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PhotoMath is an app that wants to do your students’ math homework for them. Its demo video was tweeted at me a dozen times yesterday and it is a trending search in the United States App Store.

In theory, you hold your cameraphone up to the math problem you want to solve. It detects the problem, solves it, and shows you the steps, so you can write them down for your math teacher who insists you always need to show your steps.

We should be so lucky. The initial reviews seem to comprise loads of people who are thrilled the app exists (“I really wish I had something like this when I was in school.”) while those who seem to have actually downloaded the app are underwhelmed. (“Didn’t work with anything I fed it.”) A glowing Yahoo Tech review includes as evidence of PhotoMath’s awesomeness this example of PhotoMath choking dramatically on a simple problem.

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But we should wish PhotoMath abundant success – perfect character recognition and downloads on every student’s smartphone. Because the only problems PhotoMath could conceivably solve are the ones that are boring and over-represented in our math textbooks.

It’s conceivable PhotoMath could be great for problems with verbs like “compute,” “solve,” and “evaluate.” In some alternate universe where technology didn’t disappoint and PhotoMath worked perfectly, all the most fun verbs would then be left behind: “justify,” “argue,” “model,” “generalize,” “estimate,” “construct,” etc. In that alternate universe, we could quickly evaluate the value of our assignments:

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

2014 Oct 22. Glenn Waddell seizes this moment to write an open letter to his math department.

2014 Oct 22. David Petro posts a couple of pretty disastrous screenshots of PhotoMath in action.

2014 Oct 23. John Scammell puts PhotoMath to work on tests throughout grade 7-12. More disaster.

2014 Oct 24. New York Daily News interviewed me about PhotoMath.

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Kathy Henderson gets the app to recognize a problem but its solution is mystifying:

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!

M Ruppel:

I 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 -16t2+400t+987=0, and factoring/solving by hand would take forever.

Sander Claassen:

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.

Kenneth Tilton:

ps. Photomath is just a “stupid pet trick” they did to market their recognition engine.

This is a talk I gave awhile ago looking at why students hate word problems, posing five ways to improve them, and introducing this thing called “three-act math.”

My 2015 Speaking Schedule

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Here is my speaking calendar for 2015 in case anybody is interested in attending Dan’s Blog: The Unplugged Experience. Some of these sessions are private, others have open registration pages (see the links), and others have waiting lists. Feel free to send an e-mail to dan@mrmeyer.com with inquiries about any of them. It’d be a treat to see you at a workshop or a conference.

BTW. Delaware, Idaho, Nebraska, Rhode Island, Tennessee, West Virginia, and Wyoming will complete my United States bingo card. If you’re the sort of person who schedules these kinds of sessions for a school or district or conference in any of those states, please get in touch.

From Pearson’s Common Core Algebra 2 text (and everyone else’s Algebra 2 text for that matter):

Mark has 42 coins consisting of dimes and quarters. The total value of his coins is $6. How many of each type of coin does he have? Show all your work and explain what method you used to solve the problem.

The only math students who like these problems are the ones who grow up to be math teachers.

One fix here is to locate a context that is more relevant to students than this contrivance about coins, which is a flimsy hangar for the skill of “solving systems of equations” if I ever saw one. The other fix recognizes that the work is fake also, that “solving a system of equations” is dull, formal, and procedural where “setting up a system of equations” is more interesting, informal, and relational.

Here is that fix. Show this brief clip:

Ask students to write down their best estimates of a) what kinds of coins there are, b) how many total coins there are, c) what the coins are worth.

The work in the original problem is pitched at such a formal level you’ll have students raising their hands around the room asking you how to start. In our revision, which of your students will struggle to participate?

Now tell them the coins are worth $62.00. Find out who guessed closest. Now ask them to find out what could be the answer – a number of quarters and pennies that adds up to $62.00. Write all the possibilities on the board. Do we all have the same pair? No? Then we need to know more information.

Now tell them there are 1,400 coins. Find out who guessed closest. Ask them if they think there are more quarters or pennies and how they know. Ask them now to find out what could be the answer – the coins still have to add up to $62.00 and now we know there are 1,400 of them.

This will be more challenging, but the challenge will motivate your instruction. As students guess and check and guess and check, they may experience the “need for computation“. So step in then and help them develop their ability to compute the solution of a system of equations. And once students locate an answer (200 quarters and 1200 pennies) don’t be quick to confirm it’s the only possible answer. Play coy. Sow doubt. Start a fight. “Find another possibility,” you can free to tell your fast finishers, knowing full well they’ve found the only possibility. “Okay, fine,” you can say when they call you on your ruse. “Prove that’s the only possible solution. How do you know?”

Again, I’m asking us to look at the work and not just the world. When students are bored with these coin problems, the answer isn’t to change the story from coins to mobile phones. The answer isn’t just that, anyway. The answer is to look first at what students are doing with the coins – just solving a system of equations – and add more interesting work – estimating, arguing about, and formulating a system of equations first, and then solving it.

This is a series about “developing the question” in math class.

Featured Tweets

I asked for help making the original problem better on Twitter. Here is a selection of helpful responses:

2014 Oct 20. Michael Gier used this approach in class.

Featured Comments

Isaac D:

One of the challenges for the teacher is to guide the discussion back to the more interesting and important questions. Why does this technique (constructing systems of equations) work? Where else could we use similar strategies? Are there other ways to construct these equations that might be more useful in certain contexts?

Christopher Danielson and Megan Schmidt have both written recently and compellingly about the trouble students have when taught that math is a series of correct “steps.”

Danielson, doing his best Howard Beale:

THE STEPS WIN, PEOPLE! The steps trump thinking. The steps trump number sense. The steps triumph over all.

Let me add to the conversation the category of “steps that are correct but useless.” These are great. They come from a conversation I had, like, fifteen minutes ago with a teacher named Leah Temes here at NWMC 2014.

Leah teaches Algebra II. We were talking about solving systems of equations. It’s really easy to teach the solution to a system like this as a series of correct, useful steps:

2x + 3y = 10
5x – 3y = 4

  1. Add the second equation to the first one.
  2. Solve for x.
  3. Substitute x in either equation to solve for y.
  4. Check that pair in the other equation for full credit.

Leah said she was tired of seeing her students mimic those correct steps without understanding why they worked. So instead of showing her students steps that were useful and correct, she asked them if she was allowed to add the following two equations:

2x + 3y = 10
5 = 5

To get:

2x + 3y + 5 = 10 + 5

Is everything still correct? Yes.

Was that useful? No.

This experience awakened her students to a category of steps in addition to the correct and useful ones they’re supposed to memorize and the incorrect and useless ones they’re supposed to avoid – correct and useless steps.

Alerting your students to that category of steps may make math seem less intimidating and more interesting. Math isn’t any longer a matter of staying on the right side of a line between the incorrect and correct steps. There’s another region out there, one that’s a bit less tame, a place for explorers, a place where the worst thing that can happen is you did something right but it just wasn’t useful. That category of steps also requires justification – “how do you know this is correct?” – which can help bend the student away from memorization and back towards understanding.

BTW. All of this implies a fourth category of steps – incorrect but useful. Can anybody give an example?

Featured Comments

Cathy Yenca:

I do a similar thing when solving equations in one variable by asking students if I can add 1,000,000, let’s say, to each side of an equation… or if I can subtract 27 from both sides… or divide both sides by 200… etc. etc. We talk about what is “legal” (have we followed the rules of algebra and the concept of “balance” and equivalence?) and what is “helpful” (have we done something “legal” that helps us isolate the variable so we can solve this thing?”) Exaggerated examples like adding 1,000,000 to both sides seem to make an impression on kids.

David Petro:

I have long been a fan of deliberately sabotaging a solution to something that I might be doing on the board so that somewhere down the road things become obviously wrong. This is so students can start to develop strategies for what to do when this happens.

Many will tell you that it’s important for students to make mistakes (in fact, that they learn the most when they do). But that sometimes runs counter to what they see in class. That is, a teacher demonstrating flawless execution of mathematics. Even some of our best students often won’t even attempt a problem unless they are sure they will get it correct. If they are ever going to become comfortable with making mistakes as part of the normal process then we have to include managing those mistakes as part of our day to day in class.

Moana Evans:

[It's] incorrect but useful to estimate things like area problems, in order to find out a ballpark figure and check if you’ve done the math right.

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