The Dishes Problem

 My solution using conventional algebra:

My solution without symbolic algebra:

If 2 people eat one dish of rice, 3 people eat one dish of broth, and 4 people eat one dish of meat, then each person eats 1/2 a dish of rice, 1/3 a dish of broth, and 1/4 a dish of meat. Each person then needs 1/2+1/3+1/4 dishes, or 13/12. There are 65 dishes and 13/12 dishes per person, so there must be 65/(13/12) people, or 60. 

I'm not certain whether or not this method would still be considered algebra, since the mathematical procedure is in many ways the same, but it seemed like a reasonable approach provided the solver has an understanding of proportions and adding fractions. I think that the context of a story puzzle can sometimes feel contrived, but placing a puzzle in its historical context avoids that feeling. Something about using the same story that was originally used lends it validity, at least in my eyes. I think it is hugely important to offer problems (and solutions) from many cultures and traditions in mathematics. Students are primed through social dialogue and prevailing educational culture to think of mathematics as an activity done by only a small subset of a single group (that being White, European men). This perception means that students who don't fall into similar demographics are being told that math isn't for them, a message they probably hear far too much as it is.

As for enjoyment, I think that I do get some additional satisfaction in approaching the problem because of the story, provided it doesn't feel too contrived, as I addressed above. I believe wholeheartedly in the motivating and imaginative power of story, and I especially think that story can be helpful in prompting students to look for other ways of solving the problem. I'm reminded of a study that was done with child street vendors (I can't remember in what country, but somewhere in Central America). The children were sat in a math classroom and given tests on simple arithmetic, on which they did very poorly. Then, the students were given the same problems but in the context of making change and purchasing goods (something they did daily). In this new context, the children performed far above what we would expect at their age level. Context is everything.