The following problem has the internet all abuzz.
The problem comes from the Edexcel Maths GCSE paper—which, as I understand it is an examination for UK 16-year-olds with aspirations of studying at a university.
The buzz is fascinating for a number of reasons: some positive; some not so positive. But all are worth thinking about. First off, many students found the problem to be very difficult. I’m not sure why this is, but I’ll offer some speculations as we move through this post.
(I’ve had to convert my Word document into graphics because the equation objects wouldn’t paste into the WordPress format.)
Why I Like the Problem
There are a few very nice things about this problem. First, it asks students to demonstrate something, rather than to find a single solution. As a math teacher, I am always looking for demonstrations of understanding, rather than the consequence of a search that may or may not simply be a matter of “getting lucky”. On that note, I am pleased that this question requires a human marker. No multiple choice or numerical response item could replicate this. To demonstrate understanding, students must use natural language (i.e. English) and structured symbolic representation (i.e. numbers and letters with mathematical connectives) to convince another human mind that they understand. This is good.
Second, there are at least four straightforward methods of solution available to an average post-secondary-bound high school student. No one is bound to a single method. If I were to use this question in class, I would have choices for how to approach it, depending on the goals of the lesson.
Solution #1—Elementary Probability
This is probably the most straightforward way to approach the problem. We use two pieces of basic knowledge.
This is not hard, and it involves very easy manipulation. I’ll go over the next two solutions quickly, as they are very similar in form, but not in conception.
Solution #2—The Fundamental Counting Principle
This is a close cousin to #1. The idea is that if you have n ways to do one thing, and m ways to do another distinct thing, then you have n×m ways to do both. So if you have 3 pairs of pants and 4 shirts, you have 12 different outfits you can wear. We now see that Hannah has n(n-1) ways to pick two sweets out of n, (She ate the first sweet, so she only has n-1 left for her second choice.) By similar reasoning she has 6×5=30 ways to pick and eat two orange sweets. From here, the arithmetic and algebra are the same as #1.
Solution #4—All Guts no Math
One of the standard complaints about this problem (and others like it) is that it is so artificial. Sure it gives us a protagonist, a storyline and some talk of yummy sweets, but this is a million miles from real-world mathematics.
I hear this kind of objection all the time, and frankly I don’t get it. One of the other complaints about mathematics is its coldness and distance from human creativity. Well this is a silly story. It’s a flight of fancy; it’s a goofy narrative that paints a picture. Yes, very few people will do math when eating candy; we all know that. But no one ever complains that fiction is unlike real life, because it reminds us of real life and gives us something to think about. As do silly problems like this one.
Now it is often easy to create a problem that can be stated with or without the narrative. Indeed, it is often the case that the narrative obscures the mathematics. That isn’t the case here. A scenario free statement of this question would be quite difficult to write without absolutely obscuring the clear and simple question being asked.
So why did students find it hard? My guess is because the question is surprising. Students are often lulled into calculating numbers, solving equations, concretely answering problems. In this case, they were given a concrete problem and were asked to show that the problem leads to an unexpected equation. The work isn’t difficult, but the interpretation and strategy-selection were surprising. At least that’s my supposition at this time.
Whew. This is more concrete mathematics than I aim for in this blog. Hope it brought some thought and (dare I say it?) some fun to your day.
If any students who wrote the paper happen to stumble onto this blog, I’d love to hear your reflections on the problem.