About:

Kristen Tripet, Mathematics Education Consultant

Tripet demonstrates that Mathematics is a creative subject in which ideas can be generated, tested and refined. She believes that working mathematically should be incorporated into every lesson as a natural part of mathematical investigation. Tripet provides practical strategies that teachers could employ and emphasises that small group collaboration is highly beneficial as it is through communication that mathematicians explore ideas, proofs, solutions, problems and conjectures.

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Kristen Tripet: What is maths? What is creativity? Can mathematics really be creative?

Teacher: Boys and girls what is maths? Rachel.

Student: Maths is adding and subtracting, multiplying and dividing. Maths is the equation of multiple numbers. Maths is the way of solving everyday problems using equations and theories.

Teacher: Alright, boys and girls, what is creativity?

Student: It is using your imagination in the real world. It is something that comes from you and it’s unique and no one else has it but you.

Kristen: Hello, my name is Kristen Tripet. If you were to walk down the street and asked anyone who passed you by the question, ‘What counts as maths?’, most people would respond using words such as ‘numbers’, ‘sums’, ‘rules’ and ‘formula’. Many people’s memory of mathematics at school revolves around lessons comprised of drill of facts and the practice of procedures that held little meaning. Questions were posed and there was often one correct answer and only one correct method to use to obtain that answer. So then, what is ‘creativity’? Defining creativity is a really difficult task. A research group in England, chaired by Sir Ken Robinson explained that creativity is: ‘Thinking and behaving imaginatively in a way that is purposeful and valuable and that generates something original’. If our previous definition given to mathematics is true, how on earth can maths be creative? I hope you will agree that the earlier definition of mathematics is exceptionally limited. It is not a subject that should be broken down into segments that are taught in isolation, with clear right or wrong answers. Rather, maths is the seeking and solving of problems. It is a way to describe patterns and relationships. It involves measuring and sorting. It provides the tools to abstract our imagined world. Mathematics is a creative subject in which ideas can be generated, tested and refined. Imagination is vital as maths requires a great deal of abstract thought and abstract thought has its roots in imagination. Teaching maths creatively is about appreciating its creative nature and seeing maths in the world around us. I encourage teachers to see that at the heart of mathematics is raising questions and then the solving of problems. Children are natural problem solvers, using strong reasoning and creative strategies. And then, they come to school and sit through passive maths lessons and have all the creativity knocked out of them. Sir Ken Robinson stated that children don’t grow into creativity, they grow out of it. To nurture children’s creative thought we have to allow them to play and experiment as they problem solve and then to select and use their own strategies.

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Part 1

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Kristen Tripet: In 1999, a video study was conducted as part of the Third International Mathematics and Science Study. Researchers observed classrooms around the world to compare the different approaches to teaching mathematics. When looking at the Australian classrooms, the researchers concluded that our “students would benefit from more exposure to less repetitive, higher order problems; more discussions of alternative solutions; and more opportunity to explain their thinking”; and that “there is an over-emphasis on ‘correct’ use of the ‘correct’ procedure to obtain ‘the’ correct answer”. If this is the way we are teaching in our classrooms, then maybe we are killing our students' creativity. So then, how could we be teaching? We can adopt a problem solving approach to teaching mathematics creatively. The benefits to this way of teaching are numerous. We will not just be nurturing creative thinkers. We will also be building proficient mathematicians. How is this achieved? There are so many ways to answer this question! Let me give you three main reasons. Firstly, it focuses classroom activity on working mathematically. The working mathematically strand in our syllabus is designed to be embedded into the content that we teach. It is not something that can be taught in isolation nor should it become that strand which addresses a token problem solving lesson on Fridays. Working mathematically should be incorporated into every lesson as a natural part of mathematical investigations. Another reason to teach mathematics through problem solving is that it is a great way to differentiate instruction. Let me illustrate this through a problem. I have a rectangle. It has a perimeter of 30cm. What is its area?

Teacher: So, Caroline before you move on I noticed that you’ve only got six squares there. Can you tell me what each of those squares represent.

Student: Okay, so we thought that there were no squares that we could use anymore so we thought that would shorten it. So, that would be three there instead of one and three and three would be nine. And then here three and three would be six. And so six times nine is fifty four.

Teacher: So, are you saying that each of those squares, coloured squares is three units by three units?

Student: Yes. .

Teacher: What’s the perimeter of this rectangle?

Student: Thirty units, thirty centimetres.

Teacher: And how did you calculate that? Can you show me how you calculated it?

Student: Firstly, I did six centimetres which should be here with these two and then it would be nine here and then nine, eighteen because it just has to equal these two sides. It equals twelve and twelve plus eighteen is thirty.

Kristen: This question does not give the students a picture of a rectangle to measure and use, rather it poses the problem in words with no measurements given, apart from the distance around the shape. It is a very open-ended problem with which the students will need to experiment, before they will come to any solutions. To make this question accessible to all of the students in the class, the teacher can provide opportunities for students to select and use various concrete materials. Some students might find a geo-board useful, while others might like to use square tiles, while still others might like to use grid paper to help them solve the problem. There is likely to be another group of students who are comfortable working with more abstract methods. With such an open-ended task, there are many possible solutions! If we limit ourselves to whole number answers, there are 14 possible answers. As the teacher, I can allow this investigation to differentiate itself. There will be students who work hard to discover and record two or three rectangles and others who discover many more. What about the more able students? Well, they might like to investigate decimal measurements or perhaps they could go on to graph the length of a side against the area to see if they can discover patterns or to make any generalisations. A further challenge for these students might even be to consider, what would happen if it was not a rectangle? What if it was any shape? Such an open-ended problem allows for very effective differentiation.

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Part 2

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Kristen Tripet: Another reason to use problem solving to teach mathematical concepts is that it provides rich assessment for teachers. In our previous problem, it would be very easy to see those students who had a firm grasp of the concepts of perimeter and area and those students who did not. Observation of students working and their work samples of their recordings provide a clear indication of what the students are able to do and what areas still need further teaching. If we are to adopt a problem solving approach to the teaching of mathematics, then we need to be able to equip students with the strategies to work through a problem. Many teachers find that students are hesitant when they are posed with a problem or that they give up very quickly and say, ‘I can’t do it!’ Students should be encouraged to take pleasure in the challenge of solving problems appropriate for their ability level. They should enjoy those moments that produce crinkly eyebrows and to see this as ‘hard fun’. George Polya, a mathematician, wrote a book on problem solving in 1957 called ‘How to Solve It’ and his work is still being used today. He devised a four-step plan to help guide students when they approach a problem. He said, first of all, you have to understand the problem: What is the unknown in the question? What are the data? What is the question asking me to do? Are there any maths words that I need to know the meaning of? The second stage in the problem solving process is to devise a plan. What computations might I need to do? Have I done a similar problem before? How might I get to a solution? Have I considered all the different parts of the problem in my plan? The third stage is to go ahead and carry out the plan. Check each step as you go. Can you see clearly that each step has been done correctly? Can you prove your answer? The final stage is to look back and examine the solution that you have obtained. Can you check your result? Ask yourself: ‘Is this the solution or is it just a solution? Is this the only possible way to get to the answer or could I do the problem another way? Can I use my result or method to help me answer another question?’ Let’s watch some children using this process to help them solve some mathematical problems.

Teacher: Okay, these are your shapes and I want you to use these shapes to show me which one will tessellate. . Soraya what have you made?

Student: I’ve made a trapezium, a rhombus and a hexagon.

Teacher: Okay, do they all tessellate?

Student: Yeah.

Teacher: How do you know they all tessellate? How do you know these …

Student: Because …

Teacher: … this shape tessellates?

Student: … they don’t leave any gaps.

Teacher: They don’t leave any gaps, excellent. And what else?

Student: And they’re closed shapes.

Teacher: They’re all closed shapes, yeah. Do they overlap?

Student: No.

Teacher: No, they don’t overlap. Okay, so they’re they all tessellates. Well done, excellent.

Kristen: So then, how should a lesson be structured using a problem solving approach? Many teachers pose word problems and investigations towards the end of the lesson after first learning or practising a new procedure. There are a number of concerns associated with this structure. Firstly, problems become an extension activity and our less advanced learners do not get to experience this creative maths. Another issue to consider is that when students come to a problem at the end of a lesson, they perform the calculation they have been using at the start of the lesson. They don’t read and interpret the problem and do not use creative thinking. This prevents students from linking problems with problem solving. So what needs to happen is that the lesson needs to be turned upside down. Try starting a lesson with a problem. Allow students to creatively select and apply strategies to solve the problem posed. Students should be able to then share their strategies in a class reflection time. The teacher can then use the remaining part of the lesson to refine students’ strategies through different activities.

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Part 3

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Kristen Tripet: Let’s consider another problem. We can use somebody’s 10th birthday as a clear problem solving opportunity. Assuming someone has had a birthday cake with candles every year, how many candles has this person had on all of their birthday cakes if we add them all together? This problem is really focusing on pairs of numbers that add to 10 and also considers an important mathematical idea, that is, the associativity of addition which means that you can add numbers together in any order to produce the correct result. In this task, I would like to build students’ ability to recognise compatible numbers that add to 10. As students problem solve in pairs, they have opportunities to experiment and apply different strategies. And they are all working mathematically. Students should be encouraged to talk to each other as they problem solve. It is through communication that mathematicians explore ideas, proofs, solutions, problems and conjectures. The teacher has the opportunity to move around the classroom and observe students’ working and to question and to prompt where needed. After students have had enough working time, they can share some of the solutions that have been used. Consider the purpose of your lesson’s reflection time. As the focus of this lesson is compatible numbers to 10, this strategy can be shared at the end of the reflection time and then all students can be given the opportunity to use this strategy in different ways. Some students might need to practise their 10 facts and so they may play a game of 10 fact ‘Snap’, snapping the cards when they add to 10. Another group might roll a collection of dice, pair compatible 10 facts together to make totaling the dice easier. There might be another group who are proficient with their 10 facts and so the teacher might like to extend their thinking by exploring numbers that add to 20, 50 or 100. As I conclude, I would like to ask you the question that I posed at the start of the presentation, ‘What counts as maths? Do you believe that mathematics is a subject that allows creativity? Do you see it as a beautiful and elegant subject? Do you see that mathematics is in the world all ‘round us? Are you a seeker and solver of problems? Do you see yourself as a mathematician? As teachers, our values and attitudes toward maths will influence the way we teach the subject. Then this, in turn, will mould our students’ values and attitudes. Mathematics is a subject that helps us make meaning of the world in which we live. Galileo, the famous philosopher and mathematician, stated: ‘The universe cannot be read until we have learnt the language and become familiar with the characters with which it is written. It is written in mathematical language and the letters are triangles, circles and other geometrical figures, without which means it is humanly impossible to comprehend a single word’. Music .

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Part 4