Two Things I Learned in My Teacher Training that I Still Use Today

Fourteen years ago I was starting my teacher training course. I had already been teaching a little (two undergraduate courses at university) and been to a mathematics education conference and numerous seminars. But I felt as though I knew nothing about teaching as a discipline and a career. After leaving my teacher training there were still so many things I did not know!

It’s impossible for teacher training courses to completely prepare teachers for what lies ahead. Mostly I hear other teachers say that they learned nothing of use in their training. But there are two things I still remember and draw from daily or weekly, fourteen years later.

1. Students need to experience an idea in order to learn it.

I was greatly impacted by reading Experience and Education by John Dewey. (It was a set text in my Philosophy of Education class.) In it, I learned that students need to play around with ideas in their pursuit of understanding. They need to create, investigate, take apart, put together, describe, and explain. Students shouldn’t be told what to think but come to think it for themselves. Additionally, students need to come at a concept from many angles before they gain a deep understanding of it. (Some of the other ideas in Experience and Education don’t sit so well with me anymore. That’s another blog post.)

For several years after first becoming a teacher I was struggling to keep pace with my workload and found it really hard to help students experience ideas. But I decided to make sure I thought about this in detail during my planning time once a week, or twice a week, or more often as I became able. Now it is the founding statement of my education platform: students must experience mathematics in order to learn it. I ask myself, how could a student come across this idea? How could them come to investigate it?

Recently I’ve added a new mantra to this one: the first time students see an idea they don’t have to produce exam-ready solutions. Getting to grips with an idea is important first, then coming to a deeper understanding, then being able to apply that learning. Exam-ready answers to exam-style questions on that topic can be introduced along the way, but aren’t needed at first.

2. The context for a mathematics problem can be “real world” or just plain “intriguing”.

My mentor and biggest influence during teacher training was a professor who studied and celebrated problem solving in the classroom. He didn’t care if a problem was based on some real life context or just really interesting.

In fact, working with him led me to prefer a puzzling situation over an over-simplified, meaningless “context”.

Thanks, John Grant McLoughlin, for embedding puzzling problems into my teaching. (He is still at the University of New Brunswick. See him in action in this video.) John is a huge fan of problem solving and non-routine questions and caused me to not really worry about real world connections; my students are intrigued by the problems and they are engaged in solving them, so I’m happy they are thinking about the concepts that I want them to learn.

Recently, I’ve been revisiting this idea thanks to reading about Peter Liljedahl‘s work on Building Thinking Classrooms. One of his recommended practices is to start each lesson with a quality problem.

I reinvigorated my love for problems this year; for the last term I set myself a goal of using a problem in every year 8 lesson that I taught. Here’s one of the problems I used. (It’s from a collection of problems devised for AQA GCSE [the link is a pdf].)

My students loved this problem and we spent most of a lesson talking about it. I used my visualiser to show several of the students’ solutions on the interactive whiteboard. They had lots of ideas about how to show the required idea – and this was one of the first times they used algebra to prove something. It was thrilling!

Do you still use ideas that you first encountered in your teacher training?

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 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.)

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.

Completing the Square: Starter Question

Here’s an idea that worked for me last week. My A-level class learned how to complete the square earlier in the year and I wanted to know if they still remembered how to do it. Instead of asking them to just do a couple of completing the square questions, I put this up on the board.

complete the square

This appeals to me as a starter question because it’s more broadly accessible than a couple of examples and also allows for more in-depth extension work.

1. For the student who sees this and thinks, “Oh no! I can’t remember how to complete the square,” there’s a quadratic expression there for them to have a go at. While I’m circulating I can give tips to these students.

2. For the student who thinks, “Oh yes! I know how to complete the square,” there is a quadratic expression there for them to try and also permission to make up any quadratic expression at all to try. While I’m circulating, I’ll encourage them to proceed to generalisation.

3. For the student who thinks, “Oh yes! It is possible no matter what the quadratic expression is,” there is the permission there to say why and give evidence. While I’m circulating I can ask questions and push them to generalise with justification or evidence.

a mathematics lesson that worked


When I was writing this question I was trying to create something that would push forward my students’ thinking no matter what they could remember about completing the square. I think this worked for me and my students and I hope I can think of more good questions like this one!

What maths question worked for you this week?


Peer Assessment: The “Production Line”

My grade 9 students did an individual investigation last week and I wanted to involve them in the process of marking it. I heard about a peer assessment structure called the production line from a colleague last year. In brief, the students mark each other’s work in groups, and each group concentrates on one aspect of feedback. The assignments travel around the group, gaining lots of detailed feedback.

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The investigation task already had an assessment rubric. It is based on the MYP year 4 criteria. The rubric focused on Investigating Patterns (MYP criterion B) and Communication in Mathematics (MYP criterion C). I divided the task specific criteria into themes – five in total:

  • B1: Finding and stating patterns
  • B2: Problem-solving techniques
  • B3: General rules
  • C1: Language and representation
  • C2: Reasoning

The Lesson

I rearranged the tables in my room and assigned groups of three students to sit together. I gave each group the details of one of the criteria. An example is shown below (and you can get them all in the investigation file).

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I had some instructions on the board (shown below) and we talked about what we were going to do. I had students help me staple an extra sheet of paper to the back of each task so that we could use it give feedback.

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I gave each group three tasks and asked them to read them, as a group, one at a time. Then they were to discuss the criterion and give some feedback. When they were finished with a task, I passed it on to another group.

Reflection on the Lesson

Since the rubric was already set up for this task, the preparation was quite easy for me. Making a task-specific rubric is a job that can take some time! Thankfully it only needs to be done once.

The number of criteria I wanted to mark didn’t match the number of groups I made in the class. I opted to have three criteria marked by two groups each and two criteria marked by one group each. This turned out to not work very well. The repeated criteria groups finished all the class tasks much more quickly (obviously!) and the other groups had much more to do. When I realised this, I asked two groups to take on a different criterion so take up slack from the inundated groups. Next time, I need to think more carefully about this. I didn’t want to make bigger groups because I felt like they would not work together well. In that case, I should have more criteria so that each criterion is only assessed by a single group.

I just managed (in 80 minutes) to get all the tasks marked by all the criteria groups. Next time I will not need to explain the production line in as much detail and I expect it will be more comfortable in terms of time.

The feedback that was given was extensive, though, I think I need to talk with the students about what the criteria is looking for. I can see the value of a class discussion about what constitutes good reasoning or communication or pattern stating.

The students gained a much more clear view of what criteria they are marked against. (The same criteria are in use throughout MYP maths.)

How do you use peer assessment in your classroom?

Algebraic Fractions Quiz-Quiz-Trade Activity

One of my classes of students sat an exam lately and I realized they need more practice with adding and subtracting algebraic fractions. There were a collection of misconceptions on the exam, including not taking a common denominator, trying to cross multiply, and cancelling incorrectly. I made a set of cards (split over two files) to use to help practice this tricky manipulation.

Screen Shot 2013-12-13 at 11.28.03 AM

In this quiz-quiz-trade activity, students start with one question each and make sure they can simplify it. I give them a few minutes to check with their partner or with me. My students like to use their mini whiteboards because it lets them change any errors easily.

Screen Shot 2013-12-13 at 11.28.45 AM

Then they get out of their seats and meet someone from another table. They take their mini whiteboards (or their notebooks) with them. Meeting someone else, they quiz each other. After they are satisfied that they both got their questions correct, they trade cards, thus leaving with a different card then the one with which they arrived.

While the students are quizzing each other, I am able to circulate and address misconceptions. The students are quite good at helping each other. After this activity I think they will feel a lot more confident with adding and subtracting algebraic fractions.

When students sit back down, I ask them to do a few of the algebraic calculations in their notebooks so they have a record of what they learned.

I made these cards using Tarsia software, designed for making mathematics activities. Here are two files (pdf) of the cards.


“Naturally Occurring” Algebra

My grade nine extended class (similar to GCSE top set) have an algebra unit at the beginning of the year. I am assigning all the skills work for homework: simplifying, expanding brackets, the mechanics of solving equations, and so on. In class I am trying to give experiences that answer the unit question, “What did algebra ever do for anyone?” I’m trying to show how algebra’s power to solve problems is naturally arising from good problems.

Background: We have spent two lessons talking about types of sequences and formulas (such as 4k +12) that can lead to sequences.

Today I asked them to watch as I demonstrated something on the board in silence. I did the same thing you see in this video. In one minute, I demonstrated in silence how to find the sum 1 + 2 + 3 + … + 9 + 10.

Then I asked them to describe to their partner what they saw. That took another one minute.

Next I asked them a series of questions. First, could they adapt my method to find the sum of the first 100 positive integers?

Then, I put this list up on the board. What I like about this list (also from nrich) is that there is no time wasted on easy repetition. Each item is a little harder and provides a lot to talk about.

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After about 20 minutes, I asked two students to come up to the board and put their solutions to the first two problems. As the lesson ended, some students were starting to attack the last part. What a beautiful and naturally occurring use of algebra!

Do you have any classroom problems that show how powerful algebra can be?

Intersections of Curves Treasure Hunt

My year 13 class were ready to practice finding points of intersections of two curves using their calculators. Instead of a boring textbook exercise, I made this treasure hunt for them.


There are ten orange cards stuck to the walls, windows, chairs, and surfaces of my classroom. Each one has a question. Students travel around the room in pairs, solving the questions. Then they look for the answer on another card.


The students know they have finished when they have visited all ten cards and solved ten problems. They need to tell me the card numbers of the loop they followed, which allows me to check their work.

Treasure hunts are easy to make and are a clever way of turning a worksheet or textbook exercise into a more social exercise that students enjoy. The cards are available here as a document and as a pdf.

Equations of Lines, and Other Coordinate Geometry

This is just a simple worksheet with twelve questions about finding the equations of straight lines and about gradients and the distance between two points. It’s laid out as twelve questions and I asked my students to work through them in any order.

A few of my students really like to cut out the questions individually and paste them in, writing their answers below. It helps them keep their work neat. I like anything that makes a worksheet more interesting for them! Even just giving them the choice of what order to do the questions in seems to make them feel more resilient.

The worksheet is available here in pdf format.

True or False Sorting Cards for Arithmetic Sequences and Series


I have noticed that my students get a bit confused sometimes with arithmetic sequences. They think that the statement above is true: the sixth term in a sequence can be found by doubling the third term. So I made up a set of true/false sorting cards that highlighted some common misconceptions. There are seven statements like the one above that I ask students to sort and then we discuss them as a class.


I made these cards with Tarsia; a pdf is available here.