Are the two kinds of understanding distinct/separable?
If I were to define relational and instrumental understanding, relational understanding entails depth and explores the 'why' something works, whereas instrumental understanding is algorithmic and/or procedural mastery. We came to the conclusion that they are certainly two distinct terms on a semantic level, but they are also closely related under the umbrella term of "understanding". They are not mutually exclusive, but complement each other. Instrumental understanding can be improved by relational understanding, and there are also cases of the reverse being true, especially in more technical disciplines. For example, you could learn all the theory behind the usage of a woodshop tool, but it only proves beneficial once you gain an instrumental understanding of how to use it.
Is there a "best" order to teach them?
I believe that this is very dependent on the content that we are trying to get students to understand. Let us take the case of learning the formula for a circle's area, which involves the number pi. Of course, the relational aspect is to understand why every term is where it is, whereas the instrumental aspect seeks to train students to apply the formula masterfully. One could argue that instrumental understanding is better to introduce first, as it builds familiarity with the concept that would make relational understanding easier to acquire. There is also an argument of whether the students have the base knowledge to understand what "pi" is before being taught the 'why it works' for the formula... though a counterargument would be that pi is a simple enough concept to teach together with the relational understanding of the circle's area, As an example of relational understanding being better to teach first, take exponent laws. Showing how one law is derived can be akin to teaching students how to fish, as they can derive the rest of the laws by applying the same methods.
What kinds of activities promote each one?
For instrumental understanding, conventional exercises come to mind first. The repetition of using the tools and formulas given is designed to build mastery in using them. Such repetitions may be administered through worksheets, homework and the like. Simple tool demonstrations (example questions) are another activity that may be used.
For relational understanding, I was able to come up with a few ideas. First, I noted the sparsity of proofs in high school mathematics, especially compared to university. One activity could be to get small groups to try and derive a formula on their own (such as the exponent laws, which are an easier exercise due to how the laws are derived similarly). Another set of activities deal with the tangible; matchstick exercises to visualize concepts, or graphing calculator playground labs (such as Desmos, which has functionality with sliders and variable setting). In high school, I fondly remember a Desmos lab exercise where we played with function transformations to see how a graph changes as certain values are changed. This gave me a relational understanding of transformations, whereas simply knowing that "c = horizontal movement" and "a = vertical stretch" would've been helpful, but shallow.
How to assess understanding?
For instrumental understanding, I turn to the conventional once again. Much of conventional testing is what I like to call "applied regurgitation"; while the student doesn't simply repeat answers with exact values, they are repeating the steps needed to use their instruments properly. In a vacuum focused on instrumental understanding, this is a fine way to assess.
However, relational understanding has been considered more challenging to assess. My discussion group seemed to agree that talking and questioning the student was the best medium to assess relational understanding. Regarding the implementation, I had an idea that I liked: if mathematics is a language, then why not take a page from how modern language courses are taught? I recalled my old French 12 class where the teacher included a "casual conversation in French" section in the final exam process, and was scalable with a class of thirty. As such, the idea was this: once in a while, why not allow students to show their deeper understanding in conversation? We could set a rubric for what sort of things we look for in our conversation, and assess accordingly not just on their specific relational understanding of whatever is covered at the time, but also their general capacity to mathematically reason.
As far as "understanding" in a general sense goes, this is a greyer issue that depends on the set student learning outcomes. Do we only want them to understand instrumentally? Do we want them to have deeper understanding on top of their instrumental mastery? Once this is decided, then the reflection loops back to the previous two paragraphs on how to assess specific types of understanding. There is also a question of how much to weigh each type, but to me that is again a question of SLOs.
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