Explaining subjects whose justification requires demanding technical content

Question

This is my first question and I hope it's appropriate.

Often in the process of teaching a subject I start with examples of a phenomenon, exhibiting similar properties between the examples and building an abstract concept of something. Turns out that the concept underlying the technical definition is much more powerful than the definition itself and becomes a hindrance when the theory is not strong enough to support the idea in its full generality. Students come up with examples compatible with the idea but not the technical definition.

For instance, when teaching functions I discuss that for each street in a city there is a name to it (here in Brazil our streets are rarely numbered but have the name/date of important events or people). For each name we can compute the length of the word and therefore we can associate with each street the length of its name. This illustrates what I would later name function composition. This goes on for sometime until I define function as a relation of two sets satisfying certain properties.

A student could ask me if associating a set with its size is an example of a function and I answer: "Although the idea is correct, we can't do that because our function would have no domain, because there is no such thing as the set of all sets."

The student is often disappointed because concepts he thought he understood (set as the extension of a property, like Frege's original idea, and a function as I taught) are not achievable by the mathematical entity called function and set.

How can I deal with similar situations in a encouraging way for the students, while mathematically rigorous and concrete?

(I apologize for eventual mistakes in the text as English is not my native language.)

Answer 1

You could reply something like this:

Yes, associating each subset of the integers with its size would be an example of a function. And, in fact this does not just work for the integers but for all the subsets of any set. But note that to do it really for all sets would cause some problems as "the set of all sets" is a problematic notion. However, this does not really concern us here and it goes a bit beyond our current scope, but we can discuss it after the class if you like.

What I feel is key is to give some clear positive feedback, whence the "yes" at the start (leaving it out the reply could come over as "While something similar is true, what you said was wrong.") It was a good idea, and this should be made clear. But I feel it is also important not to say something false.

Generally, in such a situation I try to turn what the student said into something that is right, and only mention the need for a correction or addition in passing or indirectly.

Answer 2

A function is a "weird" thing. You must emphasize this first, if you are to explain functions. My example is a function is an oven: you take milk, baking soda, flour, etc. and you make biscuits. The function is that you take those input variables (milk, baking soda, flour, etc,) and do something with them and out comes a biscuit (something that is not the sum of its parts)!

A problem I see of many high school curricula, is that a function is taught as an explicit function, i.e. $y = f(x) =ax + b$ where you simply put in the value of $x$ and you get the function's value. Even worse, I see that a function is taught as $y = f(x)$--which, while it serves the purpose of the objectives, is not what a function is!

Answer 3

This kind of interaction is one reason why I'm personally not fond of the classroom pattern of presenting examples first, and then prompting students to intuit the definition as an exercise (perhaps this a pedagogical anti-pattern?). The one advantage to that approach that I can think of is that it somewhat resembles the investigatory practices of mathematicians; but even then you're more frequently working with pre-existing definitions. A major downside is that you're spending lots of time on an exercise that confuses students as to what end-goal of the course is; likely they are not going to be given assignments or tests that ask them to invent new mathematical definitions.

I would recommend avoiding these situations entirely by simply presenting the definitions (to some workable level of formality) as the very first thing in a presentation; and then subsequently exploring motivations, affirmative and negative examples, patterns and theorems around those constructs, etc. Advantages:

1. It maximizes the time that the proper definition is hitting the students' visual receptors.
2. It models professional mathematical practice of laying out definitions first by fiat in a paper, and then exploring their ramifications.
3. It avoids the frustration and confusion of students thinking that maybe the primary activity they'll be responsible for is creating new definitions.
4. It permits more time on questions and skills that students will be responsible for; like recognizing what entities do and do not qualify for that definition, resulting theorems, proofs, applications, etc.

Now, I will say that one of my most memorable classes in my undergraduate program was the day in senior-level intro to analysis when the professor challenged us to precisely describe what it means for a function to be continuous. That was a useful exploration of the possible pitfalls, for which all the parts of the epsilon-delta definition really are required. But: We all actually had seen the formal definition prior to this. And I don't think it would be a good use of time to do that every day. Usually, my recollection as a student is to be very frustrated and feel like it was a waste to be spending time on examples or being asked for speculate on answers before a definition-of-terms was presented for the topic in question. Most of the time it really feels like flailing around to no good purpose.

Coincidentally, I was just at an academic conference a few days ago where this happened in numerous sessions -- presenters wanted to do a "think-pair-share" exercise before presenting their findings, and in most cases there was an interesting pushback from the professional academic audience. One attendee said, "You're the expert, I'd prefer to spend more time hearing from you", which I think is a cogent analysis.

Answer 4

Most of the time, technical considerations are what allow us to make our ideas more general. For example, the radial definition of a circle allows us to consider "circles" in generalized metric spaces, while the less technical "round and symmetric" definition does not. For the theory to be unable to capture the ideas in full generality, as with functions, is very rare. When this happens, it is usually due to technical reasons that can be sidestepped with even more technical constructions.

For a function that takes sets as inputs, here are some ways to do so:

1. Considering a function which maps a set to it's size is fine, as you can always consider a set of finite sets. We can even allow the set of all finite sets to be a set, or even the set of all countable sets to be a set! As long as there is a set that is larger in size than the sets we are considering, it's just fine.

2. Even when we actually want a function whose domain is any set (no matter how large), we still have a trick to talk about them. They are called class functions, and can be formally expressed through formulae. For example, if we want to talk about the function $a \to \{a\}$, we can let the formula $"\phi(a)" =\ " \exists_{x}x=\{a\}"$. Then, if we want to say $\phi(a)=\{\varnothing\}$, we would just substitute $\exists_{x}x=\{a\}$ for $\phi(a)$, to get "$\exists_{x}x=\{a\}\wedge x=\{\varnothing\}$". Thus we can always talk about these functions even if they aren't sets.

3. While I am not sure how suitable such theories are to actually doing mathematics, there are "weak" set theories where the set of all sets is also a set!