Well, oops, took a little break from blogging there. But I’m back now.

In the course of working on an article with the same title as this blog post for a publication about Felix Klein, I did a Google Image search on the word “function,” with the following results.

I find this fascinating for a number of reason. First, notice the proliferation of representations: graphs, tables, formulas, input-output machines, and arrow diagrams (those blobs in the second row). A corresponding search in a French, German, or Japanese gives a very different result. Here is the Japanese one (where I searched google.co.jp for “関数”).

This has a much greater focus on graphs, and no arrow diagrams. The French and German ones are even more focused on graphs; a search in Spanish, however, gives results similar to English.

What does all this mean? Well, I’ll be discussing this more in the article I’m writing, but for now I have just a couple of observations.

First, the English language search reveals a preoccupation with examples of relations that are not functions, with examples of graphs, arrow diagrams, and tables. I have always found this preoccupation perverse: the examples are artificial, and there is very little to be gained from them except an ability to answer questions about them on tests. This preoccupation is not evident in the searches in other languages.

Second, take a look at these two representations, from the English and Japanese language searches:

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Together they represent a case study in designing representations. Can you see what’s wrong with the one on the top? A function is supposed to have only one output for every input. Does an apple slicing machine have that property?* The representation on the bottom, on the other hand, clearly represents how to think of addition as a function with two inputs.

I’m interested to hear readers’ thoughts on the representations that come up in these and other searches. Maybe someone who can navigate Baidu can tell us what the Chinese results are.

*To be fair, the authors of the web page with the apple slicing machine are clearly aware of the problem. But their contortions to get around it only emphasize the fundamental flaws in the representation.

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Thoughts around the school math definitions of a relation and a function have been rattling around in my head for the last few months. If you were to ask 100 high school students or math teachers for an example of a relation, I fear we’d get 95 trivial examples and maybe a handful of any interest. This despite the fact that students have been working with interesting and important relations – equality, similarity, congruence, order relations and such – since kindergarten. This working definition of a relation as a function with something wrong is troubling and typifies the gulf between math, the discipline, and math, the high school class.