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Welcome back to the Alcoa Maths Enrichment Program Blog.
For the most part, this blog focusses on lessons from the Alcoa Champions of Maths program. For an overview of the program, you can read the first blog post from 2019 here and to understand what we achieved in the first year of delivery, you can read about the results in the final 2019 blog post here.
If we haven’t met before, I’m Shyam Drury and I coordinate the program, delivering workshops and providing coaching and consultation for participating teachers. I’m working with a new group of 8 teachers this year from Bungaree Primary School, Ocean Road Primary School, St Joseph’s Primary School (Pinjarra) and Oakwood Primary School. You will hear primarily from them for the remainder of the year.
We continue to focus on developing problem-solving and reasoning abilities in the students, using mobile whiteboards, random groups, and the 5 practises for Orchestrating Productive Mathematics Discussions. All of this is in line with a connectionist approach to teaching, shown to be the most effective in achieving high student gains. You can get more detail on that from our first 2019 blog. Last year we saw a significant shift in the teacher’s belief-set towards a connectionist orientation and strong student gains for the participating classes.
Ok, preamble done. What’s happened this year? We’ve had the first few rounds of coaching sessions and workshops, and I thought I would share with you a couple of themes that came up during our first sessions.
The importance of a clear instruction
The problem-solving lesson format we follow runs in three phases: Launch, Explore, Discuss. During the Launch phase we give students instructions for the task and organise them into groups. We aim to keep this very concise to get the students on to task quickly so they have plenty of time for the Explore and Discuss phases. Also, we don’t want to start modelling answers, or going through examples at this point, as this may take away from thinking that students would do on their own (and hence learn more deeply.) However, although we don’t spend a lot of time delivering the Launch phase, we do need to spend some time beforehand thinking about how to make our instructions as clear as possible so that students investigate what we want them to explore.
The importance of recording methods
In problems where students are looking at multiple cases and comparing them, the way they record their test cases matters. For example, let’s consider the following problem:
I have a blue, red and a white car and I park them all side by side. How many different orders can I park them?
One of the first things we’ve noticed is that younger students, with less experience in this kind of problem solving, will spend a lot of time drawing cars. Perhaps not organised in any particular way, just a lot of very cool cars. Obviously, they are not going to make a lot of progress toward the answer to the problem as they are spending most of their time on something that is a distraction to the essence of the problem.
Some students will still draw cars but will organise them in some sort of system. Like below:
Then we may have students who use an abstraction, a representation, to be more efficient in their recording:
Clearly, the last way of recording is most efficient and supports finding an answer to the problem most quickly. These are things that we should discuss with our students. We may need to model methods in the absence of anyone showing a systematic or abstracted recording method. However ideally, we would aim to facilitate discussion, between students and with the teacher that clarify the usefulness of this type of recording. We might show students these different methods and begin a discussion with the question: “Which method of recording would be most helpful in finding the answer to this problem? Why?” Students would discuss their thoughts with each other before sharing answers, from which we would build on to clarify how a systematic, abstract recording method helps us keep track of which options we’ve tried, and which takes less time to write or draw.
What is a systematic approach?
This next point is very connected to the last. When we talk about being systematic, we mean changing only one aspect of the trial case at a time. For example, in the picture above, the student has kept the colour of the first car the same and changed the order of the remaining two cars. Following a systematic approach stops us from doubling up, lets us know with certainty when we’ve explored all options and allows us to see patterns more easily.
A more structured approach to the Explore phase.
As mentioned briefly here, and discussed more in previous articles, our lessons follow a Launch, Explore, Discuss format. During the Explore phase we typically have groups of four students working at vertical whiteboards. We can add some extra structure to this phase to help scaffold the problem-solving process (which starts with making a plan) and induce more collaboration. We’ve been discussing the following structure:
Introduce the problem to all students and give them a minute or so to consider the problem on their own, in silence.
Send students to their groups.
Allocate 1 – 2 minutes in the beginning for planning. Instruct students not to write anything at this stage, instead they must discuss with their team members an approach to the problem they all agree on.
Allow ample time for groups to work on the problem. It’s ok to change plans so if a student sees a reason to deviate from the agreed plan, they should discuss this with their group and explain why.
At the end of the Explore phase, instruct all students to put pens down and hold a discussion. The point of this discussion is to make sure that everyone in the group understands what they’ve written on their boards and how they got there.
During the Discuss phase, when a group is presenting their work, do not ask for a volunteer to share, instead pick which student will explain (and you can move this around as you cover different ideas.) Every member of the group should be prepared to report on their work if they have followed the steps above. Sticking to this approach consistently will build accountability. If a student struggles in explaining some point, they should not be made to feel stressed or ashamed and other members of their team can be invited to help them explain. Not everyone will understand everything, and we aim to support each other as we build our mathematical understanding together.
If you’ve been following the blog and working to implement these methods so far, hopefully today’s tips will help build on what you’ve been doing so far. If you’re reading for the first time I’d recommend going back through the blog and reading previous posts to get a full picture of our problem solving approach which has been showing remarkably strong results so far.