5 Maths Gems #72

Welcome to my 72^nd gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. Solve Me Puzzles
Back in Gems 17 I shared the excellent Mobiles problems on solveme.edc.org. People got quite excited about these at the time. Thanks to a recent tweet from Daniel Finkel (@MathforLove), I discovered that this website has expanded. The new section, Who Am I?, consists of hundreds of delightful little number puzzles which start off easy and get much harder. These are adorable.

The Mystery Grids are also excellent. Do explore - it's a great website.

2. Underground Maths
There has been a lot of buzz about Underground Maths lately. When I attended Maths in the Sticks last week I got the chance to have a go at some great problems, including the lovely Integral Chasing and Discriminating activities.

Underground Maths have now produced an incredibly helpful spreadsheet linking all their resources to the new A level content specification. This is perfect timing as many schools will be starting work on their new A level schemes of work in a few weeks.

They've also produced a nice set of classroom posters.

3. MathsPad
I've been using lots of MathsPad resources lately. Yesterday I used their prime factorisation activity in a revision lesson with Year 11.

Extract from 'Using Prime Factorisation'

If you don't already subscribe to MathsPad, I recommend getting it set up for September. They add lots of new resources every month. Some of their resources are available for free - including some recent additions: a workbook on mental calculation strategies, a linear graph interactive tool, a quadratic inequalities interactive tool and a tangent drawing tool. The tangent drawing tool is fun because you get to see how close you were to the correct gradient.

4. The Essence of Calculus
Grant Sanderson (@3Blue1Brown) has added ten new videos on calculus. They explain the underlying concepts beautifully. Even for experienced calculus teachers, these offer incredible subject knowledge development. It's worth making time to watch these.



5. Exam Preparation 
I really like the slides that Mel from justmaths.co.uk has produced for Year 11 in their final countdown to exams. She blogged about them here. I've attempted to make various versions of this over the years but from now on I'll use Mel's slides because they're way better than my attempts! A great resource, perfectly timed.

Update
It's been a really busy few weeks! School is absolutely manic with exam preparation. I launched my #summaths event, hosted a #mathscpdchat, wrote a Schools Week article and attended Maths in the Sticks. In case you missed anything, here are some of my most recent blog posts:

• Year 7 Topics
• Structured Revision Lessons
• Gem Awards 2017
• Maths Anxiety

My 9 - 1 Revision Resources post has been popular lately, reaching almost 20,000 views. Do check out my A level Revision Resources post too - I've added a few new activities.

As part of the TES Maths Panel I helped pull together a new GCSE resources feature which is well worth checking out. There are four recommended TES resources listed for every single GCSE topic. It's an impressive collection! I think these pages will really speed up the process of finding good resources.

If you want to come to my #summaths event at Bletchley Park then book soon! In only three days I've sold 24 out of 80 tickets. Exciting!

Book now for #summaths

I really enjoyed running #christmaths events in 2015 and 2016 but this year I'm doing something a bit different. My annual social and enrichment event for maths teachers has moved to the summer holidays!

#summaths will take place on the Sunday of the August bank holiday weekend. It's just before most of us start the new school year, so perfect timing to be inspired.

If you've never been to Bletchley Park before then you are in for a treat. It's brilliant there - I absolutely love it. I've negotiated discounted entry, plus you'll be able to attend Enigma workshops while you're there. Later in the day you can visit The National Museum of Computing next door which is full of the most fascinating geekiest stuff you'll ever see. 

Socialising is an important part of my events! If you can stay for the evening, join us in the pub. It's a bank holiday so no one has to wake up early the next morning! Book a room if you're coming from a long way away - it's less than £50. I'll meet you for a hungover breakfast in the morning...

All the details can be found at summaths.weebly.com. Book your ticket quickly - there are only 80 places available so I expect this to sell out. Everyone welcome!

#mathscpdchat - Year 7 Topics

I'm hosting #mathscpdchat between 7pm and 8pm on Tuesday 9th May. We'll be discussing what topics are taught in Year 7, and in what order. To join in the discussion, please tweet your thoughts, making sure you include the hashtag #mathscpdchat in your tweet. I'd love to see a screenshot of your school's Year 7 curriculum and hear what works well and what doesn't.

To start us off, here are a few examples:

Don Steward sent me his Year 7 curriculum, shown below.

Don believes that we should ensure that Year 7 is not just a repeat of Year 6. What strikes me most about his curriculum is the explorations that feature throughout the year (notably in the first two weeks), and the wide range of attainment topics. From a student's perspective, there's lots of new, exciting mathematics here.

The White Rose Maths Hub has produced a very different Year 7 Programme of Study - the overview is shown below. Much of Year 7 is spent developing key number concepts. The idea is that students who are successful with number are more confident mathematicians and better equipped to tackle the topics that follow. With 'mastery' being all the rage at the moment, I think that a lot of schools have started moving towards this approach.

A few years ago I read about the teaching order that Bruno Reddy developed during his time at King Solomon Academy (he blogged about it here). At the time I was struck by the contrast between this curriculum and the Key Stage 3 curriculum I was used to. Calculating the mean features in multiplication and division. Proportion is paired with pie charts. Bruno wrote,

"You should notice that more time is given to number work at the beginning of year 7 (especially times tables), we spend one half-term at a time teaching a topic rather than 2 weeks, some things are ‘missing’ (because they’re taught in KS4), we’ve tried to separate minimally different concepts and we’ve thought carefully about the order things are taught in so that all the while we’re building on top of prior learning."

Kangaroo Maths is a popular source of Schemes of Works. The example below is for Year 7s working at the minimum expected standard at the end of Key Stage 2. This is just the unit overview, you can see a lot more detail here.

In my own school we are currently developing our Year 7 curriculum. We tried something new this year but we still need to make some tweaks. I felt that negatives numbers came too late in the year, featuring at the end of the summer term. Changing a Year 7 curriculum is a big job as it has a knock-on affect on the Schemes of Work of all other year groups. Assessments, and possibly other resources like homework booklets, also need to be re-written, which is really time consuming.

I'd love to hear about your experiences in developing and delivering Year 7 curricula. Tell me how you start the year, tell me what topics link well together, and tell me how you develop enthusiastic, inquisitive mathematicians. I'd also like to hear how you assess Year 7 and at what stage you do any baseline assessment. What works well for you? Please join in #mathscpdchat on Tuesday to let me know your thoughts.

Schools Week Article

I wrote an opinion piece for Schools Week - do have a read and let me know what you think.

In his dream to build a country of mathematicians, Gove might have done more harm than good, says @mathsjem https://t.co/LRRZCEe403

— Schools Week (@SchoolsWeek) 1 May 2017

This follows on from my article about maths teacher shortages and the new GCSE that was published in Schools Week in December 2014.

Structured Revision Lessons

Revision season is upon us! I mainly teach exam classes (Year 11, 12 and 13) so exam preparation is currently a big focus for me. For Year 12 I feel a sense of urgency - I only finished teaching the C2 specification yesterday and their C1 exam is fast approaching. Eek.

Thankfully I finished teaching my Year 11s the GCSE specification at the end of last term, leaving me a good 20 lessons for revision. I teach a top set - their target grades range from 5 to 8 and I think their current grades probably range from 4 to 9.

Throughout the year I've been drawing up a list of topics that I need to reteach. The list is very long! It includes indices, constructions, bounds, congruence, linear graphs and algebraic fractions. This information mostly came from marking mocks, plus the weekly quizzes we've done over the last two years. I also used AQA's revision list to check that I hadn't missed anything on the specification. 

Unlike previous years, I'm following a set format in my Year 11 revision lessons. Routine works well with my students. 

On arrival to each lesson they find a Corbett Maths 5-a-day on their desk. This is printed on A5, double sided. On the front is a set of Higher questions, on the back is Higher Plus. This combination is exactly the right pitch for my students - even the strongest are finding the Higher Plus questions sufficiently challenging, so no one is getting bored. These 5-a-day exercises are perfect for getting students to revise a mixture of topics, including the topics that I can't devote an entire revision lesson to. When they get stuck they ask me for help or confer with a friend. It takes quite a while for my students to work through these questions - a good 15 to 20 minutes to complete both sides. Then I go through the answers, briefly summarising the key points for each of the topics covered.

This leaves me half the lesson to focus on a particular topic - for example one lesson last week focused on circle theorems, another on similarity. I speak for about 10 to 15 minutes, reminding my students of the key points and common misconceptions, and running through an exam question or two. Some years ago I realised the importance of this teacher-led instructional element of revision lessons. If they're struggling with a topic, they won't magically get better without some additional teaching.

I then give them 15 to 20 minutes worth of practice questions focused on that topic. For example in my revision lesson on compound measures they completed these density questions. The questions by topic from JustMaths are particularly useful.

I think this lesson format is working well. Now I've established a set structure, these lessons are pretty quick to plan. My students get through a lot of revision in each lesson, and I feel that we're revising all the high priority topics in sufficient detail. There are no gimmicks. Importantly, I'm addressing common misconceptions and reminding students of key facts and procedures.

I occasionally do a Churchill Paper lesson too, though the majority of practice papers are done at home and in Papers Society. Churchill papers are quite challenging. Next week I intend to start quizzing my class on facts and formulae. But most of my lessons will continue to follow the format described above.

Three weeks and counting until the first GCSE exam!

Maths Exam Meme Posters from Paul Collins

Gem Awards 2017

This week it's the third anniversary of resourceaholic.com. It's become a tradition for me to mark the anniversary of my blog by publishing an annual 'Gem Awards' post. Here I look back at all the ideas I've shared in my gems posts over the last year and choose some of my favourites.

1. CPD Award
This award goes to Craig Barton, whose brilliant podcasts have been a very welcome addition to the world of maths education over the last year. In his epic podcasts, Craig talks to a wide range of people about teaching maths, from classroom teachers to world famous education experts. Every episode gives us lots to think about.

Special mention to La Salle Education for continuing to run affordable, accessible, high quality conferences all year round. I really love their conferences and am very grateful for everything they contribute to the development of the maths teaching community.

2. Best New Resources
The winner of this award is @taylorda01 who has produced a large collection of incredibly useful 'Increasingly Difficult Questions'. I first featured these in Gems 59 and have since added all the links to my resource libraries. These well written questions are easy to use, printing nicely onto A5, and come with answers.

Special mention to Edexcel who really came to our rescue for some new GCSE topics with their set of supporting resources, as featured in Gems 64.

I should also mention my all time favourite resource makers Don Steward and MathsPad, who have continued to produce brilliant resources all year round.

3. Subject Knowledge Award
I think that one of the most important things a maths teacher can do is continually enhance their subject knowledge, no matter how experienced they are. This award goes to teacher trainer Ed Southall, who is leading the way on developing maths teacher subject knowledge. His recently published book 'Yes, But Why? Teaching for Understanding in Mathematics' has had unprecedented sales and rave reviews. I wrote a full review of it here. Ed frequently delivers subject knowledge presentations at conferences and has also recently started making videos which are well worth a look.



4. Best TES Resource Author
This award goes to Dan Walker, maker of outstanding resources for Key Stage 3 to 5. I wrote all about his resources in this post so do have a read to see lots of wonderful examples.

5. Best New Website
This award goes to Clarissa Grandi - the artiest maths teacher I know. I wrote about her beautiful website artfulmaths.com in Gems 57. It features classroom displays, creative lesson ideas and extra-curricular activities.

Special mention to Darren Carter for his website mrcartermaths.com. It is super slick and easy to navigate, instantly providing questions and answers for busy maths teachers. And because the answers are so accessible, it's a good revision tool for students to use at home too.

6. Best Video
This award goes to the 'Why the Metric System Matters' video that I featured in Gems 60. I showed it to my Year 7 class and they had a million questions about units afterwards! I'd never seen them so interested in anything before.



Special mention to 3Blue1Brown for their excellent range of videos, including the vectors video that I enjoyed sharing with my Year 13s.

7. Best Problems
This award goes to Underground Mathematics for continuing to expand their collection of excellent A level problems.

I should also mention brilliant.org, who won my Best Problems Award in the Gem Awards 2015. I still enjoy following their Facebook posts - many of their problems are suitable for my students.

8. Best Teacher Support
The White Rose Maths Hub definitely deserves an award for the huge amount of support that they offer to schools all over the country. Their schemes of work for both primary and secondary schools are really useful. Their assessments are also excellent, and they have recently launched a collection of multiple choice questions on one of my favourite websites - diagnosticquestions.com. These questions are aimed at primary schools but I regularly use them with my Year 7s.

Special mention goes to the Mathematical Association - their Twitter account is an absolute 'must follow' for maths teachers, keeping us all up-to-date on news and developments in maths education.

9. Best Interactive Tool
This award goes to MathsPad for their excellent exterior angles tool. MathsPad has a great range of quality interactive tools for demonstrating concepts.

#mathschat #mtbos Take a look at our new free interactive tool for exploring exterior angles https://t.co/lPy5lvOUVf pic.twitter.com/nHrt19Ai56

— James Pearce (@MathsPadJames) 9 June 2016

Special mention to the zoomable number line on mathsisfun.com - my Year 7s were genuinely excited when I used it in a place value lesson.

10. Lifetime Achievement
John Corbett is definitely too young to receive a Lifetime Achievement Award! But I'm giving him one anyway. His awesome website corbettmaths.com continues to go from strength to strength. I regularly use his textbook exercises and send my students links to his videos. I'm using his Higher and Higher Plus 5-a-day in all of my Year 11 revision lessons. His lovely revision cards have been very popular too. I wrote more about corbettmaths.com in this post. Thank you John - you're amazing!

That's it for the 2017 Gem Awards! What a fantastic collection of ideas and resources. Thank you to everyone who tweets about what they've tried in their classroom. It's so inspirational. If you want to read more Maths Gems, there's an index here. For highlights, check out Gem Awards 2016 and Gem Awards 2015 too.

Finally, while we're on the subject of awards, I'd like to say an enormous thank you to all of my readers for their ongoing support. I'm incredibly pleased to say that I won the Individual Education category at the UK Blog Awards 2017. I think this may be one of my greatest achievements ever, and I couldn't be more happy. I'm truly grateful to all my readers.

Maths Anxiety

A lot has been written about maths anxiety in children. I have a Year 7 student who always complains that he 'can't do it' before he's even tried. It's clear to me where his anxiety stems from - unlike his peers, he doesn't know his times tables well, which makes many simple tasks (long multiplication, simplifying fractions etc) really difficult for him.

But this post isn't about maths anxiety in students. It's about maths anxiety in maths teachers. Whatever your job, when you stand up and say you're an expert at something, you naturally feel some anxiety that you'll get caught out. This isn't something unique to teaching. In my previous career in banking I often worried that I'd be asked a question that I couldn't answer. 

I experience most of my maths anxiety on Twitter, where I worry about saying something stupid and being publicly criticised by clever mathematicians. But I also sometimes feel a bit of maths anxiety in the classroom. I'm willing to bet that some of you do too.

1. The 'stupid mistake on the board' anxiety

I do this way more than I should! It's harder than it looks to simultaneously solve a mathematical problem, write on the board, address a class of 34 students, manage behaviour, and a thousand other things all at the same time. It's fairly common for me to say the number five as I write the number eight, and other equally silly things. When I taught at a girls' grammar school my students didn't have much tolerance for silly mistakes so I used to let it get to me, but now my lovely students just laugh with me, so I have more of a sense of humour about it.

It's good for students to see a real person doing real maths, and making mistakes in the same way they do. It will probably make them feel better about their own anxieties.

The worst thing is when no one notices, because that suggests that no one was paying attention! In a recent Year 12 lesson I made a mess of my solutions to a trigonometric equation (temporary craziness with the symmetry of a cosine graph) and no one said anything until later in the lesson when a student was looking back over his notes. Was no one even following what I was doing on the board? That's more worrying than me making a mistake in the first place!

2. The 'genius in the class' anxiety
It's taken me a while to accept that some students may be better at maths than me! Last year I taught a really clever Year 13 Further Maths class and they certainly kept me on my toes. Twice I had to tweet during a lesson to ask for help on questions they'd asked me!

Over the years I've got a lot better at saying 'I'll think about it and get back to you' if I'm asked a particularly tricky question. Then it's really satisfying when I work it out and am able to give them a good answer. 

3. The 'mind blank' anxiety
Often when I'm out for a meal with friends, they'll ask me to split the bill. Inevitably I'll have a total mind blank and completely forget how to do a simple division. It can be hard to do mental maths under pressure.

Of course, maths teachers are used to having maths problems sprung on them out of the blue. It's common for students to suddenly ask you to check an answer or figure out where they went wrong. It's absolutely fine to take a couple of minutes to look through their work. We put unnecessary pressure on ourselves.

4. The 'I have no idea how to do this question' anxiety

A lovely colleague of mine was recently teaching algebraic proof to Year 11 when she got stumped on this example, which she'd taken from the AQA Teaching Guidance:

Prove that the product of three consecutive positive integers must always be a multiple of 6

She wrote n(n+1)(n+2) on the board and expanded it, but then she got stuck. Her students waited while she struggled to work out what to do next. She gave up and admitted she didn't know, and they said, "Miss, if you can't do it, what chance have we got?". She was embarrassed and frustrated after the lesson. She asked me how to do it - I only knew because I've seen a similar question before. Things get easier with experience.

5. The 'I've never taught this before' anxiety

I was seriously rubbish the first time I taught C4 integration. Now I'm on my fifth time teaching it, I'm a million times better - in fact I now really enjoy it.

The first time we teach a new topic, we can feel like we're only one step ahead of our students. There are some new GCSE topics that many teachers haven't taught before. In this case it might be helpful to watch a video (eg Corbett Maths or Hegarty Maths) in advance - hearing another teacher explain a topic is really good preparation for teaching.

If you suffer from any maths teacher anxiety, I can assure you that you're not alone. Here are a few words of advice: 

• Admit to any silly mistakes you make in lessons, and be relaxed about it. Encourage your students to tell you as soon as they spot something.
• I'm sure you're awesome at maths. Of course you are. It's crazy to even doubt that. Being good at maths isn't something that goes away with age. Sure, we forget things, but we're smart enough to pick things up quickly. One of the joys of maths is getting the right answers, and you get plenty of right answers, so don't let the odd silly mistake suck the fun out of it.
• Mental arithmetic mind blanks in lessons are easily dealt with - just throw the question out to the class. Someone will get it right, and if not then you've bought enough time to work it out.
• If you genuinely aren't confident with a topic and you make errors that go beyond 'silly mistakes', then you do need to fix that as a matter of priority. Perhaps a bit more practice in advance of the lesson would be a good idea. The first time I taught FP1 I completed all the examples and exercises before every lesson. It's time consuming but important.
• Don't let your students know if you lack confidence or you don't know a topic well. It's really important that they trust you. I know a teacher who recently told every parent at Parents Evening that she's an NQT and not entirely sure what she's doing. This is not a good idea! It undermines your authority. Even if you don't feel confident and enthusiastic, you must come across as confident and enthusiastic during lessons.
• Know that it gets better with experience, and that if things don't always go to plan now, don't worry - in a few years it will be easier. You'll still make mistakes in lessons sometimes, because you're human. And that's ok.

5 Maths Gems #71

Welcome to my 71^st gems post. This is where I share some of the latest news, ideas and resources for maths teachers.

1. GCSE Revision Resources
Grant (@AccessMaths) has recently published loads of great GCSE revision resources. You can check out the full collection at accessmaths.co.uk.

Resources include Crossover Problems, Octagon Revision Mats and Pentagon Problems. These all work well printed on A3.

2. Desmos Geometry
The awesome people at Desmos have created Desmos Geometry. It looks like it will be just as slick, accessible and user-friendly as their graphing calculator, so this is exciting news. It's currently in Beta and you can read more about it here. I'm looking forward to seeing how it develops.

Do check out Desmos's classroom activities if you haven't discovered them yet. If you have access to a functioning IT room, or a class set of tablets, these make great lessons - I wrote about Polygraph a couple of years ago, and have since used Waterline with great success.

3. Always, Sometimes, Never
Some classic maths activities never get old. I remember the first time I did an Always, Sometimes, Never activity from the Standards Unit in my NQT year. It generated brilliant mathematical discussion and was a really worthwhile lesson. Sarah Carter recently shared another great example of an Always, Sometimes, Never activity for teaching averages:

Tried Always, Sometimes, Never for the first time today. Awesome discussions! #teach180 pic.twitter.com/GHUMcvtrfm

— Sarah Carter (@mathequalslove) 24 March 2017

There are some good Always, Sometimes, Never activities on MathsPad, Nrich and TES. And here are some examples that were shared on Twitter by Mark McCourt (@EmathsUK) a while ago:

4. Pick a Card
The Underground Mathematics team at Cambridge have been busy expanding their collection of A level resources. In this 'Pick a card...' exercise, the content of one card is revealed by clicking on it and students have to decide whether they can work out the rest of the answers. There are some lovely follow up questions to consider, such as "Which card would be the easiest to start from?" and "Does each card always give enough information to uniquely identify the quadratic function?". If you teach A level do check out these excellent resources.

5. Perpendicular Gradient
I shared an animation in Gems 60 which demonstrates the relationship between the gradients of perpendicular lines. I've now found a video which shows the same thing. This is really clear and useful. Thanks to @MathWithMonkeys for sharing it.

Update
Hurrah for the holidays! I've been away to sunny Devon with my family over the last few days. I'm going back to school on 18th April for the final exam countdown with my Year 11s, 12s and 13s.

Since my last gems post I've blogged five times - here's what I've written, in case you missed anything:

• Yes, But Why? which features extracts from Ed Southall's new book
• Update! which provides an update on recent improvements to my resource libraries
• Papers Society which is about something I'm trying with my Year 11s this year - this post appeared in Schools Week's 'Top Blogs of the Week' column
• Ultra Outreach Project which is about an opportunity for free enrichment in high-PP schools 
• #mathsconf9 which is a write-up of the Bristol maths conference - this seems like ages ago now!

I feature as a special guest co-host in the next episode of Colin Beveridge and Dave Gale's podcast Wrong, But Useful. It's out later this week so do have a listen. Speaking of podcasts, check out Craig Barton's latest podcast with Dani Quinn - his most controversial one yet!

Don Steward has been busy posting new resources lately, including some tasks for new GCSE topics which will be added to my resource libraries this week. I've now added all of @taylorda01's 'Increasingly Difficult Questions' to my libraries too. I also made two new resources of my own:

• Equating coefficients in identities (this came up in the recent AQA mock - I couldn't find any resources for it whatsoever)
• Linear graph exam questions (useful for revision - this includes plenty of questions on perpendicular lines)

All of my resources are available to download from TES, and appear in my resource libraries.

Quadratic Points from Don Steward

Look out for another round of my Annual Gem Awards later this month - the third anniversary of my blog is fast approaching.

Yes, But Why?

I'm sure you've already heard about Ed Southall's new book 'Yes, But Why? Teaching for Understanding in Mathematics'. It was published less than a month ago and is already on its third print run. It has been incredibly well received by both primary and secondary teachers - everyday I see tweets about how useful it is.

I remember a couple of years ago when Ed (@solvemymaths) first had the idea of writing this book. There was no subject knowledge handbook for trainee maths teachers - it was a clear gap in the market. By writing one, Ed has provided something that maths education badly needed - an expertly written reference manual for mathematics teaching.

People assume that a maths degree is sufficient subject knowledge for teaching secondary mathematics, but there's a lot more to it than that. In maths, subject knowledge includes knowing common misconceptions, effective teaching approaches, alternative methods, underlying concepts, links between topics, interesting maths facts and stories, and more... Even the most experienced maths teachers with high levels of subject-specific expertise can develop their knowledge further.

Since reading Ed's book I've noticed a subtle shift in my teaching style. For example I found myself excitedly talking to my Year 11 class about the difference between an ellipse and an oval. They were surprisingly fascinated - one of them was really pleased with himself when he spotted the link to the word ovary. And in a recent lesson on geometric proof, I talked for a while about Euclid - it was the longest period of silence I've had from that class! They genuinely seemed enthralled. A highly knowledgeable teacher is confident and engaging. That's the kind of teacher I want to be.

In this post I've featured a few extracts from 'Yes, But Why? Teaching for Understanding in Mathematics' to give you an idea of why it's worth buying.

Statistical Graphs
Histograms are really unpopular amongst GCSE teachers. I'm a statistician and even I admit that I don't look forward to teaching this topic. Ed explains very clearly why histograms use frequency density instead of frequency - in fact I plan to use his examples next time I teach it.

Ed's book tells me that histograms were developed by English mathematician Karl Pearson in around 1910. This is of personal interest to me because Karl Pearson was the man who founded the world's first university statistics department at University College London in 1911 (the very same department where I did my statistics degree ninety years later).

Given that maths teachers often find statistics difficult to teach (or even, dare I say, not enjoyable...), I think that the statistics chapter of Ed's book will be incredibly useful. In the section on pie charts, Ed shares a good example of a useless pie chart, and encourages teachers to discuss the pros and cons of each type of statistical graph with students.

Teacher Tips
Practical tips feature regularly throughout the book. This tip regarding area is very sensible - asking students to find the area of a parallelogram when only the base and perpendicular height have been provided seems a little silly.

Deriving y = mx + c
I'm guilty of telling students that the equation of a straight line is y = mx + c without telling them why. If they know that gradient is 'change in y ÷ change in x', then it's pretty simple to derive y = mx + c, as shown below.

This is obvious, if you stop and think, yet I've never shared this with my students. I guess the problem is that for teachers, right from day one of their training, there is no time to stop and think.

Measurements
The book features a list of unusual units of measure, and lots of other wonderfully geeky mathematical trivia. I love this stuff!

'Yes, But Why? Teaching for Understanding in Mathematics' is informative, witty, enjoyable and incredibly useful. It's available to buy now. If you're still undecided, you can preview the first chapter here. Let me know what you think.