Issue 49 – July 2023

Welcome to our email roundup of all things Cambridge Mathematics

Dear reader

Marhaba,

George Polya, considered by many as a pioneer in the field of problem solving, wrote several books on the topic and created his renowned four-step process for problem solving: 1) understand the problem, 2) devise a plan, 3) carry out the plan, and 4) look back. This process is still widely used, whether people know they are applying it explicitly or not, and it is still considered a staple in problem solving. One could ask: do we always have to use such an approach? What does understanding the problem really imply? Do all students need a plan, and if so, should they follow the same one? What role do giftedness and creativity play (a lot has been written about the definition of each and the link between the two, as in this paper)? What about "thinking outside the box"?

It is uncertain where the phrase "think outside the box" originated. The most common story though is that it is closely related to the nine-dot puzzle. This puzzle involves nine squarely arranged dots; one is asked to use a pen to connect all the dots with four (or fewer) straight lines without lifting the pen off the paper. Try it! The secret to solving the puzzle (spoiler alert: a hint will be given next!) was believed to be the ability to notice that the lines can go beyond the outer boundary of the largest square, but since many ignored thinking outside this boundary, they presumably failed to find the solution. Or so it was believed for many years. In later studies, it was shown that even if those attempting to solve the puzzle knew that going beyond the outer border was an option, it did not significantly affect their ability to solve the puzzle. 

Let us pause here for a while, rewind, and before talking about hints and how they affect a solution, consider the problem itself. Let us investigate its wording, its formulation, the conditions and hypotheses within, and let us ask a few preliminary questions, because their answers could affect our understanding of the problem and alter our plan for looking for a solution. They might even take the problem to a different place from the one intended by the person who wrote it. Does it matter what the size of a dot is? Are these dots circular in shape, and if so, should the lines we are supposed to draw pass through the centre of each dot? Can the lines be as thick as we want? Are we solving this problem in Euclidean geometry? Should lines stay on the same planar level and can folding the paper be an option? Can lines intersect or pass through the same dot multiple times? Based on the answers to these questions, and many more, a lot could change, even leading to the possibility of solving the puzzle with one line only (I will leave you the pleasure of finding both the questions and answers that could lead to such a solution). 

As educators, we certainly know that learners keep surprising us. While algorithms and solving processes are important, and while some students think outside the box and others stay inside it, many go beyond it and follow paths that we never thought existed, sometimes even creating new boxes! Bertrand Russell once said that "the pure mathematician, like the musician, is a free creator of his [sic] world of ordered beauty" (p. 33). Sofya Kovalevskaya stated that "it is impossible to be a mathematician without being a poet in soul" (p. 316) following in the footsteps of her mentor Karl Weierstrass who said that "it is true that a mathematician who is not somewhat of a poet, will never be a perfect mathematician". So, like a musician and a poet, allow students enough room to create and let their imaginations roam free, let them question everything, and remember that while some non-orthodox methods lead to correct answers and others do not, and while traditional approaches are preferred by some but not all, we each learn differently. It is important to ensure that all methods used by students are catered for, all thinking processes are considered and discussed, and all attempts are appreciated and encouraged. There is no pre-defined nor unique road to success, and the path to great discoveries is not always a straight one and is often filled with failed attempts. 

In this issue, speaking of boxes, we have chosen for you a blog by Darren and Lucy showing us how to build a Corsi-Rosenthal box, because the air quality in a classroom matters. We also interviewed Elena Nardi who shares with us her experience of mathematics at an early age. The icing on the cake is a new Espresso by Fran and Lucy, who build up and break down shapes! 

Finally, before I wish you all a fun and relaxing summer, I must tell you about our various fascinating resources which we are looking forward to sharing with you when you are back from the well-deserved break. We are especially excited that we will be bringing JourneyMaths (our innovative Professional Learning environment) back into the spotlight with a free webinar. This will take you on a guided exploration of the mathematical terrain, explore how to get the most from JourneyMaths, and give you a taste of some of the new features and developments we have planned – so stay tuned for a special September event! I would also like to take this opportunity to thank all those who filled in the form that we included in our past two newsletters; we will share their contributions with you soon. 

Best wishes, 

Rachad

P.S. Please send us your comments and suggestions on our blogs, Espressos and on this newsletter by replying to this email, and don't forget to follow us on Facebook, LinkedIn and Twitter!

Espresso

Mathematical Salad

These are the Mathematical Salad items which were published on our website in June: