There Is No Best

I now know why everyone loves using a John Mason quote. I was listening to a Mr Barton Maths podcast where Craig (@mrbartonmaths) is interviewing Anne Watson and John Mason. These two are like the equivalent of a Jedi in Mathematics education. This particular episode caught my attention because of a book John co-authored, Thinking Mathematically, which is sitting in my Amazon basket – according to Amie Albrecht (@nomad_penguin), I just HAVE to get the 2nd Edition. If you haven’t listened to the podcast, do yourself a favour and head to Craig’s website to listen to it now. If you’re in a public space, play it nice and loud as John and Anne’s advice is pure gold. Or you can just read on.

I’ve loved every quote I’ve seen people extract from John Mason’s work – I know Dan Meyer isn’t afraid to throw John’s face up on his slides. One particular one that has stuck with me is this:

So, when John interpolates when a question is posed to Anne from Craig (around about an hour into the show) about a practical takeaway for supporting students to form conjectures in the classroom, I was curious about what he had to say…

Anne: What do you think would happen if… is the basic conjecturing sort of question.

Craig: …What have you found to be the most effective way to get the most out of these conjectures, to get the most out of the mathematical thinking we need our students to have?

John: I don’t think there is a best way. There are different ways of stimulating conjecturing. There are different ways at the beginning of the term in setting up an ethos in which it becomes part of the classroom practice that you make a conjecture and that you don’t believe your conjecture. That’s important. So, you learn to challenge other people’s conjectures, so that you can learn to challenge your own conjectures. So, conjecturing is a way of being. There isn’t a sort of a best way to make conjectures.

A little bit later on, Anne then talked about how she used an area of a kite lesson with some pre-service teachers (on a side note, there were many parallels with how she talked about the running of the lesson and the Five Practices). The focus of the lesson was to pay particular attention to the various methods used to determine which kites were bigger than others. This then prompted Craig to ask the question on everyone’s minds:

Craig: Would there be a period of so-called Direct Instruction? Where, for example, you modelled, explained and gave kids practice on how to actually work out the area of a kite and, if so, where would it have come in this sequence? Would it have come before this activity, or would it have come after this activity?… Where does Direct Instruction fit into this, if anywhere?

John: I could use that task at the beginning, or fairly early on, or in the middle, or fairly late on, or even at the end of the topic. I can actually use it anywhere, it depends on my relationship with the students, their relationship with mathematical thinking, or at least my reading, my interpretation of their relationship to thinking mathematically, and it might also depend on the weather. Seriously, if it’s raining and miserable outside, I might act differently than if it’s sunny. There are a lot of factors which could lead me to choose make use of a task at different points of their experience.

John: I don’t have a pre-this is how I do it. It’s what comes to me in the moment with my various sensitivities or insensitivities which makes me choose do it one way or another.

Anne: It depends on what your pedagogic aim is.

What really shifted my thinking here was when I considered how predetermined my teaching is. How do I know if a task is a better as an introduction to Pythagoras Theorem than it is an application lesson before I’ve even met my students? Or even that, just because it worked well for last year’s class, it ought to work well again for this year’s? I then started thinking about my lesson and unit planning as a whole, and have been since perplexed by the question, “how much is my own prejudice determining how my students learn and see mathematics?”

By considering John and Anne’s perspective of “there is no best”, it helps me bring everything I have to planning a sequence of lessons and be prepared to use the things at my disposal to support students to think mathematically and communicate effectively. Earlier on in the episode, John essentially says that he believes that he can support a student to think mathematically with pretty much any task, which further leads me to think more about what I am having the students do with what I give them, rather than thinking too much about what I give them. Instead I need to be spending my energy on considering whether I am creating a culture in my class where I’m encouraging my students to make conjectures they’re willing to test and disprove. Am I giving students time and space to think freely? Being open to simply make things work as best as possible for the conditions at the time is a far less stressful place to be – I can only talk for myself here because I know this isn’t how everyone feels about it.

This notion that, although we may think some tasks are better at the beginning, middle or end of a unit, it depends more on the context of the class than your preference of explaining before, during or after has really big implications on my practice, I feel. Hmm… lots to think about!

Now, time to click on the “checkout” button on Amazon to get this book. Happy now, Amie?


8 thoughts on “There Is No Best

  1. I should really get back to Thinking Mathematically (I got the ebook after it was recommended to me earlier in the
    year, probably by Amie). I found it needed time to read and think with a piece of paper at hand, which I really haven’t had the time for lately.

    My current struggle is a tendency to talk too much and not give opportunity for my students to think. I think this stems from the perceived pressure to cover content at a relentless pace to keep up with a predetermined scheduled followed by other classes. Taking the time and care to encourage that mathematical thinking is what I should try and focus on, just finding that hard to right now.

    You talk about planning, and a change in focus, but that does still require you to have tasks planned, and better yet, variety in the tasks you have available for any particular topic. That way they become tools that as a skilled educator you can pull out when you’re thinking about the specific needs of a given class as you have that focus on developing that conjecture culture and their mathematical thinking.

    My problem is not having had the time or experience to have built that toolbox so as to be able to have that flexibility to find that best way for the students at hand…


    • Thanks for your reflection, Jeremy, I’m always far more interested in hearing what others are thinking, so I’m glad you’ve provided your thoughts here. I think experience does serve as a big advantage, but perhaps more in the confidence that you have in the classroom. I don’t feel like I have “5 years worth of tasks”, but I am probably more aware of where to find the types of tasks that help my students work in the way that I see as purposeful than I was 2 or 3 years ago. In terms of the “talking too much”, I’ve certainly learnt a lot from teaching that students may listen, but not comprehend what you’re saying as well as you think they are. This is for quite a number of reasons, but my advice to the first year teacher version of myself is probably something like, “give the students a reason to listen or want to learn more”. This could be from a task you’ve just done an example for, and know you’ll need to re-explain to some students or something else that helps develop the need to learn or simply ask questions. You might not talk less, but rather feel like the students are taking more in.


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  6. John
    There is a Mason quote that had such an impact on me as an educator and it played a key role in my (sadly) unfinished dissertation.

    Genuine enquiry is an important state for students to recognize and internalize as socially valid. Consequently it is an important state for teachers to enact. But it is difficult to enquire genuinely about the answer to problems or tasks which have well-known answers and have been used every year. However, it is possible to be genuinely interested in how students are thinking, in what they are attending to, in what they are stressing (and consequently ignoring). Thus it is almost always possible to ask genuine questions of students, to engage with them, and to display intelligent directed enquiry. For if students are never in the presence of genuine enquiry, but always in the presence of experts who know all the answers, then students are likely to form the impression that there is an enormous amount to know, and that experts already know it all, when what society wants (or claims to want) is that each individual learn to enquire, weigh up, to analyse, to conjecture, and to draw and justify conclusions.

    John Mason


    • I love that, especially this part: “students are likely to form the impression that there is an enormous amount to know”. I’ve noticed that students are interested and excited to share the novel things they discover and I am equally interested and excited to hear it (even if I have just heard the same discovery from five other interested and excited students).


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