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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?

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    $\begingroup$ Are you teaching "significantly advanced Mathematics"? Are your students meant to be synthesising concepts at this point in their mathematical education? Your questions should require a level of synthesis that is appropriate for the level of course you are teaching. The level of a course is not just about how much material is on the course or how complex it is, but also how deeply the student is expected to understand and manipulate the content. $\endgroup$ – MadScientist Jan 15 at 16:02
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    $\begingroup$ An interesting question. I will answer it by saying that I know very little about how people teach children fundamental arithmetic and that I would probably have a different opinion about doing so. However, once a student reaches the point of making abstractions ala algebraic expressions, then I think I would answer your question with my own: Is there ever a point where a student should not be synthesizing concepts? That you are asking suggests that you consider the answer to this to be 'Yes', but I cannot think of any. That said, I teach an "Advanced" class at a math/science magnet school. $\endgroup$ – Mark B Jan 15 at 16:16
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    $\begingroup$ What do you mean by "The function f has domain (2,4)"? Do you mean $f$ consists of the single ordered pair $(2,4)$ (in which case, the domain of $f$ is $\{2\})$ or $f$ is a $2$-variable function whose domain consists of the single ordered pair $(2,4)$ (in which case, the domain of $f$ is $\{(2,4)\})$ or something else? If you're going to deal with formalisms at the level of "functions are a set of ordered pairs", then you should be more precise in your statements. (moments later) OK, I think you mean "interval $(2,4)$". Maybe insert the word "interval" just before $(2,4).$ $\endgroup$ – Dave L Renfro Jan 15 at 17:50
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    $\begingroup$ I think the question is very appropriate, but students should have seen examples of this type of question before. My experience has been that these kinds of things are good, but they should be done during lecture or in guided worksheets, not sprung on students in a graded test (unless maybe as an extra credit problem, which I did quite often for things like this). $\endgroup$ – Dave L Renfro Jan 15 at 17:53
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    $\begingroup$ It is normal to expect students to synthesize concept C out of A and B ("high-order thinking" has been a trendy buzzword for decades) provided that A and B are clearly defined, and the domain of the expected concepts is known to the students. As worded, there is not enough info to give an answer. Do you expect (4, 6)? Why not (5, 5.5)? Or do you expect "any interval enclosed by (4, 6)"? Is it a multiple-choice test? Do you accept different answers? Do you curve the results? If you do (which I think is a cop-out), then the issue will take care of itself if no one solves it ;-) $\endgroup$ – Rusty Core Jan 15 at 18:05
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I think the given example is highly appropriate. You cannot cover every possible combination of ideas in class. Students display understanding of a concept (rather than "recipe following") by showing the ability to adapt at least a little bit to novel conditions. I think the problem you gave is a great homework problem.

I personally like homework to be a place for students to grow their understanding, which means engaging with novel and creative ideas.

Exams, with their time pressure, are not appropriate places for such questions however. I like to think of exams as "basic content mastery checks".

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  • $\begingroup$ It depends on the exam. In my day, the final paper of the final year math exam at Cambridge (UK) usually had a few open-ended questions of the general form "here is the definition of a math structure you probably haven't seen before. Prove some interesting theorems about it." (General instructions: "Time allowed: 3 hours. Attempt as many questions as you wish, but good answers to two or three questions will gain more credit than superficial answers to more"). $\endgroup$ – alephzero Jan 16 at 7:03
  • $\begingroup$ I agree that having it be untimed would be best, but then it becomes difficult to ensure that the work belongs to the student when they can take it home to think on. Still, I think this is the most applicable and straightforward answer. Thanks. $\endgroup$ – Mark B Jan 16 at 20:00
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I would frame this issue a little differently than you have. I think it's unreasonable, at least in the context of courses which aren't well into a math major, to ask students to do something they have not been taught to do. That is, the problems on an assessment should be the same as problems they've seen already.

The catch is that "the same" is actually a really ambiguous term, and it depends on how abstractly you think about the steps - and, more importantly, how abstractly it's appropriate to ask your students to think about them.

To make that concrete, suppose that in class we did one of those calculus optimization problems where someone's building a fence out of a couple materials with different prices, and you want to maximize area given a certain budget. "The same" task could mean:

  • literally the same problem,
  • the same words, but the numbers have changed,
  • the same setup, but this time we want to minimize price with a fixed area,
  • the same setup, but the geometry of the fence is a little different,
  • a similar setup, but this time the geometry of the fence is changed in a way that means the algebra needed to solve the problem looks different,
  • a different word problem that leads to similar equations,
  • a different word problem that leads to unrelated equations (i.e. any optimization word problem).

An assessment is typically going to have some range of levels of abstraction, and what that range should be depends on the specifics of the class (and maybe the specific assessment - I might ask for more flexibility in approaching a subject on a unit test than on a quiz in the middle or on a final exam).

Getting the top of that range right is difficult, especially because it's easy for us, as specialists, to underestimate how unfamiliar a problem will look to our students. When we aim too high, students start to see math as an endless series of inexplicable magic tricks they could never reproduce. When we aim too low, we end up with cookbook math, where students stress themselves out memorizing fifty algorithms and then plow their way through the exam with it.

To bring all that around to your specific question, I suspect that you're currently aiming a little too high, and you're struggling with that because the only alternative you're considering is too low. At the level of class you're describing, I think you probably should be asking them to solve problems where you haven't shown them how to do that exact task in class.

But if you want them to synthesize ideas from class, you probably need to have taught them how to synthesize those ideas in a pretty explicit way: if you want them to solve a problem by combining A and B, you probably need to have done a problem in class where they needed to combine A and D, and also a problem where they needed to combine E and B.

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    $\begingroup$ "in the context of a course for non-majors" Where do you live, that high school students have majors? This question is tagged "secondary-education". $\endgroup$ – nick012000 Jan 16 at 5:49
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My advice is to minimize the amount of such synthesis required. Don't make it a large fraction of your tests, if at all. Teach the students the methods you expect them to display on the exam. Not something requiring some spark of creativity. Program for success. Creativity is tough in general and even tougher under test conditions.

If you push too much creativity, you're basically testing IQ more and more and subject matter mastery less and less. (I acknowledge there is always some mixture of the two, talking relative amounts.) The smart kids have enough advantages as it is.

This is not to say there is no value in synthesis or connections. And I quite enjoyed the piling example. But it may be more appropriate to derivations, lesson plans, etc. (Consider, would you really expect kids to derive the next lesson, not yet covered/worked, under test conditions?)

Maybe a way to indulge yourself is to restrict these synthesis questions to a single "extra credit" question on your tests. Keeps the smart kids interested. And psychologically will be more acceptable to all the students (that it is "extra", not core).

EDIT: Given you are teaching pre-calc (inferred) with excellent students, some degree of puzzley synthesis may be allowable or even enjoyable. But I would still caution you to not overdo it.

If you are really going to be using a lot of "never seen it before" questions on tests, than you need to spend the time to train the students with a lot of puzzley drill so they exercise that faculty. Probably also need to make sure they have very solid knowledge of the different aspects of all the topics also (to support handling later synthesis). I just wonder if you have time for that sort of thing or if some of the basic calculational method familiarity will then suffer by not being covered enough.

Consider also that sometimes acceleration may be more beneficial than enrichment. This is a somewhat contrarian stance. But I got WAY more benefit in a STEM career from having a very solid AP Calculus BC background, than all the tricky stuff in "functions" (pre-calc) class. Of course, it's a balance. But just at least consider this concern.

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A compromise approach could be to give a problem (with parts), as you might on a guided worksheet. Such as:

A function $f$ has domain $(2,4)$. We define $g$ by $g(x) = f(x-2)$.

a) Is $g(3)$ defined?

b) Is $g(5)$ defined?

c) What is the domain of $g$?

You might even give a follow up problem (perhaps as extra credit) along the lines of:

A function $f$ has domain $[3,7)$. Define $g$ by $g(x)=f(2x-5)$. What is the domain of $g$.

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