I’m doing a lot of ACT prep tutoring right now, getting students ready for Saturday’s test date. While I can’t say I endorse the idea of standardized testing, it’s been interesting to re-visit the process almost 10 years after I did it myself. In particular, it’s interesting to read through the questions on the math section through the lens of a math degree. The ACT math questions are all answerable with a high school understanding of mathematics, but their mathematical foundations are far deeper, and I find myself appreciating the ideas on which the question is founded, even as my students can only skim the surface of what they mean. The following question is a good example:
The 12 numbers on a circular clock are equally spaced around the edges of the clock. Belinda chooses an integer, n, that is greater than 1. Beginning at a randomly chosen number, Belinda goes around the circle counter clockwise and paints in every nth number. She continues going around and around the clock, painting in every nth number until all twelve numbers on the clock are painted. Which of the following could have been Belinda’s integer n?
You don’t need an advanced math degree to answer this question. A few circle diagrams and trial and error should produce the right answer; a quicker thinker might notice the pattern that numbers like 2 or 3 won’t work because they divide into 12 and so the answer is likely D or E.
Meanwhile, as the all-knowing tutor with a master’s degree in math, I see this question for what it really is: group theory. This is a question about generators of the cyclic group Z_12. Generators of Z_n must be coprime with n, so that answer has to be 7.
This is, however, not the sort of thing that you can explain to your tutee, especially when they get the question right without the benefit of a course in Abstract Algebra. It gives me pause however: If students can puzzle their way through a problem like this, do they also have the capacity to handle group theory? If not, where does the bridge break between the problem and the theory…and why?