PhotoMath is for the most part a gimmick that nonetheless highlights questions about technology vs. paper-and-pencil computational math. It barely touches, if at all, on what mathematicians do on a daily basis. And those whose work involves primarily loads of computations, particularly if large numbers are involved, are using technology extensively to probe further into what’s out there. But of course they use the technology to pursue questions that no technology is going to ask or evaluate in the first place.

Exercises as a means to an end of conceptual understanding of numbers and operations sounds great. That’s true even of basic computational exercises. Exercises as an end unto themselves is what I hope Hypothetical PhotoMath will undo.

I’m totally on board with that. My worry is that that’s a fine distinction, and that the false dichotomy of “computation bad, concepts good” may mean that the baby gets thrown out with the bathwater.

All I ask is that those heralding PhotoMath and all of its siblings and eventual descendants just take a minute to try to imagine what these new technologies might undo.

Again, I do imagine what they might undo and I hope for it.

But I think M Ruppell and Michael Pershan have helped me see my final kiss-off to problems that Hypothetical-PhotoMath-That-Doesn’t-Suck can do was too strong. Exercises as a means to an end of conceptual understanding of numbers and operations sounds great. That’s true even of basic computational exercises. Exercises as an end unto themselves is what I hope Hypothetical PhotoMath will undo.

Lots of quotes favoring practice here: http://www.goodreads.com/quotes/tag/practice

This one is short and sweet: “For the things we have to learn before we can do them, we learn by doing them.” ― Aristotle, The Nicomachean Ethics

The thing about sufficient (not excessive) practice is that there is a difference between consciously knowing something like the distributive property and having internalized it through practice. The latter is necessary to perform at a higher level where the skill is taken for granted, and indeed I think that is what we mean by learning.

Anything hard we can do requires us first to do in our sleep what goes into it. Successful hockey players do not think much about how to skate, i like to say.

Conrad Wolfram discusses the issues in great depth here http://www.ted.com/talks/conrad_wolfram_teaching_kids_real_math_with_computers?language=en, coming down in favor of using computers wherever possible (except for number facts).

Of course he sells a product that does Algebra for us and I sell a product that helps us learn to do Algebra ourselves, so no surprise on our disagreement. :)

I am about to start a section in my Discrete Math class on graph coloring and chromatic numbers. I am presenting my students with around 20 “blank” graphs, and having them try to figure out on their own what the chromatic number of each graph is. I imagine that, five years down the road, they’ll be able to snap a picture of a graph and have an app determine the chromatic number. So why should I make my students “suffer” and try to figure it out on their own? Because I hope to have them try to give words to any patterns or rules they are noticing, so that we can write them down, organize them, debate them, and maybe prove them. That is doing math. That is why I don’t want them to “just Google it” or “just [insert spiffy new app] it”. Then again, these are conversations I have with my students regularly. I ask my Advanced Topics students to not use calculators or computers unless they get expressed permission from me. Obviously I have no control over what they do outside of class (or even inside it), but once they understand my motives, they are amenable.

I’m a big fan of this quote from Neil Postman (in Technopoly):

We are currently surrounded by one-eyed prophets who see only what new technologies can do and are incapable of imagining what they will undo.

All I ask is that those heralding PhotoMath and all of its siblings and eventual descendants just take a minute to try to imagine what these new technologies might undo.