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?

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?

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! :)

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.

So, what’s the answer?

Just joking.

exactly the mindset we need to be in… i love the 3 questions…

grazie..

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.

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…

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?

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.

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.

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:
http://web.mac.com/ihor12/CMDB75/bus/bus_s2.html

Bus problem — teacher page:
http://web.mac.com/ihor12/CMDB75/bus/busproblem_t.html

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,

http://web.mac.com/ihor12/CMDB75/roadsign/roadsign_s.html

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.) ☺

-Ihor

You’ve obviously never taken Algebra with Dr. David E. Conroy

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.”

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.”

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.)

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.

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.

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?

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

[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….”

and

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

and

“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…..”

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.

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.

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.

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.)

“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.

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.

I may need an example of mathematically augmented reality to see if that’s different from what I referred to as “the so-called reality.”

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.

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.

@Maria:

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.

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.

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).

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

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”…

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.