5 Maths Gems #7

When I write these weekly gems posts I have to stop myself from going on and on about all the brilliant blog posts I've been reading. Each week I read quite a few blogs about maths teaching and wider issues in education (often via tweets from The Echo Chamber). These blogs prompt me to reflect on the way I do things and make changes to my teaching practice. Two examples this week were posts from @ImSporticus, a PE teacher who shares words of wisdom from experienced colleagues in 'Common practice of successful departments' and 'What I learnt today' - both are well worth a read, particularly for teachers of A level maths.

A lot of the inspiration behind today's maths gems has come from teachers' blogs - I'm incredibly grateful to those who take the time to share their ideas and experiences for the benefit of others.

1. Interactive Areas
Computer-based tools have made it easier to explain some mathematical concepts and generate interesting questions. It's fantastic when technology enhances and enriches mathematics education.

I recently tweeted this colourful illustration which shows that we can cut a circle into sectors and rearrange them to determine the area (taken from @edfromo's Google+ post). @fmaths42 replied with this brilliant applet which illustrates the same thing in very clear interactive steps. Never before have I seen the formula for the area of a circle explained so clearly (and did you see the 'proof without words' video in Maths Gems 4? That was good too).

Another excellent interactive tool is 'Investigating Triangle Area' (via @mathslinks). My students are always happy to accept that a right angled triangle is half a rectangle, but are not so easily convinced for non-right angled triangles. The tool illustrates the fact by allowing students to rearrange triangles into rectangles (click on subdivide and then rotate and translate the top sections).

2. Estimation

Examples from estimation180.com

I've seen a lot of American teachers tweet about Estimation 180. This is a great website created by Andrew Stadel - each day of the school year he presents his students with an estimation challenge. The aim is to help students improve both their number sense and problem solving skills. The website contains a huge range of themed estimation activities including heights, weights, distances and lengths of songs. A template worksheet is included to encourage a methodical approach to estimation. Answers are provided so students can calculate their percentage error.

Extract from Estimation180 worksheet

Another estimation resource from our friends across the pond is Fry's Bank by the legendary Dan Meyer (thanks to @katebook for sharing this). This is one of his three-act tasks. Students are shown a clip from Futurama in which Fry checks his bank balance. His 93c balance has accumulated interest over 1,000 years and students estimate (and then calculate) the account balance.

If you're going to use this activity, it's worth reading Mr Meyer's description of how he intends his three-act tasks to be used.

3. Negatives 
This week there were lots of tweets and posts about negative numbers. The first post I read was 'Why?' from @JemmaPDuck. This lovely post is about the progress her students have made in articulating their reasoning. I noticed that her students had been taught negative numbers in a way that I've not seen before. To answer the question 5 - -4, a student said "subtracting -4 from 5 is like taking 4 colds out from something which will make it warmer so 5 gets bigger by 4″. I like the temperature analogy and by the sound of it, so do the students.

We should avoid simply telling students that two negatives make a positive because it can cause confusion (see 'Damaging short cuts' by @srcav). Here's some more alternatives:

MathedUp.co.uk

• @MathedUp shared his post 'Adding subtracting negative numbers' in which he recommends 'the Happiness Model' from Nrich. This approach involves adding positivity or taking away negativity. The excellent Nrich article 'Making Sense of Positives and Negatives' suggests a number of alternative approaches. 

• @cpip1601 shared a really interesting method which involves using the numbers 1 and -1 to physically show why subtracting a negative works. 

If you're looking for a new way to teach negative numbers then you'd benefit from reading @MrReddyMaths's post 'How we teach addition & subtraction of negative numbers' and James Tanton's take on Negative Number Arithmetic.

Piles and holes by James Tanton

4. Cheat Sheets
Another gem from @JemmaPDuck's blog - this one is about review activities. To quote her directly: 'Occasionally I will finish a topic and give pupils time to create a cheat sheet which would be helpful if sitting an assessment on the topic. Twist is that I then give them a wee test/quiz and give them someone else’s cheat sheet. They then give feedback on how helpful, clear, understandable and useful the cheat sheet was. Good activity for clarifying thoughts on a whole unit as well'.

What a good revision idea - not only do they revise when making their own cheat sheet, they also get to try out someone else's cheat sheet - this will help all students make better revision notes in the future. A great idea. Related to this is the post 'How tests teach and motivate' from @Kris_Boulton's blog. I like the idea of regular low-stakes open-book quizzes to test the application of knowledge.

Going back to @JemmaPDuck's post where her students gave each other feedback - I don't do much peer assessment in my lessons so I wondered how to do it effectively. I stumbled across an unusual website this week - teachinghow2s.com. This website contains 'visual explanations that transform research into easy-to-follow, step-by-step guides'. One of their free examples is 'Two Medals and a Mission' - check out the infographic to see how it works. Many steps in these instructions seem obvious but some of these infographics might be helpful, especially for new teachers, much like Dan Meyer's explanation of how to teach with three-act tasks.

Source: teachinghow2s.com

5. Whiteboards, big and small

Source: teachinghow2s.com

There are mixed opinions on the benefits of mini-whiteboards in maths. They can be useful for assessment but they have their limitations. If half the class hold up the same wrong answer then you’ve identified a common misconception that you can address immediately, so they are a useful tool for the teacher if not the students. Mini-whiteboards work best with topics where little or no workings are required and all students can work out the answers fairly quickly – beware of putting students under unnecessary time pressure.

The main benefit of mini-whiteboards is that the teacher can glance around the room and instantly assess the general level of understanding, as well as identifying which individual students will need help. So is there an alternative to mini-whiteboards in which students have space for workings and can go at their pace whilst still allowing the teacher to easily assess understanding and intervene where necessary? Yes - a full size whiteboard for each student. In this post, 'Every Math Teacher in the World Should Do This...Right Now!' by @nathankraft1 (shared with me by @DianeMaths) you can see students working on whiteboards around the classroom (I wish my classroom was this big!).

Nathan enthusiastically says, 'When students are working on the whiteboards, I can see everything happening at once. It's like I'm looking at the freaking Matrix. With a quick glance, I can see which students got it, which students are making minor mistakes, and which students have no idea what's going on. I can quickly identify errors for students. I can ask a stronger student to help a struggling one. Once a student has the correct answer, I yell, "Great! Erase it! Next problem!"'. His enthusiasm has rubbed off on me. I want to try it! And before you disregard this idea because you don't have a room full of whiteboards, bear in mind that any vertical surface will do - try writing on windows or using Magic Whiteboards or flipcharts. A related post summarises the benefits of 'Vertical Non-Permanent Surfaces':

• Improves visibility.
• Allows for transfer of knowledge around the room.
• Non-Permanent removes fear of writing.
• Formative assessment by teacher at all times.
• Teacher can answer questions by having students look at others work.

A related idea comes from @Jeremy_Denton in his post 'Improving students' collaboration using acrylic sheets'. Jeremy suggests investing in clear acrylic sheets that students can write on. Diagrams, pictures, axes etc can be put underneath. Read his post for more on this, it really is a brilliant idea.

Next week
A lot of my readers will be at La Salle Education's maths conference next week. I’m really looking forward to it. Please come and say hello if you spot me. Bear in mind that I have a three month old baby who rarely sleeps, so if you see me doze off at any point then just give me a nudge.

I still hope to write a maths gems post next week, it will be a day late but hopefully full of ideas shared at the conference.

By the way, if you didn't catch my last post 'Practical tips for a (newly) qualified maths teacher' then do have a look.  It was a popular post in which I shared some ideas for getting organised, so it's not just for NQTs.