Obedience vs. Risk Taking in Mathematics

[Christine Newell]

Discuss or write about obedience versus risk taking in mathematics. What came to mind when you read this passage? What are you thinking now? What new questions do you have?

[Amie]

Christine: I’m glad you opened a thread on this! I was struck by this section in the book. I had never twigged that one of the ways in which we as teachers stifle risk taking is by promoting obedience.

While reading that section, I had a strong realisation that as a student (from primary to university-level) I was predominantly an obedient mathematician. The shift to being a risk taker really only happened when I became a research mathematician.

On page 53, Tracy says: ‘A cautious, fearful, obedient, or passive mathematician will not be a mathematician for long.’ I want to write more about this in a maths autobiography at some stage, but I think I enjoyed mathematics as a student partly *because* I liked following algorithms and getting the right answer. I wasn’t cautious or fearful (although I was a bit obsessed with having only neat, correct work in my book), I just thought that successfully following procedures was what doing mathematics was all about. I also don’t think that my teachers overtly demanded obedience, but there was an absence of risk-taking habits on show. I don’t recall seeing my teachers model the kind of organic, exploratory thinking that typifies the real mathematical habits that we want our students to develop.

In retrospect, I think my transition to being a mathematical risk taker was a bit rocky in places because I hadn’t been given opportunities to practice having my own ideas and seeing where they might lead. In fact, I probably viewed constructing my own understanding as an inadequacy. For example, I can recall an experience where I couldn’t remember index rules, so I would try a few specific examples and generalise from there. I now see this as a positive way to approach mathematical ideas, but at the time I was embarrassed that I needed to do this because I couldn’t remember the rules.

Again, looking back, I think that starting to work on mathematical research problems was quite a freeing experience — I realised that no-one knows the ‘right’ way to tackle an unresolved mathematical question, so we all need to muddle about  with half-developed, unfinished ideas while we work it out.

Later, I realised that this applied across all my mathematical experiences, and that the only person who really mattered was me. Was it an unresolved mathematical question for me? Discovering that I could apply the same exploratory approach, irrespective of how others might have previously tackled it, meant that my experiences with mathematics became so much richer.

I wonder often about how to set up the conditions so that my own students have these ‘research-type’ experiences with mathematics. The table on pages 51-53, which contrasts specific examples of practices that promote obedience and practices that encourage risk, is exceedingly helpful in identifying which behaviours contribute to which of these two climates.

In some areas of my teaching, I think I’m fairly successful in encouraging risk taking. But I uncomfortably recognise that some of my classrooms have more instances from the left-hand column than I would like (but never humiliating people; that’s just unacceptable). My challenge now is to work out how to move those classes further towards the right-hand column. Thankfully there is a wide range of resources (including this wonderful book) to help spark ideas.

So that is my new question: what are the specific actions (small or large) required to make this change?

[Sarah Caban]

While I was re-reading your response, Aime, I was thinking about how I was never really encouraged – or more importantly supported – to think outside the box in math class – to explore a solution pathway that led to a wrong answer.  I am trying to figure out how risk taking  has to be supported – not blind.  I was a student who didn’t get procedures. I have awful recall ability. If I don’t understand it, I won’t remember it.  I was always getting wrong answers – in a way – I was taking risks – I was trying not to blindly follow the procedures because I hated them, but I didn’t have enough understanding to “go out on my own.” So… the feedback I was getting was that I didn’t understand and I wasn’t able to find the right answer on my own. The only way to find the right answer was to remember the procedure and I couldn’t.  What is fascinating to me is how “the feeling of dread” has stayed with me – it is so deeply embedded in my intuition. I have to fight it. It has definitely diminished, but sometimes, when I take a risk or make an “old mistake” – the immediate feedback my body experiences is dread.  I have to talk myself out of it – retrain myself. I can’t help but wonder – If it has taken me so long and I have to invest so much energy to reconfigure my intuition from experiencing dread to wonder – how long will it take some of the teachers I work with who aren’t even yet aware of how their math intuition is effecting their instruction?

[tzager]

Just popping in to say I absolutely love this conversation. Thank you for starting it, Christine, and thank you for all your thoughts, Amie! So interesting. I’m glad to hear the table helps you put teaching moves in relief on this question.

[Sarah Caban]

This is such a great question.  I have been thinking about it a lot. Recently, I have been thinking about obedience vs. risk taking in the context of precision and formality vs. creativity and wonder.  I don’t think any of those things are mutually exclusive, but I am wondering about the relationship among all of those words.  Recently, a new #MTBOS friend and I started exploring a question with students – What is a shape? Telanna’s blog post really got me thinking about a lot.  She asked several different aged students to tell her what the definition of “shape” is.  Here are my thoughts (copied and pasted from comment on her blog post). I thought this forum might be another place where they could live. Would love to hear other peoples thoughts:

I love this post. It leaves me wondering so many things! You mentioned intuition and that is what I was wondering about the whole time I was reading. The description that spoke to me the most – the one that I identified with the most was from a Kindergarten student – “Everything is a shape, but I don’t know what shape is?”. This is what I was thinking when I originally asked the question.

“Shape” isn’t a math term, but our collective understanding of it- as a concept- encompasses so much math thinking. I can’t help but wonder what would have happened if we had asked the same question about the word “polygon”. I suspect the answers would be more consistent, but less creative?? What does that mean for us as mathematicians and facilitators of math thinking? It reminds me of T. Zager’s book. I just finished the chapter that discusses how to balance honoring mistakes with teaching precision. I wonder if the same thing applies here – how do we honor the use of precise vocabulary with cultivating intuition and ownership of ideas? I kind of want to argue that the K student is the most intuitively connected to math. That is probably a stretch because some of the 3rd and 5th students answers are also intuitive. One said “A shape is a thing that makes everything.” Maybe I am not using the right the word to describe what I am noticing – maybe it isn’t intuition – maybe it is wonder. The K student was the only one who had a question in her/his answer, I think? That is fascinating to me because it is where I am now. As an adult, I am trying to strip myself of my formal relationship with math – not the formality – or precision – of the subject itself, but the formality of my relationship with it. I don’t want to have such a formal relationship with math anymore. I want us to be family – so comfortable with each other that we can laugh, cry, argue, fight, play, maybe not talk to each other for awhile, but then pick up right where we left off. That might sound silly, but there is truth in it. How do I keep the formal understanding (there is beauty and truth in structure and patterns), but ditch the formality? Does that make sense?

Thanks so much for sharing this!! Let’s keep exploring. I’ll keep you posted about what the kids in my neck of the woods say about shape.

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