Can you give me an example?

Probably the most powerful question during tutorials and lectures to ask of students is “Can you give me an example?”.

Students often don’t see the assumptions in a theorem — their focus is on the conclusion — and so they miss what the theorem actually applies to. Thus after stating the theorem it is a good idea not to launch immediately into the proof. Far better to ask for examples of the objects in the assumptions by asking “Can you give me an example?” This will help bring the statement to life. Does it apply to lots of examples we know? Is it telling us something about examples we know or ones we don’t know?

Just asking the question forces students to think about the statement and observe what is there. Only things that are consciously observed are remembered. Also, forces them to retrieve examples and retrieval is, of course, good for learning.

It is a good idea to push students for plenty of examples or else they will believe that the theorem only applies to the discussed examples. So, the assumption that $f$ is a differentiable function should produce constants, polynomials and even smooth non-analytic functions from students.

The question works for definitions as well. Consider linear independence — a definition that traditionally students find difficult. The condition within the definition, $\lambda _1 v_1 + \lambda _2 v_2 + \dots \lambda _n v_n = 0 \implies \lambda _1= \lambda _2 =\dots = \lambda _n=0,$
has a lot going on. (In particular, those zeros are different types of zeros — one is a vector, the other an element of a field!)
To help students we can ask “Can you give me an example of $v_1$?” Hopefully students will have no problem coming up with examples such as $(1,4,-2)$. But how many would think of $v_1$ as being a function? By repeatedly asking them for examples such as functions they get to see the point of the general notion of vector spaces. Too often students are only thinking in terms of vectors in $\mathbb{R} ^3$.

Asking for examples when we don’t have many examples of the assumptions is also worthwhile. Consider the classic from group theory: Let $p$ be a prime and let $G$ be a group of order $p^2$. Then, $G$ is abelian.

Asking for an examples in this case is great. When this theorem is stated students will probably only have examples of order $2^2$ and (I would imagine) very few lecturers will have let students see groups of order $3^2$. However, this is still a fruitful exercise as it will bring home that the theorem is telling us something about lots of new examples!