Concept-Based Teaching in Mathematics

October 20, 2015

Some members of my team and I were recently meeting with the IB subject manager for Maths HL assessment. She mentioned that one possible development for the next round of curriculum updates (pdf, IB OCC login required) is a greater focus on inquiry learning and concept-based teaching. When we started discussing this I realised I don’t really know what concept-based teaching looks like in maths. Do you?

I am just beginning my learning journey about concept-based teaching in maths. Here are my stops so far.

Jennifer Wathall has written a book about concept-based maths – it will be published in 2016. I knew Jennifer when I worked in Hong Kong and I attended workshops given by her. I think it is likely to be a very practical and useful book.

This brief blog post by Jeff Hadad argues that the idea of concept-based learning is a good one for maths teachers.

The blog post above links to a book called How Children Learn: Mathematics in the Classroom by the US National Research Council (free download).

Here is a short article by an MYP teacher that gives an example of concept-based learning using simultaneous equations.

What can you share about concept-based mathematics?

What Went Well Bookmark for Peer Assessment

September 9, 2015

Peer assessment can be a bit bland if students don’t know what to look for in their friend’s maths work. Today @tesmaths tweeted a resource to help with this: these What Went Well bookmarks (free, sign-in required). I adapted them slightly since I’m more interested in students’ mathematical communication than in their neatness. Here is my version of the What Went Well bookmarks.

www bookmarkebi bookmark

These are in a PowerPoint file and are double sided, five to a sheet.

Reflection Time

August 27, 2015

Last week we had a professional development session with Andy Hind (@andyhind_es4s) about deep learning. One thing that stuck out to me was the value of reflection time in order to deepen learning.

  1. Reflection time for students.
  2. Reflection time for me.

Students need time to reflect on their learning in order to embed it and connect it to their existing knowledge. One strategy Andy used which I will use in lessons was a small picture of a nutshell that popped up about twenty minutes into a session. Andy said, “Tell your partner everything that has happened so far, in a nutshell.”

He said that students should reflect at four points in a one hour lesson. At the beginning (thinking back to the last lesson), after twenty minutes, after forty minutes, and at the end. I scribbled down this time line.

reflection times

Tomorrow I’m incorporating reflection time into my lessons twice. At the beginning of one of my lessons we are going to recap the last session with the instructions on this slide.

recap last lesson

In another lesson I want to ask students to reflect at the end of the lesson, and I’m using the slide below. It’s a feedback structure I have used since even before I was a school teacher. I learned about it from my colleague Richard Hoshino while I was a lecturer at university.

3 min feedback

The “3 Minute Feedback” questions always follow the same pattern. The first question is related to today’s lesson and allows me to see if students have succeeded with the objective of the lesson. The second question relates to my teaching. The third one is always worded exactly as above and gives students a chance to share anything on their minds. I like to respond to these via Edmodo after the lesson by giving the class an idea of the proportion of responses of each type and by answering the questions.

I need reflection time in order to become a better teacher. I used to blog more regularly and this was a good method of reflection for me. But Andy also suggested a private reflection journal and I’ve started one this week. I set an alarm for half an hour before I want to go home. I use 15 minutes for writing reflection and 15 minutes for tidying up my desk. I haven’t managed to do it every day this week, but I’m pleased that I have done it three out of the last six workdays. I’m going to either write in my journal or on this blog during my afternoon reflection times this year.

How do you include student reflection into your lessons?

Stereotypes of Mathematicians

February 12, 2015

I was reading on the NCETM website about films that have mathematicians as main characters. Four films are mentioned:

  1. A Beautiful Mind
  2. Pi
  3. Good Will Hunting
  4. Enigma

One commentator says that these films contribute to stereotypes about maths since they are all about men, men who are unsociable, and that they are uncomfortable in their roles. Films are not helping maths break away from a “nerdy” stereotype.

Then another commentator goes on to lambaste this idea by saying:

the first three films are about mental illness, not mathematics: the characters happen to be mathematicians, their profession is incidental to the drama that arises from their malfunctioning brain chemistry. The negative, frightening “nutter” stereotype they perpetrate is far more reprehensible, and dangerous, than any “nerd” stereotype.”

The article ends by asking what schools are doing to counteract these stereotypes. For our part, we have named ten of our classrooms after mathematicians. Nine are men and one is a woman: Sophie Germain. They are not all dysfunctional; though Georg Cantor did go insane.

2015-02-12 17.33.09

My classroom is named after Paul Erdös but I have yet to capitalise on this with my students. There are so many good stories about Uncle Paul and his love of maths. He was a bit nutty though, so I am not sure he refutes any stereotypes. The truth is, a lot of mathematicians are men and a lot of them are a little odd. (I say this as a proud nerd.) I think this is especially true among academics, and perhaps less so among mathematicians in industry.

Does your school do anything to counter stereotypes of mathematicians?

A Nice Activity for Statistics and Data Representation: Estimating One Minute

January 20, 2015


My year 8 students were learning about working with grouped data. I used an activity that I pull out regularly when I need some data from the class to work with.

Students worked in pairs and one partner timed the other while they were estimating one minute. I asked the estimating student to close their eyes, say start, and then sit quietly until they think one minute has passed. Their partner used a stopwatch (on their iPad this time) to time the duration.

We recorded on the board all the times from the class. In my year 8s we took several tries each since we are a small group and I wanted to have about 30 pieces of data.


Then we went on with our lesson about grouped data. We got to consider all the authentic data questions–for example, how broad should the classes be to record this data? Also our outliers seemed very far from the bulk of the data. Students naturally asked about these. I like it when there is less need for me to point out these things. Because it is their own data they are intensely interested.

a mathematics lesson that worked

We had a bit of a giggle because one of the biggest outliers was my own estimate for one minute. Once I had my eyes closed I vastly overestimated how long one minute was. My teaching & learning assistant said I was having a nap for 114 seconds!

What data collection activities do you like to use?

Logarithm Questions Around the Room

December 15, 2014


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?

Productivity: Never Wait Again

November 22, 2014

Waiting is a great pain of my work life. In my day, I regularly wait: for the bus, for the photocopier to finish, at the printer for my documents, by the microwave for my lunch to heat, for a doctor’s appointment, for someone who is late, for my tea to brew, and many other things.

I have decided that I will wait no longer. No, I can’t hurry up the photocopier or tardy person, but I can use my time more usefully.

how to get stuff done as a teacher

When the waiting is a bit longer (like for an appointment or the bus), pull some reading out of your briefcase. Could you always carry an article or book or your eReader? While at the hospital this week I made it though this article about promoting deep learning (pdf).

When the waiting is a minute or two (like for the microwave or tea to brew), do a little job. Could you refill the tea canister while you are waiting or wipe down the countertops? Is the shared work fridge in need of a little sorting out? While making hot beverages in the staff room I usually wash my hands and tidy the dishes.

When the waiting time is unknown (like waiting for a colleague or queuing), try to do a few discreet stretches. Calf raises are easy; while standing, push up onto your toes for a few seconds, then release down again. Tense and release muscles up and down your legs and arms. Or, if there is anyone nearby, use it as a chance to catch up and build a stronger relationship.

And even if there is nothing to do when waiting, thinking is something you can do! Plan and visualise how the rest of your day or week can be successful. You could think ahead to the next lesson you will teach, or mentally run through the next series of lessons to see if one will lead to the other successfully. I often think ahead to my next meeting in the week and make notes about how I want to prepare.

This also reminds me about research years ago that figetity people burn more calories, which is reported on here. I think doing something while waiting is a mental version of fidgeting. It keeps your mind healthy.

What do you do while waiting?

Completing the Square: Starter Question

November 16, 2014

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?



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