I am a math teacher and I have been for a decade now. One of the foundations of my philosophical approach to teaching has to do with Synthesis. For the purposes of this query, I consider Synthesis to be defined as The process of putting discrete ideas together to form a new idea. Consider that you could teach someone to dig holes, and you could teach that same person to pour concrete. Then, they might employ synthesis to put those two skills together; they dig a hole and then pour in concrete to make the foundation for a pylon. This is technically different from either skill that you taught them, but it represents a combination of the skills in question.
As a math teacher, I teach students about fundamental skills and concepts. The domain of a relation represented as a graph is the set of the x-coordinates of the points on the graph. You can translate a graph two units to the right using an algebraic form. To synthesize these two fundamentals, I might offer an equation of a transformed graph and ask about the domain of it.
But, here's the rub! I ask this, but I've never asked them to do this specific task in class. That is, I have never given them: "The function f has domain (2,4). The function g is defined by g(x)=f(x-2). What is the domain of g?" as an in-class problem. But, I have taught what domain means and what transformations are.
Is it okay for me to ask this sort of question on a graded assessment?
I believe that the answer is a resounding 'Yes!' but so many people have questioned my stance on this - people whom I respect - that now I'm wondering if I'm just being stubborn.
To be clear: I teach skill/concept A. I teach skill/concept B. Skill C requires the use of both A and B in concert with one another. I ask a question requiring skill C on a quiz but have never asked a skill C question to these students before. Am I being unfair?
My argument typically focuses around the idea that sufficiently advanced mathematics requires students to write proofs, and those proofs are typically unique to that specific theorem (even though many structural elements will be similar to other proofs, there will be details that are different). So, advanced mathematics requires its students to assemble concepts from different places to answer questions in an open-ended format. Hence, I think it should be a necessary part of any assessment.
But again, am I just being stubborn?