There are four questions listed on the blog, I will answer each question independently.
1. A general principle could be expressed as a verbal procedure (or like a recipe?) rather than a formula with symbols. For instance:
x^2 + 7x = 60
can be expressed as:
Take a quantity, multiply it by itself, then add seven times itself, and the result is 60
The Babylonians often used terms like "length" or "breadth" or "area" to represent unknown quantities / relationships, showing that algebraic expressions can exist without x or y:
Take a length, multiply it by itself, then add seven times itself, and the result is 60
2. Not really? Mathematics HAS generalization and abstraction, but it can still start with things like measurements, visual diagrams, or patterns. Again, the examples shown in the text (say example 4.7) were still solved using practical ideas like length and width, with no signs of "let this be x and let that be y" anywhere in the solutions.
3. Said topics can certainly be expressed through words only. Take this example from geometry:
a^2 + b^2 = c^2
the words equivalent would be:
the squared of longest side of a right triangle has the same length as the 2 other sides' squared
Although one thing is definitely noticeable: words are a lot more cumbersome, where a few symbols might translate to multiple sentences in words.
4. Our students 100% do! Take this example:
x + 5 = 12
I think both us teachers & students might first express this equation like this:
A number plus five equals twelve
which can be viewed as:
? + 5 = 12
that looks like some kind of syncopated algebra.
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