Test Champions

This idea was recently shared with me by Pietro Tozzi. He is a maths teacher at Gumley House Convent School in London but also works for Pearson (the Edexcel exam board) two days a week. We brought him out to our school in Singapore to do some training for the team about the new A-level in maths. (By the way, if you are looking for an A-level or (I)GCSE trainer, I would highly recommend him.)

During our time together, he shared this idea for returning a test. As it turns out I had a KS3 test to return and tried it out. As I marked I kept track of the best answers for each question. When I compiled this table, I made sure that every student was listed and students with lower scores were listed more frequently, if possible.

The class score represents the fact that among all the students, they have expertise to get full marks; someone scored perfectly on every question.

After passing back the tests, I asked students to get up and meet others who could help them correct their errors. They spent about half an hour up and about, talking over their questions, and making notations (using a coloured pen) on their tests. While they were doing this, I was able to work around the room and talk to those who had difficulties that their peers couldn’t solve.

They enjoyed this way of reviewing and correcting their tests. Having them out of their seats was good for their concentration (for the most part!) and also allowed me to be less conspicuous in my help of a few key students.

I completed this lesson with a fill-in sheet for students to list strengths and targets based on the test. We spent the following lesson with students doing individual work on their targets before we moved on to our next topics of study.

The Best Compliment I’ve Received this Year

I was in my year 13 A-level lesson. Students were working away on some integration questions that use partial fractions. One student called me over, asked for a hint, and after my explanation he nodded. “Lit,” he said. And went back to work.

That’s the best compliment I’ve had in ages. And, gosh, lingo changes so frequently, doesn’t it?

Association of Teachers of Mathematics – Singapore Branch meeting

Last year we set up a branch of the Association of Teachers of Mathematics (ATM) in Singapore. The ATM is a UK professional organisation and I used to go to their meetings in the UK and in Hong Kong.

Last year, one of my colleagues in the Infant school decided to set up an ATM branch in Singapore. (Setting up a branch is free and astoundingly easy.) So far the ATM branch has met four times with about 50 teachers each time.


We had a meeting this week and it was brilliant to meet both primary and secondary colleagues from around Singapore to talk about maths teaching ideas. And eat cakes and fruit drink wine!


The theme for our meeting was shape and geometry teaching ideas. One good idea that was shared in the Secondary teachers chat was a gift-wrapped sphere with a note that said, “This Secret Santa present is yours if you can estimate its volume to within 50 cm2.” I made an estimate (without a calculator) but apparently I was 52 cm2 off the correct answer! Doh. I never did find out what was in that package.

I shared an idea for teaching about the surface area of a sphere using an orange that you unpeel. It’s described on this wonderful blog by William Emeny. (His blog’s name is eerily similar to my own!)

If you work in Singapore, please come along to our next meeting! We meet once a term (three times a year) and email invites are sent out.

If you don’t live in Singapore, consider joining a professional group near you. I highly recommend it for meeting new friends, hearing about other schools, and sharing ideas.

Animal Algebra: General Terms of Sequences

Use linking cubes and make these three animals.

algebra animals

Ask students to discuss:

  • How do they see these animals growing?
  • What does the next animal look like? How many cubes will be needed for its {head/body/front leg/hind leg}?

Using these answers, bring students to see the general term for the number of cubes used. I’ve used this picture to help:


Now do a similar exercise using growing L-shapes, growing T-shapes, growing Z-shapes, growing animals of their creation.

I have used this lesson countless times and it continues to be a favourite. I like that general terms are introduced without the method of making a table of values. I want to avoid reducing geometric sequences to a meaningless sequence of numbers.

Excellent similar ideas are found at visualpatterns.org and this nrich problem about cable bundles. (The nrich problem has sample student work, making it good for teacher workshops, as well.)

Does anyone know where this idea is from? It’s not original to me and the second image in this post is a screenshot from a long lost book. I have also seen that Colin Foster had an idea like this in his (amazingly free) book, Instant Maths Ideas for Key Stage 3 Teachers: Number and Algebra.

a mathematics lesson that worked


Thanks to Colin, I think the original idea is from Paul Andrews’ book, Linking Cubes and the Learning of Mathematics. It’s available for sale from the ATM and I highly recommend it. (I have just bought a new copy.)

Learning Names at the Beginning of the New School Year

I am a slow but persistent learner when it comes to names. This year I asked my students to use a sticky note and put it on their desk with their name. When I’m walking around I use their names as often as I can. Then at the end of the lesson, I asked them to put their sticky note on the wall near the door.

Then the next lesson they take down their name to use again on their desk.

wall of names

In between lessons, I’ve been using the wall of names as a self-test. I ask myself which class someone is from, or interrogate myself to say something about them. So far, so good.

I teach one pair of siblings and the older sister found her brother’s name and decided to put it really high up the wall where she thought he couldn’t reach it. Ha!

Do you find learning names easy? Do you have some tricks to share?


Which is More Valuable: Getting or Giving Feedback?

Students can get feedback from me, their teacher, when I mark their work or talk to them in class.

Students can give feedback to me when I ask them about their learning.

Which is more valuable for student learning? According to an article I was reading today, getting feedback from students is more powerful than giving students feedback. Cris Tovani argues that when teachers obtain feedback from students they can make changes easily to subsequent lessons, and this leads readily to improvement in the students’ performance. Tovani is a reading teacher, so here are a few ideas from me to help get feedback from students.

Three Minute Feedback

I do love this strategy and it’s the only thing I’ve continued doing since the very first teaching I did at university. (Sometimes I forget to use it, though, for months at a time. Does anyone else have this problem, even with great ideas?!) See here for an example when my IB SL students were preparing for a test; see here for an example when students were learning to expand brackets. Here’s one I prepared for my year 9s for tomorrow; they are revising for a test. Tovani says that after she spots the patterns in her exit tickets, she throws them out – it was freeing to read that.

3 minute feedback revision

Looking for Themes in Book Marking

When I take in a set of books (or tests), I jot notes to myself about commonalities among the students’ work to see on which topics they need help. If it’s just a small group of students who need help with factorising, for example, I might invite them all around one table when the class are working on something.

Quick Quizzes

Short quizzes with only a few questions that can be done at the end of class let me know if a concept from earlier is still secure. For example, I gave a four question trigonometry quiz to my year 10s a few weeks ago (and discovered that I need to refresh their memories about the difference between trigonometry and Pythagoras). I would like to get into the habit of using more mini quizzes.


Giving and Getting Feedback

Each of these examples allow students to give me feedback about their learning but they are also a means of the students getting feedback from me. In the case of the Three Minute Feedback, they have the chance to reflect on their learning and identify what they need to do next. With book marking, I have been challenging myself to only write questions to prompt thinking that will help students improve. In the case of quick quizzes, I provide detailed exemplar solutions afterwards for them to see and analyse. Thus students know where they are now and how to improve.

Which do you find easier in the classroom: giving students feedback or getting it? Tweet me (@mathsfeedback) or comment below.

Fingers, Maths, and Music

To return to Trninic’s idea about body movement in maths, I was reading an article by Jo Boaler which mentioned students who count on their fingers. Boaler quoted a study which said that students see a representation of their fingers in their minds when calculating. Another study looked at finger perception – whether students know what finger is being touched when they can’t see. Amazingly, young students who get training on how to perceive and represent their own fingers go on to do better in maths. In fact, in six-year-old students, finger perception was a better indicator of maths performance than tests of cognitive processing. Boaler posits that this is also why some people are who are good at music are also good at maths.

[Sidebar: I’ve certainly known a lot of people who are great at both maths and music. I have quite a few in my department now. I had a friend who did joint Honours at university in maths and music and wrote a thesis that connected both. Recently I’ve seen some good IB Internal Assessments about harmonics and chord structure. Does this ring true with you also? (Punny!)]

Boaler has developed some activities for young children (pre-kindergarten/nursery) to increase finger perception. I would love to hear from anyone who works with younger children and tries them.

finger perception Boaler

Exploratory Practice

Recently I have been reading and learning about exploratory practice, thanks to a very interesting talk and a few articles.

Movement in Maths and Exploratory Practice

Last week I went to a talk at the Singapore National Institute of Education. Dragan Trninic was talking about how maths can be learned through bodily movements. He has done some work on proportional reasoning in which students raise their two arms to different heights above the desk while looking at a coloured screen. The screen turns green if the students raise their hands in a certain proportion, for example, if the right is double the height above the table as the left. (Read more here [pdf].)

exploratory practice notes_opt

Trninic was explaining that he designed his study to see if the physical experience would help students understand proportions better. (It’s such a hard thing to teach and understand.) He wondered if practicing in this way – he called it exploratory practice – would prove valuable. And it did.

By responding to questioning from the researchers, the students were able to voice their findings. This helped them develop an understanding about proportions.

Trninic linked this kind of exploratory practice to the way people learn dance or martial arts. In those disciplines, students learn through a collection of sequenced movements, making improvements as they go. Trninic was careful to distinguish this from the kind of repetitive practice of a single skill that we sometimes use in maths classrooms; that kind of practice doesn’t include any variation and hence students learn efficiency and speed for the atomic skill. Exploratory practice, on the other hand, is set up by the teacher in a way that students are asked to learn as they go by trying to generalise.

Shanghai Maths and Procedural Variation

This reminded me of some reading I have been doing about procedural variation. In the UK, there has recently been a two-year-long teacher exchange with Shanghai. Sue Pope and Mike Ollerton wrote in Mathematics Teaching 250 (ATM members’ access required) about their experiences with secondary maths teachers from Shanghai in the UK. They recount that they were asked to read an article in advance: “Teaching with Procedural Variation: A Chinese Way of Promoting Deep Understanding of Mathematics” by Lai and Murray (pdf, free).

The article by Lai and Murray quotes international maths comparisons that show that Chinese learners have a very secure understanding of the mathematics they have learned, and that they can apply it. Yet some Western onlookers say that mathematics education in China is characterised by rote learning or passive transmission. Without an explicit focus on conceptual understanding, why can Chinese students understand and apply maths so well? Lai and Murray aim to illuminate the “paradox of the Chinese learner” by describing the teaching as procedural variation.

Carefully Constructed Exercises

When teachers in China use procedural variation, they set up a carefully constructed set of examples or experiences for students, who are then expected to attend to essential features and notice connections. The tasks are chosen help learners to create generalisations, and are sequenced to help this occur.

The article includes an example about the teaching of division involving decimal numbers. This series addresses the misconception that “division makes smaller” and invites students to form a new conclusion about division.

Problem 1: There are 9L of apple juice and every 3L is put in a jar. How many jars are needed?
Problem 2: There are 9L of apple juice and every 1L is put in a jar. How many jars are needed?
Problem 3: There are 9L of apple juice and every 0.3L is put in a jar. How many jars are needed?
Problem 4: There are 9L of apple juice and every 0.1L is put in a jar. How many jars are needed?
Problem 5: There are 9L of apple juice and every 0.05L is put in a jar. How many jars are needed?

In this series of tasks, the total amount of apple juice was kept constant while the amount in a jar was varied from a whole litre to less than a litre. This exercise might be considered rote drilling if computing for a correct answer is the focus. However, an experienced mathematics teacher will organise this series of tasks hierarchically and provide scaffolding to illustrate and generalize… mathematical ideas.

This type of procedural variation involves varying the problem. The variation is created by changing a constraint or feature of the the problem while other parts remain the same. (This reminds me of the what-if-not technique that I first read about in the book Adapting and Extending Secondary Mathematics Activities by Pat Perks and Stephanie Prestage. It’s also something Watson and Mason talk about in an article called “Seeing an Exercise as a Single Mathematical Object: Using Variation to Structure Sense-Making“.) There are two other types of procedural variation: examining multiple methods of solving problems and using a single mathematical method in varied applications. Lai and Murray use examples to describe these also: first, simultaneous equations can be solved using multiple methods, which are then compared and contrasted; secondly, a combinatorial method can be applied to various problems including handshaking, crossroads, and diagonals.

This type of practice is what what I would describe as exploratory practice: the questions are designed to lead students to a greater understanding of an underlying idea or structure.

How Does This Apply in My Classroom?

A good question, and one which I have not fully answered yet. I am still reading about exploratory practice, thanks to the Watson & Mason article linked two paragraphs above.

First, teaching through movement. I have been encouraged by a recent Jo Boaler article to use movement and gestures more. For example, when talking about transformations of shapes, we can use our hands to show reflection from palms up to palms down. (I wonder if this also extends to use of manipulatives?)

conditional probability

Boaler’s article also encouraged me to talk to students about how they see their fingers or bodies moving in their mind’s eye. But Trninic’s research is more about using movement to expose mathematical structure directly. He mentioned that he wants to work on conditional probability next. I am looking forward to hearing about it.

Regarding exploratory practice, creating or sourcing the exercises will be an interesting task. For number-sense exercises, I have used Pamela Weber Harris’ book Building Powerful Numeracy for Middle and High School Students. She presents problem strings which are sets of questions that lead a learner to see patterns and make generalisations about number.

Do you use exploratory practice in the classroom and have some resources to share? Tweet me (@mathsfeedback) or comment below.

Five Superb Maths Lesson Ideas #2

1. Pythagoras and Trigonometry Revision Activity

I love activities that get students out of their seats. This task (designed by Steel1989) asks students to distinguish between Pythagoras and trig questions. Yet instead of a worksheet, the questions are designed to be printed out and stuck around the room on sheets of paper. Students get one to work on, answer it (in their book or on a mini whiteboard) and then write the answer on the back of the sheet. Then they put the sheet back up on the wall. When another student answers the same question, they check their answer with the one already written there. If the answers differ, they students need to talk to each other to discover which is correct.

pythagoras or trig

2. Polygraph Desmos Activity

Oh, wow, I’ve discovered a great one here and maybe you’ve heard the hype already. Desmos has introduced a teacher section that allows you to run class-side activities. I tired out the Polygraph: Lines activity with one of my classes. Have a look at the teacher guidance to learn more. Only you as the teacher needs to create an account; you give students a code to join the game. One student chooses a linear graph and their assigned partner has to ask yes/no questions to guess which graph it is. Meanwhile, as a teacher you can see all the questions and answer being given, who has been successful with the task (or not). I called one of my students over when I saw that she had typed “Does your graph go through the point y = 2x?”. I was able to clear up a misconception I didn’t even know she had until then.

The student’s view is shown in the screenshot below. Desmos is adding to the collection of class activities and I’m sure I’ll use them all in time!

polygraph lines.PNG

3. Tree Diagrams Challenge

A few of my year 11 students are ready to take on the challenge of those nasty tree diagrams questions that lead to quadratics. Fortunately, tonycarter45 has created this lovely sheet with probability extension questions. The sheet includes the answers.

Tony (who works at my school) has produced quite a few nice worksheets and you can see them on TES Resources. He specialises in thought-provoking questions. I like that his investigative worksheets often remove scaffolding parts as the questions progress.

tree diagrams


4. Two is the Magic Number worksheets

Three activities called “Two is the Magic Number” from Just Maths. Each one is a collection of cards solving a short problem, only two of which are done correctly. The rest show common errors and misconceptions. The cards generally cover number and algebra skills such as simplifying terms, using indices, and calculating with fractions. Depending on what you have taught your students, there may be a few topics that they haven’t learned, so check first. (My bottom set year 8 need to practice like terms, but they can’t do a conversion between meters squared and millimeters squared.) These sheets are great for checking students’ misconceptions.


5. IB DP Maths Resource Collection

I have a former colleague, Andrew Clarke, who is a brilliant resource collector. He has now started three curated collections of maths teaching ideas for IB teachers. The one that is most relevant to me is Teaching Diploma Program Mathematics. He has collected all kinds of teaching ideas for Maths HL, SL, and Studies SL. One item that caught my eye is an investigation about using calculus to describe concavity, which is one topic I have never found a good way of introducing.

Andrew’s other two sites may interest you: Teaching MYP Maths and Teaching PYP Mathematics.

What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.

Five Superb Maths Lesson Ideas #1

1. Areas of Flags

Areas of Flags (from Owen134866 on TES Resources). One of my colleagues introduced me to this brilliant series of worksheets (and powerpoints) that use flags as a context for finding areas of rectangles, triangles, parallelograms, and trapeziums. There is also a further activity with circles.

areas of flags

2. BC Numeracy Tasks

I was browsing on the website of Peter Liljedahl from Simon Fraser University, Canada. (I was reading a paper of his about task design.) I discovered that he was on a team to develop tasks to assess students’ numeracy in British Columbia. They look as though they are lovely, well thought out tasks. However, there aren’t any solutions that I can see, likely because these are in use as assessment tasks in BC. I note that some of them are too Canadian, though! “Last week I went out crabbing with a friend. We took my canoe and paddled out to a point just off Belcarra Park and threw in our trap.” I’m not sure my city-dwelling, mostly expat students would know what to make of this. However, there are lots of great tasks here and I reckon I will try some of them out soon.

crab trap.PNG

3. GCSE Five a Day Sheets

These GCSE starter sheets, Five a Day, by Corbett Maths. Each sheet has five questions. They are available for numeracy, Foundation, and Higher, and answers are provided. One sheet for every day of the year. I have asked some of my students to use them at home on weekends, too.


4. “Think of a Number” Lesson for HCF and LCM

I’m planning to use this lovely lesson about highest common factor (HCF) and lowest common multiple (LCM) from the Mathematics Assessment Project. I like that it provides a pre-test (which could be used as homework) to help me plan the lesson. The main tasks are really well explained in the teacher notes and include a whole class discussion with mini whiteboard responses, and a card sorting activity. Then there’s a post-test to see what students have learned. All 100 of the lessons in this series are designed with a pre-test and a post-test; I love that it makes it easy to see how students have improved.

The only downside of this lesson is its American vocabulary. I am going to need to use white-out to correct greatest common factor (GCF) to HCF throughout!

hcf lcm shell map

5. Shakespeare and Numbers

Our Head of English has started talking about upcoming celebrations for the 400th anniversary of Shakespeare’s death (23 April 2016). I have been thinking about what we might do in maths to celebrate. So far I found this Numberphile video about the numbers in Shakespeare’s sonnets. I will continue hunting for some other things to use in lessons but this video (duration 4:36) will be a nice ender for lessons on that day.


What superb lesson resources have you seen or used recently? Comment below or tweet me @mathsfeedback.