A one-way problem is a problem that requires strong conceptual understanding of a topic to solve, but the converse is not true — skilled students may not be able to solve them because they require an insight or application beyond the scope of the concept. These tasks hit hard on CC.SMP.1, Make sense of problems and persevere in solving them.

These are questions that are incredibly valuable in pushing students’ conceptual understanding, and can create the “a-ha” moments that are critical in building flexible mathematical knowledge. However, these are not questions that are useful to assess students because a student can understand a concept and not be able to solve a one-way problem.

My favorite examples:

Five Triangles is full of beautiful one-way problems. This one took me two months of trying, failing, then trying some more.

It can be solved solely with the sum of interior angles formula and a nifty use of auxiliary lines. Give it a try.

Another favorite of mine is this guy, from MathNotations, given at the lower high school level (pre-logarithms):

Or this one:

Is an isosceles Pythagorean triplet (three integers that satisfy the Pythagorean Theorem) possible?

If a student can solve it, they have an excellent understanding of the structure of exponents. However, this isn’t useful to assess a student’s understanding if they get it wrong. It’s a great puzzle to play with and reason about, but that’s where its use ends for me.

Finally, Martin Gardner’s books are a gold mine of challenging puzzles and problems, many of which can function as one-way problems at the appropriate level–not to mention they are a ton of fun.

I love one-way problems. I try to include one or two every day, usually from the internet, to challenge students who finish early — or who are ready for it and hungry for a challenge. The challenge is structuring a class where students can try — and often fail — one-way questions as a part of learning, and to understand that trying and failing is just as important to their learning as a simple problem they can solve successfully.

More than that, the Common Core’s emphasis on going deeper rather than faster pushes teachers to find more rigorous extension material for students who show mastery faster. One-way problems are a great way to push students to deepen their understanding and challenge themselves without moving faster.

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Five Triangles>[one way questions are] not questions that are useful to assess students because a student can understand a concept and not be able to solve a one-way problem

We find it unfortunate in the education reform era that educators and students must conform to the assessment paradigm, when there are better problems that, as you say, are “critical in building flexible mathematical knowledge”. We’d consider it a far greater accomplishment if a student, over the course of a year, after “trying, failing, then trying some more”, finally figured out how to solve just one “one-way problem”, rather than getting perfect scores on every concepts test.