A Vignette About Subtraction 2020! Cartoon Prompts, Number Lines and Bar Models

Welcome to 2020!

I sit here after a great three days of professional learning in Melbourne, excited to share some ‘vignettes’ about the work my team and I have been doing exploring addition and subtraction with our 5/6 students. As I write, I am thankful for all the great experiences through the PMSS course and love how the program provides the opportunity for a second go at implementing, exploring and reviewing.

I understand the program is intended to enable school wide consistency and leading a maths revolution, but for me, I need to learn and build my own skills through trial and error and seeing the results of my students in order to help others in my learning community. This post is a reflection on the tools and strategies I’ve tried as a grade 5/6 teacher exploring ways of helping my students develop a deeper and longer lasting set of strategies to help with the dreaded subtraction involving renaming. Through the sequence you will notice the influence of the PMSS workshops and how well the ideas gel together.

Cartoon Prompts

I’ve played with these types of prompts after trying Doug Clarkes ‘Helping Bert with division’ Rich task (from the red book) and Jill Browns workshop. This year, with a brand new class in front of me I’ve had some interesting reactions to using cartoon prompts as both launch prompts and exit tickets.

My 5/6 class has been working on addition and subtraction. Early examples demonstrated some clear challenges with renaming and a fixed mindset on using the traditional vertical process. Students who displayed good mental computation and strategies that show flexibility with numbers were not estimating or spotting their errors because they thought the process of the traditional method was solid. *It was very relevant when Doug shared the story of the girl with the fantastic strategies and being flexible with her thinking at the end of grade 2, and after being ‘taught’ the traditional vertical addition method was less willing to be flexible (Doug told this much better, I’m just paraphrasing the ideas).

*I tried the ‘Which problems can you do without pencil and paper?’ activity with my students and was blown away. What was particularly interesting was talking with the students who had initial problems with the pretesting questions. They had some great strategies when working with numbers and implementing the ‘counting up’ methods we had explored previously. It was also interesting to note that the students who were confident at using the traditional method had a harder time thinking flexibly. They would describe it to me as doing the same method in their head, just without the paper and pencil. So they would mentally try to slot the numbers underneath and justify to me that it was an easy way to do it.

This led to the creation of some cartoon prompts, with the characters demonstrating the same challenges. It was interesting observing how these prompts broke down some barriers for certain students and how they were willing to offer ‘help’ and advice to these fictional characters and the realisation that they had also been making the same error.

The other interesting part was how they reacted to see subtraction problems with the answers and working out already completed and being asked if it was right or wrong. The subtle shift enabled my students to enter the task and they felt empowered to use other strategies

Reflection

• The prompt positions the student to take on the role of a problem solver. The traditional ‘what is the answer’ has gone out the window and they are faced with a new type of question. Students were asked to analyse what they saw and what they think the person has done before they can make recommendations. Seeing an answer with the prompt caused a bit of confusion at the beginning but as PB would say, ‘the goalposts had changed’ and the students responded.

• Students are asked to help. I didn’t really understand the significance of this until this year. Rather than answer this question, students are asked to help someone. That someone might have already made a mistake so it’s ok. For my group the initial hesitation and barriers about subtraction with renaming became ‘Oh that’s what they did’. Observing my students, they were happy to ‘help’ someone because it created a sense of collaboration and they could identify with how this person is feeling. This also prompted discussion and justifications. I think they did this… Maybe they thought this ….

• This was a positive way of identifying misconceptions. Using the misconceptions of the students as the prompts enabled targeted help. In most of the cases it was the students who made the initial mistakes who were the ones sharing what went wrong.

• Students enjoyed the story. I played with drawing the prompts first and drawing them with the students watching and both worked. They related to the person. ‘Oh I used to do that’ or they had small laugh at how the answer couldn’t be right because they used estimation or seeing the reasonableness of the answer.

• It enabled clear links with the strategies and methods we had worked on during class. This was another opportunity to use the method and share with their peers. My students were able to make links with a strategy they had been working on and apply it to traditionally set out subtraction.

Number lines

After being part of an awesome PD run by Sam who referenced some work by Mark Chubb, I was inspired to dig deeper and found a video by Dr. Alex Lawson. This led to focusing our teaching and learning of subtraction and addition around the implementation of the number lines tool. We sorted the methods into ways to count up, count down and decompose / compose numbers.

Prior to this prompt, we had explored some ways to use the number line tool. An effective part of the tool is that it enabled me to hear what the student was saying and I was able to clarify and show the students using a visual representation. Linking this with the Matt Sextons article on ‘Talk moves’, I found it really powerful to use the prompts from the talking cards as I’m visually showing it using the whiteboard or ipad, or getting the students to visually show it as they are speaking. Observing my students, they were more involved in the sharing time, more willing to discuss and elaborate on their strategies and students felt good about their peers valuing their thinking.

It was interesting to see how students still only did jumps they were confident in doing. You will notice the student who went down by 10’s and 1’s and the student who when viewing this added on that it would be more efficient to make a bigger jump.

The images below were created during our class discussion.

It was interesting seeing the reaction of the students when we explored ‘counting on’ as a subtraction strategy. This was particularly evident in the equations like 1003 – 997 and 400 – 63.

After the last workshop with Doug, we discussed the need to change our assessment documents and scope and sequence away from the traditional equation setup.

eg.

40 to 40 - 16 =

– 16

This was emphasised by the way my students approached the assessment. Even though I had created a prompt saying ‘I’m not good at these equations but I do have a strategy that works, what should I do?’ and telling the class that they could choose the best strategy for them, I had students still attempting the traditional method, even though we had up-skilled them to use a number line and they could use it efficiently and effectively. After seeing their page, I spoke to them and said ‘I’m not saying these are right or wrong, but would you like some more time to check using another method we have looked at?’ It was a nice little reflection to see that some habits are lasting, even if it didn’t work for them. My next step is to trial updating the assessment and seeing if it changes the way they approach the question.

Bar Model

Our pre-assessment (BOOKER subtraction test) showed clear uncertainties with students misinterpreting subtraction problems involving stories with additional information.

The problem wasn’t that they couldn’t do the equation, they were making errors in understanding the question and using the right information.

Adding to the work with number lines, we began exploring bar models as a tool to help make sense of word problems. I have to say, the resolve website is amazing! I was able to read the notes, use the prompts and work with the students to explore the benefits of this tool.

https://resolve.edu.au/teaching-resources

I’ve collated some of our work using the resolve resources in a way of reviewing and sharing.

Here some examples of the prompts my team has used.

Reflection

• Some students wanted to avoid doing the bar model picture and go straight to the equation. ‘What’s the point?’ ‘It only slows me down’ – this was particularly evident in the students who were confidence in the traditional method.

• Students would write the equation and going back to draw the picture (not to check but just to tick of that part of the lesson)

• They were representing the model like an equation but in picture form.

• I found the drawing of the question before any equations were written out was important to model. Reading the information together and then making the bar model a priority with a question mark proved useful for my students. ‘Now we know what they question is asking, what equation do we need to do?’ You will see the difference in the pictures as students begin to show confidence in the method.

• As we progressed to questions that required two or more parts, the students could see why the method was being used and they enjoyed the challenge of the problems I was putting in front of them.

• Students were able to explain and justify their bar models with each other. This proved to be a really valuable part of the learning. Just like in all of our exploring, the deeper learning happened when they were challenged and were encouraged by their peers to share how and why they sorted the information in a particular way

• This method had a major impact on the post test results with a noticeable difference between the pre and post testing.