A place for gimmicks?

A friend recently used the word 'gimmicks' in reference to some of the ideas I share on this website. I got a bit defensive but I think it raises an important point.

I'm well aware that role of teachers is to impart knowledge. I'm paid to fill children's minds with mathematics. I need no equipment, no facilities, no tools - I just need my brain, my voice and an audience. I haven't lost sight of that. But if I have tools at my disposal that help make my explanations clearer, help inspire and engage my students, improve my assessment and feedback, and make my teaching experience more enjoyable, then I will utilise those tools.

What is a gimmick?
The word gimmick originally meant a piece of magician's apparatus. It now refers to 'a trick or device intended to attract attention, publicity, or trade'. In teaching you could see it as anything that is there primarily for the purpose of engagement, enjoyment or novelty value. Examples may include writing on desks, post-it note activities, Plickers, stickers and stamps. So I admit that some of the ideas and resources I feature in my weekly gems posts are a bit gimmicky. I accept that. Some of my posts focus on the process of imparting mathematical knowledge (like my posts on trigonometry, Pythagoras, calculus and fractions) and others are more concerned with techniques to engage students and make the learning process more enjoyable.

Gimmicks gone wrong
In my NQT year, I took on a smart but chatty Year 11 class. We finished the GCSE syllabus by Christmas and they were all on track for A*s. After their mock exams, I had to find a way to keep them engaged and focussed for a whole term. I started by reviewing topics from Key Stage 3 but I felt that I was patronising them. We did some past papers but they didn't focus. By February half-term I was dreading every lesson so I asked my colleagues for advice. I was advised to get my students to teach some revision lessons. So I did - they worked in groups and took it in turns to deliver maths lessons over a three week period. To be frank, these lessons turned out to be absolutely useless. No student made any progress whatsoever. At the end of the process no-one in the class knew any more mathematics than they'd known three weeks earlier. The only person who'd learnt anything was me - I learnt what it felt like to sit in a really badly taught lesson. This was a classic example of a gimmick gone wrong - I'd prioritised entertainment over mathematics.

I was very aware that the whole endeavour had been a huge waste of time. Even worse, it may have actually been damaging - I'd managed to undermine my own authority by delegating my role. I never managed to get them to focus again. Quite a few students from that class got a grade A in their GCSE when they were capable of getting an A*. I still feel responsible for that.

Fast-forward a couple of years and I found myself in the same predicament. This time I'd done some brilliant revision lessons with my new Year 11 class, particularly in reviewing the topics that are often overlooked in GCSE revision. But again I had a lot of lessons to fill, and again I was advised by my colleagues to get the students to teach the lessons. I was told that the reason it hadn't worked before was because I hadn't done it right. I decided to try it again but with a new approach. I shared the criteria against which I'd assess the lessons (for example they had to design their own worksheet) and I gave my students a list of things to consider:

I gave them more time to plan their lessons and more support and guidance in the planning process. Did it work? Well, it was better than the first time. The lessons and resources were a much higher quality. The students enjoyed doing the teaching and everyone was engaged in the lessons. Some progress was made (a negligible amount - and that was mainly when I interrupted lessons to highlight misconceptions). They learnt far far less maths in the three weeks of student-taught lessons than they would have learnt if I'd taught those lessons.

I do know teachers who insist that this approach works well if properly managed. They think these lessons work well because the students are engaged and enjoying themselves and developing 'soft skills'. I say these lessons don't work well because no mathematics is learnt.

These two things are not mutually exclusive - students can both enjoy maths lessons and learn mathematics. The pleasure of mathematics comes in learning something new and applying it to solve a problem.

So what should I have done with these classes to fill that time between the mocks and their final exams? Well I could have given them some rich tasks that consolidate skills and knowledge across a number of topics (Don Steward has some excellent examples). In addition I could have enriched their education with interesting mathematics that is not on the GCSE syllabus. I could have talked about the history of mathematics, etymology, peculiarities... I could have explored number systems, Fermat's Last Theorem, primes, matrices, set theory... Such a wasted opportunity. Next year - no more gimmicks for me.

Don Steward GCSE revision

A place for gimmicks?
If I had to pick a side in the traditional vs progressive teaching debate, I'd pick traditional. That means I see a lot of value in teachers giving clear explanations and students developing fluency through independent practice. I see less value in group work and inquiry approaches. However, one of the things I enjoy about teaching is that I have opportunities to be creative in my approaches. I enjoy experimenting with innovative teaching tools and technology. But in doing so I don't lose sight of my main priority, which is teaching mathematics.

If I want to use dots and stickers for marking, it may or may not be adding value but it's not obstructing learning is it? If I use Plickers for assessment, it doesn't mean I've stopped imparting knowledge does it? These 'gimmicks' may be useful. They're certainly not detrimental. Other gimmicks, like getting students to do my job for me, are detrimental.

I'm trying to find my way to becoming an 'excellent' teacher and I'll try a wide range of tools and teaching approaches until I get there. I'm learning from my mistakes as I go.

So, I've decided - I'll keep writing my regular gems posts, where I share teaching ideas I've seen on Twitter. And yes, some of them will be gimmicky. Some ideas will be more useful than others. But I'll also focus my thinking (and my writing) on the important questions: what mathematical concepts should we be teaching and how can we best explain those concepts? First up: circle theorems. Watch this space.

Too traditional...?

5 Maths Gems #11

I've got some brilliant gems for you this week. As usual Twitter has been on top form.

1. Quick Key
Last week I wrote about Plickers, which is an app you can use in the classoom for formative assessment - you need a device but your students don't. Today I'm suggesting another app that you can use for assessment - Quick Key. With this app you can set multiple choice homeworks or tests and mark them in seconds by scanning the answer sheets ('Quick Tickets') with your phone.

Quick Key is the invention of American teacher Walter Duncan (@4_teachers). It "turns your phone into a scanner and eliminates hand-grading of assessments, even for teachers working in paper-based classrooms without a computer or internet. Analytics and data exports are easy, so you can focus on your students".

My initial thoughts were that although this app could be very useful (less time spent marking = more time spent planning quality lessons), it won't allow me to check students' methods, which limits opportunities for useful feedback. But for straightforward knowledge checks, to inform me how much my students know, it would work very well.

But then I read this post from Kristian Still (@KristianStill) and now I'm more convinced of the effectiveness of multiple choice questions - he points out that they give you objective and precise information and free up time to spend on remedial learning and feedback (ie you quickly identify which students need support, then you have time to examine their methods and address misconceptions).

Do multiple choice questions work in maths? They do indeed. The Scottish Qualifications Authority uses multiple choice questions in their Objective Tests and some of the excellent FMSP resources are multiple choice. My school has loads of multiple choice questions (extract below) on file which I think may have originated from Edexcel. These would work well in conjunction with Quick Key, as would Diagnostic Questions transferred to test format.

Geoffrey Slack (@slackmath) has produced a quadratics quiz (suitable for Year 12) which incorporates a scannable 'Quick Ticket'. I followed his lead and made one on significant figure rounding.  It would be great if there were a large bank of multiple choice maths test papers (perhaps this will be my next project!).

If you try Quick Key, do let me know how it goes.

2. Exercise books
This week I've seen loads of creative ideas for students' exercise books.

First up is Bruce Ferrington's (@BruceFerrington) reflection stickers. Students stick a 'Not yet yeti' or 'I got it' sticker in their book to let you know whether they've understood the work.

Speaking of stickers, Adil Jaffer (@adil_3) shared his fantastic feedback stickers so you don't have to keep writing the same thing all the time when you're marking.

One of the post popular tweets I saw this week was from @MathedUp who shared his fantastic 'request a selfie' idea. I can see this really improving the quality of students' work - many of my students would love the opportunity to impress their parents. If you use iDoceo then it's easy to take a picture on your iPad and email it to a parent.

Did you see these exercise books, tweeted by @dandesignthink? The boxes at the bottom of the pages contain a checklist (accuracy, spelling, grammar), key words, further questions, key points, what went well, and feedback. An 'Exercise book 2.0' costs 50p - full details are available here. There's also a maths version - Maths book 2.0 is equally awesome. I wish I had a budget to spend because I'm desperate to order these!

The more I use Twitter, the more inclined I am to hashtag everything I write (or say!). I have to restrain myself from putting hashtags in text messages to my husband ('what do you want for dinner tonight? #pleasesaytakeaway'). It's nice to see that I'm not alone - Cathy Yenca (@mathycathy) uses them in class (read her post 'Just Playin''). #twitteraddict

3. Find the Factors
Ed Southall (@solvemymaths) discovered a brilliant and engaging activity for practising times tables - here's his post about it. Find the Factors is designed by @IvaSallay who says it's an excellent way for children and adults to review multiplication facts, use logic, and strengthen brain power. The puzzles are based on a multiplication grid where most of the numbers have been removed. The grid can be completed using logic and knowledge of factors - in the example below you could start by completing the two circled cells. The puzzles come in varying levels of difficulty, as explained here. Find the Factors has already been very popular amongst tweeters who've tried it this week.

4. Chalk Pens

Kev Lister ‏(@ListerKev) tweeted about chalk pens. I've written about writing on desks, windows and walls before, but these are new to me. Kev says the chalk wipes off non-porous surfaces easily with a damp cloth. He uses these pens to write targeted questions or provide scaffolding for selected students. I think my boss would be happier with me writing on desks with these rather than whiteboard markers.

5. Alternative Number Line
I'm reluctant to include two ideas from Ed Southall (@solvemymaths) in one post, but he's a legend so I'll allow it... In the same way the maths clock in my kitchen makes a great talking point for guests to my house, his alternative number line should be displayed in A level classrooms - students (and teachers!) will be intrigued. Mr G Walton (@mr_g_walton) has made a fantastic header to go with it.

Stuart Price (@sxpmaths) suggests making an activity out of this - cut the number line up and give it to your Year 13s so they can sort it into order. They may not be familiar with some of the notation but it's a good opportunity for interesting discussions (I found out how to read ternary to understand Ed's choice for number 5. I'd also not seen floor and ceiling notation before).

Stuart also suggests that students design their own number lines. I love this idea - it would work with students of any age, from primary school to university. It's similar to an activity I suggested in my Open Evening post where you hold a 'design a maths clock' competition. The clock pictures below are all taken from sbcrafts.net where you can check out a big range of clock number ideas that could also be used on number lines.

And finally...
Algebraic long division is fun when you know how to do it, but it can be a bit tricky to teach to Year 12. It's nice to know there are alternative methods. I've factorised polynomials by inspection before but not using a grid as suggested by @jamestanton in Tanton's Take on Polynomial Division. Have a look at his method, you might prefer it to long division.

That's it for this week. Remember to let me know if you use any of these ideas at school. As I mentioned in my last post 'The Art of Collaboration', it's always great to hear when these ideas make it into classrooms. Have a great half-term break everyone!

The Art of Collaboration

There's been a lot of talk lately about the power of collaboration. The best thing a teacher can do to improve their practice is learn from other teachers.

The main lesson coming from Shanghai is that their teachers get much more time to collaborate than we do. So, given that we spend the majority of our time (at work and at home) marking and planning lessons, how can we make time to learn from others? 

Twitter is brilliant but I've tried to encourage my colleagues to join Twitter and they're not buying it! So although I'll start by talking about Twitter in this post, I'll also talk about other mechanisms for sharing ideas - conferences, TeachMeets, hubs, department meetings and blogs.

When Twitter works well
I want to start by showing you an example of Twitter working really well.

Over the last couple of months a lot of American teachers have tweeted about Plickers - it was demonstrated at Twitter Math Camp over summer. I recently got round to reading a few blog posts about Plickers. I like the idea so I wrote about it in my last gems post. I tweeted a link to my followers (see tweet below) and so far over 500 people have read that post.

I love it when teachers tell me that they've tried ideas and resources from my blog, so I was pleased to get this tweet the next day:

Great stuff. Twitter doing its job! The next day I got another tweet from Michelle:

Can see you how effective Twitter has been as a sharing mechanism here? The idea came from a summer teaching conference in America, across the Atlantic to me via Twitter, and I've shared it with my followers. A UK teacher tried it, loved it and will probably invite her colleagues to pop into a lesson when she's using it so they can see it in action. Plickers isn't everyone's cup of tea but perhaps one or two of Michelle's colleagues will try it too and - ta dah! We collaborated, we benefitted. The power of Twitter is clear.

Making the most out of Twitter

So you've joined Twitter and don't know what to do next. Find a maths teacher and follow the same people they follow. Aim to follow a hundred or so to start with - this number will grow quickly. I follow a hell of a lot of maths teachers and organisations involved in maths education (around 1,400!).

It's important to have a short 'bio' on your profile page. Say who you are and what you're there for. I've included six examples of useful bios at the bottom of this post - if you don't have a bio at all, base your bio on these. Preferably have a photo or maths picture too. I believe I can learn from everyone so am trying to follow every single UK maths teacher (I'm missing some because they don't have bios).

Start off with retweeting and favouriting tweets that you that find helpful. Reply to tweets if you have a question or point to raise or want to give some feedback (I love it when people tell me that they enjoy my posts). Soon you'll be sharing your own ideas and engaging in helpful conversations. You may become a prolific tweeter and you may write your own blog. But if you don't have time for those things, you'll still benefit if you just pop onto Twitter every now and then to pick up some ideas. The important thing is you have access to inspiration.

If you do a lot of tweets that aren't related to education (eg a running commentary of football matches) then consider having separate professional and personal accounts.

There are plenty of websites that discuss the benefits of Twitter for teachers and provide guides on how to use it so I won't repeat all that here. Just have a go and see what happens. I think you'll love it.

Conferences
There are a great many training courses and conferences for maths teachers every year. But unfortunately many of us work in schools that don't give teachers the money or time off to attend these events. La Salle Education has come to the rescue with maths conferences on Saturdays that are free to attend. These conferences are enjoyable and informative. Some teachers wouldn't dream of giving up a Saturday to attend a conference but they really are missing out. I hope to see you all in Birmingham on 14th March for the next conference.

TeachMeets
For a long time, proactive innovative teachers have been running events to share teaching ideas. Until I joined Twitter, I had no idea these events existed. If your school is not on the mailing list then it's easy to miss out. Here's some examples of the sort of events that people are involved in:

• TeachMeets ('teachers sharing ideas with teachers') are a fantastic idea. TeachMeets are advertised though this website. It's a bit hard to navigate but is a good place to start if you're looking for local events. Sometimes there are specific 'MathsMeets' too. I expect the new Maths Hubs will get involved in these events (see below).
• FMSP. The Further Maths Support Programme runs network events for maths teachers at Key Stage 4 and 5. Here's an example of a recent event: Supporting the transition from GCSE to AS Mathematics. For information on your local events, look at your regional page on the FMSP website.
• Edexcel Hubs. I've only just heard about the 'Mathematics Collaborative Networks' - this appears to be another fantastic opportunity to collaborate with local schools.

Maths Hubs
I'm not entirely clear on what the new Maths Hubs are going to do, but I believe one of their roles will be to offer support to local schools, partly by organising and leading events in which information and ideas are shared. I think this will be particularly helpful in strengthening connections between primary and secondary schools. Some people are sceptical that the hubs will have any impact - I say be patient give them a chance to get organised. It won't work if we don't support them. The hubs have funding and time set aside to take the lead on collaboration, so find out what they're doing and get involved. Some lead schools have already held launch events but many events are coming up in November. At the moment the communication from hubs is a bit patchy - contacting hundreds of schools is probably harder than it sounds - so make a start by following the hubs on Twitter and making contact via the NCETM website.

Department Meetings
Many maths department meetings are dominated by administration and dissemination of school policies so there's little time left to discuss teaching ideas. The most useful conversations are about pedagogy eg 'how do you teach circle theorems?' (see Ed Southall's post about this).

If you have too much administration to do, then the Head of Maths should talk to SLT - something along the lines of, "you know that item on the agenda for the next Inset that's a big waste of everyone's time? Let's replace that with departmental collaboration time" (or perhaps something a bit more tactful!). You could even arrange to visit maths departments at other schools so you can bring back fresh ideas to share (Cav recently wrote a post about this). If your Headteacher wants grades to improve then every department needs time to focus on developing their teaching.

Blogs

There are so many good blogs that I can't even begin to list them all here. If you're on Twitter and you follow maths teachers who write blogs then they will tweet about their new posts. Alternatively you could set up a feed to receive new posts by email. Another good idea is to follow the The Echo Chamber - through this you'll be able to read a good selection of UK education blogs (not just maths blogs) so you can keep up-to-date on the latest issues and opinions.

Twitter chats

Finally, back to Twitter. There are two weekly UK maths Twitter chats. This is where maths teachers discuss specific topics at a specific time.

• #mathscpdchat is 7pm every Tuesday. Click here for details. Follow @mathscpdchat for information about discussion topics and links to summaries of previous discussions.
• #mathschat is 8pm every Wednesday. Click here and follow @BetterMaths for details. 

There are other chats you may be interested in such as NQTChat, UKEdChat and SLTchat. #PedagooFriday is also a great place to share ideas.

Opportunities to Collaborate?

Even with all these opportunities to develop, there are still huge numbers of maths teachers who feel isolated or who simply don't have the time or inclination to engage with other teachers. Hopefully there is at least one idea here that they will take on board. League tables put schools in competition with each other but it shouldn't be like that - everyone in education should work together for the greater good.

Examples of helpful Twitter bios

Lessons from Training

I'm a little embarrassed to admit this... Yesterday I went on a Speed Awareness Course. Teachers are meant to be good role models and upstanding citizens. But I encourage my students to admit when they make mistakes and learn from those mistakes, so I'm leading by example. I was caught (marginally) exceeding the speed limit. Still, all is not lost. By doing the course, I not only have a clean driving licence, I'm also a far better driver. Fifteen years after I passed my driving test, four hours of driver education was well worth it.

It's been a long time since my last training course. Since qualifying as a teacher I've only been on two external courses - one was about teaching Additional Mathematics and the other was about improving results at Key Stage 5. That's a big contrast to my previous career when I went on training courses every couple of months. I never paid much attention to the format and delivery of the courses I went on. But this time I did.

In this post I want to discuss some of my observations about how this training course was run and whether the ideas transfer to educating children.

Observation 1: Voting

My eyes lit up when I saw interactive voting devices on the tables yesterday. In my last Gems post I said that I'd never had the opportunity to use these so couldn't comment on their effectiveness. But yesterday I got the chance to experience them as a learner.

I loved them! Throughout the four hour session we used them on five occasions - they were used for knowledge checks eg 'what is the national speed limit on a dual carriageway?' (I got that wrong), to gather information eg 'where are you least likely to speed?' ('outside schools' was the most popular answer) and to collect feedback eg 'was this course effective?'. As a learner, I found them utterly engaging. I paid attention to the course content because I knew I'd have to vote. I particularly liked it that I could answer anonymously when I wasn't confident of my answers.

Lesson learnt: Interactive voting is so engaging! And, if used well, it can be a valuable part of the learning process. These devices are expensive but the free app Plickers should be just as effective.

Observation 2: Variation
The course was well structured, starting with a painless ice-breaker followed by a talk about expectations, aims and outline, followed by the delivery of the course content. In a four hour course it was important to deliver the content in a variety of ways. We did a bit of everything: listened to the trainers, contributed ideas, watched videos, answered questions, had small group discussions and filled in worksheets. In a mixed group of 24 delegates, a handful of middle-aged men answered most of the questions. I listened with interest. I got a lot out the small group discussions. When I looked around the room, everyone seemed to be engaged for the majority of the course. This may have been because we were told we'd be tested on the Highway Code at the end (we weren't).

Lesson learnt: Don't force everyone to participate in every part of the lesson - just because they're not vocal, it doesn't mean they're not learning. Ensure activities are varied - some teaching, some discussion, some independent practice etc. Your lessons don't need to be all singing all dancing to engage students, but variation helps. Also, don't underestimate the value of regular low-stakes tests (read @Kris_Boulton's post 'How tests teach and motivate').


Observation 3: Resources
There was only one worksheet on this course and it was simple and well-designed. It was a neat size (two sides of A5), colourful and easy to complete. In one section I had to enter numerical answers. There were two columns - one for my answers and one for the correct answers. I actually showed my husband my completed worksheet when I got home (poor guy!).

Lesson learnt: Make worksheets short, sweet and easy to complete. That's not to say the questions have to be easy - but think about the format they are presented in (read @solvemymaths' post 'Make Better Resources (Part 2: Appearance and Challenge')

Observation 4: Comfort
This training course was 12.30pm to 4.30pm with one short break. I left my house at 11.15am so had no lunch. At the break there weren't even any biscuits! Absolutely nothing to eat. I was starving! My hunger caused my concentration to lapse in the second half of the course. I had no energy to sit and listen. I watched the clock, my stomach rumbling. All I could think about was biscuits.

Lesson learnt: Don't try to teach hungry or uncomfortable children. If they've worked or played sport throughout lunch and are looking weak, let them have a snack (have a bunch of bananas on your desk?). If they look pale, ask them what's wrong! Allow toilet breaks in lessons. It's impossible to concentrate if you need to go to the toilet! If a student looks sleepy or fidgety, get them to come up to the board and answer a question. We're not there to mother these children but it helps if they're in the right physical condition to learn. Oh - and if you're a Head of Department - cake in a department meeting is pretty much essential.

Observation 5: Attitude
I was reluctant to attend the course but I enjoyed it in the end. I learnt absolutely loads of new stuff and I came home with a list of things that I will genuinely do differently. At the break I overheard a delegate ask, 'Anyone else bored witless yet?'. I was quite disappointed to hear that. The trainers were doing a really good job of making a potentially dull subject very interesting.

Lesson learnt: No matter how good your content and delivery, some learners are simply resistant to learning. Those attitudes can be changed, but it will take time and effort.

Valuable lessons?
I was pleased to see there was some expectation of mathematical understanding on this course. We were shown percentages, graphs and probabilities and we calculated stopping distances. This was encouraging but, alas, there was a point at which the trainer was asked to work something out and she said, quite unashamedly, 'my maths is a bit rubbish at the moment'. A few people chuckled. These bad 'mathitudes' drive me crazy.

Overall, the course was very well run. Trainers are just like teachers. We all have to engage learners in order to impart our knowledge.

Next time you go on a training course, think about what works well and whether you can take any teaching ideas from the trainers. Being a learner for an afternoon was very helpful - a shift in perspective is a useful development tool.

5 Maths Gems #10

Have you ever wondered where the word mathematics comes from? Me neither. But this is actually quite interesting:

Latin mathematica was a plural noun, which is why mathematics has an -s at the end even though we use it as a singular noun. Latin had taken the word from Greek mathematikos, which in turn was based on mathesis. That word, which was also borrowed into English but is now archaic, meant "mental discipline" or "learning", especially mathematical learning. The Indo-European root is mendh- "to learn." Plato believed no one could be considered educated without learning mathematics. A polymath is a person who has learned many things, not just mathematics (source: Math Forum).

So my fellow polymaths, here's my weekly maths gems. Creative maths teaching ideas, inspired by the great minds of Twitter.

1. Etymology

On Tuesday @letsgetmathing led a great #mathscpdchat about literacy in mathematics. There's a useful summary of the chat here. I've written a post about developing maths vocabulary before but now I have a new idea to share.

There's great value in talking to students about the etymology (origin) of mathematical words as they arise. It helps students make connections and builds on their understanding of mathematics. Our role as teachers is to enrich our students' lives with knowledge, and knowledge of etymology is both interesting and important. This fantastic article (a must read!) says "Etymology provides a safety net of de-mystification. When all the words you hear are new and confusing, or when those around you put old words to strange purposes, a grounding in etymology may help".

Thanks to @CParkinson535 for telling me about etymonline.com. I've taken the following excerpts from this website, just to give you an idea of the kind of thing you could share with your students:

• hexagon (n.) 1560s, from Latin hexagonum, from Greek hexagonon, from hex "six" + gonia "angle" (see knee).
• locus (n.) (plural loci), 1715, "locality," from Latin locus "a place, spot, position," from Old Latin stlocus, literally "where something is placed," Mathematical sense by 1750.
• vector (n.) "quantity having magnitude and direction," 1846; earlier "line joining a fixed point and a variable point," 1704, from Latin vector "one who carries or conveys, carrier" (also "one who rides"), agent noun from past participle stem of vehere "carry, convey" (see vehicle). 
• binomial 1550s (n.); 1560s (adj.), from Late Latin binomius "having two personal names," a hybrid from bi- (see bi-) + nomius, from nomen. Taken up 16c. in the algebraic sense "consisting of two terms."

Thanks also to @the_chalkface for sharing the list of words below, most of which I'd never heard of. Although we should always endeavor to use accurate mathematical vocabulary when we're teaching, some of these words may be a step too far!

2. Plickers

I've never paid much attention to interactive voting systems before because my school doesn't own the necessary devices. The idea is that students simultaneously respond to a multiple choice question or opinion poll using a wireless system. The teacher can gather and analyse their responses instantly.

I can't give my own opinion on their effectiveness as a teaching tool, but benefits are listed here as: improved attentiveness, increased knowledge retention; poll anonymously (unlike a show of hands); track individual responses; display polling results immediately; create an interactive and fun learning environment; confirm audience understanding of key points immediately; gather data for reporting and analysis. So they are similar to mini-whiteboards for formative assessment, but they have some clear advantages. The main disadvantage is the purchase and maintenance cost. 

I've been reading a lot of tweets and posts about Plickers lately and am keen to try it out. Plickers is a free app for your phone or tablet. Print and distribute voting cards and ask students to hold up their card in the correct orientation to indicate their answer. Scan the room with your phone to collect the data. You can load up questions and students' names on the user-friendly Plickers website. 

I've read some great ideas to make this work well in maths. One is to permanently stick the cards to the back of students' books (from this post) and the other is to use a separate PowerPoint for the questions to give you more flexibility (from this post) - this means you can use diagnostic questions from @MathsDQs. 

I'm going to try using Plickers at school soon and am thinking about running a session at the National Mathematics Teacher Conference on Pi Day 2015 demonstrating Plickers (amongst other things). Please let me know if you think this would be helpful.

3. Misconception board

@amyjscudder shared her interactive display for tackling misconceptions in this blog post. It works like this: create a true/false board in your classroom and stick a statement on it that represents a common misconception. Underneath the question are two small whiteboards - one for students to write their name if they think the statement is true, the other for students who think the statement is false. What I like about this idea is that students can easily switch their answer from one board to the other if they learn something that changes their mind. At the end of the week the answer is revealed and discussed. Helpfully, Amy's post provides some example statements to get you started.

4. Feedback

Now that Ofsted have helpfully clarified that schools don't need to micromanage teachers by imposing prescriptive and time-consuming marking policies, let's have a look at some practical tools for feedback that you can use during maths lessons. This brilliant blog post, written last year by @dan_brinton, is full of great ideas. You may have seen some of these before (like fantastic feedback plasters) - here I'm featuring two ideas that are new to me.

First - Dot Round, an idea which came from @Doug_Lemov’s post 'Has Anyone Tried a “Dot Round”?' and also featured in @LearningSpy's post “Marking is an act of love”. The feedback method is described in Dan's slide below. I think this could work really well in maths.

Second - Verbal Feedback Stampers. I've seen stamps used before but not in the way described here. The key point is that when you give your students feedback during lessons - whether it be about their method, their workings, a mistake or a misconception, you stamp their work and they write down what you said. They then correct their work. The idea is from @shaun_allison's post 'Verbal Feedback Given...'. I like this because it means my students will really have to listen to my feedback and acknoweldge it - no more empty nodding - and the stamp will act as a reminder of the conversation.

@shaun_allison's blog is always full of great ideas. In his most recent post 'Success with low ability students' I like the idea of a 'portfolio of excellence' in which students display examples of their best work (this could be work they really struggled with).

5. More Post-its!
The amount I talk about posts-its, you'd think I'm sponsored by 3M! Here's two more post-it based activities I've spotted this week.

The first idea is inspired by something I saw on Pinterest (I'm not sure where it originated so apologies to whoever took the photo shown). In a Pythagoras lesson, or any topic, you could put questions on posters around the classroom. Students answer on post-its and stick their answers under the question. This has similar advantages to the 'Stuck Post-its' I featured last week in that you'll be able to see which questions remain unanswered, but in addition you'll be easily be able to keep an eye on workings to check understanding. 

The second idea is from @eatf105. At the start of the year put up a large 'literacy board' showing the letters A to Z. If a student hears a new keyword during a maths lesson, they can take a post-it and add the keyword and definition to the A to Z board. Because post-its often fall off, this might work better on a pinboard - students could pin up an index card instead. By the end of the year you'll have a student-made wall of keywords and definitions.

What I've been up to
My post on teaching trigonometry has been very popular this week.

I've also started writing Bitesize Gems (should have called them Midget Gems - thanks @taylorda01!). These are designed to be printed onto postcards to provide flashes of inspiration to busy teachers who don't read my blog. I intend to convert all of my previous gems posts to Bitesize Gems and am aiming for a collection of 100.

I've also been exploring my new subscription to MathsPad. I love their cleverly designed resources, like this new rounding worksheet.

I'll leave you with a fantastic joke shared by my good friend (and maths rival) from school @MikeMJHarris who is now developing software for mathematics education. I’ll be sharing this joke with my Year 12s when we cover convergent geometric series.

Teaching Trigonometry

In today's post I focus on ideas and resources for teaching trigonometry. A recent conversation with a student made me aware that I need to change my approach to teaching this topic.

What's up with Hannah?
Hannah is a Year 12 who I privately tutor. She is bright, articulate and hard-working. I was her maths teacher in Years 10 and 11 (she has now moved to another school). Last week I asked Hannah to factorise a quadratic. This is what she did:

x² + 13x + 36

= x² + 4x + 9x + 36

= x(x + 4) + 9(x + 4)

= (x + 9)(x + 4)

I'm sure you'll agree that this is a rather convoluted method. At some point (I suspect from a previous private tutor), Hannah learnt a method for factorising quadratics with a > 1 and has chosen to apply this method to all quadratics. She gets the right answer, so there's nothing wrong with this method (albeit a little slow), but when I saw her do this it raised an important issue. What I realised, to my horror, was that I'd taught Hannah for two years and had no idea that this was how she factorised quadratics. She'd been getting the answers right so I hadn't examined her methods in detail. In a class of 24 GCSE students who I only see for three hours a week, I suppose I focus most of my efforts on helping students when they get answers wrong. But my lack of awareness of Hannah's unusual method highlights gaps in my formative assessment.

What's this got to do with trigonometry? Well in a subsequent conversation with Hannah she asked me about the upcoming changes to GCSEs, which will affect her younger sister. I told her that her sister will need to know how to find exact values of trig ratios such as sin30 and cos45. Hannah didn't understand what I meant by 'trig ratio'. She saw no link between what she saw as two distinct topics - ratio and trigonometry. She had absolutely no understanding of what sin30 is, though she is very competent at solving GCSE trigonometry problems. Again, this set off alarm bells in my head. I need to teach trigonometry differently.

Does it matter?
My primary role is to ensure that my students possess a toolkit of mathematical methods with which to solve problems accurately and efficiently, and an understanding of the mathematical concepts that underlie those methods.

Hannah got a good A* in her GCSE - this suggests I succeeded in equipping her with the required skills and knowledge for that qualification. But from my recent conversations with her, I realise that I had mixed success with the underlying concepts. I'm not going to beat myself up about it, but it gives me a focus - I need to think more about how I teach and assess for conceptual understanding.

Introducing trigonometry with similar triangles
When I introduce trigonometry I usually get my students to measure triangles and look for patterns (like this activity from Teachit Maths). Next time I introduce trigonometry, I'm going to try something a bit different. I'll show students these three triangles and ask what they have in common:

I hope they'll spot that they are similar triangles and that the ratio of the height to the diagonal is 1:2 (this would be a good time to introduce the terminology opposite:hypotenuse).

Then I'll show them the triangle below and ask for the length of the hypotenuse. I want them to realise that because it is similar (ie equiangular) to the three triangles above, we know that the ratio opposite:hypotenuse is 1:2. So the length of the hypotenuse must be 70.

We could say 'the ratio of opposite to hypotenuse in any right-angled with an angle of 30 degrees is 1:2'. This is a bit of a mouthful so instead mathematicians say 'sin30 = ½' (there's an interesting article here about the origins of terms sine, cosine and tangent).

I'll write sin30 = ½ on the board, then repeat this process for triangles with different angles, starting with 50^o:

I'll ask my students to find the length of the opposite side in the third triangle and I hope they'll work it out based on their realisation that when the angle is 50^o, the ratio opposite:hypotenuse is 0.766:1.

I'll continue with more sets of examples and end up with a list on the board that looks a bit like this:
sin10 = 0.174      sin30 = 0.5     sin50 = 0.776     sin 60 = 0.866    sin80 = 0.985   etc

Then I'll bring a 30^o triangle back into play. I'll ask them to work out the height of this triangle

Source: openlibrary.org

Hopefully someone will give the correct answer of 40cm - and then comes the important questioning. How did they know? Did they remember that the ratio is 1:2 when the angle is 30^o? Or did they check the list on the board? Would they be able to do it from memory if the angle is 50^o?  That ratio is much harder to remember. What if the angle was 52^o? We haven't worked that one out yet. Well, what we need is a reference list of all the ratios, like a bigger version of our list on the board. Then we'll be able to work out lengths in any right-angled triangle.

Here I have a choice of where to go next with this lesson: either make a mini-project out of this (where my class make their own book of trigonometric ratios) or just show them a set of trigonometric tables ("here's one I made earlier!").

I think students will gain a better understanding of trigonometric ratios if they use tables, instead of calculators, to solve trigonometric problems for a few lessons (click here to see an example of a student's workings using this method). There's an online version of the tables here. Once students really understand what the ratios are and how to use them, reveal that all the numbers from these tables are saved in the memory of a scientific calculator.

I haven't tried this approach yet but I know other teachers do something similar. For example, the writer of this blog has his class create their own trigonometric tables. He says, 'I have found that by using a trig table my students concentrate on the concepts being studied rather than the calculator'.

The key point is that students need to understand that trigonometric ratios represent the ratios of the sides in right-angled triangles.

I'm interested to hear how others introduce trigonometry so please comment below or tweet me.

Resources resources resources
Now let's look at some good resources for teaching trigonometry. In my resource library I've made some recommendations so if you're planning any lessons on trigonometry, do have a look there. I've also found a few extras for you today. Did I mention that I love resources?!

Dan Walker has produced a brilliant PowerPoint on right-angled triangle trigonometry which starts off by introducing the ratios as I've described above. The whole PowerPoint is excellent quality and well worth a look.

Resources guru Don Steward gives us a range of fantastic activities. To practise calculating sides and angles in right-angled triangles, I like resources like these:

Bearings
My students have a melt-down when I give them a problem that involves trigonometry and bearings. I suspect this is because my school doesn't teach bearings very well at Key Stage 3 (up until recently it wasn't even on our scheme of work), so they never know which angle they're meant to be calculating. In the question below they need to use alternate angles and trigonometry to calculate the bearing of town A from town B:

And in this more challenging question, they're asked to calculate the bearing of C from A. It can be done using right-angled triangles but it's much quicker to use the Sine and Cosine Rules.

The Sine Rule
MathsPad has a good range of trigonometry resources. I don't normally recommend resources that aren't free but I'm a big fan of MathsPad - a subscription costs £3 a month (please don't pay out of your own pocket - ask your boss). I particularly like the Sine Rule Codebreaker - it contains lots of practice questions in a more engaging format than standard worksheets.

TROL
If you haven't seen this resource before, written by Frank Tapson for TROL (teacher resources online), do have a look through it. It contains lots of helpful practice questions, including some on 3D trigonometry.

Spaghetti Graphs
Finally, I'm looking forward to the next time I teach trigonometric graphs after watching Chris Smith's brilliant video in which he makes a spaghetti sine graph. I can't wait to try this.




Well that's it from me - I hope that's given you some ideas for teaching trigonometry. Please do let me know if you have any great ideas to share.