They do. The realization that something can be applied in other contexts is what takes them from feeling overwhelmed that each step is something discrete and isolated to memorize to being excited about making it make sense.
This also happens after teaching them how to use their calculators. When they begin to be able to visualize what the approximate bounds are by understanding eg- an expression and what it applies to, it's the difference from someone who only reads word by word and someone who reads with respect to the content of the arc or chapter.
So much of math is contingent on parallels to other subjects, and I've always felt not enough was done to relate to students the similarities between the formal rules.
Ie- when teaching linguistic notation for grammar and syntax, you can have students reduce sentences to expressions (a la Backus–Naur form) and then "solve" for them using, eg- a semantic net. This is often much more comfortable to the numerically adverse, since the logic of language is often understood more intuitively. It can also be taught at any level using simple substitution, parts of speech, etc with the added benefit as a primer for its applications to machine learning.
This also happens after teaching them how to use their calculators. When they begin to be able to visualize what the approximate bounds are by understanding eg- an expression and what it applies to, it's the difference from someone who only reads word by word and someone who reads with respect to the content of the arc or chapter.
So much of math is contingent on parallels to other subjects, and I've always felt not enough was done to relate to students the similarities between the formal rules.
Ie- when teaching linguistic notation for grammar and syntax, you can have students reduce sentences to expressions (a la Backus–Naur form) and then "solve" for them using, eg- a semantic net. This is often much more comfortable to the numerically adverse, since the logic of language is often understood more intuitively. It can also be taught at any level using simple substitution, parts of speech, etc with the added benefit as a primer for its applications to machine learning.