This quarter, we’re exploring the 2008 ISTE Teacher Standards (which I will be abbreviating ISTE-TS), and for module 1 we’re looking at ISTE-TS 1: Facilitate and inspire student learning and creativity – “teachers use their knowledge of subject matter, teaching and learning, and technology to facilitate experiences that advance student learning, creativity and innovation in both face-to-face and virtual environments.” What really stood out to me when reading through the standard and its indicators was the idea of creativity. Along with creativity being mentioned in the standard itself, two of the indicators mention it: Indicator 1a – “promote, support, and model creative and innovative thinking and inventiveness” and Indicator 1c – “promote student reflection using collaborative tools to reveal and clarify students’ conceptual understanding and thinking, planning, and creative process.” This emphasis on creativity led me to ask:
What is mathematical creativity? What does the research say about how to conceptualize mathematical creativity, and how to identify and foster it in students?
A Treasure Trove of Conference Papers
I felt that I needed to know more about this before I could fully see ISTE-TS 1. While searching for answers, I immediately stumbled upon a fantastic webpage, The 11th International Congress on Mathematical Education, Discussion Group 9: Promoting Creativity for All Students in Mathematics Education. This webpage has about 40 conference papers that speak to one of the following questions:
1. What is mathematical creativity and which mathematics students can and should be creative?
2. What is the role of the teacher and others in recognizing and promoting mathematical creativity? What is the goal in doing this?
3. How might mathematical problems be used to develop mathematical creativity? How might mathematical creativity be assessed? How do we evaluate our success in developing mathematical creativity in all students?
4. How do technology, other resources, and the environment affect the mathematical creativity of the student?
I only scratched the surface of information this website has to offer, reading just two of the papers from question 1 above.
Aralas’ (2008) paper, Mathematical creativity and its connection with mathematical imagination, helped me pull the idea of imagination into the construct of mathematical creativity, and the word “imagination” struck a chord with me instantly. Being able to picture the math – or imagine the math – has always been really important to me, even if all I’m imagining is rearranging numbers with animation. But it’s a picture I strive to see. It’s a need really. I need to be able to see it. Picture it. Imagine it.
There’s a notion lurking around that goes something like, “The math that students are learning has already been discovered. It’s already done. Where’s the room for creativity and inventiveness?” I don’t exactly know why I feel like this notion is lurking around. It’s probably a collection of things I’ve heard or thought. But Mina’s (2008) paper, Promoting creativity for all students in mathematics education, helped me feel like I could settle into the idea that the math is new to the learner, and being mathematically creative isn’t necessarily about inventing new math, but also about the way we understand the math we are being taught.
Bringing Your Imagination to Life
With these two things in mind, I came to the conclusion that students can be mathematically creative in how they imagine the math they are learning. I am picturing a project where I ask students to show what they are imagining for whatever math idea they choose, through whatever way could best convey what they imagine.
To try and make it more clear what I mean, I did the activity myself. I wanted to try and show you what I imagine when I imagine the distance formula: . I really wanted to make an animation because I imagine a moving-picture-like scene. I used PowToons to make my animation. (I’ve never used PowToons before, and I think it deserves its own investigation and blog post somewhere down the line – it was awesome, check it out!) My animation doesn’t perfectly depict how I imagine this, in the way that cartoons don’t perfectly depict 3D and real life, but it’s darn close! And it definitely represents what I really am imagining.
As a first time user of PowToons, this animation took me about 8 hours.
Of course, bringing to life what you imagine doesn’t have to be done through an animation. It could be a drawing, or a story, or something you build. It could be a demonstration. Maybe even a skit. I think one of the great things about asking students to show what they’re imagining, is that it would give students the opportunity to think about their toolbox of resources, and to search for the best way to bring their imagination to life (which is a rare in a college math class, in my experience).
Thinking in terms of my student-identity, creating this animation made me reflect on my mathematical idea and it pushed me to clarify my own thinking – there couldn’t be any fuzzy places in my mental movie. This didn’t have the collaborative element for me, but otherwise I felt like it hit strongly on ISTE-TS Indicator 1c.
With the way I’m thinking about this activity, it’s very important to focus on allowing students the space to convey their ideas, whether the ideas are canonically correct or not. I am not thinking that their imagination needs to be free of mathematical errors. However, what they create can be used to discuss any errors that emerge, or limitations to a way of thinking, but the point of the activity is not to create a “correct picture” – the point is to illustrate what they are thinking, whatever that may be.
I would love to assign this project one day and/or see what your students create if you decide to give this a try!
Aralas, D. (2008). Mathematical creativity and its connection with mathematical imagination. In Proceedings The 11th International Congress on Mathematical Education, Discussion Group 9: Promoting Creativity for All Students in Mathematics Education. Retrieved from http://dg.icme11.org/tsg/show/10#inner-documents
ISTE: International Society for Technology in Education. (2017). ISTE standards for teachers (2008). Retrieved from https://www.iste.org/standards/standards/standards-for-teachers
Ming, F. (2008). Promoting creativity for all students in mathematics education. In Proceedings The 11th International Congress on Mathematical Education, Discussion Group 9: Promoting Creativity for All Students in Mathematics Education. Retrieved from http://dg.icme11.org/tsg/show/10#inner-documents
PowToons. (2017). Retrieved from https://www.powtoon.com