Problem Based Learning in Mathematics – My Reflection on Back-to-Front Maths

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The other day, I was lucky enough to observe Kathy Palmer demonstrate Back-to-Front Maths, a problem-based approach which focuses on identifying and working through misconceptions. Whereas a lot of traditional pedagogical practices are about fluency, Back-to-Front is about developing a deep understanding. Going beyond learning by memory and association to understanding the why behind it all. As Tierney Kennedy explains on the website:

Back-to-Front Maths begins with problem-solving, where students explore brand-new concepts and then use their findings to derive algorithms and formulae.  It works by creating light-bulb moments for students and enabling them to discover for themselves underlying mathematical principles, rather than providing explanation-and-practice pedagogy.

The challenge is often in finding a way to disrupt students’ usual thinking so that they don’t just get the ‘answer’, but the deep process behind it. All this while at the same time making students feel that they are valuable, that their opinion matters and that it is not only OK to be wrong, but an essential part of learning.

I had previously had some experience with Back-to-Front, having used some of the tasks and activities when I ran intervention. However, it was a lot different actually seeing it being demonstrated, rather than simply having it explained in theory. Personally, I had made the error of meticulously following each step outlined in the tasks. What Palmer demonstrated was the importance of having a curious and inquiring mindset above all else. If that means picking out just part of an activity and leaving the rest then that’s fine, because what is more important is depth not breadth. This also allows for more flexibility in regards to adjusting activities based on feedback.

I had a similar experience with thinking strategies. A few years ago we had a staff meeting where we were all told that we were teachers of numeracy and given a list of strategies to support. As a English/Humanities teachers, I felt a little bit lost and although the posters went up into the classroom, I did not really know what to do with them. Something that stood out with Palmer’s demonstration was a reference to the various strategies as she taught. They were not an ‘explicit’ focus, rather they were celebrated any time a student demonstrated it, followed with the comment ‘that’s what great mathematicians do’. We get so caught up how and when to teach interdisciplinary subjects, complaining of a ‘crowded curriculum’, when really we often engage with them each and every day. The challenge, in my view, is actually being confident with the different skills and strategies ourselves so that we can clearly call them out in the classroom. At the heart of this is language and instruction.

Unlike the traditional conception of problem-based learning, which is associated with resolving a big question or problem, Back-to-Front is about providing tasks and problems which provide enough ambiguity for students to find their own way. Although the focus maybe on ‘number’ or ‘measurement’, lessons involve students coming upon their own discoveries. What becomes important then is language and how we use it. Although many of us have the tendency to answer questions with ‘yes or no’ and correct student misconceptions, the challenge is to use language to help students clarify why they think the way they do. Sometimes the best thing to do is to simply start with the initial instructions and recount a student’s explanation of things. Not only does this allow the student in question to think through their own problem, but it also allows other students who may be confused to come on board. In addition to verbalising learning, emphasis is given to non-verbal forms of explanation, such as visualising things, physically jumping them out and using different materials to make things. Having said this, Palmer made the point that you can’t put out the spot fires of misconception all the time. Sometimes you need to let a misconception through to the keeper and come back to them later with a different perspective.


In the end, my take-aways were:

  • Celebrate vocabulary, thinking and strategies in the moment.
  • Stop sometimes and do a quick vox pop to reassess where people are at.
  • Come back to the explanation of the task in a short and sharp manner whenever possible to maintain focus.
  • Sometimes it is best to come back to some spot fires later in a focus group using a different task.
  • Emphasise process over product, that is celebrate having a go, putting in effort, identifying errors and misconception, because “that’s what great mathematicians do”.

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Aaron Davis

I am an Australian educator supporting schools with the integration of technology and pedagogical innovation. I have an interest in how together we can work to make a better world.

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Problem Based Learning in Mathematics – My Reflection on Back-to-Front Maths by Aaron Davis is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

1 thought on “Problem Based Learning in Mathematics – My Reflection on Back-to-Front Maths

  1. I’m glad you had such a positive experience with Kathy. I love working with her too.
    Thanks also for seeing the aim behind Back to Front Maths and adjusting it to suit your kids.
    You might like to check out our new online learning modules (no-cost): backtofrontmaths.academy/lms

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