I think using algebra or single letter notation in math helpes solving problems in a much easier fashion and also easier to present and understand. However, I don't think it's necessarily a must in terms of solving math problems, a lot of math questions can be represented in words. So why do we have all the notations that we are using now? From the book Crest of the Peacock, we know that the fundamentals of mathematics were gathered from different countries/regions, in order for them to communicate notations had to be invented/created. It made more sense if the communication of math only happens in the same region, but communicating with others from outside the region might bring confusing. I would rather think of notations as the "regulations" for the math world. It's a universal tools for people to communicate in mathematical language at a ease.
As for whether math is all about abstractions and generalization, I don't agree with that. As the concept of math is so broad and everyone learns/understands math differently. However, I do think abstraction is a great tool for students to ease into math as it gives the students the freedom to potray math as they like. Generilzation is also important as it gives the students some foundations/facts to imply math in a abstract way. Both are curial in math learning.
I think using pictures/illustration would be a good way to state some math problems like geometry, or easy number theories. Stating calculus without algebra would be very hard. I still think notation's excitence has its purpose there.
Lovely, Zoe! I think you're the first person in the class to bring the idea of mathematical communication (across languages and cultures) to the discussion of agreed-upon algebraic notation. That idea about regulating the symbols we use is important too, so that we can understand one another. Very interesting discussion about visual representations like pictures and illustrations too. Could we use several of these means of notation together, as teachers?
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