When the recepient of one of the most prestigious prizes in mathematics recently credited his success to playing with a Rubik's cube, it reinforced the importance of encouraging students to think visually.
Fields medallist and number theorist Manjul Bhargava explained to our colleagues at Plus how exploring his Rubik's cube led to his award-winning mathematical breakthrough:
"I was in my dorm room getting ready to go to bed, I had these Rubik's cubes in my room and I was just looking at one. I just have this memory of thinking, "What happens if you put numbers on the corners of this Rubik's cube? Instead of just thinking of it as a simple cube, think of it with numbers on the corners". I put numbers on the corners of that cube and I did some manipulations, and I saw three quadratic forms coming out, three quadratic expressions coming out. I decided not to go to sleep, that I'd figure out what these three quadratic expressions [were], how were they related."
There's a student-friendly version of the interview with Plus, and a teacher version.
A willingness to work flexibly and explore different approaches is deemed a highly desirable quality by some of our top ranking university mathematics departments (e.g. Mathematics at Cambridge: Guidance on admission selection). Adopting a visual approach can enable students to spot patterns which might not be immediately apparent as they progress through a number curriculum packed with primes, patterns, and finite and infinite series. Nevertheless, many of the solutions we receive at NRICH come from students who mainly rely on algebraic approaches, rather than drawing on the power of visual imagery to convince their readers and justify their thinking.
Working visually not only enables students to grasp increasingly complex ideas, it has other benefits too. Visuals can also provide valuable opportunities to inspire students with the beauty of some of those images. Consider for example the power of a Sierpinski Triange for engaging students and prompting their mathematical curiosity:
Visual representations also offer a powerful approach for supporting students to recall key mathematical ideas, as noted by Lynn Steen (joint editor of 'Mathematics Magazine'):
"The various relationships embedded on [sic] a good diagram represent real mathematics awaiting recognition and verbalization. So as a device to help students learn and remember mathematics, proofs without words are often more accurate than (mis-remembered) proofs with words."
We would argue that the value of encouraging students to visualise is not in question. However, we do need to consider its overall role in the reasoning process and whether a visual image is acceptable as a proof. Consider the following conjecture: 'The sums of two odd number is always even'. Although this is a very general statement, many students understandably begin exploring it by playing with different number pairs to see if it holds. Although this approach, which mathematics educator John Mason refers to as 'specialising', enables students to try out a few examples to convince themselves about the results, to reach the level of a proof they need to go further and generalise their ideas. Their image need not be overly complex, but it does need to capture the underlying mathematics. Indeed, simplicity is often the key. After all, "successful visual representations tend to be spartan in their detail." (Borwein & Jorgensen, 2001). Consider this visual proof published by Rick Mabry (1999):
This image does appear very convincing and we would welcome more students to submit solutions which include a visual representation. However, perhaps we also need to consider the meaning of proof to determine the role of visual representations. The authors of Proof Without Words and Beyond (Doyle et al., 2014) have suggested that there are two key perspectives to consider relating to visual proofs. Firstly, the authors noted those who believe that "a proof must be expressed so as to 'explicitly draw logical connections between mathematical propositions". Since many visual representions illustrate a specific case rather than a generalisation, as is also the case when specialising with numbers, then it could be argued that visual representations are perhaps well suited for engaging and convincing others when working with large data sets, but often insufficient to satisfy the demands of a rigorous proof. Alternatively, the authors noted the alternative viewpoint that visual representations "can be far more rapidly and deeply convincing than traditional, propositional mathematical argumentation, and are therefore (in such cases) perfectly acceptable, even occasionally preferable proofs.'" Either way, it appears that visual representations are becoming increasingly important vehicle for engaging with mathematical ideas.
Visual representations enable students to access problems, and learn and recall their mathematics. They can also lead to mathematical breakthroughs, such as with Bhargava, and inspire others to begin to understand the awe and wonder of mathematics, whether through the beauty of Sierpinski's triangle or other powerful images. As computers enable visual representations to play an increasingly key role in communicating mathematical ideas, we need to ensure that our students frequently engage with visual representations, use them to support their mathematical thinking and enable them to provide convincing arguments. We look forward to receiving and publishing many more student submissions which feature visual representations in the coming months.
The NRICH secondary team
Borwein, P. and Jorgensen, L. (2001). Visible structures in number theory. Retrieved from https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/Borwein897-910.pdf
Doyle, T., Kutler, L., Miller, R., and Schueller, A. (nd). Proof without words and beyond. Retrieved from https://www.maa.org/press/periodicals/convergence/proofs-without-words-and-beyond
Mabry, R. (1999). Proof Without Words: (1/4)+(1/4)2+(1/4)3+â‹¯=1/3,"(1/4)+(1/4)2+(1/4)3+â‹¯=1/3. Mathematics Magazine, 72(1), 63.