5 Maths Gems #2

Last week I promised I'd write a weekly post summarising the new teaching ideas I'd seen on Twitter. Afterwards I worried whether I'd have enough material to write about, but this week I wasn't disappointed. Plenty of ideas were shared - here's five of them.

1. Revision Whispers. As I've said before, I'm always on the look out for engaging ideas for revision lessons. Billy Adamson (@Billyads_47) shared this fantastic idea 'Maths Whispers!' in which pupils are given a fact or formula to communicate with their classmates, 'Chinese Whispers' style. Billy says that when his students were walking down the corridor afterwards, they were excitedly talking about what each statement was initially and what it became. I love this idea. It would work well in other subjects too. Check out Billy's resource on TES. Here's a variation I might try for A level revision: split the class into teams. The teams form lines from the back of the classroom up to the board. The pupils at the back are given a message to read. Here's an example from S2:

All teams have the same message. The message is simultaneously passed along the lines in whispers. The last student in each line writes the message on the board. The most accurate team wins! Then shuffle the students and try another message. My A level revision cards have plenty of ideas for messages - definitions and formulae would work well.

2. Post-its. Mr Allan (@mrallanmaths) wrote this blog post about his use of post-it notes in lessons. He displays questions on boards around his classroom. Students (in pairs or teams) work their way around the room answering the questions on post-it notes. It's a quick and easy way of assessing understanding and finding misconceptions. It's also very engaging, even though the questions are just from a textbook. The idea originated from a tweet by Chris McGrane (@ChrisMcGrane84) back in January. Shelley Smith (@saxsmith27) successfully tried it out this week.

Speaking of post-it notes, I like this 'What stuck with you?' idea, shared by @RemindHQ. It could be a nice plenary activity once a week, or even at the end of every lesson.

If you want more post-it note teaching ideas then check out 'Post-it addict?' on Number Loving.

3. Visually appealing resources. I’m a sucker for good graphics. I know it’s low on the ‘what make a good resource' list but to be honest if a resource doesn’t look good then I’m reluctant to use it. Kristin (@mathminds) wrote a post about using Aurasma in maths lessons. I haven't explored Aurasma in detail yet - I'm in awe of teachers who use all the latest technologies in their lessons. But what jumped out at me were the visually appealing questions created by Kristin's students (two examples below). I love the simplicity. I'm going to start making my starters look like this.

While I was doing research for my fractions blog post this week I discovered the blog Authentic Inquiry Maths which is full of good ideas, like T-charts for highlighting similarities and differences. Pupils use ICT to create their charts and you get a nice mix of presentation styles, like the example below.

Ed Southall's (@edsouthall) tricky maths problems on his website solvemymaths.com are also really visually appealing and I look forward to using them to challenge and extend my students next year.

Area problem #2 - solvemymaths.com

Here's another 'attractive' activity I spotted on Twitter this week. This one was shared by Khan Academy (@khanacademy).  It's an engaging activity because it's visually striking, accessible and has a good level of challenge for pupils who've studied Pythagoras' Theorem. Shame I don't have the budget for colour photocopying!

4. Starting a new school year. With only two weeks left in the summer holidays, I've started to see blog posts about things to do in first lessons of the school year. I wrote a couple myself a few weeks ago - one on your first lesson with Year 7 and one on kicking off Year 12.  Here's a few more I've spotted recently that are worth a read:

• This 'First Day of School Activities' post on the blog Existential Vertigo (shared by @math4everybody) has some great ideas - I particularly like the suggestion of asking students questions such as "What is the least probable (but still possible) event that you can think of?" and "What is the largest number you can write in the space below?". The writer also references Dan Meyer's clever Personality Coordinates Icebreaker.

Biggest numbers!

• Ben Rouse (@MrB_Rouse) wrote an article 'First lesson: Getting off to the right start'.
• Colleen Young (@ColleenYoung) shared her post about learning names.
• Kristin (@mathminds) collected some great ideas from Twitter in her post 'First Day(s) of School Planning Begins'
• Rachel Jones (@rlj1981) shared her ideas for an 'identity homework'. It's not mathematical but we all benefit from getting to know our pupils.

5. Wisdom (and sweeping statements) from Jo Boaler.  Linking with my earlier post, Deliberate Mistakes, Jo Boaler (@joboaler) talked this week about the value of mistakes in ‘Setting up Positive Norms in Maths Class'. In this document, she suggests that we make a habit of inviting pupils to the board to share their conceptual mistakes. She also suggests that pupils crumple a piece of paper and throw it at the board - just once, in a symbolic kind of way - not every time they make a mistake... Though I wonder if that would be worth trying too, to establish a classroom culture of ‘everyone makes mistakes’ and to encourage risk-taking. Can you imagine if your pupils threw all their mistakes at the board? It might get a bit messy. Fun though.

Jo makes a number of other useful suggestions in her article, such as writing students' good questions on posters around the room. She also talks about the importance of valuing depth over speed. Many of my students arrive in secondary school with the impression that it’s important to do maths quickly. ‘Beat the clock’ type activities encourage this. What’s the rush?

So, some helpful thoughts from Jo Boaler. Shame about the negative article she wrote this week about maths teaching in the UK, but yay to the NCETM for defending us! It’s nice to know someone has our backs.

And finally...
I'll finish off today's post with three recommended reads:

• Interactive Bulletin Boards for Secondary School Mathematics
• Using Fibonacci to convert from km to miles
• 'Teachers – The 10 Stages of Twitter'

Have you made yourself a #Twordle yet? Utterly pointless, but five minutes worth of entertainment (instructions here). By the way, all my weekly Maths Gems posts will be filed on this page, which you can access from the 'Extras' menu on my blog. Thanks for reading.

Thoughts on Teaching Fractions

Fraction misconceptions - MathsPad

Yuvraj (@YuviLite) asked me to write a post about teaching fraction addition. I like a challenge! Fractions are notoriously difficult to teach, riddled with misconceptions and crucial to get right. If I were to be thorough on this topic, I’d need to do a hell of a lot of research and write a dissertation length blog post.

I certainly haven't been that rigorous, but I have found a few ideas. First read what Yuvraj has written about difficulties in teaching fractions, then have a look at my ideas at the end.


When I hear students utter the phrase, ‘uh no not fractions’ a small voice buried deep in my subconscious feels the same thing. Fractions is one of those topics which seems to be on the scheme of work for EVERY year and yet there are still so many students who are befuddled by them and this is one of the reasons I have been thinking about how I teach the topic.

I am a teacher who believes in developing understanding and not just simply reducing mathematics to a series of procedures. I have always thought this and it was drummed into me on my PGCE and Masters. Yet fractions has brought me to a crossroads. Before I tell you why, let us look at different strategies of teaching the addition and subtraction of fractions. 

The Cross Multiplying (Butterfly Method). This is the one I was taught at school. I had no understanding of why it worked but it got me the right answer and at that age that is all I cared about. To add two thirds and three quarters we would multiply the denominators together then cross multiply the numerators and denominators as shown below.

The Grid Method. I recently discovered this alternative procedural method. Draw a grid as shown below. The first fraction is written on the top and the second along the side. A plus sign is put in the first box as this is the operation we want to carry out. We put the products of the other numbers into the grid. We then draw around the highlighted boxes and this would give us our numerator, in this case 8 + 9 = 17. The white box would be our denominator.

Fraction Strips. In NixTheTricks, the use of fractions strips is advocated to teach the addition and subtraction of fractions. For example to add a half and a quarter we would start with visual representations as shown below.

When we put the half and quarter together we can't express this as a single fraction without cutting the pieces into equal parts (quarters in this case). The idea here is that the understanding of a common denominator is built up and linked to the method for adding fractions.

Paper Folding. This method is from Mike Ollerton’s excellent ‘Getting The Buggers To Add Up’ - read the explanation here (or buy the book!). 

Fraction Wall. I have also seen teachers use fraction walls to help students as a method of identifying equivalent fractions. Some people use this or shapes to get the idea of equivalent fractions across.

My Dilemma
I have always gone down the conceptual route. I have championed the conceptual route, until I got to algebraic fractions. The conceptual route always seems to lead back to the more procedural route, especially for fractions where the denominators are two different prime numbers, and for algebraic fractions. If students are more used to the procedural route, they should in theory be able to able this procedure for algebraic fractions.

Which has led me to question: is it worth the pain of rewiring students who always seemed to have heard of the cross multiplication method even before they have reached me (even the Year 7s)? I have spent many hours struggling with folding bits of paper and sharing cakes to embed the idea of equivalent fractions. 

Is it enough just to explain why the cross multiplication or grid method works and then ensure fluency in the skill?


Thanks Yuvraj. I wonder the same thing. In many topics we struggle to find the right balance of conceptual understanding vs procedural methods. Us poor maths teachers are forever fretting about it.

My preferred procedural method, for both numerical and algebraic fractions, is the no-frills version (find the lowest common multiple of the two denominators, then convert both fractions to equivalent fractions with that denominator, then add the numerators). It works for me, but I teach high ability pupils so I have a relatively easy life.

Now for a few teaching ideas. I’m not trying to solve the world’s fraction problems, but I hope you may find some interesting activities here for your fractions lessons.

Don Steward is a Genius
Often, Don Steward's clever activities develop conceptual understanding and fluency at the same time. There's loads of nice fractions stuff on his website (such as these fraction triangles). I particularly like this subtracting fractions activity in which students look at the diagram below and figure out what's going on. As Mr Steward says, this gives the teacher time to drink their tea! It's a really clear way to explain fraction subtraction using common denominators.

Misconceptions and Assessment
This blog has some brilliant ideas for identifying fraction misconceptions which might be particularly helpful when you start tackling fractions with a Year 7 class. For example, the writer asked his students to explain what's wrong in each of the three pictures below and got some interesting responses.

You could then ask students to use the same shapes and find an accurate way to represent the fractions indicated. Other ideas in this blog include asking the students to complete the sentences 'Fractions are...', 'When you add two fractions you have to...' and 'You use fractions when you...'. Or even ask pupils to write down 'Everything I Know about Fractions' and write a list of what they'd like to learn. These activities will help you determine your starting point in Year 7. The blog is well worth a read.

Manipulatives and Visual Aids
Yuvraj already touched on visual aids. I remember being told during my PGCE that I simply must use Cuisenaire Rods, but not actually having a clue what to do with them. There's whole books written about this! I found explanations here and here - the method seems rather overcomplicated. But if they appeal to you, then Nrich provides this online tool for demonstrating Cuisenaire Rods on the Interactive Whiteboard.

This website has a lot of good stuff on fractions and visual aids, including downloadable fraction circles.

‘Number one in disguise’
Look at the 'NOID' poster below. Yes, it is quite ridiculous. But it’s very easy to follow the thinking here and I wonder if this explanation (or similar) would help develop students' understanding. If you’re looking something a little less NOIDy for teaching improper fractions then this blog post has some excellent ideas using manipulatives.

From cindywhitebcms.blogspot.com

Division
'Ours is not to reason why, ours is but to inverse and multiply'. This made me laugh because I've never attempted to explain fraction division to my pupils beyond the 'KFC' rule (keep, flip, change). Don Steward has written an excellent post about ways to explain why it works - this one from moveitmaththesource.com particularly appealed to me. I'll keep this in my back pocket in case ever asked the question.

Final thoughts
I'm afraid I don't have the answer to how best to develop conceptual understanding, but there may be some ideas here that you haven't seen before. In terms of resources, there's tonnes of worksheets, games and activities online - see my resources library for some recommendations. For more teaching ideas, here's some further reading:

• The Guardian's 'How to teach fractions' 
• Bruno Reddy's 'The multiple personalities of fractions'
• Education Week's 'With Fractions, Common-Core Training Goes Beyond 'Invert and Multiply''
• James Tanton's 'Troublesome Fractions'.

Thank you so much to Yuvraj for his helpful input to this post. Comments and tweets would be much appreciated - we'd love to hear your ideas.

Fraction hopscotch in the maths corridor - from cisdmathmusings.blogspot.com

5 Maths Gems #1

I’m fairly new to Twitter. My absence over the last few years means I’ve missed out on thousands of excellent teaching ideas. Twitter is a source of endless inspiration. So I have a plan. I’m going to write a weekly post highlighting five great new ideas I’ve gleaned from Twitter. These posts will help me remember things. And hopefully my readers will benefit too.

Today I’ve exceeded my limit and am presenting a list of 6 ideas from my week on Twitter (not a good start! I’ll aim for 5 next week).

1. Practical ideas from Maryse. It sounds like Maryse (@AllThingsMaths) has done an amazing job transforming mathematics teaching in her school and enthusing her students with tonnes of creative ideas. Here’s a selection of those she’s shared this week:

• Play catch with eggs and work out the relative frequency of smashing.
• Get an old tyre, paint it and roll it to demonstrate circumference. 
• Have a number line painted in the playground.
• Have discussions about going back in time and Doctor Who with travel graphs. 

I love these ideas. Follow @AllThingsMaths for lots more.

2. Maths lies. In response to my post about the Mistake Game, Colin Beveridge (@icecolbeveridge) shared this awesome story ‘My Favorite Liar’. You must read this! It prompted a discussion on Twitter about whether the idea would work in maths. I think it would. In your first lesson with a new class, tell them that you'll tell one lie per lesson. Your pupils have to identify your lies. In the discussion on Twitter we came up with a good list of lies and misconceptions, such as 'multiplication makes numbers bigger', 'if a number has 6 zeros then it's in the millions' and 'one is a prime number'. The idea is that you slip these lies into your teaching and hope that your pupils will immediately challenge you. And if no-one challenges you then it will make a nice plenary discussion. Mr Allan (@mrallanmaths) is going to try it out - I’m really looking forward to hearing how effective it is. I think, like the mistake game, it will be incredibly engaging.

3. Box plots party. In response to my post about teaching box plots, Pete Sides of the South Yorkshire Maths Hub (@SYMathsHub) suggested a lovely activity in which pupils discuss the ages of guests at different types of party, to help them understand the concept of spread. For example you could ask them to draw a box plot representing the ages of guests at an 18th birthday party, a wedding, a child’s party etc. I like this idea so I turned it into a worksheet and added it to my blog post.

4. 100 people. I came across these visuals ‘If the world were a village of 100 people'. Maryse (@AllThingsMaths) told me of the related video, which reminded me of another visually striking video, Debtris. I thought these ideas might be nice for form time or PSHE but Mark Greenaway (@suffolkmaths) then shared this data and this blog about the psychology of percentages. The blog is well worth a read, it’s an interesting idea for introducing the concept of percentages in maths. @El_Timbre shared this display work resource which she created to help students with the visualisation.

5. Percentage triangles. I’ve been using speed distance time triangles for years – in fact I think I was taught them when I was at school. Trigonometry triangles are similar. I’ve never given much thought to the fact that my students are missing an opportunity to practise rearranging formulae. I understand the reason for mixed views on these methods, but I do rather like the triangle shared by Malc Henderson (@malc_henderson) for solving reverse percentage problems. It’s nice to have a range of tools to support students when they’re preparing for their GCSEs.

6. Display ideas. Clarissa Grandi (@c0mplexnumber) posted a picture of her new ‘vwllss’ corridor display (inspiration from @El_Timbre). It looks fantastic - what a good idea!

Other good display ideas I’ve seen recently include circle formulae by Dawn (@mrsdenyer), a square root clock by Duncan Smart (@duncsmart), and an interactive number puzzle for a maths corridor or classroom, shared by Mr C Ward (@MrWardMaths).

There you go, six maths gems from @mathsjem (my initials are JEM by the way, it’s not a spelling mistake!). This is just a small selection of the good ideas I’ve seen on Twitter this week. More to come next week!

I’m working on a few posts at the moment, including one on introducing algebra and one on introducing differentiation – if you have any great ideas then please email resourceaholic@gmail.com or tweet me.

Image sources: lightbulb tampawebdesigner.net; birds techwyse.com

Animations and Simulations

Last night's #mathschat discussion about images gave me so many good teaching ideas, I woke up at 2am to write a list! One thing that came up was animations. I love a good animation that demonstrates a concept beautifully, so here's a few examples.

I posted this radians animation on Twitter last night (originally featured in this blog post) and judging by the number of retweets, I guess others like it as much as I do. There's more gifs like this here. 

The animation below shows how the exterior angles of a polygon sum to 360^o. Sometimes it's better to use an animation that you can pause and talk through though, so I prefer this animation for exterior angles of a polygon. Absorb Learning is an excellent website for animations.

Another example of an animation you can pause during teaching is this 'bisecting an angle' tool. It comes with both a clear instruction sheet for students and associated proofs. There are similar pages for every possible construction! As I mentioned in a previous post, I hate teaching constructions, so I find this very helpful.

Here's another maths gif. This one is for demonstrating the properties of parabolas, as taught in Further Maths. This can also be demonstrated nicely using Desmos.

I also want to mention PhET simulations. I don't teach mechanics but I'm told that PhET has lots of great interactive simulations for demonstrating concepts. 'Balancing Act' is a good example. They also have simulations for Key Stage 3 topics such as fractions - see the full maths selection here. The simulations can be downloaded and run from your school network. 

Balancing Act - PhET

Aren't we lucky to have such excellent technology to enhance our teaching?

If you know of any other good animations or interactive tools, please comment below or tweet me.

Teaching Box and Whisker Plots

Image: ruthmaas.com

When I wrote Long Live Stem and Leaf, I'd been challenged to find a practical application of stem and leaf diagrams outside the maths classroom. After extensive googling - and helpful input from twitter - the only examples I could find were bus and train timetables. But I argued that real life application isn't the be all and end all.

Stem and leaf diagrams were invented in the 1970s by John Tukey, who also invented box and whisker plots. As a matter of interest, I've also investigated real life applications of box plots. I found that box plots are widely used in research papers and analyses. Unlike stem and leaf diagrams, they are a statistical tool genuinely used by statisticians. Because box plots are commonly used, much has been written about their effectiveness. As this blogger says, “a box plot is a simple yet powerful tool, with a truly great design - a universally beautiful thing that stands the test of time”.

In this post, I’m going to focus on teaching ideas and resources.

Practical activities
I’ve read a lot of blog posts about human box plots. For example, 11 students line up at the front of the classroom in height order. Their classmates determine who is the median, lower hinge and upper hinge (yes, I’m using the word hinge instead of quartile. Read my blog post on Consistent Mathematics if you want to know why). This is a lovely activity in a lesson introducing the idea of median, but not so effective in a box plot lesson. Because even though you can use a bedsheet and string to get the students to look like a box plot, if they’re not appropriately spaced out (ie if there’s no scale) then it’s not a box plot. If you want a proper box plot you’ll have to actually measure the heights and draw a scale on the floor.

An alternative idea (inspired by this post) involves paper aeroplanes. You’ll need a big space for this - the school hall or playground. Tape (or draw in chalk) a long scale on the floor. Have each of your pupils make a paper aeroplane, write their name on it, and throw it from the start of the scale. Leave the aeroplanes where they land. Once all pupils have thrown their aeroplane, you can draw or tape a box plot on the floor around the planes to represent the distance flown. Pupils will be able to see which quartile their plane landed in. Hopefully one future engineer will throw an amazing outlier which will make a good discussion point! You could even split the class in half and do two box plots on the floor side by side, then the class could discuss which team had better aeroplanes and which team’s planes were more consistent.

If you like practical activities, here’s two more:

• This simple paper folding activity is a nice introduction to median and quartiles.

Source: bigideasmath.com

• This Cheerios activity explores mean, median, mode and box plots.

Resources
There’s surprisingly few good box plot resources online. But Don Steward never lets us down. His website has a number of box plot activities - my favourite is this true or false activity.

I also like the activity below from bigideasmath.com:

If you’re looking to generate box plots on your interactive whiteboard for class discussion, this online tool is nice – it has pre-populated data or you can input your own.

I've listed some more good box plot resources below. I’ve found that many resources ask questions about skewness, which we don’t cover until S1 (but perhaps we should cover earlier). Treatment of outliers is also not covered at GCSE but makes for a nice discussion.

• Standards unit activities S5 and S6, which can be found on Mr Barton’s website. 
• Illustrative Mathematics has loads of excellent statistics questions, like Haircut Costs and Speed Trap.
• Interpreting Box Plots Treasure Hunt from numberloving.co.uk.
• Representing Data Using Box Plots from Mathematics Assessment Project (activities at the back)
• Lots of exercises from CIMT in this chapter – I particularly like the comparison questions on the last few pages 
• Using NBA Statistics for Box and Whisker Plots from NCTM 
• The variability exercise (starting on page 8) in this lesson plan could be helpful when teaching the concept of spread. 
• My Describing Box Plots activity on TES helps pupils practise reading and drawing box plots and emphasises the use of the correct vocabulary.
• My Party Box Plots activity on TES helps pupils understand the concept of spread.

If you want to show your class some ‘real world’ box plots then these box plots showing the ages of World Cup players could prompt some good discussions about comparisons. Read @loumeracy’s blog post for more on this.

If you want to ‘wow’ your class with an example of how box plots can be used to compare huge amounts of data in a small space, show them this example which shows the age distribution of Olympics athletes (more of this here).

Final thoughts
In French the box plot is called boîte à moustaches (box with a moustache). The writer of this article attempts to create a box and beard plot!

There are many variations to the box and whisker plot, such as the bean plot and violin plot. Look these up on google images for lots of examples.

I read an interesting idea here that I’d never thought of before: “One way to help you interpret box plots is to imagine that the way a data set looks as a histogram is something like a mountain viewed from ground level and a box and whisker diagram is something like a contour map of that mountain as viewed from above”.

Well, there you are – a 'box plot blog post'. Try saying that five times out loud. I've invented a new tongue twister.

Deliberate Mistakes

Image: weandserendipity.com

I'm always on the lookout for new ideas for GCSE revision lessons. If I write about this idea now then hopefully I'll remember it come Spring.

This year I had great success with relay races and carousels in my revision lessons. I also did some pupil-led lessons which were a lot less successful.

This 'deliberate mistakes' idea is something I've adapted from this blog post, so thank you to @bkdidact for sharing the link and @kellyoshea for the inspiration. I've never seen this in maths before but it may be that many of you already do it, if so then I'd like to hear your feedback.

The Mistake Game
Students, in pairs or small groups, are given a problem to solve and explain to the class. In their explanation, they must include a deliberate mistake. Not just a numerical slip, but something that highlights a possible misconception that their classmates might have.

If I use this for GCSE revision then past exam questions (the longer, unstructured types) would work best - there's some examples in this excellent set of questions from Suffolk Maths. But this idea could also work when teaching specific topics. I think I may try it this year when teaching both trigonometry and probability trees.

For example, if you use the Mistake Game when teaching Pythagoras' Theorem, you could use this question (solve for x):

The common mistake here is to not square the 4 (though students may come up with something else).

Here's another example, from this worksheet from thechalkface.net.

There's numerous things they could (deliberately) do wrong in this one!

When each group goes through their question on the board, their classmates will be listening carefully to spot the mistake, so it's great for getting students to focus. I'm not sure how best to organise the responses. Perhaps students could put their hand up as soon as they spot the mistake so it can be corrected immediately and then the rest of the solution can be done correctly. Or perhaps when the group have finished presenting their full explanation, a volunteer could come up to the board and correct their work. Or maybe after each presentation the students could discuss the problem in their groups, decide what was wrong and re-do the question correctly.

This is a bit like 'tick or trash' exercises - it's helpful for pupils to be aware of common mistakes. I've also used these 'Classic Mistakes' in previous revision lessons.

The Mistake Game is also a good opportunity for students to see each other’s approaches to solving problems and discuss the effectiveness of different methods.

I think this activity will engage the whole class - I'll try it out and let you know! If you've ever tried anything like this then please share your thoughts. I recommend reading about the experience of the teacher who wrote the original post. As she says, ‘we all learn more when a wrong answer goes up than when we’re simply watching perfection’.

Image: bigthink.com

Consistent Mathematics

‘But Miss, that’s not what our last teacher said’

I read an interesting article this week about mavericks in schools. The article talks about the need for consistency in behaviour management - this got me thinking about consistency in mathematics teaching. Us maths teachers all have our own teaching style. We even have preferences for particular mathematical methods. I think it's fine that we all structure our lessons differently - pupils adapt quickly to their new teachers at beginning of each school year. Children are far better at adapting than adults. But it's worth thinking about consistency in our mathematical approaches, terminology and standards, and that's what I want to explore in this post.

Some of the things we teach, or the way we teach them, are confusing. Some of these things we can change, and some we can't. People are generally resistant to change, but now is a good time to make changes because many of us are implementing new schemes of work.

Changing how we teach the order of operations
Whether ‘BODMAS’ is a good teaching method isn’t really up for debate anymore. We all know it doesn’t work. It gives students the impression that division should come before multiplication, and addition before subtraction. There’s been lots of excellent posts about this topic - like this post 'Poorly Executed Mnemonics Definitely Addle Students' by Bill Shillito - so I won’t go through it all again, but let’s think about how to fix it.

As a minimum, we need all teachers in a school to be on board with a new approach. In an ideal situation we’d get all maths teachers in the country – no actually, in the world – on board. Now, in a rather tiny step towards changing the world – I’ll start by discussing this in my department meeting on the first day of term in September.

If our pupils have met BODMAS at primary school, we’ll have to tell them to disregard it, and we’ll have to show them why by demonstrating the misconceptions it may lead to. Like many others, I’m an advocate of teaching GEMA or GEMS instead. Some American maths teachers have already written blog posts about this approach. It’s a pain that most of my lovely resources reference the term BODMAS though. If I get time over summer I’ll make some alternative GEMA resources.

Changing quartile terminology
Now I don’t think I’m going to make any progress with this one because it doesn’t result in any serious misconceptions. Am I the only person bothered by the fact that we use the term ‘lower quartile’ to describe both the bottom 25% of the data and the cut-off value 25% of the way through the data? John Tukey, inventor of the box plot, used the word ‘hinge’ instead ie the lower quartile consists of the values between the minimum value and the lower hinge. I prefer this terminology, and I’ve seen it used in a few places, but I doubt it will be widely adopted.

Maths we want to change but can’t
When we introduce the idea of squaring sinx at A level, we tell pupils to write sin²x instead of sinx² (to clarify that it’s not the x that’s being squared). But we have to carefully explain that sin^-1x doesn't mean the reciprocal of sinx, but actually the inverse ie arcsin. The obvious solution to this confusing notation is to stop using -1 to indicate inverse functions – but then we’d need to redesign calculator buttons! Why didn’t calculator designers write arcsin, arccos and arctan on the inverse trig buttons?

The use of -1 to mean both a reciprocal and an inverse function is one of the many mathematical annoyances discussed in this post on Reddit ‘What Mathematical Conventions would you change?’. You might also be interested in this article ‘The most common errors in undergraduate mathematics’.

Inconsistent mathematical methods
I don’t think it’s a problem if pupils use a variety of methods to answer the same question. Multiplication methods are a good example of this. There’s been some discussion about multiplication methods recently on twitter - most people agree that if we get the right answer then it shouldn't matter what method we use.

As teachers we have our own preferences. And pupils may have their own methods too, learnt from teachers, parents, tutors, Kumon (grrr...) and so on. A good example is the way we find a highest common factor - there’s many different methods, all equally valid (I'm convinced I've finally found the best method though!). Another example is the approach we take to substitution and rearranging. When I use the Cosine Rule to find an angle, I substitute the numbers into the formula a² = b² + c² - 2bcCosA then I rearrange. But I’ve noticed that most of my colleagues tell students to rearrange the algebra before they substitute. It might confuse students when they first see a new approach, but this opens up a useful dialogue about the equivalence of different methods.

Here’s a few more examples of where teachers take different approaches:

• When teaching trigonometry in right-angled triangles, not everyone says ‘SOHCAHTOA’. At school I learnt a mnemonic instead. I’ve recently discovered trigonometry triangles too.
• I’ve already written about differing methods for finding the equation of a straight line in my linear graphs post.
• I once had a Year 11 class who’d been taught by their previous teacher to find the nth term of a sequence using the a + (n-1)d formula that we normally teach at A level. It worked well.
• I don’t use the ‘FOIL’ order when I expand brackets on the board and this confuses some of my pupils who seem to think that FOIL is the only correct method.

Inconsistent vocabulary and pronunciation
I say parab-ola, my colleague says para-bola. I’m pretty sure I say it right (in fact, I just checked here to make sure!). But I had a moment of doubt when my pupils all laughed at my pronunciation because they’d heard it pronounced differently the previous year. Now this really isn’t the end of the world, but it made me wonder about my pronunciation of other words, such as cyclic (as in cyclic quadrilateral), alternate (as in alternate angles) and ln (as in natural log).

As for vocabulary, I switch between the terms 'index' and 'power' all the time. This is just one example of my inconsistencies. I don't think it's a problem to use a variety of words, in fact it might be a good thing, as long as all new vocabulary is explained to students.

Inconsistent standards
I taught a Year 12 last year whose work was set out so badly it was almost impossible to follow. She scattered her workings all over the page, sometimes showing me where to look next with flowery arrows. She put multiple equals signs on the same line of working, and her final answer was always hidden away somewhere. Her work was also mathematically incorrect. Although I commented - many times - on the state of her workings, I decided it was more important to focus our limited time together on fixing her mathematical misconceptions. I felt frustrated that her appalling approach to setting out her work hadn’t been dealt with at an earlier age.

I remember when I was a student in the Upper Sixth Form, my maths teacher wrote on a homework that I’d used the symbols for 'equals' and 'therefore' interchangeably. I’d seen her use a double arrow symbol on the board and misunderstood its meaning. I was utterly confused (and embarrassed!) because I’d been doing it wrong for over a year and she’d never mentioned it before!

I also think we could do a lot more to encourage clearer mathematical handwriting (both numbers and algebra). We all have our own styles though. I’m guilty of writing diagonal lines in fractions on the board even though I know I shouldn’t.

A case study
It’s worth reading this good practice report from Ofsted: “In many schools, students’ long-term progress in mathematics is slowed by inconsistent teaching. At Archbishop Temple School, very effective leadership of a strong team of teachers ensures that students experience teaching that is consistently good or better. Students in all groups benefit because different teachers use common approaches and teaching styles”. There are lots of good ideas in this case study, such as the formal sharing of resources between teachers and ‘book swapping’ – not in the critical ‘how much marking do you do?’ sort of way, but with a view to encouraging consistent practice.

I think that new products and technologies like Complete Mathematics by La Salle Education will do a lot to support teachers, improve consistency and raise teaching standards. The flipped classroom approach, which has been gaining popularity recently, may also have an impact on consistency.

I’d be interested to hear your views on whether we need to improve consistency in maths teaching and how we could go about doing so.

Image: mentalfloss.com