Today in our second callback day we had a PD session from CeRME and Massey University on developing maths inquiry. Our session started with a talk about the traditional accessibility of maths and mathematical development. We need to promote a love of maths as lifelong learners, often children have not had the opportunity to access certain educational opportunities.
Discussion on the Numeracy Project- ability groups were detrimental to learning as it kept them down a stage because they were not able to solve a problem in a certain way. Too many different strategies were learnt and children got confused. Numeracy project focused on number to the detriment of strand, the fun part.
Measurement learning should always start with non-standard units. Gave example of new-entrants class using non-standard ‘shoes’ units to measure a mat. Using a sheet of paper to teach right angles, fold to teach 45 degrees etc.
The 7 Mathematical Practices that were discussed were: Making a claim, Developing mathematical explanations, Justifying thinking, Constructing viable arguments, Generalising a Mathematical Idea, Representing Maths ideas using numbers, pictures and materials (gave an example of building with uni-fix blocks to learn volume), Using Maths language. Mistakes are powerful learning opportunities, being numerate involves having a good concept of estimation, knowing that the reversibilty of addition and multiplication does not translate to subtraction- something that I have already identified in my classroom this year.
Much of what he said were things I had already heard in University, a decade ago. I think it is interesting that the same conversations are happening after 10 years, for instance don’t tell kids the answer, don’t dumb things down, introduce algebra as addition problems with a missing digit.
Introducing Problems: Present problem, read it, ask someone to explain what is happening in the problem and get someone else to re-voice it. Part of the planning has to involve all the ways the children could get it wrong- plan for misconceptions. When sharing answers, don’t forget to re-state the problem, e.g. 29 x 15 is 29 leis x 15 people. Insteead of the close-ended “any questions?” try “what questions could we ask here?” to elicit more conversations.
Knowing when to park problems- for instance “add a zero” instead of times by ten- and when to address it is a balancing act, you want to harness the teachable moment but not get derailed by going off on a tangent. When introducing a new solution, e.g. the grid multiplication method, you should allow a tangent of giving at least another 2 problems to consolidate. Visual representation cannot be underestimated, e.g drawing the problem to clarify the final step when solving a problem like 29 x 15 using rounding and compensation (take away 1 or take away 15).
There are strong similarities with the White Rose Project in the UK. Higher ability students need to explain it to the rest of their group and try to find another way to solve the SAME problem, before moving on to solve a different. Limit the time in senior classes to about 15 minutes of problem solving in groups before moving on. Make explicit the links to Mathematical Practices when you see them in action in the class, highlighting the behaviours when you see them so that students see the value in the solving of the problem.
Setting Up the Class: Seniors, split class into halves and see each half on alternate days. Strength based social grouping, groups of 4, one challenging task and allocate roles. !0 minutes warm up, 5-10 minutes for launch 15 mins group activity, 15 mins large group discussion, 10 mins making connections to big ideas. Juniors split class into halves but groupings are in twos or perhaps threes if there is a non-verbal (or non-counter) student. Other class doing task rotations, e.g. make 10 with counters, make 6 etc. Warm ups can be a problem or ain introduction to a new activity to be used by the non-taught half of the class. Teacher role- anticipate, monitor, select, sequence, connect.
Older children do one problem a day, younger ones do three or so versions of the same problem. Older children have a more complex problem. Ask the children who you saw solve the problem using the correct strategy to share, and only them, not the ones who have not got to that stage. After they have shared, get all the children to repeat the strategy to make sure they have followed it. When moving to next iteration of the problem, make clear that you would like to see this strategy used to solve it. Then move to an open ended task or problematic task.
Independent work should be purposeful. Make the practice related to previous problems, e.g use problems from previous day, week or last term, refer students to solved problems from the wall. Can use mathletics, games usually end up being about winning than about maths learning, and it usually gets noisy. Have some open-eended quesitons e.g what things in the class have 4 sides? How many ways can you make 15? Sorting shapes, finding objects larger/smaller than 1 metre etc, weighs more than 1kg. There were 24 legs in the park, what animals were there?
Talk Moves: Restate, Repeat, Reasoning, Add On, Waiting. Restate is Teacher verifying what a student has said and clarifying anything. This gives a bit of think space for all students and makes the idea accessible. Repeat-teacher asks another student to repeat what the last student said. Reasoning- teacher asking other students whether they agree or disagree with another’s answer, can be asked to prove it. Add on extends the explanation, would someone like to add on to that? Wait Time- total silence, teacher counts to 10 in head, communicates expectation that everyone has something important to contribute.
Using questions to justify their answers instead of accepting correct answers or stating that an answer is incorrect. Begin the term by asking: What makes a good mathematician? Compare answers like “Knowing your timestables” with “Taking risks, justifying answers” which should replace the former after some time spent seeing maths in this way.
During the launch, make sure the problem is visible to all students. During small group discussion, provide individual think time before sharing answers. Make sure the topic is relevant to the learner’s own experience- don’t assume children will be able to relate to the situation, e.g. filling up a car with petrol.
Trevor also introduced us the the concept of a Quick Image, an image that is flashed quickly across the screen and students have to say how many dots there are on a screen. The focus is not on how many dots there are, but how you worked out how many there were, for instance 7 dots flashed up as 2 dots, 2 dots and 3 dots scattered across the screen.