“Could PhotoMath solve this? Then why are we wasting our time?”

In the future, I’ll give the same shpiel for algebra that I currently offer for arithmetic in the early grades. Technology replaces paper/pencil calculations but never mental calculations. Any calculator can handle 8 + 6, but if a kid can’t do that in his head he’s going to have a hard time noticing patterns, justifying, explaining and doing all that other good stuff.

Ditto with solving equations. In the future I’ll do a lot of mental equation solving, worry much less about written algorithms for solving equations.

That’s why I think we need a slightly different line than “If technology can do it then why should we bother with it?” Maybe the slogan should be “Technology can never replace thinking.”

Ditto Prof. Wright.

Is our primary goal as doers of math to “get answers” as efficiently as possible? If so, then go, PhotoMath, go.

Shame about the example. Once upon a time x was written in a curly font and multiply was x. Now we have x for ex (how else can I write it?) and a big fat dot for multiply, which the kids will copy as a normal dot, and then interpret as a decimal point, or vice versa. This will keep the machines at bay for a bit longer. Yes, Dan, I’m all for it, but they’ve got some work to do yet.

New style homework question: Do this problem with Photomath and explain exactly how it arrived at a garbage solution.

Dan,
Just tried the app with basic, one-step problems I’ve been giving my seventh graders. Wish I could post the screenshots to show the app.

-16=d+21

It gave the correct answer, but when asked to show the steps, it showed:

-d=21+16
-d=37
d=-37

I find this one of the most convoluted methods to solve this problem! I may show my seventh graders some screen shots from the app tomorrow and ask them what they think of this solution – a teachable moment from a poorly written app!

@Yolanda. I agree completely. Once we start examining questions, we are taking the right steps. Questions worth answering are generally not answered quickly…or with little effort.

Three things
1) They ripped this app idea off of The Big Bang Theory :-) http://bigbangtheory.wikia.com/wiki/The_Lenwoloppali_Differential_Equation_Scanner
2) I tried it out with some easy and tougher equations/expressions. There are some issues with the text recognition. But this one was odd. It was a question that had exponents and fractions over fractions. It correctly interpreted the text and started to simplify it correctly but then it dropped a set of brackets as seen in these two steps (https://drive.google.com/open?id=0B0-WU3yuHa7GQWpjeVhZUk01YWs&authuser=0) but interpreted it correctly for the exponent but incorrectly for the 2 out front as seen in these two steps (https://drive.google.com/open?id=0B0-WU3yuHa7GUURlWnhqZjFnQ2c&authuser=0). So this has potential but still needs some work.
3) Even though it is not perfect I do think it’s shots fired. There are some teachers who see things like this as an apparatus to replace teachers. My statement to them is that if you think this kind of thing can replace you then perhaps you should be replaced. Harsh but I think true.

In this rave article, the same problem shows up in their example image. It doesn’t solve for ‘x’, just uses it as a multiplication symbol: http://techcrunch.com/2014/10/23/disrupt-london-finalist-photomath-rockets-to-the-top-of-the-app-store/?utm_campaign=sfgplus&%3Fncid=sfgplus

But if I’m doing something like a physics acceleration problem, after I already understand the underlying concepts, I might use this app to save myself the time of typing a long equation with exponents and parentheticals into a calculator.

No doubt that the reality right now is underwhelming. Shouldn’t be that hard to combine better recognition tech with powerful CAS to provide what the Vimeo hype suggests is already here.

That said, we’ve seen this time and time again. I was at an NCTM regional in Cleveland in 1997 at which Bert Waits gave a talk about “What Will You Change in Your Algebra Curriculum in the Face of THIS?” “This” was the about to be released TI-89, which he had in his shirt pocket.

I was still in graduate school at U of Michigan, had not yet taught high school mathematics (not counting teaching everything from remedial basic math through precalculus at a community college in NYC for two years), but I could already see the tremendous positive potential for improving how we spent time in classrooms (“we” being both teachers and students). Oddly, the rest of the audience, which I suspect comprised mostly actual h.s. math teachers, seemed less than thrilled.

Waits told a great anecdote in which he was teaching trigonometric interpolation to a class of Ohio State students back in the late ’60s/early ’70s, when a student in the back raised his hand and said, “Waits, you’re an idiot” (I asked Bert to confirm that when I last saw him in Philadelphia in 2004, which he did). The student had a scientific calculator and was doing in seconds what would take considerably longer with the method Waits was teaching the class.

And of course, we no longer teach trigonometric interpolation. We no longer have fat tables of trig values, square roots, or logarithms in the back of our textbooks because they aren’t needed (sadly, what those pages have been replaced with is often not worth the paper its printed on, but that’s another conversation. Suffice it to say that I first heard about this PhotoMath ap when Kirby Urner posted about it under the subject line “textbook-obsoleting technology”).

There are, I’m afraid, a lot of K-12 math teachers, as well as many teaching lower-level mathematics in post-secondary institutions, who like teaching all the dull, mindless arithmetic and algebra that they and their predecessors have taught for. . . ever. And technology annoys and frightens the bejeezus out of them. The reaction to Kirby’s post on the math-teach list serve is enlightening, even though it doesn’t have many actual teachers on it The sentiments expressed there are I suspect, quite typical of the current view of educational technology in some quarters.

For my money, Dan is right, just as Bert Waits was right, just as Kirby Urner is right. Sure, kids need to have number sense and a strong grasp of arithmetic and algebra, but likely do not need to be bogged down for 13 years, more or less, practicing it at the expense of anything and everything else they could be doing in math classrooms. Today’s “absolute requirements” will mostly go the way of trigonometric interpolation and all those tables in the back of the book. And that’s a good thing.

The point in this case isn’t how well the character recognition is. Or how correct the solutions are. Because it’s just a matter of time before apps like these solve handwritten algebra problems perfectly in seconds, providing a clear description of all steps taken.

The point is: who provides the equation to be solved by the app? I have never seen an algebraic equation that presented itself miraculously to me in daily life. In a real world problem I have to make a translation from the situation at hand to something mathematical. I suppose the 20/80 rule applies here too: 80% of the work to be done in solving a real world problem lies in asking the right questions, making some sort of mathematical model and (after doing some “magimatics” in between) interpreting the outcome: is the solution plausible, what does it mean?

The 20% of algebra halfway is something apps like this one could do perfectly well. As long as we give the right input to the app, and as long as we can verify the correctness of the app’s output, I think everybody would be happy to leave the boring stuff of calculations to a computer.

So, although there remains an important task to teach children the rules of algebra (to understand what’s going on in the app’s black box), more important is to teach kids the other 80% of the job. Because that’s something I don’t believe computers will be able to do for us for a long long time.

And knowing whether your input to the computer actually corresponds with the problem to be solved, and deciding whether the output is correct is something very complicated. Anyone who has ever programmed knows that most of the time in writing a computer program is spent in finetuning the model. Because even seemingly simple tasks often turn out to be quite complicated when they have to be presented in code for a computer. That’s why learning to program is so valuable. Maybe even more important than learning the rules of algebra.

So, summarizing, let’s spend more time in math education on modeling and interpreting the plausibility of outcomes. And teach kids to program.

I think PhotoMath and Wolfram Alpha are useful tools to have us reflect on our classroom assessments. Take a look at the last math assessment you gave: What percent of the questions could be solved immediately by these tools? What percent of the assessment is evaluating computational skills vs. conceptual skills and applications?

That question leads to a more important question: What percent of questions on our assessments should be able to be solved by tools like these?

What I should note is that, if we are structuring this the right way, kids (a) won’t use the app when developing the concept, (b) have a degree of comfort with doing it themselves after developing the concept and (c) take the app out when they end up with something crazy like -16t^2+400t+987=0, and factoring/solving by hand would take forever…

When I saw it, I had always been thinking about this development as a good thing. It’s not to say that I don’t understand where all the angst and worry is coming from, but I think about it differently.

“sub “WolframAlpha” for “PhotoMath” and we’re already there.”

The first comment by David Cox was on the money. I don’t think calculators should be hidden from students during tests, the same way I don’t think we need to confiscate phones. Of course, I just gave a poor comparison of two items, but what I mean is that: if tests are what we are worried about, then:
1) we should be designing tests differently anyway, and include problems that have a wider range of opportunities instead of just calculations.
2) both teachers and students should be thinking about tests differently — as opportunities to demonstrate what they understand
3) Maybe tests aren’t, and has never been, the best venue for demonstrating certain aspects of understanding.

What has concerned me most, since the beginning of all of this, was the perception of the roles of math teachers in our current classrooms. Is our roles really just to show students the steps of solving an equation? It seems to be what the app description claims – that having this would be like having a math teacher in your pocket. As we have reduced the roles of math teacher to demonstrating a single way of solving algebraic equations.

Is there actually discontent amongst teachers? I imagine there would only be discomfort if that’s all the teachers are doing in the classrooms.

I actually think the launch of this app provides a good opportunity for us (those of us that haven’t already) to change our fundamental understanding of what teachers do in a classroom. Much like I thought Wolfram Alpha could have done for us when it came out a while ago.

I am showing and inviting my students to think about this today. I will share what happens there.

@Kenneth Tilton wrote in part: ” I would not say boring, but then I enjoy pure maths for itself.”

So do many of us. But I’m not sure that simple arithmetic or algebra problems that require nothing more than exercising an already-mastered skill fall into the category of “pure maths for itself.” Nothing wrong with ENJOYING nearly-mindless number-crunching for its own sake. We all have hobbies and guilty pleasures (and not-so-guilty but clearly idiosyncratic pleasures), but let’s not try to justify them by making them sound like more than they are. Even doing easy derivatives and integrals as a form of doodling shouldn’t be confused with doing real problem-solving of the pure or applied kind. To my thinking, it’s important to distinguish between a puzzle and a mystery, and between an exercise and a problem that isn’t simply the mechanical application of steps we’ve been trained to do with near-automaticity.

I would also dispute some of what you write about community college students but I’m not 100% certain what the learning problem is to which you refer or, for that matter, where you think it lies. So absent elaboration on your part I’ll hold my peace on that for the moment.

I’ve been ruminating on this for a few days now. What’s frustrating to me is that I feel I’m being lumped into the category of teachers who “like teaching all the dull, mindless arithmetic and algebra.” I don’t like teaching anything that is dull and mindless. I guess my main argument is that there is value in repeated calculation–not for its own sake, but for the sake of having students discover patterns, relationships, and shortcuts on their own.

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.

@Matt E.: I might agree with you if in fact the last 25-30 years hadn’t shown that it goes just the opposite: the thoughtful stuff gets tossed; the computational stuff dominates. With the exception of classrooms taught by people like Dan Meyer and others who blog about their practice that reveal a great deal of deep thinking about math and teaching.

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.