Craig Barton invited me to talk about something I learned last year for his podcast, which you can listen to here (my bit is around 1:21:50). This is a transcript of what I said:

"Hi, my name’s Danny, I am a maths teacher, and on twitter I am @danieltybrown. This year I have been learning the importance of tasks and activity that encourage learners to use a high level of

*initiative*.

To initiate is to begin something, to create. For me, creativity is an essential part of learning and doing mathematics. I’m not talking here about making stuff, but rather

*providing opportunities for learners to participate in the creation of mathematically meaningful experience.*

Such opportunities can be provided in a variety of ways, including learners generating their own questions, or versions of questions, or creating the examples that will form their notes. I have invited learners to generate examples numerous times throughout my career, and often abandoned it after one or two attempts with a group, not finding it a very efficient way of learning. I felt it must be efficient

*in theory*, but I never seemed to be able to make it ‘work’, consistently, in practice.

Anne Watson and John Mason’s book ‘Mathematics as a Constructive Activity’ helped me realise how to structure such activity, and this year - in my 14th year of teaching - I feel I am finally managing to incorporate these ideas into my teaching repertoire in an effective way. I think this is due in part to a renewed commitment to working in this way, and also an increasing realisation of the power of creativity - both in me, and in the learners I was teaching, over the year.

This year, I taught a class of 8 to 11 year olds. At the beginning, this group did not respond particularly enthusiastically to some of the mathematical tasks I offered, particularly those with lots of written examples that I had generated.

I learned that these children responded more enthusiastically when I shifted the generation of the mathematics from me, on to them. This is not to say we worked on any old thing. It was about finding structures, often afforded by the materials we worked with, within which the children could be creative. It was about setting up situations in which the children had to make

*decisions*, sometimes about the mathematics they would do, sometimes about how they would do it.

This group of children loved the opportunity to ask and answer their own and each other’s questions. They loved it when authority was subverted, whether that was using the whiteboard, or when deciding whether a certain quadrilateral was to be a quadrilateral. There was much energy generated when they ‘gained contol’ of the whiteboard, or got to work with other media such as video. They really enjoyed making the materials with which they would do mathematics, for example building their own geoboards.

If this all sounds a bit woolly, I must stress that we were all always working with purpose, on mathematics from the curriculum, such as working on place value (in different bases), geometric classification, equivalence of fractions and decimals… You can see the quality of the mathematics they produced at my blog, squeaktime.com.

I have come to realise that learning can be very efficient when learners have opportunities to create, part of which is about making and discussing choices. This is not always easy, particularly when learners make choices that I might think are not the most productive. There can’t be total freedom - there must be constraints and boundaries (mathematical and otherwise). The balance between freedom and constraint is something I have always found difficult, and particularly this year when allowing these children more freedom than they might have been used to. But I felt it was important that they have a different experience of mathematics than ploughing through workbooks, in groups differentiated by ‘ability’.

This year, I also taught a class of two (!) students Higher maths (the Scottish equivalent of AS-level). I provided opportunities for them to create their own problems to work on, often variations of exam questions, doing and undoing, sometimes removing the numbers, sometimes inviting them to ask: what-if-not. Such methods draw attention to the structure of a question, and

*classes*of questions. I find this makes learners more attentive when encountering novel problems, much more effective than the injunction ‘read the question!’

I was continually exploring ways of encouraging these learners to use a high level of initiative. As well as generating their own examples, I taught these students how to use geogebra, which allowed them to create and then work with dynamic images, invaluable for economy of time and energy. I also “upgraded reflection”, encouraging them to decide when and how they wanted to record their own illustrative examples.

I was surprised time and again at how well these students could recall things we had worked on much earlier in the year, more so than many other students I have taught in the past. I can only conclude that this was due in large part to the emphasis I placed this year on using initiative. I repeatedly found that tasks on which a high level of initiative appeared to be used resulted in better retention

*and*flexibility.

I must say, however, that it is not at all obvious to what extent initiative

*is*being used - or even what this might mean - but careful observaton can reveal signs that learners are working more or less ‘consciously’. These signs are easier to identify when working with only two students. But if we are to talk of effective teaching, of efficient use of time, then for me an important consideration for the teacher is how the learners will be encouraged, through the design of tasks and activity, to use a high degree of initiative.

Saying this, it was very clear to me by observing these two learners closely that initiative cannot be

*forced*. It is fair to say (and by their own admission) that

*one*of the students was more motivated to learn mathematics than the other. I remember the day that he brought in some problems he had spontaneously created at home for me to solve. In contrast, the other student was clear that after finishing the course he would be going back to work on his farm for the rest of his life, a mindset which made it harder for him to sustain interest when things became difficult. Whilst I could design tasks and activity that encouraged the use of a high level of initiative, I became very aware that I could only provide

*opportunities*for them to engage with the mathematics at a deep level.

But I would like to think that the work we did in this classroom altered both student’s ways of thinking. At the very least, it seemed that both students learned to become more conscious of what they were doing; by the end of the course they were certainly thinking more carefully before acting on first impressions or in habitual ways. And importantly, I think they realised that they were competent mathematicians. I think this was also true for many of the children in the 8-11 year old class.

My aim in teaching both groups of learners was to encourage them to think differently, about mathematics, and themselves. I wanted to provide access to what it is to do mathematics. I wanted them to get a glimpse of what mathematics really is - a

*creative*endeavour, in the widest sense - whilst ensuring that they had opportunities to create the meanings required for further study, success in qualifications, and so on. In short, they were successfully learning what they needed to learn, whilst making decisions about how to do it, within the structures that I provided - at least, most of the time!

It is my conjecture that this is an effective way of teaching mathematics: to bring the powers learners already possess into use - to find ways to encourage learners to bring something of

*themselves*into the doing of mathematics - through the design of tasks and activity that require the use of a high level of initiative."