https://educationrealist.wordpress.com/2015/03/30/illustrating-functions/

First I have them pick a location: their house, the school, the movie theater, local stadium, whatever. We describe the location in terms of nearby major intersections, like you would if you were giving someone directions. If you can reference a location from two parts of an intersection (north, south, east, west), you’ve narrowed down the region significantly and give your friend an idea of where they’re going. Adding streets increases the accuracy of your region.

Then I give them a task, pick a bunch of locations and describe how you’d find them using at least two references. Screenshot the map, doodle on top of it, and share it. Lacking any sort of screens, print them some maps you make with various landmarks on them and ask for the same thing.

http://infinitesums.com/commentary/2013/11/8/more-inequality-explorations

Does it create a headache? Maybe a little bit. But my kids speak to each other in intersections all. the. time. when it comes to finding things in the area.

Later, get more specific with your references. Maybe the region is defined by an absolute value function and a constant. Maybe a couple quadratics. Or if it’s Algebra 1, a couple of lines. Have them graph a dozen or so points and give them some systems to draw (or you know, desmos). Give them some conditions (less than line A, greater than line B) and see what they think, then change the condition, and then get picky and ask how you’d prove that without a picture.

Harel… Need for communication… What would Clay Davis say to that? In the introduction, the situation is concrete and the disequilibrium is fleeting – easily addressed and articulated by the students with everyday language. We are wondering into mathematical la-la land if we insinuate that the students do not have sufficient language to communicate the underlying ideas in the introduction.

The problem lies in this. ” This forces the students to restrict the domain for reasons that don’t really make sense.” That’s quite right – it doesn’t make any sense. Confusing for students as they’ll undoubtedly run up against such functions that do make sense(logarithms, etc.) in different contexts later in their studies, as any year 11/12 student knows who has been bored to death by the universality of Families of Functions, of which logs, etc. are certainly members.

(This by the way, is an excellent reasons to be very cautious around this issue, since it would seemingly directly fall under Dan’s original argument regarding “Rules That Expire”: http://www.nctm.org/Publications/teaching-children-mathematics/2014/Vol21/Issue1/tcm2014-08-18a_pdf )

Far more powerful then would have been the birthday example with ALL months listed, to illustrate that(given n>12) a range may be much smaller than a domain and yet provide a function, resulting in domain pile-ups at somewhat regular range intervals(….. one might even chance to suggest a periodic distribution – hello trig).

That is, unless the entire intent was to spark an investigation into mathematical functions that have domain restrictions that DO make sense for very good reasons, but again, I’m fairly certain that was not the intent here.

“Example 3

Suppose P is the set of all the students in the classroom, and the function birthday(p) takes a person p as its input and return’s p’s birthday as the output. It is convenient to view this as a mapping from the set P (all students in the classroom) INTO the set D (all 366 possible birthdays, including February 29 for those special people who were born on that day in a leap year). The range of the function birthday – the days on which the actual birthdays of the students in the classroom happen to fall – is a subset of D.” (p. 4)

I would personally suggest that here the “bigger set” D is the range and that the range described above is the image of birthday. But that’s just quibbling over terminology. What matters is the idea that the target set is a subset of a (possibly) “larger” containing set. We map the domain either into or onto a set which might be identical to the “larger” containing set, and when they are in fact identical we call the mapping “onto” or “surjective,” in Bourbaki Talk.

As for how this reflects on Dan’s example, I’ll suggest that the notion of a “natural domain” as the Litvins and others use it reflects an implicit restriction. But that’s a matter of chance to some extent. Unless there are at least 366 students, we know that the domain may be too small to map to all 366 members of D. That’s fine, and no one should/would complain.

But Dan purposely places an unnatural restriction, arbitrarily listing a range (or image) set that is too small relative to any reasonable domain set. This forces the students to restrict the domain for reasons that don’t really make sense. And so what is described doesn’t have the feel of a function, even if we can torment our definitions to “make it so.” I think that’s what Dan was trying to come up with, and for my part I think it worked. You need to have a way to assign exactly one arrow from each member of the domain to a member of the range. Otherwise, whether you have some unassigned domain members or some domain members assigned more than one arrow (i.e, more than one corresponding member of the range), you don’t have a function.