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A lot of students complain about "never being shown that before". What's the idea called when you test multiple concepts or one or two new ones along with some old ones in a word problem, for example? I'm trying to figure out how to articulate it so I can let the students know ahead of time that I do this during exams.

Suppose, for example, I give a word problem asking to maximize profit, a calculus concept and application they're being tested on (finding the max/min of a profit function). They know that they need to come up with a profit function if one is not provided. But the way I describe how to get the revenue and cost functions is not direct either - I give them enough info to determine those two functions. And perhaps in the way that I give them that info is not like any previously assigned problems but nevertheless all they need to do is some algebra to figure out the linear or quadratic cost or revenue functions. Of course, they also need to determine a suitable domain which again I may bury in the problem somehow.

What's this idea called that I'm doing? Applying calculus concepts to novel situations or ...?

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    $\begingroup$ Perhaps multi-part math problems or multi-step math problems. $\endgroup$ Commented Jul 16, 2023 at 17:56
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    $\begingroup$ Synthesis?${}{}$ $\endgroup$
    – Xander Henderson
    Commented Jul 16, 2023 at 19:02
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    $\begingroup$ @XanderHenderson Yes, that's the word I was looking for. It's an ability to synthesize the concepts that I'm testing. $\endgroup$
    – E2R0NS
    Commented Jul 16, 2023 at 19:07
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    $\begingroup$ "integrative" is also a word to describe problems or tasks that require the student to put together (integrate) different concepts in creating a solution. This term can be applied to tasks involving different concepts in a course or different disciplines.entirely. $\endgroup$
    – user1815
    Commented Jul 17, 2023 at 2:07
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    $\begingroup$ I always find these sorts of problems really tricky to grade: I can easily imagine the example profit maximization accidentally measuring students' familiarity with economics. Even in the best case where all the skills being combined are skills I taught (quotient rule + chain rule + multi-step derivatives), if students are missing a piece skill (often the quotient rule) I don't really get a good measure of their multi-step derivative skills. $\endgroup$
    – TomKern
    Commented Aug 6, 2023 at 3:11

3 Answers 3

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The answer I am about to give is a little tangential, but I think that it will help to answer the question "How do I communicate to students that the problem I am setting for them is going to require the application of several ideas, perhaps in a novel context. My approach is related to Bloom's taxonomy of knowledge.

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When I write questions (exam questions, homework questions, worksheet questions, etc), I try to put them into one of three categories:

  1. Recall: These questions sit at the bottom of the taxonomy (roughly, "remember" and "understand"). They ask students to remember facts, regurgitate definitions, or perform very simple computations. For example:

    • Determine the derivative of the function $f$, given by $f(x) = x^3$.
    • Evaluate the indefinite integral $\int \sin(t)\,\mathrm{d}t$.
    • Complete the definition by filling in the blank: Let $f : I \to \mathbb{R}$ be a function, and let $a$ be a point in the interior of $I$. The derivative of $f$ at $a$ is ____, assuming that this limit exists.

    Note that this kind of question should be answerable almost instantaneously (either you know the answer or you don't).

  2. Analysis: These questions sit at the middle of the pyramid ("analyze" and "apply"). They ask students to complete more complicated computations, solve equations, compare and contrast different ideas, and so on. Typical questions are

    • Evaluate the definite integral $\int_{0}^{1/2} \sin(\pi x)\,\mathrm{d}x$.
    • Explain the difference between a definite integral and an indefinite integral.
    • Determine the maximum value of the function $f$, given by $f(x) = 3x^3 - 4x + 6$ on the interval $[-2, 1]$.
    • An open topped box is to be made by removing identical squares from each corner of a rectangle of cardboard which measures $24 \times 18$ inches (see the picture). Determine the size of square which must be removed in order to maximize the volume of the box. [INSERT PICTURE HERE]
  3. Synthesis: These questions are the kind asked about in the question, and try to hit the top of the taxonomy (evaluate and create). These questions are often beyond the level that I would expect students to be able to complete on their own (especially in an exam setting). I usually assign these kinds of problems as homework, and encourage collaboration. That said, there are times when they might be appropriate on an exam.

    • The force of gravity on the moon acts to accelerate objects towards the center of the moon at a constant rate of about $-1.6$ m/s$^2$. A golf ball is hit at an angle $45^\circ$ above horizontal with an initial velocity of $200$ m/s. Determine the maximum height that the ball reaches. [Note that this question would be given before discussing multivariable calculus—the students need to figure out for themselves that the velocity is "split" between a vertical and horizontal component.]
    • Recall that the cosine and sine of an angle are defined to be the $x$ and $y$ coordinates (respectively) of the point where the angle intersects the unit circle. Analogously, define the "square cosine" ($\operatorname{sqcos}$) and "square sine" ($\operatorname{sqcos}$) of an angle $\theta$ to be the $x$ and $y$ coordinates of the point where the angle $\theta$ intersects a square with vertices at the points $(1,0)$, $(0,1)$, $(-1,0)$, and $(0,-1)$. Give formulas for $\operatorname{sqcos}$ and $\operatorname{sqsin}$ in terms other known functions (polynomials, the usual trigonometric functions, exponentials and logarithms, etc).
    • Let $f: I \to \mathbb{R}$, and let $a$ be in the interior of $I$. The symmetric derivative of $f$ at $a$ is defined to be $$ \lim_{h\to 0} \frac{f(x+h) - f(x-h)}{2h}, $$ assuming that this limit exists. Show that if $f$ is differential at $a$ (in the usual sense), then the symmetric derivative of $f$ exists at $a$, and has the same value as the usual derivative.

In communicating with students, I explicitly tell them what kind of question they are answering. That is, I explicitly use language like "This is a recall question," or "This is a synthesis question—you might have to think about material from many different sections of the text in order to get to an answer."

So... to (kind of) answer the question: the kinds of problems described in the original question likely fall at the high end of "analysis", or at the low end of "synthesis" (depending on how much students are supposed to create on their own, as opposed to just compile together a bunch of ideas in the process of solving the problem).

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    $\begingroup$ @ryang The taxonomy is meant to classify knowledge itself (so calling these "knowledge" or "comprehension" questions seems kind of out of line, as these things encompass all of the taxonomy). The idea is to pick a description from the appropriate level of the taxonomy. "Recall" or "Remember" or "Description" or "Identification"... something in there. Pick your poison. $\endgroup$
    – Xander Henderson
    Commented Jul 17, 2023 at 2:26
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    $\begingroup$ Ok I just Google-searched for “bloom’s taxonomy”, and the results generally call it Taxonomy of Learning (Objectives), not Taxonomy of Knowledge. $\quad$ Furthermore, it turns out that Wikipedia even squarely agrees with my suggestion, which I’d independently developed: it says that Bloom’s original labels were (in order): Knowledge, Comprehension, Application, Analysis, Synthesis, Evaluation. $\endgroup$
    – ryang
    Commented Jul 17, 2023 at 14:16
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    $\begingroup$ Notice that your example tasks under your first section “Recall” are actually Understanding/Comprehension rather than merely Recall tasks. (And while the higher tiers build on the lower ones, it isn’t quite accurate to frame the lowest tier as “encompassing” the entire taxonomy.) $\endgroup$
    – ryang
    Commented Jul 17, 2023 at 14:16
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    $\begingroup$ @ryang From the site I linked in my answer: "Knowledge 'involves the recall of specifics and universals, the recall of methods and processes, or the recall of a pattern, structure, or setting.'" Notice the repeated use of the term "recall". I would also argue that the tasks I have given are recall tasks, as identities like $\int \sin(x) = \cos(x) + C$ ought to be rote tasks for a calculus student. $\endgroup$
    – Xander Henderson
    Commented Jul 17, 2023 at 14:21
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    $\begingroup$ It should also be noted that a lot of work has been done since Bloom, and that the modern understanding of the taxonomy (and how to use and apply it) has moved away from Bloom's original work, focusing more on "verbs," for example. See google.com/books/edition/… or cft.vanderbilt.edu/wp-content/uploads/sites/59/… . $\endgroup$
    – Xander Henderson
    Commented Jul 17, 2023 at 14:25
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Many textbooks simply identify these as "Applications".

Of course, the more bits and pieces that need to be pulled in and synthesized for a problem, the more challenging it will be. Some books mark increasing difficulty of exercises in this way in some fashion (often A-B-C level exercises or the like).

The general idea of practicing old skills interleaved with new ones is called "Mixed Practice" (or "Varied Practice"), but that more often refers to a series of small, separate exercises on a variety of skills.

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A related concept is Rich Math Tasks (https://www.doe.virginia.gov/teaching-learning-assessment/k-12-standards-instruction/mathematics/instructional-resources/rich-mathematical-task) -- activities that require math students to pull in many different levels of thinking (as Xander points out), as well as multiple areas of the course. Getting students familiar with performing these tasks before an exam can help improve student readiness for novel problems on test day.

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