I’ve had the great opportunity to work with Prof Ron Ritchhart (his website has pdf versions of most of his presentations) from Project Zero, Harvard Graduate School of Education, today.
He started with a great activity that could be replicated in department meetings. First, ask participants to write down all the actions that students engage in within mathematics classrooms. Then, people select from this list to create their own private lists of about 5 items for:
- Actions your students spend most of the time doing during mathematics classes
- Actions that are most authentic to the discipline (what is ‘doing maths’ about)
- Actions you remember from a time when you were learning new mathematics (eg at uni, teaching a unit of work with content you are not familiar with) and what you did to develop understanding
If actions arose that were not on the shared list, these could be added.
As a group, participants then created a shared list of the above. This was a great, unthreatening way of eliciting people’s underlying beliefs about what mathematics is about and what it should be about – for teachers and for students.
We then looked at 4 Useful Patterns of Thinking (whilst these are not inclusive of all types of thinking we might want to foster in our students, these are useful for teachers to nurture in the mathematics classroom):
- Speculation: What might be going on here? What are the possibilities..and the limitations? What questions, puzzles or issues are being raised?
- Generalisation: What principles, strategies or formulas can we identify from this situation that will always hold true?
- Analysis: What relationships and connections can we identify here? What are the various parts and facets? What is their role, effect or purpose?
- Proof: How can we convince both ourselves and others that our findings, assumptions and conclusions are valid?
These broad umbrella terms struck a chord with me. These fit in very nicely with what ‘working like a mathematician’ means. Wouldn’t it be great if we could plan our lessons, topics, curricula, assessment around these ‘dimensions’ of thinking? What if we made these explicit to teachers and students about what is expected in a mathematics classroom, that these were the goals we had in mind and aligned everything we did to these?
We were then given a transcript of a lesson that started with the question: What shapes can be made by a straight line and a square?
Such a simple question, such rich opportunities presented, and developed, by the teacher. Students were challenged to think about whether a 30,60,90 triangle could be formed. What congruent shapes could be formed? “How do you know?” Ron referred to this as ‘bumping up the thinking opportunities’…what I call ‘ratcheting up the thinking’. When I plan learning experiences, I use one of Project Zero’s thinking routines: Connect, Extend, Challenge. I connect to previous knowledge, extend this into the next phase of constructing knowledge and understanding about new material then challenge students to transfer this knowledge into an unfamiliar context to check for understanding. Ron mentioned that one of the most popular thinking routines in mathematics is Claim, Support, Question. He described a class where students had 3 dice numbered 0 to 5. The students were told that the aim was to create even 3-digit numbers from the digits thrown on the dice. A point was awarded for every even number created. After doing this activity for a while, the teacher asked for ‘claims’…eg “If all three digits are even there are 6 even 3-digit numbers possible”. So much mathematics, so much thinking..and the students own it…it hasn’t been passed down to them from on high like a lot of mathematics seems to be.
We ended the day by watching a teacherstv video on Revising Polygons.
An invigorating, energising day that extended and teased out my ideas on thinking and learning in mathematics.