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