Problems That Can Be Solved

Problems That Can Be Solved

Math puzzles and cryptic problems

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Henri writes about the connections between teaching math and constructing crosswords.

When I’m not constructing and solving cryptic crosswords, I am a math educator. Yet I have never quite fit into the culture that dominates math education in the US. Many people believe that learning math consists of memorizing a broad array of techniques, so that when a student is presented with a problem on a test, all they have to do is identify what sort of problem it is, remember how to solve it, and carry out the required steps in succession.

In this view, understanding is something that comes later, after “solving” many problems by rote, and good teaching consists of presenting the methods that are easiest to remember. The student is seen as a programmable entity, the teacher as the programmer, and the curriculum as a list of procedures. Successful students are the ones who are obedient, have a good memory and pay close attention to details as they execute the programs.

Alas, this approach is spectacularly ineffective. We live in a country where the vast majority of college graduates, all of whom have taken three or four years of high-school math, cannot recall anything beyond middle-school arithmetic. Such people often say things like “I just don’t have the math gene,” or “I’m not a math person.” In reality, they are the victims of a largely anti-intellectual culture which has reduced one of the great achievements of the human spirit to a boring collection of algorithms that lack any connection to actual meaning.

Bringing a puzzler’s mindset into math education is a partial answer to this sorry state of affairs. My work as a teacher and curriculum developer has been much enhanced by my experiences as a solver and constructor of cryptic crosswords. In both cases, I create puzzles for the student or the solver. In both cases, the most satisfying puzzles are the ones that initially seem too difficult. And in both cases, while I want the student or solver to be challenged, I also want them to succeed. Managing that balance is the heart of my role as a crossword constructor, and as a math educator. It involves including easier questions along the way, to offer entry points into the puzzle and to make forward motion possible.

Moreover, I want students and solvers to enjoy the experience as they tackle the puzzles. In part, this is related to maintaining another tricky balance: I want the solvers to get to know the ropes, but I also want them to be surprised sometimes. Which is why I relish the inclusion of nonstandard problems in my math classes, and unorthodox clues in my crosswords.

In short, my day job is not all that different from my Nation job—luckily, I enjoy both!

This week’s clueing challenge: MATHEMATICS. To comment (and see other readers’ comments), please click on this post’s title and scroll to the bottom of the resulting screen. And now, four links:
• The current puzzle
• Our puzzle-solving guidelines | PDF
• Our e-books (solve past puzzles on your iOS device—many hints provided by the software!)
• A Nation puzzle solver’s blog where every one of our clues is explained in detail. This is also where you can post quibbles, questions, kudos or complaints about the current puzzle, as well as ask for hints.

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