You could photograph entire books. Pages upon pages of absurd problems. From a Glencoe Pre-Algebra quiz available online:

“It costs $7.50 to park a car or van at the Family Fun Amusement Park. The admission price to the park is $22.50 per person. If a group in one vehicle spends $142.50, how many people went to the park?”

Who has vested interest here? Perhaps a math-challenged deity auditing a sect of his spawn. Or the CEO of FF Amusement trying to challenge his precocious six year-old. Maybe the sheriff’s office of ‘Town where Family Fun Is.’ Maybe there was a murder at the amusement park and all of the suspects were in this one vehicle and all they had was a receipt that read ‘142.50’ and surveillance cam footage of an overstuffed minivan.

It is a stupid problem.

It’s like when students ask, ‘When will we ever use this?’ I tacitly acknowledge their pain with a smirk and averted eyes. What I should say: ‘If you ever have my job, and only then. Otherwise, you probably wouldn’t be interested in finding the trajectory of an invented dolphin jump using a quadratic model.’

It came to me last year to not teach algebraic modeling under the guise of ‘real-life.’ Teach it without pretext.

My students had more fun with weird stuff like arithmagons than any other word problem I gave them. They’d rather analyze something obviously abstract than something speciously ‘real.’

What you and Mathalicious are trying to do is the next phase.

Um, not that I think it’s not worth calling out, or that it isn’t a problem.

It totally is.

I’m hoping your plan with this is to continue to take those psuedo contextual probems,and turn them into real problems.

We need more of that. But examples of fake pretending to be legit? I should just mail you my whole textbook.

I think that it’s unfair to blame this on textbook authors, unless one means textbook authors going back throughout the history of algebra. Take just a random example from the history of math:

“There are four pipes leading into a well. Among these each fills the well in 1/2, 1/3, 1/4 1and 1/5 of a day. In how much of a day will all of them together fill the well and each of them to what extent?”

This is from a medieval Indian text, the “Mahavira.” It’s not hard to pull out more “pseudocontext” examples from just about any early text.

So here’s my submission to your assignment, Dan: Ars Magna, and the author is Cardano: “The dowry of Fancis’ wife is 100 aurei more than Francis’ own property and the square of the dowry is 400 more than the square of his property. Find the dowry and the property.”

I think I could send you every word problem in the Saxon Algebra I and Algebra II books I have on my shelf. (I keep them for the nostalgia, mostly.) I’m pretty sure I’ve done every coin combination problem known to mankind.

Here’s where my mind always goes in this discussion: every curriculum designer runs into mathematics that “resists” being presented in an authentic, meaningful context. (Example that’s stumped me: rational expressions and equations.) We all know this happens. At what point do we stop contriving a pseudocontext and begin seriously questioning the value of the mathematics? While I don’t want to suggest that we apply mathematical Maoism (http://bit.ly/dnflxz) to the whole of school mathematics, I wonder how well we prepare ourselves to discard pieces of mathematics that we can’t apply in context.

Maybe I’m missing the point.

I don’t see what’s so deadly about pseudocontext. It can provide examples to illustrate application of a mathematical principle. It creates conditions that are sanitized where simple mathematical models can work.

The path of a object thrown in the air isn’t really a parabola in the real world. Just try throwing a hollow plastic ball far and fast. But if you throw it slow enough, the air resistance might not make much of a difference and a parabola is then a good approximation. Or you can also use a heavy enough ball. Reality can be messy.

In the real world no one calculates how long it takes a train traveling at 50mph to get somewhere and add it to departure time to find arrival time. I mean no one except the engineer or whoever is in charge of scheduling. Most people just look at the schedule.

In the real world, no train travels at 50mph the entire trip. It accelerates and decelerates in the beginning, end, and also when making turns. We assume it travels 50mph the entire time when we give word problems to show how a simple mathematical model involving rates can work. I guess one could argue that for a long trip the acceleration and deceleration is negligible, but if a student is struggling with grasping a new concept, doesn’t exposing them to the imprecision in measurement and sources of error just add to the complexity?

Take the following problem:

Train A heads to City B from City A at 20mph and Train B heads to City A from City B at 10mph on the same track. A bird flies at 30mph between the 2 trains starting by flying from Train A to Train B. If both trains depart at the same time and City A and City B is 60 miles apart, how far will the bird have traveled when the 2 trains meet?

First time I did it, I thought about the bird getting squished. Then I actually began calculating the distance traveled when the bird flies from Train A to Train B, then again from Train B to Train A while keeping track of the elapsed time. Then I realized this process would take forever and had to discover another way to calculate the distance. There’s a nice mathematical principle to be discovered here by shifting perspective (calculus or summation are not needed). It’s a problem with pseudocontext. I mean should we take the size of the bird into account?

Where would math puzzles fall?

I’m not arguing that pseudocontext doesn’t disconnect some kids from math. I can see how kids can be more engaged in math if they did analysis of a real world situation and used mathematical models to come up with results. I can see how problems where they get to apply their knowledge and experience with math can be useful. I can see how that experience can be more authentic and meaningful.

Pseudocontext can be a bit of a turn off for some kids, but I’m not seeing why pseudocontext is so bad. Maybe you can enlighten me.

PS. In stats, answers to some questions aren’t in the chapter (e.g. What’s the shape of the distribution of height at a little league game?) and often times require student knowledge and experience. The opposite happens there. Some kids don’t know enough. It’s like they were locked in a closet.

Jo Boaler’s latest book The elephant in the classroom is equally inspiring. I read it in two sittings.

Raymond made a great point earlier about how some content simply resists being taught in a contextual way. As someone who spends a lot if time writing lessons, I look at something like simplifying (4x^6y)/(8xy^3) and have no idea what to do with that. Still, insofar as states test it, I can’t ignore it.

Ultimately, though, many of the issues come out in the wash. For if you try to incorporate real-world applications into classroom instruction, then invariably you’ll end up doing more with functions, ratios & proportions, etc., and a lot less with simplifying square roots. The problem comes when teachers don’t distinguish–treat everything as equally (perhaps infinitely) important–which makes it hard to get but-in from kids who know what’s up. Still, if you try to keep it real when possible, then you’ll probably have some cred when you’ve got to hit the relatively trivial.

Finally, as many have already mentioned, there I something to be said for learning something, even though you might not have an application in mind (or know if one even exists). Again, I JUST had my first encounter with a system of equations in the real world, so good thing someone taught it to me. Also, there are a few ways of defining math which also informs how we communicate it. You can take something like the normal distribution and focus on shoe sizes, ages, etc. All good. But you can also use it as a way to better understand the way nature may work. You’ve got a kid who grows up and feels in his gut that he wants to be a painter, but the world tells him that he’s crazy. But then maybe he thinks back and says to himself, “But maybe progress and beauty and genius all happen +2 standard deviations from the mean,” and that’s what turns him into Van Gogh. Math becomes something transcendental; it becomes permission.

From the preface of Terence Tao’s ‘Solving Mathematical Problems:’ “Mathematical problems are ‘sanitized’ mathematics . . . if mathematics is likened to prospecting for gold, solving a good mathematical problem is akin to a ‘hide-and-seek’ course in gold-prospecting: you are given a nugget to find, and you know what it looks like, that it is out there somewhere, that it is not too hard to reach, that it is unearthing within your capabilities, and you have conveniently been given the right equipment to get it.”

Nobody else has mentioned it, but Underwood Dudley also calls out this particular problem in his excellent recent article What is Mathematics For?, with the more general thesis that the actual methods involved in algebraic problem solving (as opposed to the abstraction skills which are ostensibly cultivated by them) are of no use whatsoever to most people.

Avery, the busses problem you mention is the exact reason why it must be made clear that this is pseudocontext, and not a real world situation. When math people see that problem, they are not thinking about real world busses; they are thinking about platonic ideal busses that hold only a specific amount of people and no more. Since suburban students don’t have much experience with real busses, they would also be thinking about ideal platonic busses. (But not because they are thinking mathematically. They just got lucky.)

The trick is to get the student to see the problem as specifically not a real world problem. That way the student can enjoy the math part of it, without the confusion of the always nebulous real world.

As a recent Algebra 1 textbook author I have read this discussion with interest :)

Dan (via Jo Boaler) is right on that these kinds of problems (what our lead author calls “pretexts” instead of “contexts”) can lead to all sorts of issues. But “real” issues are often very messy, sometimes in ways that are debatable as to whether good mathematics is happening.

So I feel we attack this issue in a few ways (and I don’t mean to pimp our books but they are called “CME Project” if you are interested in looking through them)…

1. Work some “real” problems through completely, even if it means taking a long time to do it. The flagship problem for this in our books is the monthly payment on a car loan. In Algebra 1 students can estimate car payments and use Excel or other tools to check while learning about iteration and recursion. In Algebra 2 they can track payments as variables to build expressions for the total payment or the monthly payment. In Precalculus they use the formula for the sum of a geometric series to find the formula once and for all. This is a real context, done completely, and serves some other good mathematical purposes.

2. Use mathematics as a context. (Dan mentioned this specifically very recently.) Students can look for patterns in addition and multiplication tables to help them see why negative numbers behave the way they do. They can investigate what 2^0 and 2^(-1) might be defined as, and come to an understanding that there isn’t much choice in the matter. They can find functions that agree with input-output tables, by intuition, by common differences, by setting up systems of equations, by using matrices, by using historical methods (Lagrange Interpolation, Newton’s Difference Formula). None of this requires a pretext, and all of it feels as real to the students as any other context might. It probably feels more “real” than a lot of the contexts that appear in books, which students know are fake.

3. Acknowledge when pretexts are being used, and why. As “Mr. H” says above, we have good reasons for thinking about the parabolic path of an object, even though the real path isn’t parabolic due to resistance (and heck, even with no wind resistance, it’s actually elliptical). Sometimes a true context needs to be adjusted to be usable; I think it’s important for students to recognize or discuss when, and why, such adjustments are being made.

4. Use pretexts for legitimate algebraic growth! (What?) This is hard to pull off in a blog comment, but these “pretext” word problems are actually good breeding ground for building the algebraic skill of generalizing from repeated calculation (one of Common Core’s Math Practice standards). Do this by flipping a word problem on its head by guessing an answer then following the process to check that answer. Here’s one we used based on some problems from Egyptian mathematical papyruseses:

“A number plus its fourth equals 560. What is the number?”

When I first taught Algebra 1 this would be the part where we would tell kids to translate the sentence “into algebra”, and hope it works. What our book recommends is to guess at the answer and test it, NOT to guess-and-check for the answer, but to determine the process for checking any guess. This process becomes the equation. For example, say I guess 100 is the answer. How do I check?

100 plus its fourth… hm that’s 100 plus 25 that’s 125. Nope, it’s not 560, it’s not right.

(Note: no “it’s too small” or “it needs to be like 5 times bigger”.)

So I’ll guess again, maybe guess 8 this time. 8 plus its fourth… that’s 8 plus 2 that’s 10. Nope, it’s not 560.

Eventually, tell kids to guess “n” and follow the same process. They will see through their numerical work that the second piece is n/4, and the equation is n + n/4 = 560.

I hope this is described enough. Try it the next time you see a word problem, especially those goofball Mensa how-old-is-Billy problems. It’s slick, it actually works, it supports mathematical habits of mind, and it reduces the difficulty and “getting started” hurdles of algebra word problems significantly.

So, I’m not a fan of pretext problems, especially the type where a book puts Problem 41 at the end of a long skill set pretending it’s a real example from cartography or carpentry or carburetors. But pretext problems can actually be useful mathematically…

This has gone on far too long, and I apologize! Best wishes to all teachers for the upcoming school year.

– Bowen Kerins, one of the CME Project authors
http://www.edc.org/cmeproject

In 1990, I interviewed to be admitted to Duke University for engineering. The man asked me Mr. H’s bird question from comment # 9, except it was a fly instead of a bird. I was floored. I was taking calculus at the time, and tried to apply this to the question, looking foolish. When the man told me how easy it was to solve with simple algebra, I wanted to strangle him. I didn’t get in. I haven’t seen that question in 20 years, but it still haunts me.

Pseudocontexts have long been a pet-peeve of mine. They are a hallmark of a particular grade-band in math. The beginning algebra student is the usual victim of a pseudocontext.

“A, B, and C together have $1000, A and B have $761, B and C $675. How much has each?”

While instructive of algebra, the problem is pendantic, overly concerned with the numbers and rules and not realistic in circumstance. Perhaps what is most annoying is the destructive shift away from ‘Central Ideas’ or ‘Big Ideas’, towards math as useless puzzle solver. I’d bet that puzzle solvers like the pseudocontext just fine.

After consulting my “Ontario High School Arithmetic” (1909) I’ve found relatively few examples, perhaps since people didn’t have time for such nonsense 100 years ago – they wouldn’t put up with it!

Inquiry based learning is the way to go, so that students wouldn’t be solving pseudocontext problems unless they chose to pursue an inquiry of puzzle solving.

“A person sold A 3/4 of his land, B 4/5 of the remainder, C 5/6 of what then remained and received %40 for what he had left, at $60 per acre. Find the number of acres he had at first.”

Finally, in reference to ancient pseudocontexts, that’s a non-starter for me because bread and beer were indeed the big ideas back then! lol

I love this discussion as one of my favourite examples of a psuedocontextual problem is one where we are told “The Sydney Harbour Bridge has its platform with an area given by x^2 + 6x + 5……..” and then basically goes on to apply factoring in the actual question.

However, I am concerned with the fact that it would be nearly impossible to come up with real contexts for everything we have to teach (based on our curriculum guidelines). Though some might argue that that, in fact, is the problem – all the needless stuff we need to teach. I agree with Mark who says that much of what we do in school is exercise for the brain and that is its real purpose. Most of the stuff we actually need to know for day to day life we learned in elementary school. Of course that is no reason to not try to give real and inquiry based problems when ever we can but I think it might not be realistic to do it all the time.

Another thing is that sometimes the actual question may be non contextual but the set up could be. That is, we give a context where the math needed relates to the math being taught but may be too advanced. Someone mentioned the projectile motion problem and how to truly attack this problem you need to consider more than just gravity (that’s what the programmers for computer games with good physics engines or the modelers for CGI movies use) but we give students the “lite” problem so that the math is palatable.

For example, I wrote a question dealing with the intersections of lines and planes in 3 space. The question itself was pretty standard and could have been asked on its own with no context. But the lead in was a blurb about a new technology that allowed researchers to study where autistic people were looking in given situations by using special glasses that could track the eye movements. Its really the same context but just under far more complex circumstances.

I wish I could write questions like that everyday but there are only 24hrs in each day (for those of us who choose not to sleep).

It would be great, however, to generate a list of questions for various topics that have real contexts and aren’t contrived. And then have that list searchable and somehow have the questions go through a vetting process. Should be easy to set this up with web 2.0. This is 2010 right :-)

Michael P beat me to the punch with Ars Magna. Let me throw in Kitab Al-Jabr W-Al-Muqabala by Muhammad Ibn Musa Al-Khwarizmi (Baghdad, 820). I was surprised and amused by the number of ridiculously contrived word problems in both books. (Btw, Karim, yeah dowries are useful, and I know you were being cheeky, but just to drive the point home why would you ever square one?)

I haven’t read Liber Abaci (Leonardo Fibonacci, 1202) but for my money I bet it’s just the same. The rabbit problem is anyway, for sure.

On the other hand, the problems from those old books have the advantage of being utterly awesome because they’re old. I would totally be excited to do a contrived problem from a book that was 1200 years old on the strength of it being 1200 years old, c’mon now! It helps if the problem is acknowledged as contrived. It almost ups its coolness because then it’s like “hey students, you know how you’ve been oppressed by contrived problems your whole schooling? Check out this 1200 year old book, you are about to do the MOTHER OF ALL CONTRIVED PROBLEMS!” I especially feel this way about the Fibonacci rabbit problem because THAT PROBLEM IS WHY IT’S CALLED THE FIBONACCI SEQUENCE.

To contribute to Avery’s list of issues with pseudocontext (great word, thanks Ms. Boaler via Dan), or perhaps just to repeat what everybody has already been saying, but hopefully add a nuance (?), it seems to me that the biggest danger in pseudocontext is this. If the story, as understood through the math you’re intended to do, runs in any way counter to students’ knowledge about the world, they are being trained to suspend their reasoning faculties while in math class. This is basically the craziest most harmful thing we could ever do to their mathematical development. The test for this (worst) kind of pseudocontext is: is it possible to get the wrong answer by reasoning sensibly about the context? Let me coin the term “evil pseudocontext” for this type of story problem.

By this measure, Bowen, none of the problems you mentioned are anywhere near evil pseudocontext. On the other hand, Avery’s example about the buses absolutely is, unless the lower number of buses is also accepted as correct.

Dan, you’ve been defining a broader meaning of pseudocontext (Let me coin the term “lame, depressing pseudocontext”) where the test is “could you sub the context out for meaningless doodads without the story losing any interest?” On this measure Bowen I think the Chiko and Diophantus problems might be a little pseudocontexty (though actually in the Diophantus problem somehow the story is adding something for me; I think it helps that Diophantus was actually historically like Mr. equation-with-an-integer-answer, somehow that causes there to be something wistful and emotionally compelling about an equation of his life; students doing this problem should get to know that about him), but this is made up for by the historical thing. In a way, the context in the problem is not even the real context (esp. if the teacher is up front about the contrivance). Really it’s a straight math problem being given vitality (if not scaffolding) by its historical context.

The danger of this second kind of pseudocontext is about alienating students, teachers, and math from each other, as per your illustration Dan, since the problems and therefore the teachers are full of it. It’s insidious because it poisons the learning environment. But somehow it strikes me as not quite as bad (as evil pseudocontext), because at least it’s not directly debilitating to the students’ intellects.

Here’s a reformulation of the test for lame, depressing pseudocontext. I think it is basically equivalent to Dan’s test.
1) Does anyone want to solve the problem?
2) If anyone does, do the people who want to solve it care about the story?
If the answer to either question is “no” then this is a case of lame, depressing pseudocontext.

What I’m going for with this reformulation is a shift that I feel like a lot of folks in the blogosphere (Dan, obv, and Riley, for example) have been jointly articulating, in how to understand the question of students finding math to be relevant / irrelevant. I feel like the emerging idea is that the question “how does this get used in ‘real life’?” is a red herring. As I remember Riley writing about though I can’t find it now, understanding compound interest is something that everybody can use in real life, but if when students say “this is boring and useless” you turn around and start teaching about compound interest but don’t change anything else, it’s not going to help. Conversely, it seems to me that actually polynomial factoring is totally useful in real life to a student in high school because you need it to pass your class. That life is a lot realer than the theoretical future in which you have a bank account and a credit card.

Big.

Math problems embedded legitimately in the physical or social world have the advantage that the students’ knowledge about the physical and social world helps orient them, and provides rich substance for the above-listed processes to operate on. That’s why physical and social reality are great settings for problems (if the settings are for real i.e. not pseudocontext). Not because the math skills involved are “useful in the real world,” whatever that means. This is why “math itself is a context.” If the students have a developed enough understanding of a mathematical setting for a question, the above processes can operate on this setting just fine, and then the skills relating to this setting and this question are instantly relevant whether or not they are “useful in the real world.” (I mean, they’re totally useful if they help you answer a question you want to answer, right? What could be more useful than that?)

So, this is my answer to David Petro and other folks concerned about how there are things in the curriculum not susceptible to being motivated by “real life examples.” This is fine. There was a reason why somebody made up these ideas. That means there were people who wanted these questions answered. That means there can be again. The challenge is to find the (strictly mathematical) drama / enticement to curiosity that is at play in the ideas, and figure out how to make it visible to students. If the ideas are really important enough to stress over how to teach them, then there should be plenty of drama and enticement available.

What I’m ultimately getting at is that I think that basically the same principles are at play when you hook kids on a problem like the escalator problem or a pure-math problem, and I’m interested in all of us doing more thinking about how to carry the design principles we’ve found for each kind of pedagogy to the other.

Okay I’ll shut up now. I don’t know what possessed me.

It strikes me that the root problem is lying to students about the purpose of learning a given skill/technique/topic. Some mathematical topics DO have wonderful “real world” applications, and should be taught as such. Others are explorations into the realm of pure mathematics, and should be identified as such. We run into trouble whenever we try to sell math to our students for the wrong reasons (i.e. “imaginary numbers are useful for modeling in electrical engineering,” etc.). In my experience, being painfully honest about the reasons for studying a given topic or completing some mathematical maneuver has been met with appreciation and more engagement than I would have expected. Even if it means telling them now and then “we rationalize denominators because decades ago people had to look up roots in table form, and its easier to divide a nasty decimal than to divide BY a nasty decimal, and no one has bothered to break this trend.” I’d rather give an honest but slightly dismal answer to my students than to lie to them. My students generate a list of five reasons for doing math at the beginning of the semester, and I encourage them to ask me “why are we doing this” at any time. I can always relate it back to one of the five reasons, without overselling applications, for example.

‘Something I’m realizing in all of this is that it doesn’t matter how I reply when a student asks “when will I ever use this in real life?” I’ve already lost. Because that student isn’t asking a question about an uncertain future; she’s lodging a complaint about a frustrating present.’

This is spot on. You don’t get that question when you’re teaching rational expressions with verve. You get that question when you’re tranquilizing with busy work.

I just blogged about a student of mine who, when I asked what math is, said, “It’s stuff you use every day. Like when you need to make change.” He’s 9, by the way, and probably isn’t going around making a whole lot of change. It made me furious that kids are (still) learning to treat pseudocontext as “important” and that actually important stuff has no business in school.

Math for Love

@Mark, Thanks for the recommendations. I’ve ordered all 3 books. (The 2 volumes of the Smith book were different editions, unfortunately.)

Is this an example of pseudocontext? It certainly made me laugh and cringe at the same time on a Saturday morning…

@Breedeen:

You are a really brave person to venture into the comments in Failblog… :) Great find!

I love all the comments and responses regarding pseudocontext. I agree with so much of what was said; however, the post about the need, or lack thereof, to rationalize denominators piqued my interest. It made me wonder: How many topics (that we currently teach) are outdated or perhaps unnecessary to bother with in K-12 mathematics?
I know “pseudotopics” would not be a proper nomenclature, but I can’t help but think of all the time that is spent in high school mathematics classrooms on topics that have little contextual (pseudocontextual or not) application to the students’ future (in mathematics classes or life). It feels as though so many topics have been added, but none are ever purged.
I have taught Algebra 2 every year for nine years and I have yet to completely address every learning standard in any of my classes. I often wonder if 1.) it’s possible, and 2.) a better way to organize the topics for high school mathematics in a more manageable and meaningful way, while still maintaining the integrity of mathematics (as a context in itself).
What do people think? I get that certain topics (e.g., operations on rational expressions) are needed to simplify (say, a derivative being computed using the Quotient Rule) in higher-level mathematics; but is this something we should spend significant time on in high school courses?
I am sure everyone reading this blog knows how much students love performing operations on rational expressions! And I totally agree that it helps to develop logical reasoning and is a context in itself, but there are so many other, and dare I say more interesting, mathematical topics for high school students that will still develop reasoning and critical thinking, while perhaps maintaining their interest. Examples: discrete mathematics or number theory. Or then again, I may be slightly hypocritical for suggesting more topics.

Thanks for mentioning this book, btw. Just finished it; great read! Also, she mentioned several other books I’ve now gotten and learning a lot. Great thinkers out there :-)

Asked my differential geometry teacher for an example of a space that’s continuous but not bicontinuous. His answer: “Ummm, for example a space that satisfies the axioms of continuity but fails with at least one of the axioms of bicontinuity.”

I haven’t read all of the comments, so I’m not sure if someone already discussed this or not.

My school uses the CMP 2 curriculum, and for the most part, I really like it. It poses problems within the psuedo-contexts, and actually allows the students to explore the problems with some real world knowledge coming in to play.

However, in one unit, there are questions about the “break-even point” – defined as when the income = expenses. The only problem is that in BOTH cases they give, the number is fractional (when you solve the equations), but the items being sold are not divisible! You can’t sell 133.3333… t-shirts!! That doesn’t make sense! If you round down to 133 shirts, you make a slight loss, and if you round up to 134 shirts, you make a slight profit. So the REAL answer is that with the numbers you chose, you can NEVER actually break even!