…

Differentiation from First Principles

The Resource: https://goo.gl/NRq3fc

This is my worksheet which helps pupils to discover the rules for differenting polynomials.

I’d use this after an introductory discussion of a problem in which the gradient of a curve is sought. Often I’ve asked “How would you work out Usain Bolt’s speed 20m into a race?” This usually leads to a discussion about how speed guns work.. I inevitably sketch a distance time graph and we talk about how to find the gradient at a point. Tangents are often suggested and I may have to push students towards the idea of chords and limits, perhaps unsurprisingly, given how major an idea this was in the history of maths.

Writing this, I’ve started wondering if all this build up could be introduced through discovery, but I think the idea is just not intuitive enough.

So then the worksheet guides the students through an investigation of some standard curves before encouraging generalisation. In my experience, it works well with a range of A-level students, and allows scope for those who work more quickly through it to generalise and prove more thoroughly.

As always, I’d love any suggestions for improvements.