So what things can we do as teachers to offer opportunities for creativity and how can we embed these opportunities into our everyday practice? Piggott (2007) mentioned 3 principles to teach creative mathematics:
- how we present content
- how we model good practice
- how we encourage our students to be creative
I would argue that the above 3 principles involve ideational code switching from big-C math concepts to little-C math ideas!
Presenting content
One concern raised by many teachers, is the need to cover the content requirements of the national curriculum and exam board specifications. However, by developing problem-solving skills and using problems to explore aspects of mathematics, learners can feel empowered to “think for themselves” and, as a result, become more confident when tackling standard questions. The interactive “Tilted Squares”, published in September 2004, is based on the ability to create tilted squares on a coordinate grid and to use this to investigate the area of squares with different tilts as shown in the diagram.
| Big-C concepts |
Little-c ideas |
| Pythagora’s Theorem |
– Calculating length of slide in playground |
Some suggested questions include:
- What areas are possible?
- What areas are impossible?
- Why?
- What observations, thoughts and conclusions can you offer?
Such environments enable students to explore and work from their own level of understanding, building on this towards new understandings. For example, in the case of Tilted Squares, students have worked at a range of levels:
- some have made progress in understanding that squares do not have to be constructed with sides parallel to the edges of the paper they are drawn on;
- some have begun to identify relationships between the amount of tilt and the areas of squares;
- others have been able to generalize and offer a justification of Pythagoras’ Theorem for right-angled triangles with two short sides of integer length.
Modelling
How often do math teachers model problem solving in the mathematics classroom? How can we share with students the fact that we can also struggle with mathematics and that this is the “normal” state of affairs when meeting something new? What is important is that at any point of being stuck we acknowledge that we are stuck, and share our thought processes as we start putting our creative juices to work (Mason & Burton, 1982). At these points we should not be afraid to experiment and try ideas out – this is a common strategy we can all use. Perhaps is a good way to try this out is to walk into the classroom with a problem we have found in an old text book or mathematical activity book, such as books by Martin Gardner (Gardner, 1965), and say “Let’s look at this together” -and then spend time thinking out loud. This may push teachers out of their comfort zones, but we need to show the students that this is a fairly normal state of affairs by sharing such an experience with them from time to time.
Questioning and encouraging students to think for themselves and share their understandings are important aspects of any curriculum and a focus on problem solving and posing offers a way forward. So, what are the key features of a problem-solving curriculum? One where students and teachers:
- engage in problem solving and problem posing;
- have access to experimental opportunities (environments) to explore which have the potential to lead to particular mathematical ideas;
- are actively applying math concepts to real life (identifying the mathematics in situations);
- make connections with other mathematical experiences;
- engage in and examine other people’s mathematics;
- are not constrained by the content of the previous lessons but supported by them;
- value individuality and multiple outcomes;
- value creative representation of findings.
Examples
Rebecca Goulding (n.d.) offers some math activities to inspire inventive and creative thinking. Below I present the activities and adapted them for use in the classroom.
Arranging Utensils for Multiple Possibilities
| Big-C concepts |
Little-c ideas |
| Multiple possibilities |
– Arranging cutlery into different combinations
– Figuring out a “secret password” from a set of letters and numbers given |
Ask the students if they know how to count. “Of course,” they will respond. But the real question we want to ask is: “what is it that we are counting?”
For example, if a student is given a fork, a knife, and a spoon, it’s only three objects. How many ways can they arrange the items in a row? Here’s where it gets interesting:
- Savvy table-setters might say “Six ways”: fork, knife, spoon; knife, spoon, fork; spoon, fork, knife; spoon, knife, fork; fork, spoon, knife; and knife, fork, spoon.
- What if you allow the possibility of flipping the utensils upside down, so the handles face away from your body? The answer is then 48.
- And if you can also flip the utensils over, so they face either way? With all four possible orientations of each utensil, the answer is 196.
- Should the student be able to figure this out entirely, go ahead and add a salad fork to increase the complexity of the problem. Or, generalize to n different utensils.
- On the other hand, if it’s a little too challenging, try to solve the puzzles with only a fork and a spoon, saving the knife for another time.
Cooking and the Commutative Property
| Big-C concepts |
Little-c ideas |
| Commutative Property |
– Cooking procedure
– Baking muffin grids
– Ice cube trays |
To get more abstract, the commutative property is at play in cooking! The commutative property describes how an operation (such as addition or multiplication) is applied to numbers. The commutative property tells us that 5 x 3 = 3 x 5, and 2 + 7 = 7 + 2. In other words, the order in which the numbers appear doesn’t change the result of the operation. In contrast, subtraction is not commutative, because 5 – 3 is not the same as 3 – 5. If the concept is taught in school, it’s usually introduced in around fourth or fifth grade, but even little kids can understand it in the context of the operations of making pasta.
- Would your pasta sauce be the same if you added oregano and then basil, compared to adding basil and then oregano? (For the most part, sure!)
- Would it be the same to boil the water, then put the pasta in, compared to putting the pasta in and then boiling the water? (Definitely not!)
- But why is every young mathematical thinker so sure that 5 + 3 is the same as 3 + 5? When children first learn about multiplication, they often find it surprising that 3 x 5 (3 copies of 5) matches 5 x 3 (5 copies of 3). Why is that?
- All it takes is laying out a 3 x 5 grid of pieces of pasta to see it. Try turning the table and see that it’s also a 5 x 3 grid!
Most important, have fun with mathematics. Let your students invent crazy ideas that don’t make sense, think about questions that don’t seem so mathematical, and grapple with “basic” mathematical ideas that might seem obvious to you. Because if math is fun, then your student may actually want to think about math all the time!
References
- Boaler, J. (1997). Experiencing School Mathematics . Buckingham, Open University Press.
- Gardner, M. (1965). Mathematical Puzzles and Diversions , Penguin.
- Goulding, R. (n.d.). Creative Play with Math. Retrieved from http://www.pbs.org/parents/education/math/math-tips-for-parents/creative-play-math/
- Mason, J., L. Burton, et al. (1982). Thinking Mathematically , Prentice Hall.
- Olkin, I. and A. Schoenfeld, H. (1994). A Discussion of Bruce Reznick’s Chapter [Some Thoughts on Writing for the Putnam]. Mathematical Thinking and Problem Solving Schoenfeld, A, H. Hillside NJ, Lawrence Erlbaum: 39-51.
- Piggott, J. S. and E. M. Pumfrey (2005). Mathematics Trails – Generalising , CUP.
- Watson, A. and J. Mason (1998). Questions and Prompts for Mathematical Thinking , Association of Teachers of Mathematics.
