This semester I had to write a literature review of a topic of our choice. Last year I focused on assessment, which will be my dissertation topic. I wanted to try something a little more “lighthearted” this year, so I went with Mathematical Creativity. I’m not going to reproduce my paper here, but I did want to share a couple practical takeaways for teachers.
What is mathematical creativity?
If I were to put together all the varying, but similar definitions of mathematical creativity, from G.H. Hardy (1940) to our friend Peter Liljedahl (Liljedahl & Sriramen, 2006), I’d personally define it in the following way:
I’m sure folks could quibble with the wordsmithing there, but that’s my definition and I’m not sure anyone can achieve a perfect definition. The important thing is that students (a) think divergently, (b) think convergently, and (c) create something new or novel. There are certainly debates on the connection between mathematical creativity and just plain ol’ creativity. Does one necessitate the other? Are they correlated? A lot of this debate depends on what instrument one uses.
But enough of this, I said this post was intended to be practical, not esoteric. We can get esoteric in the comments.
How can we foster mathematical creativity?
Thankfully, research has shifted from the mid-20th Century when mathematical creativity was portrayed as static and immutable to recent scholarship that demonstrates how mathematical creativity can be fostered through problem solving tasks. Let’s look at a few examples.
Open Ended Problems and Multiple Solution Problems
Open ended problems are tasks in which there are multiple solutions or multiple paths to a solution. Relatedly, multiple solution problems are open ended problems that require students to develop multiple solutions. Here are some examples that I quite like:
Quarter the Cross
QtC is an absolute classic of the genre. I use it all the time. For all grade levels and ages. It’s great. Check out the original home of it from David. I’ll say I make two slight modifications: I require students to do it two times in two different ways. And I often get rid of the edges between the squares, so it’s a straight up plus sign.
Ask students to create a rule or identify a characteristic that would group numbers in certain ways.
Use the numbers below to construct groups of four numbers with a common characteristic. Name these groups. Create as many groups as possible.
2, 3, 4, 5, 7, 9, 10, 15, 21, 25, 28, 49
I suppose this isn’t that dissimilar from Which One Doesn’t Belong, which would also be a good mathematical creativity activity.
When you assess for mathematical creativity on a mathematical creativity test, say, you look for these three features:
- Fluency: how many solutions did a student achieve?
- Flexibility: how many different types of thinking did a student demonstrate?
- Originality: how is the students’ solution different than other students’?
It’s difficult to translate that into an assessment in a self contained class. However, you could use a … RUBRIC! (*balloons and streamers start falling, elephants with banners start marching through, a marching band plays John Phillips Souza*)
Nothing too fancy though. I think even a single point rubric will do.
For more on single point rubrics, check out Cult of Pedagogy’s post.
Emphasis Visuo-spatial representation
There appears to be a correlation between visual experiences and mathematical creativity. We’ve also never had more access to dynamic math visualization tools. I’ve recently been using Mathigon’s Polypad for nearly all my classroom visualization needs.
I also had my students use polypad to represent mathematical concepts like adding negative numbers and grouping.
Of course, we all know about Desmos. I had some students engage in an activity similar to Quarter the Cross in Desmos.
Peardeck’s drawing features (premium access only) are also natural ways to embed more visual experiences.
Craft Problem-Posing Experiences
Problem-posing is at or near the apex of mathematical creativity demonstrating experiences: if students are posing problems in addition to solving problems, they’re demonstrating and developing some serious mathematical creativity chops. However, it might be the least utilized of these three practices.
That’s probably because it’s weird to just be like, “pose a problem to your tablemate.” You need something to start from. You actually need constraints and a template. I’ve offered my problem creating template in the past, so feel free to check that out.
I also quite like the “structured problem posing” scenario (Stoyanova et al., 1996)). In this format, you give a students a problem to solve – that’ll be our reference problem. Then you ask them to come up with a (a) simpler problem, (b) similarly challenging problem, and (c) more difficult problem, based on the original reference problem. Here’s an example from ().
The original reference problem:
After solving, students came up with new problems of varying difficulty. Here are some examples of what three students came up with.
Simpler: A monkey wants to reach bananas. In the 7 x 9 square below the route is indicated. Knowing that the side of a square is 5m and the monkey crosses 19m in 5 minutes, find how long does it takes the monkey to reach the bananas.(Singer et al. 2015, p. 1077)
Similar: Ballerina lost her ribbon in the maze. The ribbon is represented in the square grid. Knowing that the side of a small square is 3m, find the length of the ribbon.
More difficult: A maze is between the houses A and B. Jack wants to go from A to B, so he takes a string to indicate his way to go. On the 11 × 11 square grid below the Jack’s travel is drawn. Knowing that a square has a side of 1.2m, find the properties of the number that shows the length of the road.
That’s pretty cool! In the study, some students kept the ribbon scenario the same, others demonstrated creativity in varying the context of the problem. In other examples, students created a non-rectangular map.
I like this structured problem posing routine better than my template from (good gravy) 2014. Perhaps a mashup of the two would be in order.
Where can I find some of these mathematically creative tasks?
Your best bet is to head to NRICHS maths. They have so many interesting mathematically creative problems for so many grade levels. Many of them are great as is, or could be modified to boost their level of mathematical creativiti-ness.
I also think that Open Middle problems would do well for a multiple solutions task.
One final note on mathematical creativity. Bishara (2016) found that experiences with mathematical creativity were particularly beneficial for students with learning disabilities. That study found that mathematically creative experiences enhanced students’ motivation, which is atypical from most remediative experiences or for math support situations. I’m currently working on a proposal that will study fostering mathematical creativity within pre-service teachers, for which there’s a recent but growing literature base (Ellerton, 2013).
Bishara, S. (2016). Creativity in unique problem-solving in mathematics and its influence on motivation for learning. Cogent Education, 3(1), 1202604. https://doi.org/10.1080/2331186X.2016.1202604
Ellerton, N. F. (2013). Engaging pre-service middle-school teacher-education students in mathematical problem posing: Development of an active learning framework. Educational Studies in Mathematics, 83(1), 87–101. https://doi.org/10.1007/s10649-012-9449-z
Hardy, G. H. (Godfrey H., 1877-1947. (1967). A mathematician’s apology. [First edition] reprinted with a foreword by C. P. Snow. London : Cambridge U.P., 1967. https://search.library.wisc.edu/catalog/999470113802121
Liljedahl, P., & Sriraman, B. (2006). Musings on Mathematical Creativity. For the Learning of Mathematics, 26(1), 17–19.
Singer, F. M., Pelczer, I., & Voica, C. (2015). Problem posing: Students between driven creativity and mathematical failure. In K. Krainer & N. Vondrová (Eds.), CERME 9—Ninth Congress of the European Society for Research in Mathematics Education (pp. 1073–1079). Charles University in Prague, Faculty of Education and ERME. https://hal.archives-ouvertes.fr/hal-01287314
Stoyanova, E. & Ellerton, N.F. (1996). A framework for research into students’ problem posing. In P. Clarkson (Ed.), Technology in mathematics education (pp. 518–525). Mathematics Education Research Group of Australasia.