## [PS] Swedish Yoghurt

Arla, a Swedish yogurt producer.

Translation:

You catch a pike but the scales are broken. The pike weighs two kilograms plus half its weight. How much does it weigh?

Pseudocontext

At this point I'm comfortable with two definitions of pseudocontext:

1. context that is flatly untrue: "a basketball team scores two points every minute for the duration of the game."
2. operations that have nothing to do with the given context: "the age of Mark's dad is three more than four times Mark's age."

We've worked hard for those two categories. We've digested some really untasty mathematics in their development. They indicate problems that aren't just boring or irritating but problems that are actually alienating, problems that disrupt a student's innate and true sense of the world.

Pseudocontext Saturdays will run their course eventually. For now, though, the intellectual challenge of identifying different levels of badness (and, in many cases, redeeming it) is too invigorating to give them up. Those of you who have invested time and effort on these features in the comments and in your submissions — thanks.

Assignment:

1. Scan an example of pseudocontext.
2. Email it to dan@mrmeyer.com
3. List the textbook title, edition, and publisher.
4. Give me your interpretation of the term "pseudocontext."
5. Let me know if you'd like credit (name, blog or twitter) or if you'd prefer anonymity.

### 27 Responses to “[PS] Swedish Yoghurt”

1. on 20 Nov 2010 at 11:01 amHawke

As I see it, here’s the more explicit version of what the picture and words above seem to be getting at.

“A designer at Aria likes math and hopes that you do, too. By the way, do you like fish? OMG, NO WAY! So does she. You two have so much in common; you should hang out. Anyway, she was thinking about numbers the other day and wonders if you can solve this challenge: Can you find the number that is two more than half of itself?”

This is an optional mathematical riddle on a yogurt container. Young Swedes who want to explore will do so; others will simply ignore it. Sure, this is pseudocontext in the literal sense (definition #2), but who cares? Would you rather have no math on the yogurt container or optional, abstract, trying-too-hard math? I’ll take the latter any day. While I agree that many textbook math problems are demotivating and ridiculous, I don’t think it’s a good idea to try to rid the world of them, but rather to replace them with better problems when appropriate.

Here, we could go through nutritional contents (if they’re into that kind of thing) and ask questions about how many cups of Aria yogurt you would have to eat in order to…, but why not let the kids play around with the math a bit? Surely, no child will read this and decide that they don’t want to be a pike fisherman after all, nor will anybody be so enamored with the thought of talking about their favorite fish that they will be disappointed when the math doesn’t match what they know…

2. on 20 Nov 2010 at 1:10 pmElizabeth S

Pseudocontext Saturdays will run their course eventually. For now, though, the intellectual challenge of identifying different levels of badness (and, in many cases, redeeming it) is too invigorating to give them up.

I have to admit — mocking pseudocontext has indeed become a guilty pleasure.

- Elizabeth (aka @cheesemonkeysf on Twitter)

3. on 21 Nov 2010 at 7:42 pmChirs Sears

Here’s an attempt at redemption for this problem.

“You are fishing and you catch a pike. Your scales are broken. In the back of your Volvo is a metre stick and a one kilo bag of condensed milk. Find a procedure to weight the fish.”

Now I’m off to put a meter stick and condensed milk in the back of my car to deflect any accusations of pseudocontext in this problem.

4. on 22 Nov 2010 at 8:37 amDave

re:Chirs Sears “Find a procedure to weight the fish.”

Fix my scale? Use someone else’s scale? Just clean, cook, and eat the fish, since it’s possible to cook without an exact measurement?

If I go too far down that road, I’ll have to admit that a mush more appropriate math problem for myself would be verifying that I received the number of fried catfish pieces that I ordered.

5. on 22 Nov 2010 at 8:46 amEric B

*NOTE I know nothing about the size/weight of Pike so these numbers would need to be adjusted.

Your first 2 pike of the day are 6 inches and 12 inches long and weighed 3 kilograms and 5 kilograms. Your next pike was 24 inches and weighed 8 kilograms before breaking your scale. How much does the 18 inch pike you just caught weigh?

6. on 22 Nov 2010 at 10:16 amEric B

This will teach me to post in a hurry.

According to wikipedia http://en.wikipedia.org/wiki/Northern_pike

There is a formula for approximating the wight of a pike in pounds based on its length in inches

W = 0.000180*L^3.096

I think I would give the students several different weights that fit the formula one of which is very large and breaks the scale then ask them if the next fish is above the weight limit and has to be thrown back.

7. on 22 Nov 2010 at 11:48 amDan Meyer
Hawke: This is an optional mathematical riddle on a yogurt container. Young Swedes who want to explore will do so; others will simply ignore it. Sure, this is pseudocontext in the literal sense (definition #2), but who cares?

I care. This has come up in previous Pseudocontext Saturday threads. “Big deal. It’s just a silly little riddle. I’d want to solve it.” And I agree. But I have a math degree and I taught math. I don’t need to be convinced that math and the real world play nicely together. Try to put yourself in the mindspace of a child who isn’t so convinced. That student’s perception of math is vulnerable and the pseudocontext attacks it.

Eric B: Your first 2 pike of the day are 6 inches and 12 inches long and weighed 3 kilograms and 5 kilograms. Your next pike was 24 inches and weighed 8 kilograms before breaking your scale. How much does the 18 inch pike you just caught weigh?

My hero. Awesome work.

8. on 22 Nov 2010 at 1:12 pmJason Dyer

Forget the pseudocontext for a second: the presentation itself would cause most to miss the interesting part of the problem, that is, that the “obvious” answer (3) is wrong.

Maybe it’s just me, but I’d much rather solve the original problem than Eric’s variant. I don’t know if it’s because there’s still the weight of pseudocontext (why can’t I use my scales again?) if it’s because there’s no mathematical “trick” as in the original problem, or if a textual presentation is too dry but it’d go off well with better design.

I’m fine with riddles that are clearly riddles. Even as a six year old I didn’t think there were actual cats and wives going to St. Ives. I think the broken-scales bit on the cup is what pushes it from a riddle-with-ok-pseudocontext into a word-problem-with-unfortunate-pseudocontext.

Here’s a puzzle from a Professor Layton game which uses a very similar “trick”. It’s written more naturally so is sort of a border case:

While walking through a market on vacation, you notice a small stand selling cameras. A camera-and-case set is selling for \$310. The seller tells you that the camera costs \$300 more than the case itself and that the case costs the price of the set minus the cost of the camera.

You decide you’d rather wait on buying a camera and opt to just buy the case alone. You hand the seller a \$100 bill and see his eyes light up. Think fast now! How much change should you be getting back?

9. on 22 Nov 2010 at 1:24 pmJason Dyer

Just for fun, here’s how I introduce the “form an exponential with two points” problem:

Devoured by Wolves

Supplies: lots of dice. Spooky music optional.

1. Students get into small groups, each group with optimally 15-20 dice. Each die represents a villager. Every night the wolves come, and some of the villagers don’t make it out alive.

On Day 0 all the villagers are alive. Students should simulate each day by throwing all the dice. The dice that roll up a 6 get sent to the graveyard.

2. After all the villagers are dead and the results recorded, the experiment should be repeated two more times and graphed.

3. (inter-discussion / work here)

4. Large village apocalypse: The entire class gets in a circle with all the dice. The total number of villagers is recorded, and then all the dice are thrown simultaneously in the center of the circle. All the sixes are gathered and put in the graveyard [I had one student calling himself the "Grim Reaper"] and the rolling continues until there are no survivors. The data this time can be put on the class white/chalkboard.

5. (more inter-discussion / work here, culminating in students finding a graph first modeling the situation experimentally and then deciding on an exact model based off of the probability)

[SHORTER CLASS VARIANT] Each student is themselves a villager and has only one single die. They start standing up and roll after each day, sitting down if the wolves get them.

10. on 23 Nov 2010 at 10:30 amEdna

Nice share, Jason. Hope we have this on our tetra packs=)

11. on 23 Nov 2010 at 1:36 pmSue VanHattum

Jason, this is way cool! Would you post this one on your blog, so people can find it more easily? (I’ve just copied to word, to store for my intermediate algebra and precalc courses.)

12. on 23 Nov 2010 at 7:13 pmJason Dyer

No problem, Sue. Comment reposted.

13. [...] of the two parts of our working definition of pseudocontext does it exemplify? Justify your answer. Anyhow, I'm looking at all these books, all these books, [...]

14. on 27 Nov 2010 at 1:14 pmSam Critchlow

@ Jason – love it, already thinking about polyhedral dice modeling different exponential functions, or what it would look like with a mixture of 4- 6- 8- 10- 12- and 20- sided dice…

15. on 03 Dec 2010 at 12:10 pmElm Tree

I’m in a teacher certification program in Washington state and spending a lot of time thinking about education issues. And, phony word problems have always bugged me. My take is that the math profession shoots itself in the foot every time it puts out a book or offers a problem to it’s students that exhibit these issues. But, before we get all worked up against the math world, I think I see the same problem in other subjects too. Do you agree?

Don’t we struggle to make certain Language Arts topics relevant to our students? Don’t they look at us and say ‘huh’ if we talk about commas before a conjunction that join independent clauses versus commas after a dependent clause in a complex sentence? (Why do I need that Mr. Smith, I don’t use commas in my text messages?)

Or, in History, don’t we struggle to explain to our students why we care about what happened when Norfolk was settled in the Virginia Colony in 1619, or why they should care about what was happening in 1819, or even 1980? (It’s all old news why do we care, Mr. Smith?) We can go on about ‘if you don’t understand history you’re doomed to repeat it …’ but does that really carry the day?

But, in math it seems worse? My theory is that Math pseudocontextualization stands out like a sore thumb, like 2+2=5. It grates on the nerves more. It stands out of the crowd more than a History or LA context problem.

So, what are we to do? I think we (and I mean the middle school teacher concentrating on math, AND the elementary teacher) need to step up to the challenge and find ways to explain how math is so connected to our lives that it becomes invisible and automatic. Much like how during a power outage we flip on light switches even when we KNOW the power is out – it’s just automatic.

16. [...] mostly content with the two-part definition I set up two weeks ago. I'm also satisfied that pseudocontext is much less of a problem than real [...]

17. [...] The above is not a problem, it is an exercise. Use the Midpoint Rule with six subintervals from 0 to 24. That’s the only part of the statement that you even have to read! The rest of it has absolutely nothing with bees, the rate of their population growth, or the net amount of population growth. A student might be turning this in to an instructor who takes off points for incorrect or missing units, and then you have to think about bees and time. Otherwise, this exercise is pure pseudocontext. [...]

18. [...] I'm clearing out my inbox, trying to tie a bow around this pseudocontext thing. Here are three problems that satisfy the first half of the working definition of pseudocontext. [...]

19. [...] original worksheet our group gave students suffers slightly from the psuedo-context problem, but I think it is still works. For instance, the Merrill Auditorium in Portland, Maine [...]

20. on 07 Jan 2011 at 9:59 pmAnna Maria

Today we did this systems problem in Algebra where we were trying to figure out when 2 trees would be the same height. When we finished the problem, I said “Who would care about this?” I think my students like it when I acknowledge the ridiculousness of some of the problems and I think that can make pseudo-context ok. Then we decided that Monk (if any of you have ever seen that show) would care because he likes things even. (Monk is an OCD Homicide detective, I’ve been watching the show on Netflix). I think I’ll show them a clip from the show the next time I do this type of problem.

21. [...] are three problems that satisfy the second half of the working definition of pseudocontext. I cop to a lot of guilt at the end of this post. [...]

22. [...] are submissions I received that didn't seem to fit the criteria. This isn't to say they're great problems. This isn't to say that I'd throw water on any of these [...]

23. [...] investigation of pseudocontext. I'm sure it would've taken me many months more to come up with my working definition of pseudocontext had you all not come through with so many [...]

24. [...] subjective, but Peter Brouwer sent in the problem that I thought satisfied both halves of the working definition of pseudocontext in the most spectacular fashion. This is it. This is as bad as it [...]

25. [...] problems” are horribly contrived (see Dan Meyer’s on-going discussion of “pseudo-context” for more on this), and the Fundamental Counting Principle is one of those topics that lends [...]

26. on 13 Feb 2011 at 8:25 pmGames in Math Education | mrgaffey

[...] I think the idea that most people came away with my presentation was the idea of context vs. pseudo-context.  I was happy to hear people ask me about the ways in which they were instructing mathematics.  I [...]

27. [...] Jean-Marc!) as was this explanation (thanks, Carmen!) but in my hands this problem verges on pseudocontext because I'm asking the students to use an operation (exponential modeling) that may or may not [...]