## [Pseudocontext Saturday] Photos

This Week’s Installment

Poll

What mathematical skill is the textbook trying to teach with this image?

Pseudocontext Saturday #4

• Reasoning with proportions (63%, 523 Votes)
• Calculating exponential growth (19%, 154 Votes)
• Simplifying expressions (18%, 151 Votes)

Total Voters: 828

(If you’re reading via email or RSS, you’ll need to click through to vote. Also, you’ll need to check that link tomorrow for the answer.)

Current Scoreboard

Team Me: 3
Team Commenters: 0

Pseudocontext Submissions

Michelle Pavlovsky

Paul Hartzer

Rules

Every Saturday, I post an image from a math textbook. It’s an image that implicitly or explicitly claims that “this is how we use math in the world!”

I post the image without its mathematical connection and offer three possibilities for that connection. One of them is the textbook’s. Two of them are decoys. You guess which connection is real.

After 24 hours, I update the post with the answer. If a plurality of the commenters picks the textbook’s connection, one point goes to Team Commenters. If a plurality picks one of my decoys, one point goes to Team Me. If you submit a mathematical question in the comments about the image that isn’t pseudocontext, collect a personal point.

(See the rationale for this exercise.)

The judges rule that this problem satisfies both criteria for pseudocontext:

Given a context, the assigned question isn’t a question most human beings would ask about it.

I invite any commenter to rationalize the constraint that exactly 15 photos must be purchased and we don’t know which of them will be small or large. More often (always?) people begin with the photos they want, or perhaps they work from a total budget. “I can only buy 15 photos and the number of large photos I purchase can vary from zero to fifteen,” said no one ever.

Given that question, the assigned method isn’t a method most human beings would use to find it.

If most human beings were going to find out the cost of five large photos and ten small photos, they’d multiply each kind of photo by its price. Variables aren’t a useful tool.

So the textbook has made the world serve the math when math should serve the world. If the world doesn’t need math’s service, then math should be gracious enough to step out of the way.

I guessed correctly. The first and third choices made too much sense. Always step up to the plate thinking curveball.

The problem here is that the customer has no use for a general equation, but the store owner might–she’s got to deal with people who call in with all kinds of crazy orders and questions. Still, it’s unlikely the store owner would write an equation for just small and large pictures. It’s much more likely that she’d come up with a pricing scenario for unusual picture sizes.

I'm Dan and this is my blog. I'm a former high school math teacher and current head of teaching at Desmos. He / him. More here.

1. #### Mr K

November 5, 2016 - 10:25 am -

My new strategy is to vote for the most batshit crazy concept that could be associated with a picture.

2. #### Rob

November 5, 2016 - 10:55 am -

The number of options has been decreasing by one each time. I feel like this is an elaborate setup for a three-act “how long until 100% of Dan’s readers are able to find the correct answer?” Or perhaps a justification for a “when are we going to use this” sidebar in your future textbook’s linear equations chapter.

3. #### Leigh Nataro

November 6, 2016 - 4:16 am -

This photo looks like something on the walls of my daughter’s orthodontist. Perhaps a better real-world question would be how does the cost of braces compare to my daily cup of coffee at Starbucks?

• #### Sarah G

November 6, 2016 - 10:03 am -

Or comparing the cost of teeth whitening to coffee drinking…those teeth look fairly white to me.

• #### Joshua

November 7, 2016 - 6:12 pm -

Dentistry/dental hygiene was also my first thought about the picture. Without the multiple choice, I wasn’t able to come up with any guesses about the textbook context.

4. #### melissa

November 6, 2016 - 8:53 am -

I used Mr K’s strategy too. I actually hope it’s exponential growth, which to me seems like the second most batshit option, because I think exponential growth is just cool. If I keep blowing up the picture at the current rate, how many times do I have to blow it up before it’s as wide as the solar system? It’s not a real-world question, but it’s something I’d be interested in anyway.

A real real-world question might be: how big can I blow up the picture before it gets too pixelated?

• #### Leslie

November 6, 2016 - 9:48 am -

I love your comment, because I was just told this week that “only a math geek would say something like ‘exponents are just cool’.” Glad to know I am not alone! :)

5. #### Jonathan

November 6, 2016 - 10:27 am -

I guessed correctly. The first and third choices made too much sense. Always step up to the plate thinking curveball.

6. #### Chester Draws

November 6, 2016 - 8:26 pm -

Is the photo there for any sort of context at all, pseudo or otherwise? I suspect it is there because modern publishers think books need to have colour photos to be “engaging”. The problem lies not at the hands of the question writer, who quite possibly never saw the photo, but the thinking that to get students engaged in Maths you should have pretty pictures in their textbooks.

If most human beings were going to find out the cost of five large photos and ten small photos, they’d multiply each kind of photo by its price. Variables aren’t a useful tool.

The problem is that in order to teach variables at the lower level the situation has to be easy. But in those easy situations students basically never actually need variables to solve them. Numerical methods are always going to be quicker and easier.

I’m intrigued as to any “real world” contexts which are easy enough for students entering Algebra to use in practice forming equations where variables make solving quicker.

Students generally find writing equations hard, and they need lots of practice at it. So they need short questions — so no deep contexts that take 15 minutes and are very engaging, but only have one practice at the skill. And because there are many different ways to write correct equations the contexts can’t have too many options, as I need to check/verify all the alternatives quickly.

• #### Dan Meyer

November 7, 2016 - 3:12 pm -

Given our track record, I’m not sure we’ll sync up on this idea, but Evan Weinberg has a lesson that’s simultaneously simple enough for an introduction to variables, but which also illustrates their power. Notably, it lacks any real-world context.

• #### Chester Draws

November 8, 2016 - 1:23 pm -

That’s a nice link Dan, and I’ll definitely use some of it. It is a nice way to introduce writing variables, which is a problematic issue.

But it isn’t what I asked. I struggle to find situations I can give my kids in class that have them write variables into an equation that is simple enough, yet cannot be solved by numerical methods. So the situations invariably end up being “maths teacher world”, where we solve with variables something they can do quicker in their heads.

If you go through pretty much any textbook looking at this topic you will find pseudo-context, because the constraints are impossibly tight. When I write worksheets on this I am perpetually annoyed how sad my exercises look, but I’ve never seen it done much better elsewhere.

7. #### Rachel

November 7, 2016 - 5:49 am -

The problem here is that the customer has no use for a general equation, but the store owner might–she’s got to deal with people who call in with all kinds of crazy orders and questions. Still, it’s unlikely the store owner would write an equation for just small and large pictures. It’s much more likely that she’d come up with a pricing scenario for unusual picture sizes. When I worked in a frame store, our price was based on the diagonal measurement of the thing to be framed. That was some fun math… We basically had a worksheet that walked us through the steps, though–and we found the diagonal by actually measuring. I’m sure that today it would be a matter of plugging numbers into a spreadsheet–but hey, someone has to set up the spreadsheet! I doubt that framing things is a super engaging context for for high schoolers, but it might work with my adult students…

• #### Dan Meyer

November 7, 2016 - 3:16 pm -

The problem here is that the customer has no use for a general equation, but the store owner might–she’s got to deal with people who call in with all kinds of crazy orders and questions. Still, it’s unlikely the store owner would write an equation for just small and large pictures. It’s much more likely that she’d come up with a pricing scenario for unusual picture sizes.

The expression is of little use to the customer, but there probably is an expression that’s of a lot of use to the owner. I’m imagining a fix-the-broken-cash-register activity. You either have to fix the register with an expression or calculate all the prices by hand.

PS. I featured this comment in the post.

8. #### Leigh Ann Mahaffie

November 7, 2016 - 8:12 am -

So, trying desperately to find a scenario that could cause this to be: “Your Mom ordered 15 pictures for Christmas gifts, and you need to decide who is getting the large ones, and who is getting the small ones. Your Mom didn’t write down how many of each size she purchased, so you have to figure it out based on the amount deducted from the checking account”…Yeah, it’s lame, but better than the example they set up…

9. #### Rebecca Gasper

November 7, 2016 - 9:09 am -

Hate to be a spoilsport, but I have just the sort of constraint that makes _n_ photos be ordered in different sizes:

School pictures have been taken, and a parent receives the proofs with an order form. Exactly 12 photos need to be ordered for relatives, the child and yourself, but you’re budget conscious. You can order different packages with small (exchange), medium (wallet), and “large” (3×5). The unit price can be figured from the add-ons. What should the price be for each package? Is the package with 9 wallets, 2 3x5s, and 1 class photo a good deal?

By the way, in my “real” example, the add-ons that increased the photo package to the next package up didn’t obey the triangle inequality in that the combination came out cheaper than the next package up. That’s why the question is “Is this a good value? What do I compute the price/value to be in each case?”

• #### Dan Meyer

November 7, 2016 - 3:18 pm -

You aren’t spoiling anything, Rebecca. You’re fixing the pseudocontext by adding the price constraint. Nice.

10. #### David Blake

November 7, 2016 - 12:21 pm -

The Literacy Coach at my school got it right because she said,”There is an expression on her face so it must be simplifying expressions.

11. #### Laurie Hailer

November 7, 2016 - 4:56 pm -

I love that you always bring this up. I also think these problems are all from my text book. I remember once going through the snowboarder one with the polynomial equation. I think you used that as an example at CMC-North last year.It was really so lame.

It would be interesting to bring in an actual photo price schedule from Costco or something. Do kids even purchase photos anymore? (I do)

Also, some of the logic behind some of the psuedo-contexts are things I actually used in banking and credit industry analytics. Because of that I sometimes tell students that while the examples may be things you’d never actually do in day-to-day life, some of the methods and logic can be used on a larger scale in a work environment. But, that’s really still not good enough. Plus, most of what we did was way more detailed and an analysis program did much of it, of course. That said, you had to understand those algorithms.

12. #### Laurie Hailer

November 7, 2016 - 4:57 pm -

Hey – also – the problem was lame. You weren’t lame for using it as an example. ;)