Lisa Tatham, Year 5 Teacher at Treendale Primary School with my second blog post. If you want to read my first one again it’s here.
As a classroom teacher, I’m always concerned about catering for the needs of all my students, especially when it comes to maths. I’ve struggled over the years with differentiating my lessons to ensure every student was targeted at the appropriate level. Sometimes this has meant work stations which students rotated through, three different versions of the same worksheet or simply targeted teaching with a small group while the rest of my class worked independently. However, my involvement with the Alcoa Maths Enrichment Program: Champions of Maths (ACoM) has shown me how one great question can be the catalyst for powerful learning by every student in the class.
For the past six months, I have been working with Shyam Drury to develop effective problem solving lessons which follow a three-part format. First, introduce the problem. Second, allow students to work in random groups of four to solve the problem while the teacher monitors discussions and notes the strategies that are being used. Finally, invite students to share their strategies with the whole class, using questioning to draw out the mathematical idea so they can make connections between different strategies and solutions.
This approach to teaching has meant I have been spending less time searching for materials and activities and more time thinking about my students. When planning, I consider what curriculum point I need to address, then find an open-ended question that targets that concept. Open-ended Maths Activities by Peter Sullivan and Pat Lilburn is a fantastic source however there are many more out there. Once I have my question, I think about how the students might solve the problem and what I can ask them to extend their thinking.
One of my most successful lessons this year asked students to consider this problem from Sullivan and Lilburn: The answer to a division question is 5. What might the question be? When I presented the lesson, I was worried that it might be too easy for many of my Year 5 students and that they wouldn’t engage in it the way I hoped. But I knew that some of them still struggled with simple division problems, not really understanding what the symbol meant or how to correctly write division equations. My learning goal was for students to use the inverse relationship between multiplication and division to solve problems. I hoped they would relate the problem to the five times table to make a list of possible answers. But I also wanted to address misconceptions some students had about the way to write division number sentences, for example believing that 5 ÷ 10 = 2.
When I introduced the problem, I asked students to identify the important words. They quickly noted answer, division and 5 as relevant information as well as the word question. One student also pointed out that might was important because it meant there were many possible answers, not just one.
With that, the students were set to work in their groups while I observed. I was pleased to see them immediately on task, not fearful about whether they would ‘get it right’. As I expected, quite a few students wrote their equations incorrectly. During the post-activity discussion, I asked one of them to explain why they had written their number sentence in that way. In the process of explaining their thinking, the student discovered their error and made the correction. Using the talk move “turn and tell the person next to you why she just changed her equation” helped other students who had made the same error to process this information and make their own connections to learning.
The most common response from students was a list of number sentences that equalled five. Some had organised their thinking, while others wrote in a more random manner. Sharing one methodical response and asking the class what do you notice about the way they organised their answers drew students’ attention to a pattern they recognised from the five times table. This was the link that I had hoped they would make.
As every teacher knows, lessons rarely go exactly the way you planned and this was no exception. Although I had considered possible responses, one student solved the problem in a way I hadn’t expected. His answer was (12 + 13) ÷ 5 = 5. I asked him to explain to the class how he knew that the answer was five. In the discussion that followed we considered the meaning of brackets in maths, extending students into algebraic thinking.
The beauty of this lesson was that no one thought that any one person was smarter than anyone else. There was no ‘bottom’ group or ‘top’ group, no ‘easy’ maths or ‘hard’ maths. A variety of ideas were considered and respected. One great question brought a powerful equality to my classroom, where every student could engage and learn at their level. As the recess bell went, I asked students to show using their hands who had a better understanding of the connection between multiplication and division. They enthusiastically agreed that they all did. One student lingered behind as everyone else moved out the door. With a big grin, she said to me “I think I got better at division today“.
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