Since the imposition of Common Core and ubiquitous propaganda that Common Core math standards require different methods of instruction (despite claims on the CC website that the standards do not prescribe how teachers are to teach), there has emerged an expression that causes me great anguish whenever I hear it. The expression is “to problem solve”, in which “problem solve” has become a new verb form.

It has caught on like wildfire and replaced older forms such as “solving problems”. The phrase has taken on specific meanings; it is a “dog whistle” of reform math (a term Tom Loveless of Brookings coined). It means a departure from the standard word problems that have been held in disdain by reformers as tedious, repetitious, “plug ‘n chug’, not relevant to students’ interest and not reflective of how math is used in the real world.

Case in point would be the type of word problems (which used to be called ‘story problems’ in simpler times) that one finds in great supply in older algebra books and diminishing to none in newer ones, about trains catching up with each other, people mowing lawns together at different rates, and so on. There has been universal agreement amongst reformers that these problems did not do anyone any good except the brighter gifted students who, as reform legend has it, would have learned math anyway by virtue of having been born that way.

The term “to problem solve” now means giving students so-called “rich” problems that cause them to “dig deep” into concepts, learn what is needed to solve the problems on a just-in-time, as needed basis, and other wonderful-sounding things that in the end are more often than not ineffective. Such problems are often open-ended, with multiple right answers, and students solving them in multiple ways. A classic example offered in this genre is “A rectangle has an area of 28 square inches; what are the possible dimensions of the rectangle?” Others are laborious one-off type problem which require intepretations of graphs that are then used to plug in to various formulae in order to answer various questions that the educators responsible think represent how people go about solving problems (collaboratively, of course) in the real world. These type generally fall into the category of problem-based, or project-based learning. There are those who make a distinction between problem- and project-based approaches. I am not one of them.

This is not to say that all such problem constructions are bad. And I recognize that there are teachers who are able to effect a good **balance** of different problem types, and hook in to prior knowledge and skills. I offer, however, something that veteran middle school teacher Vern Williams has said:

I have always stated that if a reform minded teacher produces competent, intellectually passionate students, they will absolutely escape any criticism on my part. But the opposite seems never to occur. Regardless of stellar results, the traditional teacher will always be criticized for being a self centered sage on the stage, controlling student learning and running a draconian classroom. Their students may be the happiest most accomplished students of all time but the teacher will never be good and pure until they cross over to the reform side.

In that vein, what I often hear are arguments that these “rich” problems justify the rejection of traditional type problems used over the years to teach students fundamental problem-solving skills–skills that are generalizable and transferable to many types of problems.

I happen to use the old type of problems in teaching my students how to solve problems–not problem-solve. People at my school marvel at how my students are challenged, and who show improvement in solving problems. I am often asked what I do to get such positive results. An answer I haven’t yet uttered but am tempted to do so is: “You know all those types of problems that they tell teachers we shouldn’t give students to solve? Well, that’s what I do.”

Another illustration of the wheel spinning around and around…what’s old is new again, and the reformers refuse to acknowledge that. They must rebrand the same package, so they can make a profit from unsuspecting parents and reinforce ed school’s budget and education ministries.

Nauseating.

Parents…please, if it sounds weird, it probably is. You KNOW how kids think, and what they require for a fundamental education. Insist that they obtain it, and if not, send your tutoring bills to your School District, or to your Congressman/woman. Enough is enough.

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There is another big issue with “open ended” problems. The extreme difficulty of marking them.

You can’t just write an answer, with working, up on the board. Each student has to be marked independently — and there is no way you have time to do that, (Even if you crucify yourself and take each day’s working home and mark it for hours every night, by the time the feedback has arrived the students are already onto the next work.)

So what happens is that many students aren’t marked. Some think they are right when they are wrong, so no learning is taking place. Others know they are wrong, but rather than admit it they hide it, so no learning will be taking place. Others are almost right, or on the right track, but it is very hard to correct the mistakes since everything you explain has to be qualified by “if you do that, then this, but if you do this then that” — which naturally leaves them more confused.

In the end the only students who learn are those that can more or less do it. So much for equality of outcomes for those at the bottom end then!

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Does it work? I would be their biggest fan and supporter. I want them to show me any set of students where it works – “problem solve” and transfer. They can’t even do variations of their problem-solve examples let alone transfer the skills (?) to variations of traditional combination DRT and work problems. That’s because understanding whatever limited set of concepts they deem important do not easily transfer to other problems. Doing proofs – words or formal forms – do not even get the job done. You have to do a lot of nightly individual homework variations, not limited in-class group work. It’s all silliness beyond belief. They are only concerned with what happens in class, not what happens at home unless it’s to flip the classroom and require more direct teaching by parents – as in watch direct instruction videos and having parents “practice math facts.” Their pedagogy is not about understanding, but elimiating the things they just don’t want to do in class.

All of this nonsense goes away in high school with traditional AP Calculus track classes and textbooks, and the only students to get on those tracks are ones helped at home or with tutors who focus “math facts.” I know what it took for my math brain son to prepare for traditional high school math. It was basic skills. In high school, I had to do nothing.

I agree with Tara. I don’t expect K-6 teachers will believe any of what I say, so I also talk to parents who might read this. If you don’t ensure mastery of basic skills at home or with a tutor in K-6, then it will be all over by the 7th grade math track split. Even the kids will believe that they are just bad in math. I once taught an SSAT prep class for 7th/8th graders and many said that they felt stupid. No. it’s the schools and the fact that K-6 is now officially a STEM-FREE zone. Talk of understanding and problem solving is just cover.

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I was cleaning out a closet and came across a MathLand homework (UNIT 1 – Week 2) for my son in first grade (2002). BTW, when my wife and I naively happened to tell this teacher that our son loved geography and could find any country in the world, she said “Yes, he has a lot of superficial knowledge.” Ouch! This was not the first or last preemptive parental attack we suffered. Ironically, he had to show the student teacher where Kuwait was later that year when they had a thematic unit called “Sands from around the world.” I’m not making this up. We both got the distinct impression that some K-6 teachers do NOT like smart kids. They like to say that all kids average out by fourth grade.

This is the MathLand assignment:

Dear Family, [They expect parental involvement]

“This week at school the class has been talking about favorites. The children have been making posters about class responses to questions such as: What is your favorite food? This type of activity gives children experiences with gathering and organizing data.”

[Forget learning styles. All kids had to do posters and draw pictures. It wasn’t just listing a favorite next to each name. No, they had to draw a picture. I’m sitting here looking at the old posters. This is incredibly time-wasting, like all student-led “active-learning” in class.]

Here is the homework for one entire week.

1. Your child’s homework for this week is to ask family members about their favorite ice cream flavor. Try to find a few minutes to talk together about the answers.

2. Have your child draw pictures that tell about the answers family members give.

3. Please send the pictures back to school.

I remember this in detail. Then the school switched to Everyday Math. There were still low expectations because of our full-inclusion environment. They trusted the spiral, but it was more like circling – repeated partial learning. It was up to parents to enforce mastery to make it into anything like a spiral.

Most high school teachers know that there is a problem with very bad skills, but they will not explain it to K-6 educators. They probably know that it won’t do any good. However, the head of the math department at our high school would not even criticize Everyday Math to me. That was in direct response to a request for comment I made to her. However, this teacher got kudos for setting up a 9th grade Algebra class that included a “lab” that had students work on skills. Nobody wants to question the pedagogical turf of K-6.

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“A rectangle has an area of 28 square inches; what are the possible dimensions of the rectangle?”

Like me you can’t even bring yourself to state the ‘problem-solve’ open-ended problems like the fuzzies do, as it grates for lack of precision of meaning. Most commonly this problem would be worded:

“A rectangle has an area of 28 square inches; what could its dimensions be?”

I would answer 4×7.

Period. It’s dimensions could be 4×7

Or I could answer 1×28. In fact once I figured out that answer it would be the basic form for all my answers. No sensible student should read the latter form as “find all possible dimensions”. If that is the intention then SAY SO.

And it seems that, between those two options teachers who use that form of questioning often don’t have a preconceived notion beforehand which they want or whether they want something in between.

That is regarded as a virtue. It’s elevated instruction past mere teacher expectation. Or something.

Really it’s lazy, fuzzy, and lacks structure and direction that many students would thrive on.

I think they would say that the merit of the problem is less any math that the students take away, but “engagement”. It make ’em think. Yep, and my main thought is “what the heck does the teacher want here, anyway?”

I think the teacher’s conception most of the time isn’t that little Robby says “4×7” it’s in the “rich” classroom discussion that takes place afterwards. That’s very good Robby … who else got this answer … did anyone get something else? Oh interesting Barry. 1×28 eh? Can both answers be right? (There’s the little moral lesson … you see, in this postmodern world it is absolutely essential that children are trained to think there are no right answers, it’s a rainbow-unicorn world in which, as long as you engage, anything you do or say is right, and truth? Ha! Passé!)

Then some other kid says “7×4”. Oh interesting Ze’evie. Did everyone catch that? Would you say that is the same as Robby’s answer or is it different? Why?

Finally you get 28×1. And 2×14 and 14×2. That is, if the teacher even decides it’s necessary to milk the problem. They might just stop there, with half the problem finished. Then there’s “how do you know that’s all? The REALLY advanced part. Whoo Boy. Ultimately some kid writes down all the possible lengths: 1, 2, 3, …, 28 and they figure out which ones have widths … completely missing fractions. I’ll give even odds whether the teacher is evening thinking about fractional solutions and expects them to stop with those six. But what if fractions ARE admitted? In that case, why did we stop at 28? Couldn’t one side be 29? Well, that might be a “rich” direction to go with this, but it’s not clear to me that there was any plan in the teacher’s mind to do that. It might throw some teachers and they’d lead kids away from it. And some kids might get it while others are thinking “that’s dumb!”. Because nothing in the problem suggested that … to kids that may have only rarely dealt with fractions … and that not in the setting of integer arithmetic, more in the field of pie-drawing.

Had the problem been appropriately stated, there is indeed a correct approach: Let one side of the rectangle be x. The other must be 28/x. The dimensions are x by 28/x, where x ranges over all positive numbers.

But the rich problems guys would object “that’s not what mathematicians want! They want to see the PROCESS. The THINKING”.

These guys have never met a real mathematician. I … ahem … have.

And they (the mathematicians) … I mean *we* … do not value that sloppy, ad-hoc approach. Yes, it’s “thinking” but it’s not of any particular value. Mathematician is about training the mind to think WELL, and undisciplined scattergun thinking is a poor instance of that. We want to see focus and elegance, and convergence a solution, deft use of standard notations and tools, and mastery of appropriate-level concepts and methods.

And it has long been a tradition in mathematics that you DON’T report whatever wild goose chase you went through to find a solution. You report the most efficient, elegant pathway to the solution that arises out of that stuff. Don’t air your underwear in public. Yes, a teacher can take a student through the “thinking process” but that teaching should help a student learn to always converge — on any problem that looks anything like this — on the straightforward 2-line complete solution.

I’m marking midterms of students, many of whom (through no fault of their own) came through schools that taught in this (fuzzy) way. Their solutions wander all over the page. And I’m shocked by their work on absolutely elementary arithmetic. They have been taught no disciplined, straightforward methods. So when simple arithmetic comes in the middle of a larger calculation they make the most blindingly silly errors.

One question required students to take three 2×2 determinants. Pretty easy. But one determinant, after the first step, came to -34+32. A couple of students have “simplified” this expression to 96. Go figure.

The fuzzies point to errors like this and insist it’s proof that traditional learning of algorithms doesn’t work. Sorry fuzzies, I’m laying this one at your feet. And hundreds more like it.

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Dr. Craigen’s problem is he thinks like a mathematician, rather than how non-mathematicians believe mathematicians think.

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“A rectangle has an area of 28 square inches; what could its dimensions be?”

This problem seems to come up for opprobrium every third post or so. I think in the past it’s been a problem about perimeter. Either way, I don’t understand the intensity it engenders in traditionalists.

Shouldn’t a student who has mastered rectangular perimeter and area be able to offer an answer? If a student

can’toffer an answer, wouldn’t we say she haven’t mastered rectangular perimeter and area? All of that seems to make it pedagogically valuable.LikeLike

As a regular reader of Barry’s blog, I can assure Dan that this topic does not, in fact, come up every now and then. However Dan seems to take issue with “treaditionalists” – his phrase, as well as with mathematicians and other teachers who have seen how ludicrous this topic has become.

There are many excellent, well organized teachers who recognize how math SHOULD be taught, also how to have various problems available for kids who are at different levels of their math development. Asking a kid to “explain” their work ad nauseum isn’t going to give them a better understanding of the problem, it’s just going to tune them out. I know, that’s exactly what happened to my youngest. She was whippet smart when it came to arithmetic, but over the course of her elementary years, where she was subjected to convoluted inquiry strategies and doing stupid such as lattice multiplication, she lost interest. Said math was stupid, which is a real shame, because she really loved it before she headed to Kindergarten.

I spent a couple days at the curriculum archives last December going through old textbooks and resources from 1895 to the current date, and I was shocked at how dumbed down our resources had become! No wonder so many kids struggle with basic arithmetic these days! The way it’s being presented to kids is so convoluted…it doesn’t provide additional understanding for kids, it muddles their thinking, and does not provide any constructive methods in reaching the answer in a straightforward manner. Kids need results quickly to see that they’re on the right track, and to make a quick transition to the next step. Textbooks in the olden days did that. But those written from the 1990’s onward are TERRIBLE at providing this type of instruction.

All this gibberish about what’s best for kids is sheer nonsense. There’s not one whit of cognitive science behind any of today’s math resources or curricula, which begs the question: what’s the point. All these “experts” running around who were trained in the “traditional” manner of mathematics, all jumping on bandwagons saying they would have been so much better at mathematics if they would have been exposed to an inquiry approach. Such bollocks! Our adult perspective is so much different than a child’s, but so many educrats are so busy talking AT children, they aren’t paying any attention TO children, or how they think, or process information.

So if the end result is having a bunch of kids who hate math, and skyrocketing enrolment rates at tutoring centres, what’s the point? Best get back to what we know works best, but that won’t sell many books now will it?

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Dan

I’m not aware I’ve ever talked about the area problem at this blog, let alone every third post, but I do voice objections to open-ended problems if that’s what you meant. In any event, I think what Robert Craigen, Chester Draws, and SteveH have voiced in their comments above are also my concerns with such problems.

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The two big lies are – The discovery process transfers and mastery is only about speed.

First, their discovery process is learned in class where they only have time for a few concepts, and it’s done in groups where the few who achieve a light bulb effect directly teach (poorly?) the other students. Those students clearly have not mastered the discovery process, so how can they transfer it to other problems?

Second, have they checked to see if their discovery students can do problem variations with unlimited time? With tests like the SAT and AMC, which go out of their way to find unique variations where a discovery skill might be needed, does it work? No, it doesn’t. Students prepare best by doing every single old problem they can find. That transfers better than a blank slate discovery process. I’ve seen this so many times. (I have colleagues who like to play this game. They pretend to discover a solution to something they’ve seen and studied before. People think they’re geniuses.)

The sum of the digits of a two digit number is 8 and it is 18 more than the number created by swapping the two digits. What is the number?

A general mathematical “discovery” process is to define variables, come up with an equal number of equations, and turn the crank. Math eliminates the need for discovery. As I always say: “Let the math give you the discovery.” If you’ve seen this class of problems before, then it becomes even easier.

Third, have they asked the parents of their best students what they had to do at home? They are most likely their best discovery students. Ask the parents. It’s easy. We get notes telling us to practice “math facts” at home. Duh. Is that necessary or not, and what happens to those who don’t get that help at home?

Finally, does it work? How do they know? Our school improved statistics when they changed from Mathland to Everyday Math. Slightly. Many teachers blame IQ when they have no absolute basis for calibration.

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Tara Houle, SteveH, Robert Craigen, Thank You! I couldn’t agree with you more.

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Dan has to think outside of his rote box – his coterie of equally-minded and trained colleagues who like to trash much better mathematicians working in the “real world” as if they only want what they had when they were growing up and have no clue why some students have difficulties in math. I taught college algebra for years. Many of us are very careful not to let our hypotheses define reality … and we don’t have vested interests in our positions. We see and evaluate, with our teaching, kids, and tutees, the pedagogical assumptions and entire longitudinal curriculum process, and not just a self-serving teacher-centric view of what currently walks into one classroom. Creativity and engagement are pedagogy neutral, and they are not magical top-down conceptual understanding and mastery silver bullets.

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