5 Maths Gems #4

Well the summer holidays are well and truly over. Many teachers, particularly those who are relatively new to teaching, will spend this weekend making lists, planning lessons and experiencing that unsettling feeling of nervous excitement. If you have time to read this week's set of maths gems, I hope they provide a little light relief.

1. Maths lessons for maths teachers
I devote a lot of time to thinking about how to develop my teaching but spend far less time building on my own mathematical skills and knowledge. Whatever our background and however 'good' we are at maths, there's always more to learn. For example, I’ve never studied any mechanics in my life (I did Pure Maths A level and a statistics degree) but SUVAT is on the new GCSE syllabus so I'd better find out what it is.

I feel that my knowledge of secondary school core mathematics is pretty sound but this week Ed Southall (@edsouthall) inadvertently pointed me in the direction of James Tanton's website (@jamestanton) and within minutes I was learning new things! Here's a few examples.

If asked to sketch the graph of y = x² + 4x + 5, I'd realise that it can't be factorised so I'd check the discriminant to confirm that this function has no real roots. My next step would be to complete the square to determine the vertex. I don't like completing the square. I've never thought to find the vertex using this method instead:

jamestanton.com

This seems pretty obvious now. This is my new preferred vertex-finding method! Am I the only person who didn't think of doing it like this?

I also discovered that i is not the square root of minus one (watch this video to see why) and that the order of quadrants is based on the rising and setting of the sun! (see this essay for an explanation). 

At some point I'll read all of James Tanton's enlightening material. I'm also learning a lot from Better Explained's (@betterexplained) excellent monthly newsletters which I highly recommend.

Have you seen this proof of the area of the circle? Great stuff by @minutephysics - thanks to @mathslinks for sharing.



2. Greek Weeks

utulsa.edu

I tweeted this week that I plan to start allocating each of my pupils a Greek letter to be used during maths lessons (by me and their peers) instead of their name. This will help them learn the Greek alphabet and means I can avoid trying to pronounce difficult names! Of course I was only joking and the responses on Twitter really made me laugh (@edsouthall: "Oi, omicron, look at me, not at rho").

Stuart Price (@sxpmaths) told me that he assigns a Greek letter to each week during the school year. Now that is a much more sensible idea! So I'm going to start writing a Greek letter by the date on the board. My students will write it by the date in their books and say it aloud to practise pronouciation. El Timbre's (@El_Timbre) Greek alphabet poster will adorn the wall. The first week of term is alpha week and we'll take it from there. Lovely idea.

The labelling of weeks has a very tenuous link to something else I saw on Twitter this week - in this post written by a student I found out that in Lithuania 1st September is a 'celebration of knowledge day'. Also, in the Lithuanian equivalent of our Year 13, they have a '100 day celebration' in which they have a prom to commemorate that they have only 100 days left of school. The rest of the post is mostly about clothes shopping for Lithuanian teenagers, so not quite so interesting... I'd like to find out more about traditions in education around the world.

3. Plenaries, extensions, puzzles and loyalty cards

misstait.wordpress.com

This brilliant 'Finished? Try one of these...' display got me thinking about the materials I prepare for my early finishers. The creator of this display, Kerry Tait (@misstait_85), wrote a blog post describing the review tasks contained within, which are intended to encourage students to reflect on their learning. This is a really nice idea. Kerry is a science teacher but this would work equally well in maths.

At the end of my lessons I tend to go for extension activities rather than reflection tasks. In every lesson I give extension work to the appropriate pupils at the appropriate time. My students know that I'm really impressed if they have a go at the extension material but at the same time I emphasise that maths isn't a race. There's value in occasionally planning a relatively short main exercise and an 'extension' that is accessible to all, not just the brightest or the quickest. Most people enjoy being stretched. I want all of my students to come out of maths lessons feeling like champions.

My extension work is usually a challenging question relating to the lesson topic but sometimes I use puzzles instead. For example at the end of a Key Stage 3 lesson on the order of operations I give students the classic four fours puzzle. For even an greater challenge there's this sixes puzzle shared by Chris Smith (@aap03102) (I think my sixth formers would enjoy this one), or perhaps even something on Friedman numbers.

@edfromo

Here's a few more puzzles and extensions I've come across this week. My school has a lunchtime puzzle club for Year 7s so these will come in handy there too, and I may set up a weekly puzzle board as suggested by Mr Duffy (@MrDuffyMaths).

• Garrod Musto (@GarrodMusto) shared the news that Mathster now has UKMT challenge questions in quiz format.
• These hexatrex puzzles look like fun.
• The Griddle has loads of mathsy variants of sudoku.

For the 'Finished? Try one of these...' display, the creator says that she might design a checklist and prize system to encourage students to use the full range of reflective tools. This suggestion reminded me of a couple of other ideas I saw on Twitter this week. In @CorbettMaths' post about how to use his 5-a-day questions, he suggests that students put their completed questions in a ballot box and a fortnightly prize draw takes place. He also shared @MissKMcD's loyalty card ideas which she's explained in this blog post 'Learning with Loyalty Cards'.

misskmcd.wordpress.com

4. Everyday lessons
My blog is about sharing creative ideas and engaging resources. But my lessons aren't all-singing all-dancing. If you were to observe most of my lessons you'd see a pretty standard format of: starter, explanation, discussion, practice, extension. I do create opportunities for students to do investigations and exciting activities but not in every single lesson. As you can tell by the title of my blog, it's really important to me that I pick engaging resources. Sometimes that just means a well designed worksheet. Harry Webb's (@websofsubstance) post about maths lessons and priorities is well worth a read, particularly for NQTs who will soon be drowning in lesson plans. In my NQT year each lesson took me a couple of hours to plan. Four years on, my lessons take around 30 - 45 minutes to plan, which is still totally unsustainable and inefficient but I get a bit quicker every year. Being able to get good resources from my blog should increase my planning speed as I won't have to do so much hunting for resources.

Harry's post also gave me a new idea. A box on the board for students to write down the homework questions they found hard - such a simple idea but much better than how I do it at the moment, which normally involves lots of students talking at me at once!

5. Bits and pieces
Here's a small selection of the other ideas I saw on Twitter this week:

• I love this 'things to do when you're stuck' poster from The Maths Magpie (@TheMathsMagpie).

'Things to try when you're stuck' poster - download here from TES http://t.co/l6YHVQArOb #PedagooFriday #edchat pic.twitter.com/FHjNACsSmG
— The Maths Magpie (@TheMathsMagpie) August 29, 2014

• I discovered a website for maths animations - mathimation.co.uk. New stuff is being added all the time so follow @mathimation for updates.
• Anna Hester (@TypeAMathLand) shared an infographic syllabus - I borrowed her idea and made a 'brief guide to AS level maths' infographic for my Year 12s. It was fun and now I'm looking for other things to infograph!
• I really enjoyed JemmaPDuck's (@JemmaPDuck) blog post about her first day back, lots of teaching ideas here.
• There's far too much form filling in our lives, but here's some forms I read about via Twitter this week that may be useful for behaviour management. These might not appeal to everyone, but are worth a look.
• Mr Taylor (@taylorda01) has been making some amazing video problems - this one is particularly terrifying so students will love it.
• Mr Collins (@mrprcollins) created these incredible comic resources for teaching constructions. The creativity and effort that has gone into these is outstanding. 
• @Daisyplum75 shared this colourful multiplication grid from thegriddle.net.

thegriddle.net

I found out this week that my term 'maths gems' is totally unoriginal - here's a nice 'assortment of mathematical marvels' which are also called maths gems. Speaking of nicely designed websites, do take a look at Mathigon, shared this week by Chris Watts (@watchri01) - stunning, isn't it?

So that's it for this week. My topic resource posts will return soon and I'll also work on a new Pret homeworks page for my blog. Pret (practise, recall, extend, think) homeworks were invented by Kathryn Forster (@DIRT_expert) and discussed in my last gems post. Using her template, I created a Pythagoras homework and intend to make some more. I'd love to share a big range of Pret homeworks so please email me resourceaholic@gmail.com if you make one that you're happy to share.

Pythagoras Pret Homework

Those of you going back to school next week, good luck! I'd really appreciate it if you'd mention resourceaholic.com to your colleagues.  I'll leave you with a nice maths problem for your Key Stage 3 students - this one was shared by F M Jones (@FMJonesMaths).

5 Maths Gems #3

Well it's been an emotional couple of weeks. A level and GCSE results kept many of us up at night and we experienced the usual feelings of joy, pride, relief and sometimes disappointment. But still we continue to prepare for the new school year and think about what we'll do differently.

The hardest thing about writing these weekly posts is deciding which ideas to feature. I have an increasingly long list of unused material because even when Twitter's teachers are preoccupied with results, the incredible creativity and spirit of collaboration remains. In this post I'll showcase just five of the very many ideas shared on Twitter this week.

1. Homework
Kathryn Forster (@DIRT_expert) shared a new set of homeworks that she has designed. Each homework consists of five sections: literacy, research, memory, skills and stretch. These homeworks would be followed up by starter activities such as spelling or memory tests and discussions about research findings. Kathryn has very helpfully shared four of her homeworks for us to borrow:

• Negative numbers  
• Rounding
• Fractions, Decimals and Percentages
• Wordy arithmetic

These are really good ideas.

If you're thinking about trying out some different approaches to homework I recommend reading Cav's (@srcav) blog posts Homework and Retention and Take Away Homework.

2. A Scavenger Hunt

Speaking of homework, the Mailbox (@TheMailbox) shared a 'scavenger hunt' which could be adapted as first homework for Year 7 or 8.  Alternatively you could bring a pile of newspapers and magazines into one of your first Year 7 lessons so your pupils can create a poster in class using this idea.

Wouldn't this be great for the first week of school & easy to revise for any grade level? http://t.co/lpCN5KrS88 pic.twitter.com/i9wghJFQtk
— The Mailbox (@TheMailbox) August 22, 2014

3. Oh for a room to call my own...

There's still a lot of classroom display ideas going round Twitter. One of my most retweeted retweets this week was this 'Change your words' display:

@mathequalslove what do you think? Thanks for the inspiration and the files! You. Are. Awesome! pic.twitter.com/vEeTQ2fCdl
— casey mccormick (@cmmteach) August 19, 2014

It's important to communicate these messages and a display like this is a good way of doing so. The idea originally came from Sarah Hagan's (@mathequalslove) post '2014-2015 Classroom Pics - My Most Colorful Room Yet!'. Sarah may well be one of the most enthusiastic maths teachers in the world! Her blog is very popular, and rightly so. Check out this lectern in her room - yes, a lectern! Is this standard in American classrooms? I must get one!

I saw lots of nice display ideas in Sarah's post - like the calculator posters below - and it made me long for my own classroom. Whilst in the baby-making stage of my life I'm having a brief spell as a part-timer. It's a mixed blessing. One disadvantage is that I don't have my own classroom so I teach in rooms all over the school. Between lessons I have to squeeze down crowded corridors with my huge bag of books and equipment (this was a nightmare when I was heavily pregnant!). There are lots of other disadvantages to being part-time such as missing important information from unminuted meetings and a total lack of career progression opportunities. Of course this is all outweighed by the massive advantage of getting to spend time with my children, so I'm not complaining... (much).

Another display idea I've seen this week is this classroom rules poster, shared by @TeachThought in this post. I think pupils would appreciate these very clear and reasonable messages.

On the subject of behaviour management, I enjoyed David Didau's (@learningspy) post on school routines.

If you want to make your own classroom posters then recitethis.com is great for creating pretty quotes and notices - thanks to @mathminds for sharing the link.

And if you're looking to furnish your classroom, Stephen Harris (@Stephen_H) shared a couple of awesome write-on table designs: these ones are great for collaboration and tangram tables are perfect for a maths classroom.

And finally on display ideas, Mr Allan (@mrallanmaths) shared his RAG123 display, Chris Smith (@aap03102) shared a picture of his 'Mathematician of the month' board, and Mr Taylor (@taylorda01) adapted Spiked Math's Mathagasm comic so it's suitable for display. I'm going to steal all three ideas - if I ever get my own classroom.

4. Teaching ideas from #mathschat
Weekly maths chats take place on Twitter every Wednesday at 8pm, organised by @BetterMaths. This week we discussed our favourite topics, tasks, resources and investigations, so the good ideas were coming thick and fast! I've got a long list to sort through.

There were lots of ideas about getting out of the classroom. Jon Treby (@JonTrebyAAN) suggested teaching loci outside using buildings and string. He also puts tape on the floor of the school hall when he teaches angles in parallel lines. Dawn (@mrsdenyer) has a school treasure hunt for 3D trigonometry and Martin Noon (@letsgetmathing) suggests using a clinometer to measure the height of school buildings and trees.

There was also a lot of talk about popcorn! Dawn (@mrsdenyer) gets her students to design a popcorn cone - an activity that involves sectors and arcs, Pythagoras and volume. If students can work out the volume of their cone, they can fill it with popcorn! Miss Ren (@ReynoldsBSGD) shared Dan Meyer's popcorn picker task - 'all you need is A4 paper. I just stand back and watch the kids get stuck in!'.

5. Mistakes and feedback
I've written a lot about learning from mistakes lately so this tweet was very relevant:

@Maths4ukplc @ccampbel14 Pupils copy/stick their mistake in the box and then correct it. pic.twitter.com/C8TSaVApDo
— Helen Hindle (@HelenHindle1) July 30, 2014

And here's a great idea for marking from @ChrisHildrew.

So there you go - a small selection of ideas from a week on Twitter. I hope that's helpful. If you missed the previous two Maths Gems posts, you can find them here. I'll leave you with a nice question for your students from Big Ideas Math.

The Hidden Treasures of Shell Centre

Discovering Shell Centre is like finding buried treasure. It's packed full of amazing resources but you have to spend time rifling through them. Mark McCourt says, “Shell Centre should be a staple of every maths teacher's repertoire” and, well, he's the kind of person one pays attention to. So here's a little guided tour of the highlights, to help you become better acquainted with the wonders of Shell Centre.

Go to mathshell.org and you're met with these nine options (note the amusing url by the way... maths hell? Thanks to Tilly Warden for pointing that out! Very memorable). 

If you're specifically looking for teaching resources, this is a bit overwhelming. Let me point you in the right direction...

Mathematics Assessment Project

I've been using these resources for a while and have already featured many of them in my resource recommendations. The website is a little difficult for UK teachers to navigate because it's organised according to American school grades and Common Core Standards. The two main areas of interest to us are lessons and tasks.

Lessons: There's loads of great stuff here - browse the topics in the left hand menu. Grade 6 is approximately equal to UK Year 7 so includes topics such as Mean, Median, Mode and Range. 'High School' covers GCSE and AS level materials. When you've selected a topic, scroll down to the heading 'resources'. The downloadable PDFs contain both lesson plans and printable activities. For example, the activities in 'Identifying Similar Triangles' are perfect for a Year 11 lesson on similarity.

Steps to Solving Equations - Mathematics Assessment Project

Tasks: These are also hard to navigate because instead of being organised by topic they are split into Novice, Apprentice and Expert. Novice tasks are short items focused on specific concepts or skills. Apprentice tasks are more substantial, but structured so as to ensure that all students have access to the problem. Expert Tasks are rich tasks often set in a ‘real-world’ context. There are tasks suitable for Key Stages 3, 4 and 5. The Standard Form task 'Giantburgers' is a nice example.

Giantburgers - Mathematics Assessment Project

I haven't looked at all of the lessons and tasks offered by the Mathematics Assessment Project yet but at some point I'll go through them and file them in my resource libraries.

Bowland Maths
Bowland Maths provides projects and tasks for Key Stage 3 pupils. At my school we often use the tasks as end of term activities. For example at Christmas my Year 7s do Speedy Santa in which pupils have to work out how long Santa can spend at each house delivering presents. My pupils also enjoy the task 110 years on.

The projects, such as 'Football: the beautiful game', take 2 - 5 lessons and have lovely supporting resources like video introductions. Time for projects like this should be incorporated into Key Stage 3 schemes of work, otherwise we'll never feel like we can spare the time for them.

Shell Centre for Mathematical Educational Publications

There are four downloadable publications here:

• The Language of Functions and Graphs ('Red Box') - this is now regarded as a classic book on the teaching and assessment of graph interpretation skills.
• Problems with Patterns and Numbers ('Blue Box') - this focuses on non-routine problem solving in mathematics
• The Numeracy Through Problem Solving series - modules tackling 'real-life' problems, such as 'Plan a trip', which take between 10 and 20 hours to complete.
• Extended Tasks for GCSE Mathematics - extended mathematical tasks and investigations.

The print quality isn't great because these 30 year old books have been scanned in. If you have time to go through the PDFs then they are full of great material, though it would be nicer to flick through the books themselves (sadly they are no longer available to order, but I suspect many schools have copies). Technology changes but maths is timeless.

Hurdles Race - Red Box

The excellent Standards Unit materials were also partly developed at the Shell Centre and are now available to download from Mr Barton's website. If you use these, did you know that a helpful teacher has created a load of accompanying PowerPoints? See Mr Barton's blog post about this.

Primas

This is all about inquiry approaches to teaching and learning at Key Stage 3. Inquiry based learning aims to promote curiosity, engagement and in-depth learning. This materials database contains some lovely resources but again, it's not organised by topic so is hard to navigate. Here's an excellent activity in which pupils make their own algebra pyramids.

An algebra pyramid - Primas

So there you go, a whistle-stop tour of Shell Centre. And what's even more exciting is that this page lists the projects that are currently underway to bring us more resources in the future.

Introducing Algebra

Building a solid conceptual understanding of algebra is absolutely fundamental. As with fractions, it's essential that we get it right because it underpins so much of what follows. But, as with fractions, pupils can really struggle here.

Sometimes good teaching resources inspire us when we're looking for ways to structure and deliver our teaching. This post contains some recommendations that may give you food for thought when you’re planning your lessons.

Misconceptions
Before teaching any topic for the first time it's really important to think about potential misconceptions. The best thing a student teacher or NQT can do is ask an experienced teacher. This algebra misconception summary is also helpful. 

The first time I taught algebra, I used this true/false activity to reveal any outstanding misconceptions in the last lesson of the sequence. As a newbie teacher, I was astounded by the mistakes pupils made, like putting both of these cards in the 'true' column:

This activity provided a good opportunity to have individual discussions with pupils and help them make sense of it all. So it’s useful both as a learning experience and for formative assessment.

I'm not stalking him, promise
I've featured resources from Don Steward in almost every blog post I've written. I make no apologies for this. His website is amazing. Here's a few highlights from his algebra collection:

• Lengthy expressions is a nice way to get pupils used to forming expressions containing letters and numbers.
• Expression pyramids are an engaging way to practise simplifying and this activity is a perfect extension.
• In Steps, pupils have to determine what step has been taken to get from one expression to the next (see below).
• Six expressions is a rich activity for pupils who've been taught how to expand single brackets.
• Three sets of excellent activities help students explore concepts relating to simplifying and equivalence: 'equivalence', 'same and transformed' and 'equivalent things'.

Steps - Median Don Steward

There's lots more on Median Don Steward - look under the labels algebra, simplifying, expressions and substitution.

Fawn Nguyen’s Noah’s Ark
This is a lovely activity which gets pupils used to the idea of representing values with letters (or pictures in this case).

Noah's Ark by Fawn Nguyen

Visual representations
'I'm thinking of a number' questions are a good way to get started with solving equations. MathsPad had a nice card sort that makes the link between words and algebra. MathsPad also features a worksheet that uses visual representations to build an understanding of how to solve equations. The same approach is taken to rearranging formulae - this lesson plan features their range of associated resources.

MathsPad

More ideas

• These slides from newmrsc (@_z_0_e) on TES are really good for a first look at algebra.
• For substitution, I like this code breaker from Teachit Maths (and this second code breaker, which involves harder expressions). Substitution scenarios is nice too.
• Great Maths Teaching Ideas has, well, some great maths teaching ideas... There's lots of algebra stuff on the website, like this ask ask trade activity for expanding single brackets. 
• Lindsay Porter (@L1nd54y) suggests using manipulatives when teaching expanding brackets and forming equations. For example, put two chocolate coins and 20 pence in a bag. If we have five of these bags, what do we have? Anything involving chocolate gets my vote. 

I really recommend buying the ebook How to Start on Teach First: Maths by Kris Boulton, Bruno Reddy and Bodil Isaksen. The associated 'How to teach...' documents (accessed via Dropbox) are incredibly helpful. For example 'How to Teach Collecting Like Terms' has very clear explanations of misconceptions and teaching strategies for introducing algebra.

There's loads more resource recommendations, covering every aspect of Key Stage 3 and GCSE algebra, in my algebra resource library. Do comment below or tweet me if you have any more resources or ideas to add.

Image: crotchetycomics.blogspot.com

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