ftp://ftp.ams.org/pub/tex/doc/amsmath/short-math-guide.pdf

(Then you can do a followup post on the wordpress latex quirks like requiring that everything be in one line, and use of aligned instead of align, …) .

The textbook presented solving matrix equations using the inverse by giving a formula for the determinant, then giving a formula for the inverse (of a 2 by 2 matrix), then showing how to solve a matrix equation, etc. If I had merely introduced the material in that way, almost all of the kids would have forgotten it (or worse, thought that they knew all there was to know) and I agree that it would be a waste of time, or worse. As a class, though, we derived the formula for the inverse of a 2 by 2 matrix using matrix multiplication and algebra. Thus, the determinant has meaning.

The textbook introduced solving matrices using row reduction by establishing the “row operations.” I insisted on making the connection to algebra so that they could see that all row operations are legal only because the corresponding algebra moves are legal.

I know that we only scratched the surface of the vast subject of linear algebra, I would contend that these kids will benefit greatly from the challenge of looking at real mathematics. I do agree that the surface level (i.e., looking at the surface, not scratching it) look that most textbooks give is useless.

Generally the unit circle is originally given by teachers in whole, so students don’t have to translate anything, just remember. (It doesn’t have to be.)

With the divide by zero question context matters here, but it is possible for students have to reach the level of creating new mathematics to get a solution. I don’t think the thought process is cut-and-dry, anyway.

Bloom, et al. describe synthesizing as (from my 1975 copy of the taxonomy)
“5.10 Production of a unique communication
5.20 Production of a plan, or proposed set of operations
5.30 Derivation of a set of abstract relations”

Your example “Construct a diagram of the unit circle” is none of these. Because the student isn’t constructing anything unique — but showing a commonly understood form of an abstract concept. What the objective is describing fits better under what you refer to as “Understanding” and Bloom originally calls “Comprehension”

“2.10 Translation
2.20 Interpretation
2.30 Extrapolation”

Particularly “Translation” which as Bloom, et al. writes “Translation from symbolic form to another form, or vice versa” and uses this example specifically “Given geometric concepts in verbal terms, the ability to translate into visual or spatial terms.”

You are correct that your first example “Explain why the graph of the tangent has asymptotes” is higher level than the second. It is indicative of Bloom’s level Analysis:

“4.10 Analysis of elements
4.20 Analysis of relationships
4.30 Analysis of organizational principles”

Your objective belongs to 4.20 Analysis of relationships as, I would suggest, you are asking the student to analyse (then state) ‘why’ the relationship exists.

Your example and analysis fails to make your point except that there is confusion about what the verbs can mean. You have proven that the first example is higher order. I agree with you that the taxonomy shouldn’t be seen as just a list of imprecise English verbs, but requires a deeper understanding of the system and principles themselves. So the teachers of teachers who do not evoke this are failing to provide a very useful tool.