Mathematics in School

24 Mathematics in School, November 2021 The MA website www.m-a.org.uk Decades ago I was at a social gathering when an odd incident occurred. It had emerged that I was studying maths, and one chap turned to me and said: “So, you like puzzles, then?” Incautiously, I said yes, and he went on to say: “Here’s a puzzle with a simple solution. You have a wooden sphere and you drill a circular hole through it. The hole is exactly 6 cm from the top rim to the bottom rim. What volume of wood remains?” (1) Take a moment for reflection: What was your reaction to that ? Did you immediately start to visualise the object being described? Did you reach for paper and pen and try to sketch the object? Did you start to wonder what tools you have that might help you solve the problem? How would your students react to this puzzle? Perhaps they would react similarly to how they reacted to the now infamous Hannah’s sweets question posed in the 2015 Edexcel GCSE: There are n sweets in a bag. Six of them are orange, the remainder are yellow. Hannah takes a sweet at random from the bag and eats it. Then she takes another sweet from the bag at random and eats that. The probability that Hannah eats two orange sweets is 1/3 (a) Show that n 2 – n – 90 = 0 ... Students took to social media to vent their outrage, many saying that they had prepared thoroughly for the exam and this was nothing like anything they had seen. Some even said it had reduced them to tears. It’s easy to claim that any capable student should be able to deal with the Hannah’s Sweet question. Start at the beginning, write down things you know, see what you can do with them, simplify, and the answer drops out. But the sense of outrage is very real, and should not be ignored. These students were caught by surprise, and felt that the question was deeply unfair. Many of these students will have spent hours diligently practising exam questions from past papers, and reviewing the material covered in class. • How can we help them? • What’s missing from their otherwise excellent exam preparation? The very act of having exams as an assessment method will distort the teaching (an example of “Campbell’s Law”). There are then inevitable accusations of “Teaching to the Test”, and claims that students react to trigger words to produce calculations that may or may not be relevant. However, we know that some students will react with horror at a question that looks unfamiliar, and potentially have their confidence damaged to the point of being irrecoverable. The apparently unresolvable paradox is that students need to be given exam questions they haven’t seen, but they also need to be properly prepared. By “properly prepared”, we don’t just mean to get good grades on their exams, we also mean to prepare them for using their mathematical skills in the real world, where questions and problems don’t come in neat packages that look similar to a hundred examples they’ve seen before, in a context that makes it clear exactly what technique or tool to use. The challenge at hand is to have something which: • Promotes resilience; • Increases engagement; • Practises some mathematical skills; • Is feasible in a curriculum pressured setting; • Is cost effective (time required versus benefits gained). We know that students can solve equations once they have them, and we know that they can apply a technique when they already know what technique is needed. The help they need is to develop the skills to find the equations, to identify the technique, and to make a start even though the question is unlike anything they’ve seen before. This requires practice and experience, and the resource is not past exam papers, nor a textbook. Enter the Puzzle … There are many different types of puzzle, and some of them are perfect to help students gain experience, resilience and confidence, and to get them engaged. Some puzzles require careful visualisation of the exact situation, for example: It takes 12 minutes to saw a log into 3 parts. How much time will it take to saw it into 4 parts? (2) Others require a restraint from leaping to do the obvious sums, such as: Three horses are galloping at 27 miles per hour. What is the speed of one horse? (3) Piquing Performance with Puzzles by Colin Wright