The Dishes Problem

How could you solve this puzzle without algebra (or at least, without the algebra we are used to)?

Because this problem deals with smaller numbers, it would be feasible to list numbers and visually "assign dishes" to the appropriate people. In my case, I listed numbers representing people, circled every second number to represent sharing of rice, circled every third number to represent sharing of broth, and circled every fourth number to represent sharing of meat. Afterwards, I counted all the circles in order until I reached 65 dishes.

Note: 5 and 7 aren't circled - I just got carried away with my circling!

 

In the case with bigger numbers, the LCM could also be used. For example, had the number of dishes been 975, we could use the LCM of 2, 3, and 4, which is 12, to determine the number of people present. The objective here would be to find a multiple of 12 that adding the results of dividing that number by 2, 3, and 4 would give the correct number of dishes. It's similar to our normal algebraic way to solve but more of a guess and check method that focuses on the LCM as a guide to the answer instead of adding fractions, using an unknown as a placeholder, and making explicit use of the overall number of dishes in the calculation like our algebraic method generally would. Here is an example of the LCM/guess and check method with 975:

Does it makes a difference to our students to offer examples, puzzles and histories of mathematics from diverse cultures (or from 'their' cultures!)
I believe it can make a difference to represent cultures if presented correctly. That is, it's not enough to simply give the problem without context. If this was just a problem on an exam or worksheet, most students would likely extract the numbers from the problem and solve it. I myself visualized the dishes but didn't really make the connection to Chinese culture because I ignored the context. To bring the cultural significance and representation to our students' attention, it would be more useful to introduce the problem as one from the 4th century CE in China from the Sunzi Suan Jing (Sun Zi's Mathematical Manual). We could also talk about how the Sunzi Suan Jing has various problems and interesting methods of solving them in case students are interested in looking into it. Having this short introduction would place a higher emphasis on the origin of the problem and potentially engage students more than if they just went straight to solving it. For those of Chinese background, it may have been a nice tidbit to see the problem had reference to their culture, but stopping to recognize its origin would be more meaningful because they could feel more connected to the historical significance and feel a sense of pride as well. I remember when I was in school and saw names like "Raj" and "Padma" in textbooks, I would think it was neat, but would move on quite quickly. It would have been more meaningful to me if the problem was reflective of a real life situation or something more relatable to me so that I could potentially talk more about it with my peers who were not of Indian background and have an opportunity to share my culture with them or read more about my culture's history with others who were interested.

Do the word problem or puzzle story and imagery matter? Do they make a difference to our enjoyment in solving it?

I do think one of the benefits of word problems or puzzle stories is the visualization aspect. For instance, before even thinking about the math, I imagined the plates (not the food on the plates for some reason) being shared among different people. I think it gives a tangible aspect to the question that would be missing in simple "Solve for x" questions. I personally enjoy word problems because of this tangible aspect, but certain word problems can be tricky for the same reason. For example, there was a word problem about a longhouse in one of my math classes in high school. I grew up in an area where everyone knew what a longhouse was either due to our exposure of them through school, because our local museum has a performance longhouse in town, or because they are Indigenous themselves and it's part of their culture. However, if that same problem was given to my cousin in India, she would have to look up what a longhouse was first to be able to visualize it, or simply skip over the context and solve the question. 

Another part that makes word problems or story problems enjoyable is that solving them is like solving a mystery. In this case, how many guests did the chef serve? There is a game-like aspect in solving the question that makes you strive to figure it out just so you can know the answer.