To be fair, we probably spent more time working on the Handshake Problem than was entirely acceptable, but I enjoyed the time we spent on the project immensely.

The problem asked how many hands one’s spouse shook at a party of people in couple groups, where everyone shook a different number of hands. For our class, it was a perfect way to engage in discussion and work together to find some sort of a solution.

I think the biggest reason I enjoyed the process so much was that it was a great example of how effective problem solving can be, and what ways we can modify and change a question in order to help students come up with an answer. For us, the problem was all about assumptions. You were given very little information, and, as such, needed to make assumptions in order to know where to start with the question. The two common assumptions were that one could not shake their spouses’ hand, and one could not shake their own hand. Using those two assumptions, we attempted to use a chart to solve the problem. We came up with an answer that was almost correct, we just had two individuals shaking the same number of hands. But we only came to the conclusion when we disregarded the assumption about spousal handshakes. The other group got a better answer because they made another assumption we had not – people were allowed to shake an individual’s hand more than once.

I still don’t know the exact answer for the question, but it does bring about many issues when you try and solve the problem. In class, we talked about how the entirety of how you approach solving the problem comes from what assumptions you make. For example, you could say that a person shook the hands of twelve people outside before they entered the party, and then shook ten hands inside, and so shook 22 hands. Basically, the question can have infinitely many solutions if you do not stipulate what is and is not acceptable.

The technique applies to all problem solving. If you make a question for students that is based on them coming to the correct conclusion, then there must be guidelines in place that ensure that students do not stray from the path too drastically. But, if you want to encourage discussion, and deeper thinking, perhaps having an open ended question is better, because then students have to think, and the answer that each group or individual comes to is dependent on their own creativity and mathematical process.

Both ways of problem solving, guided and open ended, I think, are very useful for very different reasons. A guided problem solving approach is useful for helping students learn a unit, or better their skills with a particular topic, while open ended is useful for creativity and enjoyment in the classroom.