(One Of Many Reasons) Why Students Hate Algebra

A youth group with 26 members is going to the beach. There will also be 5 chaperones that will each drive a van or a car. Each van seats 7 persons, including the driver. Each car seats 5 persons, including the driver. How many vans and cars will be needed?


Tuesday was an all-school professional development day. The math departments joined from two campuses to learn about the Gradual Release of Responsibility from a couple of math coaches from the next county over.

One coach modeled a GRR lesson and opened with the problem above.

I leaned into another teacher and whispered, “I’m trying to decide which would be more socially acceptable right now, letting out a loud fart or saying what I really think about this problem.”

We broke for lunch and came back to debrief. No one had commented on the problem by the end so I did.

“I see problems like this and I feel myself becoming less of a human and more of a math teacher. And I feel very lucky to teach our neediest students, students who punish me daily for problems like this one, students who are often very hard on me but who in return have helped me hold onto some of that humanity.

“I have three questions about this problem and we can discuss any of them or none of them.”

Three Questions

“One, is the problem realistic? Would a real person need to solve this problem?

“Two, is the solution realistic? Would a real person solve the problem using a system of two equations?

“Three, in what ways does this problem help our students become better problem solvers?”


I didn’t elaborate. I thought my questions were self-evident and their answers self-explanatory. I was wrong. The coach shrugged me off, saying, “Well, it’s in your textbook.” and I couldn’t disagree. None of my colleagues seemed disturbed by the opening exercise of this quote model lesson unquote, so I didn’t belabor the point. In hindsight, I wish I had soapboxed:

  1. This is a problem you will only find in a math textbook. It’s bizarre to me how many different ways just fifty words can fail to square with reality. Why does each chaperone have to drive? Why can’t we take five vans? Why do our vehicles have to seat the exact number of people in our group and no more?
  2. No youth group leader would ever solve this problem with a system of equations. I’d wager that no math teacher, if somehow faced with this completely fantastic scenario, would solve this problem with a system of equations. With 31 people, we’d just shuffle them around until they fit. Even if we insisted on the contrivances in #1, there are only [0, 5] possibilities for the vans so we’d use a table or just guess and check. ¶ I asked the coach why we were forcing the issue of systems when the easiest solution by a long shot was tables. She replied that we learned tables last class and this is the new skill we’re learning.
  3. This kind of algebra makes our students dumb, unimaginative, and scared of real problems. At the end of the model lesson, the coach put up our homework, which was a carbon copy of the original problem, new numbers swapped in for the old. ¶ I can’t describe my contempt for this arrangement. ¶ This is how we make kids stupid and impatient with irresolution, eager for contrived problems that look just like the last contrived problem, completely lost if we so much as switch around the order of a few words. “We don’t teach them problem solving skills anymore,” my department head said to me. “We teach them problem types.”

Algebra teachers sell students a cheap distortion of the real world while insisting at the same time that it really is the real world. The cognitive dissonance is obvious and terrible. Students know the difference. It cheapens my relationship to them and their relationship to mathematics when you ask me to lie to them.

It’s like offering someone lust or manipulation while insisting that it’s love. Not only are the short-term consequences devastating but it makes that person distrustful or wary of the real thing. Make no mistake. We are making an alien of algebra. We are doing real damage here.

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. Yeah, but if there’s a problem with chaperones and vans and cars on the state standardized test, doing this crap will make you look like a great teacher, right?

  2. My favourite maths word problem is :-

    “A man left Albuquerque heading east at 55 km/ph. At the same time, another man left Nashville heading west at 60 kph. After some time the 2 men meet. When they meet, which man was closer to Nashville ?”

    Not my problem, but included in “A Man Left Albuquerque Heading East : Word Problems as Genre in Maths Education” by Susan Gerofsky.

    In my opinion, the difficulty with word problems like the one at the top of your post is they lack creativity. A well worded creative maths word problem can be playfull and enjoyable, and a great opportunity for learning. To quote from the book I’ve mentioned, the homiliness and familiarity of the story’s images can be a way into the otherworldly world of maths.

    The blog http://offthehypotenuse.blogspot.com/ has some good articles as well about having open ended word problems, perhaps that’s more realistic.

  3. A few questions come to mind:

    Could it be that the coach shrugged you off because the context of the presentation was an instructional model and not necessarily the curriculum itself?

    Would the presentation have been more acceptable had it been centered on solving systems without the aid of a contrived, unrealistic scenario (eg. Solve the system: x+y=5 ; 5x+7y=31)?

    Did your department really need to “learn” about an “I do, you watch; I do, you help; you do, I help; you do, I watch” instructional model?

  4. The wording of this kind of question is pretty ridiculous. It’s not the kind of problem that would seem interesting to most people. You can probably come up with something better. However, I can’t really think of anything that’s much more interesting (though I can probably word it less ridiculously) when it comes to using system of equations.

  5. I agree with you (and am slightly embarassed since I know I have used problems similar to that in my class!) and so I am just wondering what kind of problems you would use to teach systems? In my class the more ‘realistic’ problems are comparing cell phone plans type problems which usually are solving by graphing. Seems harder for me personally to come up with better problems for them to solve algebraically. Maybe those more creative than me can give me some ideas! :)

  6. Robert Talbert

    January 29, 2010 - 3:27 am -

    I hear what you’re saying here, Dan, and I don’t mean to defend this kind of problem or the PD instructors’ robot-like responses to your objections. But could this sort of problem be useful for students in one of the following ways?

    (1) It gives students a little warm-up exercise before tackling a problem that really /does/ need to be solved by a linear system (or something similar like a simplex method optimization problem); or

    (2) It shows student that certain problems /can/ be solved by linear systems, even if they shouldn’t be; again setting the stage and giving students a touchstone for bigger problems that can’t be solved more efficiently (or at all) by brute-force.

    Again, not to defend unrealistic problems, but perhaps the problem’s in the book for one or both of those reasons.

  7. I like Robert’s comment, but what I have learned from Dan’s blog is that we can do better. Our teaching and our textbooks have over-emphasized the process over the take-away, and therefore we have courses that are so deep in teaching the process of solving a math, that kids take NOTHING away from our classes. I believe Dan’s general point here is that a few engaging problems are more important to teaching real skills that will have real impact in people’s lives. Sometimes we overvalue the generally meaningless list of techniques that we as math teachers have mastered.

    Think of being the writer of a sitcom, questions like this are like breaking the 4th wall to do a product placement. In the classrooms we need this problem seems obtuse and breaks the flow of the show.

    I personally think that a fully abstracted problem is better than this artificial word problem. But we shouldn’t lead in with an abstract problem either. The opening is a key moment in the presentation, and we should try to avoid problems like this.

  8. Mr K. hits the nail firmly on the head. Everything in schools is dictated by high-stakes tests. Your performance as a teacher is retrospectively validated or vilified based upon them.

    I think it’s time we infiltrated the exam boards. :-)

  9. If you want a realistic system of equations students will want to solve, look at cell phone plans.

    Suppose Plan A costs $20 a month and $0.25 a minute after you go over 200 minutes. Plan B costs $30 a month but only $0.20 a minute if you go over 220 minutes. Which plan should you choose and why?

    You can solve this graphically, using a system of equations, or by trial and error, but anyway you slice it, the kids are going to enjoy learning about how to analyze their cell phone plans.

    Through in some research about cell phone plans in your area and have the kids write about what they discovered and you have a fun math project.

  10. David-

    The cell phone example is a good one, especially if you recognize and embrace how open that question will be.

    I taught from a book several years ago (in Capitola, of all places) that used comparing land-line phone plans as the theme problem for linear systems. It was 2002 and my 6th graders had never paid a phone bill, and already used mobiles far more than landlines.

    I decided to pull up the Cingular webpage and try transfer the problem to cell plans. Becuase it was my first year teaching, I did this in class, by the seat of my pants.

    When we finished the question two periods later we were considering:

    Base Monthly Price
    Minutes included per month
    Cost per minute over base
    Cost per minute of roaming
    then, coverage map (Cingular was spotty through the hills at the time)
    Cost of “Free Nights/Weekend” rider
    then, “F N/W” minutes vs primetime minutes
    Cost of a SMS Rider (100/500/Unlimited)
    Cost per SMS over/without rider

    My plan had been to compare Cingular with another provider, but we never got there. By the end, my 6th graders had a sprawling psuedo-equation, with several “radio-button” sections based on their choice of SMS, F N/W, and Base Minutes.

    There was comparison, but there wasn’t a solution. Condensing the formula down into however many distinct permutations and graphing the choices against each other never came up. But they each wrote “their” version of the equation based on their own choices.

    I don’t think I helped them with the STAR questions over those days, but it was a better class than what I had planned.

  11. At first I thought, “Interesting question, why’s Dan ranting?”

    And then he tells me they want this solved by systems of equations. Yep, they must be in some crazy (I can imagine California doing this, in a nightmare scenario) place where (how do I show a Schwarzenegger accent online?) every seat must be filled.

    Why would I think it’s interesting? Because I was imagining our situation at my son’s school, and all the juggling we do whenever there’s a field trip. I’ve proposed the ‘how many cars’ problem to the kids. But that was one type of vehicle, and the idea was to model division for kids just starting to think about it.

    I always start systems of equations with jingling some coins in my pocket, and saying how much money it is, and how many coins total. “So, if it’s all dimes and nickels, can we figure out how many of each I have?” Not nearly as inspired as Dan’s lessons, but it pulls some of them in. It’s silly, in a good way, like that vans and cars problem isn’t. (Of course, guess and check still works better for some students than the algebra.)

    The hard part of a story problem is setting up the equation. With the coins, I get to have each equation represent a different sort of thing, and then I can point out how each term in one equation represents the same sort of thing as the others. Number of this type of coin plus number of the other type of coin = number of coins altogether. Amount of money in the nickel pile + amount of money in the dime pile = total amount of money. Big problem here – “I can point out…”. I want to hand it all over to them more. So I want a lesson that will pull them over this hump by intrigue, instead of me pushing. I’m not there yet…

    I think time sequence problems are good. “If this linear trend continues, when will the number of women in college surpass the number of men.” (Oops. That one has already passed its crossing point.) The problem with those is that it’s usually not accurate to express them as linear. Most growth is exponential.

    Sorry to ramble so long.

  12. What made you more angry, the bad algebra question or the fact that whoever was leading the PD totally missed on a chance to teach you by brushing away your question? Part of the problem is that the person giving the PD is not comfortable with the irresolution that you offer. The coach was not comfortable telling teachers that they need to think to do their jobs. The coach modeled the very type of teaching that you cannot stand. Closed, right answer, no skill required teaching.

  13. Welcome to my world. I am thinking of leaving teaching because thinking is considered anathema in the profession. Teachers are all too willing to have corporations do all their thinking for them – except that corporations don’t either. In my experience, the math coach would have been intimidated by the question. Math coaches like to have us all believe they are all-knowing when, in fact, they were all too often simply eager to get out of the classroom. I have yet to meet a math administrator in my large urban system who has a clue. No wonder our children hate math and are increasingly disengaged from it. This year I’ve gone off the grid. My students are thrilled and are learning math like whoa. They are testing at higher levels than ever (evil, but required) and they are getting AND LOVING the math behind the stupid tests. Even so, I’m surrounded by people like Dan describes and it is debilitating. knowak tweets keep me going, I swear…

  14. Using this problem as a “Here, this is why you need to learn systems of equations!” justification is absurd.

    But “Here is an easy problem to learn translating words to equations” is a necessary step.

    I remember awhile ago you talked about how your regular Alg 1 classes were doing fine, but a lower class was really struggling.

    Some of us work in places where our Algebra classes ARE essentially pre-algebra.

    I need to give them plain integer problems like this to develop their [non-existent] reasoning skills.

    But I *am* honest with them, and I do not say that we do this to prepare for the day when they will need to take a road trip. We do this as a baby step on the road to more complicated, real-life problems….which then I never have the time to get to. I hope their Algebra 3-4 (second year) teachers do.

    But what else can I do with students that shut down when they see a fraction, even if we’re punching it into the calculator to graph it?

    So far, in my two years of teaching and 10 years of tutoring, motivated students take a leap into reality as a challenge. Math-damaged students take that leap as a reason to sit there and give up. I have to give them something to hold on to, a memory of “Hey, I did something like this and got it right!” to motivate them to pick up their pencil again.

    That, to me, is enough validation.

  15. The cell phone example is a good one

    No its not. It doesn’t match up with any cell phone plan the kids have ever seen. Where are your free minutes? texting options? Data?

    What the hell kind of real world do math teachers live in anyways?

  16. What I find exasperating is that somebody in administration decided that this kind of “coaching” justified the expense and insult to teacher intelligence.

    What a colossal waste of resources.

  17. I went back to school a few years back to take some business classes. As a result I had to take Statistics, something I avoided in my younger years.

    What I discovered was that the equations made a LOT of sense because I could relate them to real world situations.

    I wish I could offer a suggestion with Algebra though, because as much as I’d like to I cannot recall the last time I actually used an algebra equation in my adult life.

  18. You need a big thumbs up button on your page. =)

    Have you read A Mathematician’s Lament? A quick read and really interesting.

    I think I might be more of a teacher than a mathematician but the author’s love for the abstractions of math was actually very insightful to me. Even if these abstractions were not applicable. I guess the main difference though is that he never claims them to be applicable, but bluntly states that he loves the abstractions for exactly what it is. I guess sometimes I’m so busy trying to make everything apply apply apply, that I forget that there was some enjoyment dwelling in the abstract.

    Even if from this perspective though, he worries about the same things we do as teachers, depleting the classroom of all that is true mathematics and reducing it to a bunch of boring procedures.

  19. We had a problem similar to this, but it involved a party. Our students misinterpreted it (on the state test) for two reasons:

    1. Their definition of “siblings” included half-brothers and sisters, step-brothers and sisters, etc.

    2. The question-writer failed to include the notion that the person having the party (doing the inviting) would be there.

    Until we have authentic ways to measure learning in math, we’ll end up with artificial problems in tests and textbooks.

  20. The cellphone example is great, but lets not fall into the trap of thinking that students only want to learn useful things. My students still tell me the best question I asked all year was on the bus on the way to a field trip, the Konigsberg bridge problem.

    “If we leave school and cross every bridge in Portland exactly once, can we make it back to school?”

    This is not a practical question, but it is an interesting one. I’m new to teaching, so when I taught systems of equations I stuck to the script more than I would like, but I also had students read articles about gas and electric cars in National Geographic, and tell me when driving an electric saves the extra money you spend buying one.

    I think what we have to get past is this feeling that 10min or even thirty minutes setting up a real problem is not wasted time. This is the crucial time that gets students to connect with the skills we are teaching them the rest of the time.

  21. Wow, sometimes I feel like I’m operating on a different planet. Like Sue I was a bit confused why this problem bothered you so much until I realized it was intended to be solved algebraically.

    This can be a terrific problem at the middle school level if used in context. I’ve used a variation of this problem in my teaching, talks and workshops for years and gotten great mileage out of it. It has great potential for kids to learn something about number sense. But the devil of course is in the details.

    First of all its a variation of a simpler, famous problem:

    From Alan Schoenfeld, “What’s all the fuss about metacognition”, pp. 195-6, in Cognitive Science and Mathematics Education, Alan Schoenfeld, ed. (Lawrence Erlbaum, 1987):

    One of the problems on the NAEP secondary mathematics exam, which was administered to a stratified sample of 45,000 students nationwide, was the following: An army bus holds 36 soldiers. If 1128 soldiers are being bused to their training site, how many buses are needed?
    Seventy percent of the students who took the exam set up the correct long division and performed it correctly. However, the following are the answers those students gave to the question of “how many buses are needed?”: 29% said…”31 remainder 12″; 18% said…”31″; 23% said…”32″, which is correct. (30% did not do the computation correctly).
    It’s frightening enough that fewer than one-fourth of the students got the right answer. More frightening is that almost one out of three students said that the number of buses needed is “31 remainder 12”. [our emphasis].

    First, here is my bus problem — student page:

    Bus problem — teacher page:

    Most students did almost as badly as the students in Schoenfield’s article. They never thought to draw busses and figure it out that way. But when I did this road sign problem first,


    the kids were more likely to think about the meaning of the bus problem rather than resort to grabbing for formula straws.

    I guess the bottom line is that you can’t always judge problems by what’s on their cover. Context is huge or “what you do with it”. Maybe WCYDWT? needs a WIDDWT (what I did do with this.) ☺


  22. It’s been a long week, so I did not process the heading. I read the problem and I thought, cute K-1 level counting exercise, maybe for some manipulative work, for early “pattern type” algebraic thinking. Then you started ranting, and the second question made me double-take. I was going, “Oh, are they doing THAT? Really?”

    You can’t soapbox all the time, though. “If you have to explain, you don’t have to explain” – a traditionally pessimistic Russian saying fits this situation, imo.

  23. Hi Dan! You’re right! I remember problems like that in high school and college algebra- I ALWAYS wondering what I’d ever use it for, the problems were so lame. I remember one bulletin board that showed a gas station attendant using algebra for filling up a gas tank- completely ridiculous and unrealistic. Perhaps math lovers are fine with lameness but the majority- I don’t think so.

    Lack of real world applicability leads to apathy towards algebra and the least effort possible to get a barely passing grade. That was me and probably a lot of other students too. I’m not a math fan to begin with, so making it more mysterious by not providing motivation (beyond test scores) seems counter productive. What good is it if a student can pass algebra in a state mandated test but can’t apply what they’ve learned in the real world?

    I’m an artist/designer now and always thought my math classes lacked imagination. I wondered why a subject so critical to infrastructure was taught with such a lack of real life examples. I finally realized that math culture is very ingrown/inbred – not focused on practical, applied areas and can’t see themselves in the shoes of most students who aren’t going on to be math teachers or physics PhDs.

    I like David’s suggestion of using cellphone plans to teach algebraic math skills. Great idea — fun AND students can feel like they’ve really learned something they can use for life.

  24. I agree with Sue @ 13.

    It’s not a problem with one solution unless the no extra seats constraint is added.

    My kids would answer this with, “Duh, 5 vans. Vans are more fun than cars, and you’d have more room for coolers, beach balls, etc.”

  25. I grew up in Taiwan, where I learned to solve systems of equations. These kind of problems are traditionally called “chicken and rabbits in the same cage”. The premise is that two types of animals are in the same cage. You know how many animals there are (or maybe you know some kind of ratio) and you know how many legs they have combined. And you solve for the number of each animal. The term is usually used by my parents’ generation (so you hear older teachers use that term too), where the society is more agricultural. By our generation, the scenarios and wordings have changed a little bit in these math problems, but it’s still the same thing for the most part. They just sort of swap out the things that’s counted.

  26. Totally agree. The three questions are great.

    As Paul Graham put it, “[word problems] look superficially like the application of math to real problems, but they’re not. So if anything they reinforce the impression that math is merely a complicated but pointless collection of stuff to be memorized.”

  27. From my 5th graders HW today-

    You have $100, you spent 3/10 of it and lost 1/10. How much money do you have left?

    I know it’s not an algebra problem, but it’s written with the same flaw. Since when did anyone refer to spending $30 of their money as 3/10? Not to mention a 5th grader will rarely have $100 to spend and losing 1/10 is a monumental disaster to a 5th grader.

  28. Me: “The cell phone example is a good one…”

    Mr. K : “No its not. It doesn’t match up with any cell phone plan the kids have ever seen. Where are your free minutes? texting options? Data?”

    Hence what I spent the next 4 paragraphs on. The point is that although it doesn’t graph easily, there’s still significant algebraic content in considering a large, multi-paramater equation and deciding(!) which variables are worth considering (how much might I talk in a month?) and which ones aren’t (I’m always going to send more than X00 txts.)

  29. @EricL

    There’s actually a great section in Lemony Snicket’s The Reptile Room where the 3 Baudelaire Orphan’s struggle to not be placed in a car with the nefarious Count Olaf, while everyone else argues about how to transport the kids to the airport and the corpse to the morgue. I always read it as an indictment of these problems.

    Or, as Stephin Merritt says in his song for the book:

    “In the Reptile Room, there’s an evil man / in a strange costume. Do not ride in his van.”

  30. I too was going to ask if you’d read A Mathematician’s Lament. I really responded to that article, and it really solidified some feelings I had already started having about how we teach math.

    I’ve got my response to kids’ questions of “why do we need to know this?” almost completely scripted. It includes an acknowledgment that most of them probably never will “use” the math they learn in high school. The most everyday useful math is math they already know. But I always compare math to running tires at football practice. At no point will there ever be tires on the game field, but they are training the muscles in their bodies to perform better at the tasks they will have to do in the future. So it’s really our job to teach them how to solve problems—how to look at a situation, decide on a strategy, work through a solution, and determine if that solution makes sense. At this point, I’m not upset about a lack of actual real-world math problems. I’m upset by the problems my colleagues have written for our common unit assessments that are not only contrived (not in itself a bad thing, in my opinion) but also require next to no actual thought beyond remembering a specific series of steps.

  31. Dan wrote, “It’s either that or telling them, “We learned tables in section 6.3. Now we’re learning something called ’systems.’” Which do you prefer?”

    Dan, while the pedagogical device you suggested is excellent, it’s not the only good one in existence, so some people may prefer something else, beyond these two choices. Just from the top of my head, here are a couple of ways to work with systems, typically meaningful and enjoyable for kids:

    Kids can arrive at systems from Early Algebra equation games and models, such as “Mr. X” or “Unknown hide-n-seek” or “weights and balloons balance scale” or “secret boxes.” Two unknowns enter within the same framework/roleplay, and even if solutions are obvious at first, such methods as substitutions or cancellations do come up within models.

    The graphic method is abstract and has to do with intersections of lines, not necessarily straight by the way. There are some nice Russian and Asian textbook problems for that way of introducing systems; the “rabbits and chickens” mentioned, and “traveling people” mocked here ring a bell. Such problems are introduced STARTING from graphs and their equations, with solutions found first visually (counting graph paper squares) and then numerically when counting becomes unwieldy. The fun part is moving from counting to reliable and general theoretical solutions, of course. Rabbits, chickens and travelers come up as later descriptions for equations, not than the other way around, and everybody usually jokes about negative rabbits and backward-walking travelers.

  32. Maria: The fun part is moving from counting to reliable and general theoretical solutions, of course.

    I’m with you there and I’m not discounting any of the hooks you listed but I can’t think of a more compelling reason to use systems than because the numbers get too large for other techniques to handle.

    Put yourself in the mind of a student who has struggled with and triumphed over tables. Finally. And then you tell the student to solve the same problem (chickens, vans, cars, etc.) with a newer, confusing tool with no obvious return on that investment.

    What does the student do with that?

  33. You argue that each algebra problem presented to students in class should be intrinsically valuable (i.e., that each pro lem should have some link to the real world that makes it worth solving. The main problem with your argument is that you wholly disregard the extrinsic value of solving algebra pro lens like these (i.e., that these problems need no immediate real world application in order to be valuable). The purpose of teaching a child to calculate vans and cars is not so that our children can be logistical experts on thenext youth group outing, but instead this problem is presented as a stepping stone in the mathematical and critical reasoning process. With these problems, children begin to see how a series of words can translate into a mathematical equation. Given, it isn’t the best connection but that isn’t the point of he exercise. Additionally these problems help children apply the fundamental concepts of algebra, which itself is an essential building block of more intricate and sophisticated math courses like calculas and beyond. When Einstein developed the theory of relativity, we did not wonder whether his daily algebra lessons in grade school were immediately applicable. The question, instead, is whether we are equipping our children to succeed in understanding more sophisticated math principles. Thus, asking that every algebra problem have intrinsic merit is signfcantly myopic; a much more wholistic approach should be taken with our children’s education.

  34. Dan: “Put yourself in the mind of a student who has struggled with and triumphed over tables. Finally. And then you tell the student to solve the same problem (chickens, vans, cars, etc.) with a newer, confusing tool with no obvious return on that investment.”

    There are several assumptions here. The obvious one is that the tables came first. This isn’t true in all curricula, and there’s no real reason to start with table representations vs. formula vs. manipulative vs. graph. I would start this particular topic with graphs, but that’s just me.

    The second assumption is more interesting to contemplate, though. Isn’t “triumph over something” the obvious return on investments? I would expect someone who just triumphed over one cool systematic tool or concept to be, after an appropriately long break to celebrate, increasingly enthusiastic for even more challenging triumphs.

    And there ARE more challenging triumphs to be had here – not in tables, or graphs, or solving systems algebraically, but in connecting all these methods and seeing they are one and the same. This is a cool, qualitatively higher-level understanding, and it requires working with all representations.

  35. Josh: You argue that each algebra problem presented to students in class should be intrinsically valuable.

    I don’t, but I see how you could draw that conclusion. I elaborated here.

    @Maria, I guess I fail to see the upside in using a sledgehammer to pound in a tiny nail. Why not give the student the same satisfaction of mastering the sledgehammer (or of finding the connection between both hammering tools) within a context where it’s actually useful?

  36. Hmm. What is the correct answer supposed to be? Is it three vans and two cars? The problem statement doesn’t actually say that the vehicles must be full. So why not five vans, as Dan asks? Or four vans and one car?

    Perhaps there is an unstated requirement that the number of vans should be minimized because they have higher per mile operating costs. Does that sound reasonable? Would it justify the specified solution?

    Dan also asks why each chaperone has to drive. Good question. Particularly if you’re trying to minimize costs, you wouldn’t require that each chaperone has to drive.

    So suppose the problem simply said that 5 chaperones are going who are all “available” to drive a van or car, but not all have to drive if it isn’t necessary. That seems more sensible. Would this change invalidate the specified solution?

    Suppose we change the problem to say that 7 rather than 5 chaperones are going who are all available to drive if necessary. Is this new problem a carbon copy of the original one? Does the same solution apply?

  37. Ok, so 2 questions:
    1.) Which is worse at weeding out the bad…Professional developers of education or sports broadcasters?

    2.) When can we expect the WCYDWT relating to systems?

  38. Einstein is a great example here. He did pretty bad in school

    That is, alas, a myth. Quoting from Isaacson’s biography of Einstein:

    Einstein ended his year at the Aarau school in a manner that would have seemed impressive for anyone except one of history’s great geniuses, scoring the second highest grade in his class . . .

    That qualified him to take a series of exams, written and oral, that would permit him, if he passed, to enter the Zurich Polytechnic. On his German exam, he did a perfunctory outline of a Goethe play and scored a 5. In math, he made a careless mistake, calling a number “imaginary” when he meant “irrational,” but still got a top grade. In physics, he arrived late and left early, completing the two-hour test in an hour and fifteen minutes; he got the top grade. Altogether, he came up with a 5.5, the best grade among the nine students taking the exams.

    The other oft-misappropriated piece of history is edulore is Lincoln’s early history, with a popular account describing his life as a litany of failure. Also essentially a myth.

  39. I can’t recall the last time I used a system of equations in my day-to-day so I don’t make a big show of real world applications when I teach it in class.

    I only know that the WCYDWT-style adaptation of this problem would involve the students in the creation of the problem. Which is to say, you would put up a picture/diagram of 31 people on one side of the board and a picture of a beach on the other side and talk about how we’re going to get those people over there. And you try to draw the constraints out of your students, which constraints would maybe suit a systems of equations solution. Maybe.

  40. Stacy captured my view of a math problem context:

    At this point, I’m not upset about a lack of actual real-world math problems. I’m upset by the problems…[that] require next to no actual thought beyond remembering a specific series of steps.

    Regarding the car and van problem itself, if the wording were cleaned up, it would be a good problem for the fourth grade. Seriously. Fourth graders could solve it in different ways, and I would hope to see one of the ways being a logical multistep solution that, for example, could involve these two thoughts: Why wouldn’t 5 cars work? (6 people would have no ride.) If a van carries 2 more people than a car, how many vans could carry 6 more people?

    Elizabeth captured my main objection to the PD you described:

    What I find exasperating is that somebody in administration decided that this kind of “coaching” justified the expense and insult to teacher intelligence.
    What a colossal waste of resources.

    The GRR approach is what everyone already knows, and seems to me to be diametrically opposed to “Be Less Helpful.” GRR gives the students everything up front, and requires them to copy the steps shown, with teacher support gradually withdrawn. It requires medium skill and offers low challenge, which in Csikszentmihalyi’s model leads to boredom.

    Dan’s method is to start with something real to the students where they can attempt a solution. It usually involves digital media. But what is real to students doesn’t have to be about the real world. It needs to be real in the sense that Ben Blum-Smith describes well. What the students feel they know and are able to work with, is real to them.

    The next step is to somehow let the students see that their solution is inadequate or cumbersome so they see a need to revise/refine it. Let them struggle and think through the issue with appropriate group work or hints. More layers are added, with the students seeing a need to revise their solution at each level, until the final solution is reached. This approach is consistent with the state that Csikszentmihalyi says is optimal for learning–medium skill and high challenge which causes stretching.

    Here’s one suggestion for reworking that problem into a layered approach. Start with a simple problem using only cars. How many cars are needed? Write an equation. Next add the vans and fix the number of vehicles. How do you revise the equation to fit the new situation? I would guess that this would be hard for students and would need good teacher questions that I would want to think out ahead of time. When two equations are defined, the challenge is clearly how do you solve them. I would plan the problem so that substitution is “easy” to see. The final layer would be using a problem where substitution is cumbersome, but subtracting the equations would be a simple solution.

    Can anyone see doing something along these lines? Or am I dreaming? What comes to mind here is the Japanese Lesson Study approach for cooperatively developing lessons like this. What also comes to mind is the Davydov curriculum following Vygotsky’s theory, because the students there are always challenged with problems for which their current methods are not adequate.

  41. I actually found the car-and-van problem to be quite realistic. Sure the wordings could be tightened to explicitly state the assumptions but otherwise this is exactly the kind of problem I would expect as a way to introduce system of equations: familiar domain, easy to work with numbers and a simpler trial-and-error solution for verification. Btw, the problem is also similar to the route optimization problems found in Operations Research that was the motivation to develop the field of linear algebra with multi-variable system of liner equations.

    More elaboration at http://pavaki.blogspot.com/2010/01/unjustified-criticism-of-way-math-is.html

  42. The fundamental problem I have with this kind of problem is that it seduces students who [should] [may] be capable of thinking in abstract terms back into the concrete.

    What I mean is, the students look back to concrete examples of field trips they’ve been a part of for a clue to the solution, and are distracted by what they know had to be part of the successful solution. The distraction may not be fully conscious.

    Let me illustrate:

    Ok, A group with X members & Y chaperones. There are two sets of vehicles: vans [V] seating 7 persons & cars [C] seating 5 persons.

  43. [Sorry got called away]

    So a student gets uhm distracted from the math analysis by the background:

    “Well, Hermione has to sit in front, cause of the car-sickness, but she really doesn’t like George and his mom is one of the chaperones, and Mrs. Peterson’s car always smells funny so that’s no good….”


    “You can’t put Rupert and Rodger in the same car, they’ll sit together and get into a farting contest and then…”


    “This is stupid. Last time we went on a field trip in a bunch of cars one of the moms got lost and we had to wait a whole hour for them to show up. They should just rent a bus and besides then you can park up close…..”

  44. I don’t think it’s a bad problem. The problem is in the expectancy of one and only one solution, and in the expectancy of using equations to solve it.

    Leaving the solutions open (and discussable) and not having to use equations to solve it makes the problem much more interesting.

    The correct answer is of course 4 vans, because someone will be sick and the rest can squeeze in anyway. ;)

    (Or just rent one bus.)

  45. Liz’s post reminded me that we as math teachers should try to model the real world problems in a way that maximizes our goals for student learning. We have the luxury of leaving out or avoiding distracting information that’s not terribly relevant to the problem. Not always easy to do.

  46. I have been dealing with similar issues lately. I’m teaching Algebra IB and I so badly want to tell the students that although I am the calculus teacher, I love math, I use it every day, I can see its usefulness all around me- I just don’t use systems of equations, I rarely solve for x, and I’m definitely not dealing with simplifying exponents. I keep my mouth shut about that and try to happily move them through the very traditional curriculum we have at my school without rocking the boat too much. I tell them all of this strengthens their minds and it’s fun to do but they don’t really agree with me and I can’t blame them.

    Then I came across this article by Keith Devlin entitled “Is Math a Socialist Plot?” and suddenly all of this banter on here took on a new dimension. Somehow, I teach linear algebra with no problems and I talk about how computers need to solve large systems of equations with many variables to figure out tons of intriguing problems. Let’s stop pretending that someone (outside of algebra class) solves of systems of equations by hand anymore. There are computers and calculators and what we are trying to do is get the students to understand what is happening in those situations on a smaller scale. Talk about businesses trying to figure out market share, governments trying to solve problems, Google trying to figure out which pages to show- that’s how math is used these days. We need people to understand math and know how to use it so that they can program computers to solve real problems in the world. Everyone else just needs an appreciation of the true power of mathematics.

  47. I love your passion for this, Dan. It’s surprising to me that so many people seem to be defending, if not the problem, the school of thought represented by this problem.

    Like many others, I too am embarrassed to have written problems similar to this. It felt forced, unnecessary, and cumbersome at the time. Luckily, I teach IMP now, so I don’t have to worry about this anymore. The IMP curriculum doesn’t really teach systems of equations until Year 3 of the program, and when it does teach systems, it shows students how to solve a system of 6 equations using matrices. I find that approach appealing.

  48. This is definitely the type of problem my students would groan at and ask me the same question you asked yourself. . .”why set up a system, when you can just jump in any vehicle and take a seat?”

    They often ask the question when will i ever use this again in my life? I often reply with probably never (b/c i’ve got to be honest with them. Then they ask, well why do we have to learn it? and i respond with a cliche answer which i’m tired of giving.

    I’m glad to have stumbled upon this site. You offer, as well as many of the comments, very inspiring ways to look at math in a new wasy and present it to the students to engage them.

  49. Geez, I just went to the web site for that ‘Gradual Release of Responsibility’ jazz and all the air got sucked out of my room.

    I’m a math coach, and if I sent my teachers to that site, I’m reasonably sure they’d lose any respect for me they may currently have.

    Not that the ideas there are “all wrong,” but there’s something so sterile about the site and the presentation that I wanted to go roll in mud immediately.

    As for solving that bus problem with a system of linear equations? Please.

  50. Many students in algebra don’t belong there, their math mastery having peaked with subtraction or multiplication of whole numbers.

    Rather than have them on the books as having failed ‘general math’, schools move them into algebra to fail.

    That way the schools themselves don’t appear to be the miserable failures they actually are.

  51. Physics has the same problem (physics – at least in the “for non-Physics majors” sections – seems to usually be presented as another “pointless math puzzles” class).

    I’ve come to the conclusion that the technique used for generating these “word problems” is to write out a selection of random numbers and math symbols appropriate to the type of mathematics being covered, and then hang some nouns and verbs on them.

    I’ve taken precisely 2 mathematics classes that really felt like they were worth the trouble. Like #19 up there, one of those was introductory statistics. The other was an introductory “Applied Calculus” class, many of the problems from which were taken directly from real-world scientific publications.

    I occasionally find myself thinking that having a special “Mathematics” curriculum separate from the fields wherein it is applied is kind of like having a “Grammar” curriculum separate from Language (wherein you would spend an hour or two a day diagramming utterly meaningless but grammatically-correct sentences.)

  52. “One, is the problem realistic? Would a real person need to solve this problem?

    “Two, is the solution realistic? Would a real person solve the problem using a system of two equations?

    “Three, in what ways does this problem help our students become better problem solvers?”

    With all due respect, all three Dan’s questions are utterly misplaced, which is not to detract from the fact that the problem is rather absurd.

    What is a realistic problem? To whom should it be realistic? How many realistic problems can you compose? Do you need to? Must students be taught exclusively with realistic problems? Is the power and beauty of mathematics in solving realistic problems?

    Mathematics is replete with problems that are not realistic in the context of the questions. So what? Let’s tackle this one.

    Can you reformulate the problem any how so that a modification has the same solution? Do that without solving the problem. Ask each student to come up with a different problem and then solve them all at once. This would give them a glimpse into the power of mathematics.

    First, you can get away with the chaperones. 26 astronauts are to man 5 cruisers, some with 4 and some with 6 seats. There are 5 pizzas. the smaller ones cut into 4 pieces, while the big ones are cut into 6. There are 26 kids to feed. Ask the question. An art teacher brought 5 packs of 4 and 6 brushes to distribute between 26 students. If every student got a brush and none was left over, how many of each kind of packs were there? At an animal farm, 5 residents have ordered warm socks for the coming winter. Horses ordered 2 pairs of socks each, but pigs each ordered an extra pair. They received the total of 26 socks. How many pigs were there?

    How many students do you have in your class? Will they all think of different scenarios? Regardless, what all scenarios are going to have in common? What if there were 25 students? (Just in case, at least one expected solution would result in a fractional amount of cars and vans. Could your students see that without solving the problem?)

    You may teach numeracy or you may teach mathematics. But this is a bad idea to confuse the two. Students won’t be paying electric or phone bills for years to come. Do you think they will remember, when the time comes, the lesson built around such calculations that are deemed so relevant?

    The fact is that most of the populace have very little use for mathematics during their lifetime. The present system, from the kindergarten through college, that stipulates otherwise is built on a lie. The textbook publishers and writers benefit from that, the teachers are on the paying end of the myth. The system is unlikely to change any time soon. Any way, insistance on the need to feed students relevant problems cannot and will not change anything. The real question for the teachers to decide is whether they’ll be teaching mathematics and if the answer is yes, then they must learn the art of making problems (even apparently stupid ones) relevant and not prowl for relevant problems.

  53. Alexander Mathematics is replete with problems that are not realistic in the context of the questions. So what?

    So pretending they are realistic only serves to cheapen the math that does apply to my students’ lives while simultaneously strengthening their impression that people who practice math are strange little beings who live in some dimension disconnected from their own.

    Is applied math the only important math? Certainly not. So let’s not force applications where none exist. Your approach here, converting one contrivance into many, might be fascinating and useful for students already very comfortable with abstraction, already very enthusiastic about and proficient in math. It will lay waste to the other 98%, though.

  54. No, Dan. Do not pretend they are realistic. Call them stupid, contrived – whatever you prefer. The publishers and the authors do the pretending, you do not have to. The circumstance is very unfortunate but it is systemic and won’t change any time soon. The fact is a child has to learn and learns through a sequence of contrived circumstances, sometimes through real life accidents. In itself, there is nothing wrong in artificiality of learning tools. It’s the pretence that children need to learn what is relevant and useful, since nothing else elicits their interest, this pretence is the cause of jeer, disbelief and confusion.

    If ever I felt affinity to somebody else’s view point, this is the one annunciated by James Starkey: http://www.mathteacherctk.com/blog/?p=153.

    The gist of his view is that good teachers do good job teaching regardless of the current educational fad. To which I can add that a good teacher (which you apparently are) may be easily led astray by his/her own success. It is so easy to fall in the trap: this works for me, for I meet success with students; hence this is a unversally good approach. A good teacher may make a good lesson out of buying groceries, a poor teacher will screw up the best lesson plan available.

    My point is this:

    1. It is patently wrong to expect that one can manage to teach mathematics (even numeracy, but to a lesser degree) from the so-called reality. Fro one, there are not enough uses to maths on a level a little above elementary,
    2. Mathematics deals with abstractions which are at best only approximations to the real world parameters.

    Now, as an aside, may I ask you a question about that teacher development event. What point did that coach want to make with that transportation problem? Surely the idea was not to show you how to solve word problems.

    Finally, as to what I suggested as a way to convert a stupid problem into something more engaging, I assure you that there are different opinions and different experiences, much, as usual, depending on teacher’s philosophy, enthusiasm and ability. I am pretty confident that if you try – just once – in your class, you’ll see that the 98% estimate was exaggerated. Just do not let students know your initial reaction.

  55. “It is patently wrong to expect that one can manage to teach mathematics (even numeracy, but to a lesser degree) from the so-called reality.”

    My current pet idea is “mathematically augmented reality.”

    Newbies learn from more experienced practitioners, as these masters do what they do. Learning math from reality won’t work, because reality is not a practice. Learning math from human practices that have to do with mathematical augmentation of realities, on the other hand…

  56. The simplest example of mathematical augmentation is labeling (based on previously constructed math) and aggregation of examples by labels. An accessible way to experience this is through picture web searches, which we do pretty much every math club. Here are a couple of the recent fun labels we found:


    Here are two artistic examples where reality is augmented and transformed, well, by transformations, in the first case Mobius, in the second, Droste:

  57. Bill Bradley

    May 22, 2010 - 4:30 pm -

    There are certainly several mindsets here, I’m reminded of a couple of jokes:
    1) Engineers think that equations are an approximation of reality, Physicists think that reality is an approximation of equations, and Mathematicians never made the connection. Being a Physicist by training, having an Mechanical Engineer for a father, and friends who have their degrees in pure Mathematics, I’m not sure who gets zinged worst there :^) We have to recognize that there is a place for all of those aspects of mathematics: Engineers are often use Finite Element Analysis, Physicists often work in infinite dimensional matrices attempting to diagonalize to a convergent series, and Mathematicians actually care if someone proved Fermat’s last theorem or the Poncare’ conjecture. Our classes contain some, all, or none of the above, and we should recognize and address that these are all under the umbrella of Mathematics.
    2) Imagine sawing a board in half, if you see sawdust, you’re not a mathematician! Current Mathematics education is horribly deficient in talking about applications and real problems. When I teach Geometry, I show car suspensions: Double Wishbone vs. McPherson strut; Watt’s Link vs. Swing Axle vs. DeDion vs. Trailing Arm. I also talk about ironing boards and window winders. Calculus was created to solve Physics problems, and Algebra often makes more sense when the numbers measure something rather than being just numbers. On the other hand I’ve also put in a course proposal to offer Logic and Discrete Mathematics since those are also horribly deficient in the current Math curriculum.

    We have a limited number of years and hours to be many things to many people. All of any one thing is not the solution, if anything what we should be discussing is the approximate percentages of the mixture.

  58. 1. It is patently wrong to expect that one can manage to teach mathematics (even numeracy, but to a lesser degree) from the so-called reality. For one, there are not enough uses to maths on a level a little above elementary.

    I’m not fully sure what you mean here, but I think you might be merging two things: the mathematics that the average population generally uses, and the mathematics that could be applied to gain insight on ordinary things.

    2. Mathematics deals with abstractions which are at best only approximations to the real world parameters.

    What’s wrong with approximations?

    What’s statistics for if not coping with those approximations?

  59. By the “so-called reality” I meant the aggregate abstraction implied in Dan’s three questions (realistic problem, realistic solution, real person.) Sorry if that was not clear.


    you lost me here. I’ll probably need more than an example. I am not sure Dan needs to host the possible discussion.

    @Bill Bradley:

    I remove my hat. I had to look up the terms on the web. Do you also prove anything about Double Wishbone? But, as an attention grabber, it is probably all right to just mention the device.

    “All of any one thing is not the solution, if anything what we should be discussing is the approximate percentages of the mixture”

    I think that the best approach is to let teachers make their own choices. This won’t make education better, but is certain to make more teachers more comfortable in class. Students feel very acutely when a teacher does not believe in or is not comfortable with, the subject.

    (To preempt a discussion as to why this proposal/idea won’t make education better? Because many teachers would be unhappy with the responsibility of having to make their own choice. Good teachers, like Dan, seek their own ways. Others follow curriculum the best they can.)

    @Jason Dyer:

    I meant the first of your choices simply because this is the subject of this thread and because little else would relate to a regular, average person. If by “mathematics that could be applied to gain insight on ordinary things” you mean the maths relevant to, say, bridge building, aircraft flying, or even something simpler, like pressure distribution in a tire or the workings of a gas pump, then this is certainly not something “a real person need to solve”. A discussion can branch in a fractal manner (Maria will probably like the metaphor) unless every one accepts the background implicit in the original post.

    And, of course, there is nothing wrong with approximation, per se. Even the insipid problem that so arose Dan’s ire, is only an approximation, and might have been excused on that ground, or declared as such, made a laugh of, and still be used to teach and learn the requisite skills. What I said (or meant) was that mathematics deals in abstractions that are necessarily only approxamations to the “real world” and, for this reason, it may not be a good idea to focus on “realistic problems” for which a “real person” may need to seek a “realistic solution.” Seeking such problems may be as misleading (about the concepts and methods of mathematics) as the attempts to pass for realistic the problem that started this thread.

    But then again, a good teacher does a good job with the methods of his/her choosing and in different circumstances. And that’s fine with me. I simply think that promoting one good teacher’s success as the next word in education is a bad idea. It is my understanding that this is how the history of math education and reforms in math education have been written all along. A charismatic fellow would come and (attempt to) lead the crowd on his/her path. But most would not be able to follow. So that in a while another charismatic fellow would take the lead, etc.

  60. “I simply think that promoting one good teacher’s success as the next word in education is a bad idea. It is my understanding that this is how the history of math education and reforms in math education have been written all along. A charismatic fellow would come and (attempt to) lead the crowd on his/her path. But most would not be able to follow. So that in a while another charismatic fellow would take the lead, etc.”

    Promoting ANYTHING as THE word is not a good idea. However, expanding stories within a community of practice that finds uses for them is different. Consider…

    Someone experiences excitement at a success and shares the story. A few follow the story and successfully do something similar. As they share their own successes, a stronger story emerges, which attracts a few more who are now able to relate, as well. Stories merge and grow more, as participants develop vocabularies to talk about their practice (“WCYDWT,” for example, or “augmented reality”).

    The majority can’t make any sense of it, don’t find the style compatible, and don’t speak the language. This is perfectly fine, because thousands of other charismatic guys and gals are sharing their stories at any given time. The abundance can grow enough to cover (Heine-Borel) everyone.

  61. Promoting ANYTHING as THE word is not a good idea. However, expanding stories within a community of practice that finds uses for them is different. Consider…

    No need to. I agree.

  62. Bill Bradley

    May 23, 2010 - 9:54 am -

    @Alexander Bogomolny
    I do mention the double wishbones, and bring them (suspensions) in as compass and straightedge constructions since everything is either a pivot or rigid beam, and it’s guaranteed to be of interest to at least some of the class, but it’s as abstract and boring to many. Always nice to have ’em come back the next day happy to announce what’s under their vehicles.

    Which brings up another thought. “Conversational mathematics” I don’t think that I have ever had a student start a conversation about a textbook problem (unless it’s how to do his or her homework) but that does happen on a regular basis with the outside-the-box(textbook) work.

  63. If by “mathematics that could be applied to gain insight on ordinary things” you mean the maths relevant to, say, bridge building, aircraft flying, or even something simpler, like pressure distribution in a tire or the workings of a gas pump, then this is certainly not something “a real person need to solve”.

    Of course one can go through life without knowing mathematics; many do. (Many also pay for medical quackery without knowing statistics and variable rate mortgages without knowing how to use percent, but that’s a side point.)

    However, one can make life better by the ability to be mathematically analytical about ordinary things. There is even, as Dan stated in his TED video, a hunger for this.

    My systems of equations via football lesson is, granted, an approximation; yet it *works* in making an action normally taken by intuition, mathematically analyzing it, and greatly increasing the success rate. Students understand and love this (and after the lesson I have had multiple students say I should be a football coach).

  64. @Bill Bradley

    1. This is a very enjoyable episode to have students peek under the cars in their garage. Very engaging.

    2. I don’t think that I have ever had a student start a conversation about a textbook problem. But a teacher always has a chance.

    @ Jason Dyer

    However, one can make life better by the ability to be mathematically analytical about ordinary things

    Or, say, solving puzzles on weekend nights. Absolutely.

    But it must be admitted that there are other venues to the same goals. Critical thinking skills may and are successfully taught without a mention of mathematics.

  65. I’m just glad I don’t teach in the same school as Dan. Then I’d have to feel stupid every day.

  66. I find the main problem I have with teaching algebra is the sheer power of it, like calculus. We don’t see many real world problems with it because the main use of it is for general rules and proof. To go from single problem to proof is a hell of a step really.

    No idea what my point is to be honest, apart from us needing an overhaul of the learning of algebra.

  67. Man, I love the coach’s reaction: “It’s in your textbook.” As if the textbook should somehow be the final arbiter of what should and shouldn’t be part of teaching.

    I assume, then, that this coach would advocate teaching every lesson, page, example, and problem out of the textbook! Surely there is time for that…

    The real world doesn’t have “problem types”…

  68. Algebra Student

    October 13, 2011 - 12:50 pm -

    I am a 7th grade student taking Algebra 1. Algebra can be very unrealistic and annoying, and it doesn’t help that many of the teachers aren’t very good at teaching it. My teacher often refers to things he taught to students that were also in his class last year, as he looped, but has no effort in explaining these to the rest of the class. He also doesn’t explain things very well and uses “realistic” problems to “help” us get a better understanding.