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?

Things I Learned about Marking, Students Collaborating, Diagnostic Questions, and Students’ Reasoning (Notes from the Maths TeachMeet in Kuala Lumpur)

The maths TeachMeet in I attended today at Alice Smith School in Kuala Lumpur was inspiring. I attended sessions about marking (by Denise Benson from Beacon House School), a collaborative KS3 scheme of work (Phil Welch from Alice Smith School), things learned from Craig Barton (John Cartwright from Garden International School), and ways to deepen students’ reasoning  talk (Simone Dixon from Tanglin Trust School).

I also led a workshop about career progression for teachers (materials here). One of the tips I give is about reflective journaling. I recommend starting a blog, for instance. Someone asked me about my own blog and I realise I should probably practice what I preach and write some posts. A good first goal would be once a month since I’m currently writing a lot less than that.

Smart Marking (by Denise Benson from Beacon House School)

My main takeaways were twofold, one of which is not even about marking. First, exit tickets are the way Denise marks because it provides her with feedback for the planning and teaching of the next lesson and because it’s immediately useful to the student as well to see if they understood the lesson. I was struck that it would make them easier to have a little pre-printed form that they use and then the next lesson they could stick it in their book (if I deem that it would be useful for them to keep them). The scribbles below are a mock-up from my notes; hope you can read them!

Secondly, teachers complain about marking because they don’t have time. But actually teachers give up lots of time for lots of things, for example, reading teaching blogs, writing worksheets, or going to a TeachMeet on a Saturday. The problem with marking is that it’s not always a good use of time. Denise was saying that we should find a way to make it quicker and worth the time it takes.

She left us with a brilliant question.

A collaborative KS3 scheme of work (Phil Welch from Alice Smith School)

We had some good background discussion first about what KS3 looks like in each of our schools (using a Padlet). But I shall skip forward to the thing that struck me: Phil has capitalised on a timetabling quirk in that the whole of year 7 has maths at the same time. Thus they can have one week a term in which the students do a collaborative project in mixed groups. Also, the school has pairs of big classrooms with sliding doors between so they can squish a whole year group in for introduction or closing sessions. They also have a lot of breakout/corridor space for groups to work on projects. Really, this is such a great idea that works thanks in part to the brilliant facilities they have at Alice Smith School. Collaborative projects would still work in my school but would be a bit messier.

Highlights of the things learned from Craig Barton (John Cartwright from Garden International School)

Craig Barton was already one of my maths heroes but this feeling was intensified. John attended some CPD with Barton recently and in this 45 minute workshop he shared some of the highlights. It’s clear that I have to spend more time getting to know and using the Diagnostic Questions and Mr Barton Maths websites. John even said that all the classroom examples he uses on the board are now taken (via screenshot) from Diagnostic Questions. They are so good because the four multiple choice answers each stem from a common misconception.

Ways to deepen students’ reasoning  talk (Simone Dixon from Tanglin Trust School)

I was intrigued by the idea of multiple representations today – something I thought that I had considered before, but maybe I had not! Ha. Why do the triangles I draw on the board always look the same? Why are my fraction drawings always of 2D shapes? She showed this lovely example (poor picture alert) of fractions of a cuboid.

I’ve been lucky enough to go to a similar session by Simone in the past (we work at the same school). She mentioned both times an idea that I think would work for me. Put up questions around the room and ask for ‘silly answers’ to be written on them. For my students, who are older than hers, I might rephrase this to be ‘answers which look plausible but are actually not right’. This connected in my mind with what John said earlier in the day about having his students create diagnostic questions – complete with three wrong answers that each stem from a single misconception. Students have to really think hard about a concept to understand the misconceptions that others might have about it.


Besides these four sessions, I also enjoyed good chats with lots of thoughtful maths teachers. I always feel more energised and encouraged after a day like this one. What has encouraged you lately?

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.

Logarithm Questions Around the Room


Here’s a lesson that worked for me recently. I had six logarithms questions posted around the room. I gave each pair of students some sticky notes and asked them to go around adding to the posters. Each answer had to be different, clearly. (I didn’t even specify that each answer had to be different, actually, the students just assumed that.)

I love that students were out of their seats and talking to each other. They were more energetic about these questions than they would have been about a worksheet. And they were automatically noticing generalisations as the activity went on and more answers got added to the posters. Also the group feel to an activity like this spurs lots of students to try creating an example that is a bit harder than they would suggest normally.


This was a good activity just like this, but I added a little more. Later in the lesson I took pictures of the posters with my iPad. They are set to automatically upload to iCloud, so I accessed them on my classroom computer and could show them on the screen. We talked about a couple of interesting sticky notes and students noted the ones they thought were incorrect. A few of my students like notetaking more than others, so they copied a few examples.

a mathematics lesson that worked

Since posting one of these pictures on twitter, I have been featured by another teaching blog: Resourceaholic. My idea is one of five “gems”; the other four are (also) amazing ideas!

I am glad that the sticky notes idea seems to work for others; a few others have tweeted to say they liked it. Thanks for the feedback, Emma Cox and MathSparkles! Here’s the file I used with the logarithms questions (make a copy to save it to your own Google Drive or download it); the questions are based on a resource by Susan Wall.

What worked for you recently?

Reading and Writing in Maths: a new version of maths homework

This week I will teach a lesson about estimating to my year 9s. The lesson will be a pretty standard one for me: a sequence of tasks and activities, some whole class items but mostly pair work. The lesson will involve some discussion about how and why someone might estimate the answer to a calculation. I’ll be using a few slides, a puzzle, some problems to solve (sourced from UKMT Intermediate contests), and a plenary about a poor guy whose calculator doesn’t display decimal points in answers.

But it’s the homework I want to talk about here. I plan to set a two part homework: read, then write.

1. Read

First students will read an article from earlier this year about two skiers who allegedly tied for first at the Sochi Olympics. Actually, their downhill skiing times were reported as identical due to rounding to two decimal places.

Two skiers tied for first (image: New York Times).
Two skiers tied for first (image: New York Times).

2. Write

Next, students will access a Google form that asks them three questions for which they need to write at least 300 characters (about 3 sentences). Here’s a copy of the form that you are welcome to answer “for fun”. (My students will be using a private version of this.)

I’m excited to see what students write in response to the third question: other examples (outside sport) where rounding of a measurement makes a crucial difference.

I’m interested to see what my students think about being asked to read and write for their maths homework. After reviewing literature and reflecting on my practice last school year, I decided to try a whole range of different homework options this year.

Our school offers the IB diploma for the final two years of secondary school. The maths courses each contain a 20% internal assessment that is a written report. So part of my interest in reading and writing in maths is to prepare students better for writing in maths during the IB diploma.

Have you tried out reading and writing activities in mathematics classes? Please tell me about it in the comments.

Factors Using Multilink Cubes

My year 8s (twelve year olds) have been learning about multiples and now it was time to talk about factors. Some of them have not got all their times tables memorised, which presents some difficulty for our unit of work on factors, multiples, and primes. So my teaching assistant and I doled out the multilink cubes and I asked them to make rectangles. I wanted to know if any number of cubes could be made into a rectangle shape, fully filled with cubes (no gaps).

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After the students played with this for a while, I started making a list on the board of all the sizes of rectangles they had made. Then I asked, which numbers of cubes can be made into more than one shape of rectangle? Above are two rectangles with 8 cubes and below are three rectangles with 12 cubes.

2014-09-02 11.53.36


Next I introduced the idea of a factor by talking about the sizes of the rectangles and after some discussion we listed all the factors of 12 using the rectangles in the picture above.

After that, students made their own lists of all the factors of some numbers of their choosing. My assistant and I went around to their tables, asking if they were noticing anything. Using their results, we made a table on the board of all the factors of the numbers from 1 to 15 and talked about what they noticed.

Top on their list of noticings were: 1 is always a factor of every number and the number itself is always a factor. These were not obvious statements to my students. We discussed why each was true by talking about making a long skinny rectangle of 1 x __ for any number.

Next lesson is going to be about prime numbers, so I hope that their next noticing is that some numbers have a lot more factors than others.

Do you use multilink cubes in your lessons?



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.

Screen Shot 2014-01-23 at 1.36.08 PM

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?

Number Sets Activity

Which numbers are real? Rational? Natural? Whole? Here is an easy activity to help meet the number sets objective.

I made this set of cards with numbers from the above sets. As students enter, I’ll give them a card. Then I will ask the students to walk around the room, see what numbers others have, and organise themselves into groups based on their number. The instruction is purposely vague enough that many possible groupings are possible. When students have grouped themselves, I’ll start a group discussion about the way they are grouped. 

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I anticipate (I haven’t tried this activity yet) that I may have to prompt the students to think again about the groupings and ask questions to help them explore the idea of number sets. And then maybe ask if there are some numbers that are “purer” than others. I think students have a sense that whole numbers are more “number-y” than items like 2.4 or negative four fifths.

I think it would be interesting to put students holding whole number cards in the centre of the room and then build the number sets outwards around them. I think this would elucidate the idea of natural numbers being contained in the set of integers, and rational numbers being contained in the set of real numbers.

I think it’s my responsibility to introduce the names for the number sets once students have developed the concepts behind them. I’ll write the names and symbols for them on the board to conclude our discussion.

How do you teach about number sets?

Standard Form (Scientific Notation) Sorting Cards

Each card in this set of sorting cards shows a conversion between a very big (or very small) number and its equivalent in standard form. However, quite a few of the conversions are incorrect. The students need to sort the cards into which are correct and incorrect. Then they need to correct the standard form conversions.


They could also order the numbers from smallest to biggest or find further uses for standard form to write very big or very small numbers. Students could also make up several questions and make deliberate errors for their classmates to find.

Download the cards or the instructions and answers slides.

Fractions, Decimals, and Percentages Number Line

Sometimes my students struggle with all the interconnected ideas about fractions, decimals, and percentages. I really want them to understand that these interchangeable representations of the same number. And to know that fractions can be placed on a number line, just as they are used to doing with other numbers.

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I asked my students to cut out this set of cards. Then they had to add to each card so that it had three representations of the same value. Finally, they had to draw a number line and place the cards on the line.

Both my students and I learned a lot from this activity. For my part, I learned that a few of these these particular students found drawing the number line troublesome; their first attempts weren’t evenly spaced or long enough. I learned that my class are able to translate among fractions, decimals, and percentages, though for some this is still a stuttering process. My students learned that they are able to move between the three representations for any number, even unfamiliar fractions or decimals.

The cards for this activity and a slide of instructions
are available here.