*Intriguing Mathematical Problems*, Dover Publications.

**Pseudocontext**

Ali MuÃ±iz:

I like this one for its sheer density of wrongness. In less than twenty words we get three notions that would be ridiculous in a real context: That the narrator would have a favorite orange, that we have to solve a contrived word problem to find its weight, and, the punchline, once we solve it, that this “orange” weighs nine pounds.

**Transcription**:

My favorite orange weighs nine-tenths of its weight plus nine-tenths of a pound. What does it weigh?

**Assignment**:

- Scan an example of pseudocontext.
- Email it to dan@mrmeyer.com
- List the textbook title, edition, and publisher.
- Give me your interpretation of the term “pseudocontext.”
- Let me know if you’d like credit (name, blog or twitter) or if you’d prefer anonymity.

## 31 Comments

## Colin Burnett

September 25, 2010 - 8:18 am -Yes, an intriguing problem…isn’t it?

## Andrew Nicholson

September 25, 2010 - 8:32 am -If I had an orange that weighed nine pounds, I’d have it preserved because “It’s nine pounds!”, and yes, it would become my favorite orange.

## Z. Shiner

September 25, 2010 - 8:39 am -At least we can take comfort in the fact that the problem is from a recreational math book and not a high school text book. Our children are still safe from this particular abomination.

## Rafael

September 25, 2010 - 9:05 am -Obviously, his favourite orange is actually a pumpkin…

## Zeno

September 25, 2010 - 9:07 am -I think it’s an orange of the variety they feed to lions in the Scottish Highlands, commonly known as a “MacGuffin”.

## blink

September 25, 2010 - 12:17 pm -I agree that we need to watch for pseudocontext when teaching mathematics, but come on! This problem makes no pretense to be anything but a puzzle to try for fun. Next will we banish chess for its pseudocontext of kings, queens, and knights?

## Matt E

September 25, 2010 - 12:21 pm -See, I’m not so sure we should be so quick to heap scorn upon this particular problem. If we were able to talk to its author, I’m quite certain that he/she would laugh at the idea that he/she intended it to be reflective of a “real-world” situation. It is merely a mathematical puzzle. It is also kind of silly. I don’t think the silliness takes away from the puzzle.

I agree with Z. Shiner, in that it would be more deserving of our scorn if it were a textbook’s attempt at “real-world” relevance. (I’m still resisting simply sending Dan a copy of the Geometry textbook my school is using…)

## Ian Byrd

September 25, 2010 - 12:53 pm -Confession: I have a secret crush on these types of problems. I love their ridiculousness. My students do, too. We make stories about why someone would have a favorite orange or why it could weigh nine pounds. I do teach a gifted class and my students tend to be abstract thinkers. Not defending this as a great way to inspire student to become lovers of math, but to those who already are, this is an interesting puzzle. If I put this up as a “challenge” on Monday and I bet I’ll get half my students voluntarily turning in a solution by the end of the day.

## Alex

September 25, 2010 - 3:27 pm -I find myself agreeing with both views in the comments. First of all, what a silly math problem. As if (Wayne’s World) anyone is going to have a burning desire to relate to this because of the guy, his favorite orange, or it’s eventual absurd weight.

But then I see the other point of view, and I find myself agreeing more strongly with this one. It’s just a puzzle, put into words about an orange and an obviously weird guy. But so what? When I see the problem I immediately see the unique word structure, the puzzle, and I want to solve it. However, that’s me. I do KenKen and Sudoku.

Which brings me to my resolution. It’s a horrible problem and a great problem. It’s horrible for the kids in my lower level classes who struggle, who we are trying to get to see that the angry wolverine isn’t so bad. It won’t capture their attention any more than two boats in a river will.

But it’s a great warm up for my algebra 2 kids. They’ll argue and come up with different ways to solve it, and they’ll laugh about a nine pound orange, and it will kick start my class in a positive way, and two hours later at lunch they’ll still be joking about it. Ain’t nothing wrong with all that.

## Mark Schwartzkopf

September 25, 2010 - 5:54 pm -If I had run into this problem as a student, I would be kinda pissed off at the patronizing tone. I know that the author doesn’t have a favorite orange, and that he’s just saying that to trick me into being interested. However, now, I’m not sure how else students should practice translating sentences into equations.

Doing time consuming problems like the staircase problem is perfect for grasping and reinforcing concepts, but what do you do for something boring, like practicing translating words into numbers and symbols? If the long problems are engaging enough, is practice with boring problems no longer required? That’s an exciting thought.

## Paul M.

September 25, 2010 - 6:23 pm -I agree completely with #8 and #9. With the right “story” or funny presentation (I know some teachers who could pull this off, and some who couldn’t), and with the knowledge that students could solve it, this could be fun. Of course, why not pick an object that actually weighs 9 lb, and not tell them what it is, and have some more problems to guess its identity? Volume, surface area, molecular structure, etc.

## josh g.

September 25, 2010 - 9:15 pm -I like this problem because it’s almost a haiku. Let me just go ahead and fix that “almost”:

Juicy, favorite orange;

Nine-tenths of its own weight plus

nine-tenths of a pound.

## Scott

September 25, 2010 - 11:50 pm -I agree with Matt E: the problem is not intended to be a real world application. I interpret its intent to provoke an exploration into recursion without jumping right into jargon and notation.

Also, I believe this is a traditional style of problem carried down from ancient mathematicians. Without modern notation or structure, they relied upon carefully worded riddles.

So is this really in the same category as the Dog’s bandana?

## Michael Paul Goldenberg

September 26, 2010 - 7:34 am -If you keep a 9 pounds orange around long enough, it won’t be your favorite orange. Nor will it weigh 9 pounds.

On the other hand, “That nine pound hammer that killed John Henry Ain’t a gonna kill me, ain ‘t a gonna kill me.”

## Gina Marie

September 26, 2010 - 8:47 am -I’m sorry — I think this problem is ridiculous. And stupid. And a waste of time — for the teacher to assign it; for the students to solve it. For the ink and paper that was wasted. A tree died for that problem…. (just sayin’)

## Dan Meyer

September 26, 2010 - 1:15 pm -I’m glad, I suppose, that there’s some question over whether or not this constitutes a pseudocontext. Pseudocontext Saturdays wouldn’t be much fun if we just gathered around curriculum to kick and spit on it and pat each other on the back.

On the other hand, a lot of the counterarguments here could be construed as “it isn’t pseudocontext if I can sell it” or “it isn’t pseudocontext if

mykids would enjoy it.”Some kids

willrespond positively to the absurdity of it. Some kidswillrespond positively to the historical context. Some kidswillrespond positively to their energetic teacher. Some kidswilldo anything their teacher asks them to because that’s the expectation from their families.None of that reassures me.

Two questions:

First, could someone explain to me what the orange adds to “my favorite number is nine-tenths of itself plus nine-tenths,” a problem I don’t mind in the least. If the answer is “not really all that much” then how are we not telling students that “math only has anything to do with the real world if you

forceit.”Second, if there

werea useful context for 9x/10 + 9/10 = x, something that would engage and challenge the students alienated by the orange problem, wouldn’t that be better foryourkids also, even though they’re okay with the orange problem?## Michael Paul Goldenberg

September 26, 2010 - 2:25 pm -I think what’s interesting here has nothing to do with oranges and probably putting it in a phony real-world context is likely to hide what I personally notice about the computation (yes, Virginia, we radical constructivist types DO like computation. . . just not for the sake of computation and deadening little minds). So I see no point in putting in giant rotting oranges or anything of the kind.

What I like is not seeing it algebraically, but simply as the fact that if 9/10ths of something + a given quantity equals the whole, then 10 times that given quantity must be the whole, since what was “missing” is 1/10th of the whole.

Doing this as mental arithmetic with of course the right initial observation seems particularly useful. Writing out the equation seems less sweet to me, personally, particularly if this shows up on some sort of timed test. Coming as I do out of the pre-calculator era, particularly the era before calculators were allowed on the SAT and ACT and the like, mental math and estimation are my by-words, along with process of elimination.

Big oranges are nice, but not here. And frankly, if you have a favorite orange. . .

## louise

September 26, 2010 - 3:27 pm -Oranges have 10 segments (just sayin’)

## Z. Shiner

September 26, 2010 - 4:10 pm -While I agree that this is definitely pseudocontext, I think that the location of the problem at least partially excuses its unrealistic premise. I assume that someone who buys a book of mathematical brainteasers and puzzles has a different outlook on mathematics than a student who is being required to learn new mathematical concepts. I think that the type of person who solves problems from the book this problem is from wouldn’t mind the unrealistic nature of the problem as he would have an appreciation for mathematics beyond its applicability to real world situations. Would I give this puzzle to a student as an introduction to writing word problems as mathematical equations? No. Would I give this problem to a colleague as a challenge? Maybe.

I believe pseudocontext is a bad tool for teaching, but not necessarily a dark mark on all of mathematics. Remember, the Fibonacci Sequence was first derived by a very pseudocontextual problem involving rabbits mating on a monthly basis.

## Randy

September 26, 2010 - 6:37 pm -I can’t find it right now, but somewhere I have in my collection an old Foxtrot comic that has a banana problem similar to this although it was a bit more interesting laid out with banana and banana peel. I suppose this would fall under pseudo-context, but still it was fun to solve just because it was embedded in a comic strip. Unfortunately, right now I don’t even remember the joke. Occasionally I would actually put this on a test. I’ll see if I can find it. Another one I used on a second semester test the first year I taught Algebra I and Geometry about 8 years ago was a Foxtrot comic where Jason is staring at the clock, the bell rings, and he sadly mentions how many seconds he will have to wait before school starts in the fall. I typically would ask them to convert it to the number of days (including the decimal fraction).

## Dave

September 27, 2010 - 6:22 am -This reminds me of puzzle books I loved as a kid — they had mazes and logic puzzles and tricks with words and punctuation, and occasionally a light math problem like this. I think I like “my favorite orange” because it’s a good puzzle-book math problem, not necessarily because it’s a good math class math problem.

That said, I’ve got to appreciate that the word use is very economical and that there’s some dry humor to it. I imagine this as a problem in the style of my high school physics teacher, who would have us calculate the speed of (his favorite?) velociraptors chasing people.

## Derrick Bowman

September 27, 2010 - 11:17 am -Just taught my two MATH 305 classes about pseudocontext today. Today was their introduction to writing simple division word problems for 3rd graders and one group came up with a perfect example of pseudocontext.

“A circus is in town. They have 31 exotic animals. If each golf cart can carry 4 animals at a time, how many trips does 1 golf cart have to take to get all the animals if 4 ride at a time?”

This group made the most perfect segway into the topic of the day. I love teaching that class!

## Zeno

September 27, 2010 - 11:01 pm -To explain my earlier comment, the term “MacGuffin” was used by film director Alfred Hitchcock to refer to a plot element that is pivotal as the goal motivating the characters, but the particulars of which are of no importance to the plot. The audience does not need to know what the MacGuffin is to comprehend the plot or enjoy the film. It is characteristic of a MacGuffin that it could be replaced by something else without significantly altering the plot of the film.

Like the MacGuffin in a Hitchcock film, the orange in this problem could be replaced by an apple or an automobile or anything else without changing the mathematical solution. That is typical of math problems which are intended to test specific math skills. The “real-world” context in which the problem is framed is not really important because the math does not depend on it.

## Barry Lewis

September 28, 2010 - 3:59 am -@Derrick(#23)–I don’t see the downside of inviting a 3rd grader to imagine that she is driving circus animals around. And what if 1 animal at most could ride beside her in a cart that pulls trailers that can carry up to four animals each. How many trailers would she need to hook together if she wanted to take all 31 animals to the Big Top i just one trip? Seems like a fine context to me.

## Barry Lewis

September 28, 2010 - 4:01 am -correction: […IN just one trip?]

## Barry Lewis

September 28, 2010 - 4:13 am -Additionally, what if there was a list of exotic animals and among them were cats and birds. If the cats couldn’t share a trailer with the other animals, then what is the least number of trailers she would need? I don’t see where context interferes with mathematical thinking or where it constitutes a cumbersome, wholly unintelligible scenario. It is a simple, accessible story line that actually affords a fair amount of conceptual scalability, seems to me.

## Michael Paul Goldenberg

September 28, 2010 - 6:36 pm -My son (h.s. sophomore, in 2nd year algebra) mentioned to me apropos of his homework (and with no mention by me of My Favorite Orange or pseudocontextual problems) that he hates problems where you would never actually do the math to find the answer, e.g., your store receives 14 containers of eggs. If each container has 12 cartons of eggs and each carton holds 12 eggs, how many eggs did you receive? He said, “Wouldn’t you just check your order? Wouldn’t you probably KNOW how many eggs you needed when you placed your order?”

While the specifics he mentioned may not be exactly right, the general notion he had was dead on: blah, blah, blah. . . so what is Juan’s salary? Why not ask Juan?