When I solved this problem, I initially translated the given information into equations. Once it was clear I had 4 variables and 4 unknowns, I was confident I could solve it. With these sort of tools, this question was relatively trivial.
One alternate approach one might take that still involves algebra is ‘guess and check’. A solution might look like this:
(1) Make an initial guess for the number of guests.
(2) Calculate how many dishes would have been served for this number.(If you choose some number not divisible by both 4 and 3, you’ll quickly realize that this is a requirement of the problem.)
(3) Manually calculate how many dishes are required for this number of guests. If your guess results in more or less than 65 dishes, adjust your guess accordingly and repeat 2.
This approach uses a form of algebra, but doesn’t acknowledge
that the number of guests can be written as a variable to be solved for.
I think that yes, it DOES make a difference as to whether or
not we incorporate historical material from a variety of cultures. Math is a
language / tool developed in (almost?) every culture to date – by favoring one
particular culture, we do a disserve to both the students and the subject. I
also believe there is inherent value in seeing problems that originate from your
culture. Since we will likely be teaching in multi-cultural classrooms, it is
natural that we might leverage historic problems from various cultures.
Lastly, I feel that the imagery used in these questions
contributes to the historic and cultural value of the questions. In this way, they
may serve to better inspire and immerse students. Conversely, I suspect students
who are NOT interested in the culture will simply try to parse out the needed
information. With this in mind, historical problems such as this must be
introduced strategically, ensuring they relate to the content already being
taught.
Great work and reflection, Jacob! I find it interesting that you saw the alternative “guess and check” solution as still using a form of algebra. Perhaps one way to approach the “thinking without algebra” challenge is to imagine how grade 6 students would approach this problem. What would they do? Draw pictures? Use tables? I agree that there is inherent value in seeing math problems that originate from the students’ own cultural backgrounds, and it’s also valuable for students to see problems from other students’ cultures!
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