What’s the Most Efficient Solution?

Hi Everyone!

My name is Kate Archibald and I am a Year 6 Classroom Teacher at Treendale Primary School in W.A.’s South West, and I am a participant in the Alcoa Maths Enrichment Program: Champions of Maths (ACoM). Over my career, I’ve always found Maths the hardest subject to teach. I spend many frustrating hours analysing PAT Maths Assessment results, often repeating to myself ‘they know how to answer that’ (something I’m sure many teachers can relate to!). And I found myself wondering if I had actually taught my students how to confidently apply the skills and content they were learning into these types of questions?

Too often we can easily fall into the trap of teaching children a sequence of steps to memorise in order to arrive at the correct answer. As participation in AcoM has highlighted to me, the problem with this is that many students do not know why they are following this process, or what each step ‘represents.’ Students see maths as a set of disconnected ideas and it becomes more about memorisation and less about mathematical understanding. Most students are satisfied with getting the right answer, rather than analysing a different way to get there or, more importantly, an efficient way to get there supported by convincing reasoning. 

My contribution to this blog will walk you through how I have begun to emphasise the importance of mathematical efficiency in the classroom and invest time in meaningful discussions following the ‘5 Practices for Orchestrating Productive Mathematics Discussions’ to highlight efficient methods.

When Shyam Drury first came to watch one of my lessons at the beginning of the year, my students had been learning about the ‘Bridging through’ strategy for addition and subtraction. I had given them the sum 329 + 57, and they were working with a partner to explain their answer and process. Most students in the class had chosen to split the 50 in to 30 and 20, an inefficient strategy and unnecessary step. Only one student shared the efficient method of 329 + 1 + 56, which would have been the ideal time for me to discuss which method is more efficient and why. This was a missed opportunity to encourage students to be more effective and show them that what matters is making good choices for good reasons, rather than whether they get the question right or wrong. Having someone else watch my lesson and deconstruct it with me brought my awareness towards the need to address the concept of ‘efficiency’ with my class.

As a result of Shyam’s debrief, my students and I brainstormed what efficiency was, why it was important and how to get there. I thought it was important to emphasize that what might be the most efficient for some, may not be the most efficient for others and that was okay. As a class, we decided on our own definition of efficiency and created the following poster as a reference to display with our other expectations.

Over the following weeks, students continued to engage in a variety of meaningful problem solving activities in their randomised groups of 4 with our discussions targeted at pulling out the most efficient methods.

An example of a recent lesson scaffolded for a strong discussion on efficiency to take place was when students had been given the task to work out the sum of every odd number between 1 – 99. I had completed the lesson proforma and anticipated the solutions students would be using as seen in the lesson plan below.

During the working time, students were initially quite stumped. Only one group began to write down every odd number and add systematically each time. Most other groups began by adding rows or columns of odd numbers together in a hundreds grid format, an approach I hadn’t anticipated. After 15 minutes of productive struggle and with no group moving towards a more efficient strategy, I called them all back down to give them a levelling prompt, ‘How can you group these numbers into pairs?’. One group then found pairs adding to 100, but with no routine or idea of where to next.

As students were working on their solutions, I was monitoring the use of strategies to help in selecting the order in which to present their ideas. Many students had used the columns or rows addition approach, some slightly more effectively than others. The order in which the ideas were presented and discussed were;

1)    Systematic Addition – the least efficient method

2)    Multiplication Pairs – an efficient strategy, but used inefficiently in this case. (The group shared their explanation before I led a class conversation about how we could build on what this team had done to make it a more efficient strategy.)

3)    Addition of 100’s chart columns – breaking down place value

4)    Addition of 100’s chart rows- noticing pattern of adding 75

5)    Multiplication & Square Numbers – led by teacher – linked with unit being taught at the time.

As pictured above, the whole class continued our discussion, by analysing how all these ideas were interconnected, asking questions such as “can you see the idea that Group 1 had, evident in Group 2’s explanation?” After this, one student put up his hand. His question was “out of all of these, which is the most efficient way?” Usually  this student is quite reluctant to have a go during maths. The fact that he was contributing, let alone independently seeking efficiency and genuinely questioning the methods presented, was a tremendous win. Turning his question back on the class, I asked them to discuss their answer to that question with those around them and be able to justify why. For several students, they weren’t quite ready to jump all the way to 25 x 100, but could recognise that it was the most efficient way to get the answer. In this task, every student was able to build on their process in one way or another to make it more efficient.

Had anyone told me before that I could give the same question to the whole class and still allow for every child to experience an achievable challenge, I would have doubted them. This process highlights how these ‘low floor, high ceiling’ tasks can be done.

All you really need is a good question.

During our debrief, Shyam pointed out another opportunity for discussion during this lesson – how the margin of error decreases in the more efficient methods. For example, the students who added every single number together had to do this correctly in 50 steps, whereas students who found pairs adding to 100, recognised there were 25 pairs and multiplied 25 x 100, only had to make one calculation. I have since raised this in my class, with several students immediately pointing out the direct link back to our class definition.

My participation in the ACoM program has reignited a love of teaching maths, and I have found myself excited to plan what will be coming next. For the first time in a long time, I feel confident that how I am teaching maths is having an impact on student engagement, confidence and learning. My class have begun to adopt a new definition of success in mathematics and are questioning and thinking about their process and efficiency wherever possible.

This is one of my favourite photos from the year to date, showing students engaged in a deep discussion justifying their belief about the colour coding of Math4Love’s ‘Prime Climb’ Board Game. At the beginning of the year, children were hesitant to share their thoughts and answers voluntarily, let alone engage in the respectful debate this photo shows. You know that things are going well when there is a collective sigh as the lunch bell goes and you haven’t quite gotten through everyone explaining their answer yet, or when students don’t want to go to lunch and would rather solve problems instead! I can’t wait to continue learning to be a ‘Math Champion’, and with such exciting progress already made, I’m eager to see the further impact this change in my teaching will have on student understanding.

I hope this entry will spark an urge in you to discuss efficiency in your classrooms and inspire you to probe further than just a correct answer.

Kate.

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