Monday, July 19, 2010

Math Curricula

I know that I'm inviting trouble with this, but something that Reader wrote in a comment on another thread piqued my interest. I would like to discuss only a narrow question. Please don't expand the discussion.

Writing about Everyday Math and Singapore, Reader wrote: "The fact is, the newer curricula stress more problem solving and discovery. That is, it's doing more than a lot of older curricula."

Here's my question: can problem-solving be taught?

I mean this in the nicest possible way and I don't have an answer myself. I'm not sure, I'm asking. Can people be taught or trained in problem-solving techniques or is it a talent that some people just natively have more than others? Problem solving requires a certain amount of creativity, doesn't it? It can require a flexibility of perspective, curiosity, persistence, and pattern recognition. Can these things be taught or trained?

I suppose anyone can be taught to play a musical instrument, but not everyone will do it well and not everyone will choose to pick up the instrument and play it of their own initiative. For me, problem-solving is a compulsion. And I have a knack for it. But I'm not sure if I could teach anyone how I do it. I certainly don't have many systems for it and I only resort to them when my usual perception fails me. I solve the Jumble every morning and usually couldn't explain how I found the solutions - at least not the ones I got right away. I just see them.

Setting up math problems is the same way. I just see how to set them and how to approach the solutions. I sometimes wonder if this isn't a contributing factor to the failings of Reform Math - the impossibility of teaching something that is actually a talent. There are others - and there are strengths. Let's try to focus the discussion on get this question: Can problem-solving skills be taught?

60 comments:

reader said...

If you don't believe problem solving can be taught, or at least improved, then why would you ever bother teaching math in the first place? I mean, the goal isn't to learn how to multiply... or "solve for X". The goal is to learn to how to think about problems, and how to solve them, and to appreciate them. That's what math is. In that regard, math isn't much different than other fields. I don't really think reform math has failed. As a math professional, young candidates are better prepared than ever before (including those from the USA).... despite all the whining to the contrary. And as to the "widening achievement gap"... that too has always been with us. The same people who hate all the blame the teacher stuff... seem to love the blame the book. The people who hate the scripted learning, love it when it comes to math. Why? The roots of uneven achievement are not from a book or pedagogy. The whole issue is pretty pedantic.

dan dempsey said...

The goal is the have such extensive knowledge and skills that there are no problems as everything is an exercise. -- I think that was a goal or view held by Johann Carl Friedrich Gauss.

Seriously what constitutes a problem?

As E.D. Hirsch's Core Knowledge approach indicates a large content and knowledge base coupled with experience can certainly turn what some would consider problems into exercises.

ken berry said...

What is problem solving? Good luck if all of those who care to enter the debate can agree on a common definition...

ken berry

zb said...

I agree with reader. I'm not sophisticated enough about teaching to know for certain whether problem solving can be taught. But, my first inclination would be to believe that it can, at least, be facilitated. Most human behaviors (as well as animal ones) can be shaped, with the appropriate instruction and feedback.

And, if it can't, there's not much point to teaching much of math. I'm certain that practically everyone can be taught to chant the multiplication tables. But without an understanding of how to use the information in context, at the very least it's no more math than teaching them to chant the kings of England, and probably a little less useful.

I am also unconvinced that math is worse now. I'm not so convinced that it's better, but I think that's 'cause it's just a hard subject to teach well. I do believe that a lot of what we called math learning in the old days was simple memorization; memorization that didn't transfer, and was lost almost immediately after the test.

Sahila said...

There's no such thing as a mistake... everything is a learning experience... and I'd venture to suggest that is what's wrong with education - including the divide between those who want direct instruction and those who advocate for project-based learning in math curricula...'mistakes' are 'bad', if you make a mistake you're a failure rather than a curious individual conducting an interesting experiment, and so kids are dissuaded from taking risks in case they make 'mistakes'...

And there is usually more than one answer.... again, in the YouTube video I posted on the previous thread, teachers talk about how children from low-income, minority community schools are pushed to come up with only one 'acceptable' answer, children from affluent, well-resourced schools are encouraged to ask questions, explore, investigate and problem solve - think creatively, think for themselves... as one teacher said: "and that's not fair"...

A Canadian professor lamented that kids come to college not knowing how to think for themselves...

http://www.universityaffairs.ca/has-ontario-taught-its-high-school-students-not-to-think.aspx and puts most of the blame on the fact that children have been taught to memorise content ...

Ken Robinson laments the death of creativity in education...

http://www.ted.com/talks/ken_robinson_says_schools_kill_creativity.html

and calls for a revolution:
http://www.ted.com/talks/sir_ken_robinson_bring_on_the_revolution.html...


"I have learned the novice can often see things that the expert overlooks. All that is necessary is not to be afraid of making mistakes, or of appearing naive" -
Abraham Maslow

and here's a timely piece from the Harvard Business Review
http://blogs.hbr.org/hbr/nayar/2010/07/the-miracle-of-making-mistakes.html

And Dan- I respectfully disagree with you that the goal is to have such extensive knowledge and skills that there are no problems, as everything is an exercise...

that's not living - we might as well all be machines because there would be no use for imagination, creativity if everything was merely 'an exercise'.... there would be no art, no music, no astronomy, no philosophy - no point to anything at all really...

Which I think - thank you for the trigger to understand that - explains my very deep and intense rejection of direct instruction... I think that process turns everything into an exercise in memorisation, not an experience in individual, unique unfolding and discovery ... my response to direct instruction is that it is dehumanising....

Sahila said...
This comment has been removed by the author.
MathTeacher42 said...

I do enjoy reader - consistently condescending, consistently anonymous.

I think most people could learn a lot more than they do, when you look at how much information they master to pursue their recreations. How do we get an education system that helps people develop skills to participate, and nurture skills to change the world, and helps people to do stuff they're interested in? Umm.

I think it is worthwhile to try to get people comfortable with problem solving. I think once you get past enough of the basic skill hurdle, problem solving mastery is a different mix of motivation & training & talent. Like basketball or the violin, different people will excel differently.

I think reform math has been a failure, except at employing reform math people. reader knows exactly what math is, and by reader's definition reform math hasn't failed.

If we look at just about everything else but math that humans do, in training towards mastery of complex things humans typically try mastering basic skills before mastering more complex things.

It has been my consistent experience that people in math ed who focus on the problem solving end of math really really really don't understand the lack of basic skills of those who struggle with the problem solving. They have been in the schools and have been in the programs and have been in the jobs where they're around people who can manipulate stuff like 3/4 or 12/16 or 24/32 into percentages or decimals, AND get the right answer, consistently. It has been my experience the focus-on-problem-solving-types are from backgrounds that have had little interaction with those who struggle with basic skills.

It has been my experience that the reform crowd has consistently missed the fact that the job market at the bottom is NOT the job market at the top. About 214,000,000/238,000,000 had money income under $75,000 in 2007*, about 185,000,000/238,000,000 had money income under $50,000. The worse your basic skills the more brutal the job market for those under $75k and under $50k. For those without family re$ource$, connection$, drive, and complex problem solving skills, the job market is NOT sipping latte's in cafes reading "What Color Is Your Parachute".

Because the reform crowd doesn't understand the economy, and many of the reformers lack experience with those who struggle, they're able to take this Nobler-Than-Thou approach to what math is and math education is, and to build up these complex edifices of excuses to justify emphasizing problem solving. And then they conveniently ignore the fact that our kids don't have the basic skills to just get by.

I'm sure that once the US joins the rest of the world with a huge underclass, and a dinky servant class to the powerful, students will be motivated to master whatever basics they have the opportunity to get exposed to.

Hopefully problem solving training will be available, too.

BM.

Sahila said...

I think problem solving is inherent and based in curiosity and an inner drive to expand and explore.... you see it all the time in babies from all species (though it requires space, encouragement and opportunity to develop - see the horrors of the Romanian orphanages for the flip side of that....http://whyfiles.org/087mother/4.html) ...

I think the ability to problem solve diminishes without the opportunity to try, try, try again (in an atmosphere of approval) and the way our schools are so heavily structured in many regards, they dont give kids the time, space and resources to do that.... and teachers certainly dont have the time and resources to allow 25 kids in any one class to spend real time experimenting.... think about lesson plans and how they are structured - maybe 10 minutes if you're lucky in a lesson to let a child spend time thinking, touching, smelling, hearing, speaking, exploring fully whatever is being taught, using multiple learning aids and tools? Dont see that happening except maybe still, to a small degree, in alternative schools....

MathTeacher42 said...

do enjoy reader - consistently condescending, consistently anonymous.

I think most people could learn a lot more than they do, when you look at how much information they master to

pursue their recreations. How do we get an education system that helps people develop skills to participate, and

nurture skills to change the world, and helps people to do stuff they're interested in? Umm.

I think it is worthwhile to try to get people comfortable with problem solving. I think once you get past enough of the

basic skill hurdles, problem solving mastery is a different mix of motivation & training & talent.

I think reform math has failed miserably, even though reader knows exactly what math is, and by reader's

definition reform math hasn't failed.

If we look at just about everything that humans do, in training towards mastery of complex things humans typically

try mastering basic skills before mastering more complex things - until we get to reform math.

It has been my consistent experience that the reform types who focus on the problem solving end of math really

really really don't understand the lack of basic skills of those who struggle with the problem solving. The reformers

typically have been in the schools and have been in the programs and have been in the jobs where they're around

people who can manipulate stuff like 3/4 or 12/16 or 24/32 into percentages or decimals, AND, get the right

answer, consistently. It has been my experience the focus-on-problem-solving-types are from backgrounds that

have had little interaction with those who struggle with basic skills.

con't

MathTeacher42 said...

con't

It has also been my experience that the reform crowd has consistently missed the fact that the job market at the

bottom is NOT the job market at the top. About 214,000,000/238,000,000 had money income under $75,000 in

2007*, about 185,000,000/238,000,000 had money income under $50,000. The worse your basic skills the more

brutal the job market for those under $75k and under $50k. For those without family re$ource$, connection$,

drive, and complex problem solving skills, the job market is NOT sipping latte's in cafes reading "What Color Is

Your Parachute".

Because the reform crowd won't acknowledge the economy, and many of them lack experience with those who

struggle, they're able to take this Nobler-Than-Thou approach to what math is and math education is, and to build

up these complex edifices of excuses to justify emphasizing problem solving. And then they conveniently ignore

the fact that our kids don't have the basic skills to just get by.

I'm sure that once the US joins the rest of the world with a huge underclass, and a dinky servant class to the

powerful, students will be motivated to master whatever basics they have the opportunity to get exposed to. Hopefully they'll have access to problem solving.

BM.

MathTeacher42 said...

UGH! on this program for editing!

Sahila said...

I dont disagree that children are all better served if they master the basics and that there is a progression in learning from the foundational to the complex...

I'm saying that that can be achieved in ways other than direct instruction...

Two people who stayed at the end of the Recall meeting at my house yesterday were discussing just that ... its the approach - how to engage a child's interest so that the motivation for learning, for trying is ignited, fostered, preserved not vanquished by boredom and perceptions of failure....

And speaking anecdotally - my elder daughter, almost 30 - has a university degree, has held managerial positions, has travelled the world and STILL HAS TO COUNT ON HER FINGERS WHEN SHE'S WORKING OUT THE TIP FOR THE LUNCH BILL... she's got mixed laterality and cannot remember number patterns/sequences such as those that are the basis for adding, subtracting, multiplying and dividing...

This wasnt diagnosed until her first (and only private) high school year, and school allowed her to use a calculator... then she began having success in math... but all that damage done to her confidence in all those years in primary school - insistence on direct instruction and not giving credit for understanding the process, only giving credit for the right answer - just destroyed the love for numbers she had in kindergarten ...

And the irony is that she's both an artist, skilled craftswoman and a business person using numbers daily, including geometry, ratio, proportions, dimensions, all the time in production management in the fashion industry...

I wish people would get to the place where life and learning doesnt have to be an 'either/or' proposition....

Bird said...

It'd be cool if someone would actually teach "problem solving". And by teach, I mean offer conscious instruction in how to approach and improve a person's "problem solving" techniques.

I've never seen anyone do that, except perhaps in books (Polya?)

Instead, folks use math to teach "problem solving" like they teach "working in groups" (which could also benefit from some conscious analysis!).

They just hand out some problems and expect the experience of working on problems in groups will lead to people learning how to better "problem solve" and "work with people". It's silly. Can't we speed things along with the application of analysis and experience?

Ah, but I suppose that would be "direct instruction". The horror!

kprugman said...

Mathematics is a branch of problem-solving, more or less a tool for visualizing problems in a simpler, logical framework that can then be readily solved with deduction, as with algebra.

Singapore emphasizes the integration of algebra and geometry. How else can you rapidly visualize a math problem?

Reform math uses a radically different approach for solving problems - with statistics and making visual interpretations of data. There is less importance on rigor (getting exact answers), hence the name fuzzy math.

Singapore is a much better written textbook and its conciseness, especially its treatment of traditional algorithms satisfies most people's expectations about what children should be learning in school, including most college math professors.

Calculators are not usually used in other countries, until college. Americans are easy marks for anything goes in education so long as it costs more money.

kprugman said...

Reading a reform textbook, like discovering or core plus, is more like putting together a puzzle with a few parlor tricks. For instance, getting children to draw a line is done iteratively or with recursion, as in each successive point following another point. The first trick is deciding what point to start with, and if you know something about math, then you know this is really a discrete line that approaches a limit (one continuous line). I would prefer writing lines as a function of x and y, not start with a seed and successive iterations. Try to do that with parabolas and then you'll realize why the authors avoid it altogether, except in a few sections that 'discuss' falling projectiles or the Earth's orbit. Without quadratics, how far can a student really advance in any science.

Finding the volumes of solids is a trick that was probably ripped off from a math olympics contest handout. The authors couldn't write to save themselves.

Try reading their manual for using a graphing calculator. The manuals were not written by teachers for teachers. So why pretend this stuff works? Its stupid.

Melissa Westbrook said...

And Bird, this was a consistent theme at the Arne Duncan event. The ability to "problem-solve, think critically and work in groups" was mentioned several times. Are teachers taught how to bring these to the fore in how they teach? I honestly don't know. Is that in professional development? Isn't the act of life working on problem solving (or escapism or denial if you don't want to problem solve)?

Reader, why do you think young math candidates are better prepared than ever?

Sahila said...

Well, you know what... I had direct instruction math in New Zealand in the 60s and very early 70s...

I'm a reasonably bright person... I was on an academic stream, learning math, science, english, french, latin and history (six subjects was the max you could take, four was the minimum and five was the average)

I passed my University Entrance exams in math...

And I dont know how the hell I managed that cos I completely lost the plot sometime two years earlier - none of it made sense anymore...

And the first time none of it made sense was when we got into negative numbers.... I always thought that was a con... a party trick.... never saw the logic or the value and still dont...

Know the algorithms for adding/multiplying negative and positive numbers but havent a clue what the hell its for, what use it is...

And that's thanks to direct instruction not giving me any context for learning this stuff...

Thank god there are countries in the world where you dont need math to go to university, or science, or a language... you can get a law degree without math and science... you can get a science degree without English or literature or french or whatever...

What you do here is needlessly convoluted and sets people up for failure as much as it does for success....

Charlie Mas said...

Ah! So this has been a fruitful discussion after all. It ploughed up this nugget:

Bird: "It'd be cool if someone would actually teach 'problem solving'. And by teach, I mean offer conscious instruction in how to approach and improve a person's 'problem solving' techniques."

Which lead to this one:

Melissa Westbrook: "The ability to 'problem-solve, think critically and work in groups' was mentioned several times. Are teachers taught how to bring these to the fore in how they teach?"

Instead of providing any real instruction in these "21st century skills" (as if no one had these skills or valued them prior to this century), do we just give kids practice with these challenges and presume that they will acquire the skills?

MathTeacher42 said...

I've about 4 responses to many of you.

I'll start with Charlie at 7/20/10 5:59 AM.

I mentioned yesterday in the diary on professional development what I wanted from coaching, pd, training, ...

In my math education certification courses, my professional development, some of my coaching (a few years ago), my math training ... how many more names do we need... by the way?...

We almost NEVER deal with the nitty gritty details of how to get through / execute / present / enhance a lesson on a particular topic. We'll sit around for hours doing jigsaws and all kinds of group stuff with other teachers, discussing how important it is for kids to question each other and for kids to discuss and for kids to probe and and and and and we will NOT figure out nitty gritty details of how to get through / execute / present / enhance a lesson on a particular topic. We won't figure out a variety of questions to ask every few minutes, or questions the kids can ask each other, and we certainly don't review suggested questions that might work, cuz there aren't any!

I suppose that is because then teachers will howl that they're creativity is being standardized? Or, the university types are just too high level to figure out those trivial details of the enlitenment they've brought down from The Mount to us lowly dirt farmers?

(In the last year, this has changed for the better at some trainings.)

O.K. ... maybe later I'll get a chance to attempt to answer some the others. TTFN.

BM.

bhammer said...

Problem solving can not be taught separate from domain knowledge. As cognitive scientist Daniel Willingham points out, "Prior knowledge is important not only for recognizing problem structure, but also for successfully deploying critical thinking skills."

The problem with reform curricula like Investigationns and Everyday Math is that they have omitted key math procedures and concepts that are needed in order to apply problem solving skills. Students can not be successful problem solvers unless they have a strong mathematical foundation first.

Rabbit said...
This comment has been removed by the author.
Rabbit said...

I wonder if doing logic puzzles and problems would help kids learn the art of problem solving? I say this because I got my kids logic problem magazines (instead of crossword puzzle magazines) for long airplane rides years ago. At first they were very frustrated, and it could take days of working and reworking a puzzle to figure it out. I wouldn't let them go on to another puzzle until they figured out the one they started (even if it meant helping them out sometimes), so they definitely learned persistence. And motivation was built in (there was nothing else to do on a 5 hour plane ride).

Now they love buying logic puzzle magazines and have fun with them at home, on the bus, on long car rides. They can work to solve them pretty quickly, even if they do occasionally still need some help. It's funny because we brought a couple of logic problem books camping recently and several of our (very intelligent) adult friends wanted to try them. Sadly, most gave up, frustrated, after an hour or so.

Bird said...

MathTeacher42,

My mom was a teacher, and she still often talks about the woman at the UW who she took education classes from back in the 50s. That person was nothing if not full of nitty gritty details, all, as I've heard many times, extremely useful in my mother's teaching career.

I'd have to check, but I do believe that person had a long career chock full of classroom experience, and they imparted as much helpful advice from this as they could.

I'm not sure how many UW Ed department faculty have that sort of background now. I suspect, from what I've seen, many of them are far more experienced in navigating a college system and working in abstracted theory.

But, of course, the biggest barrier to the kind of help you sound like you would like are the "brain" theories that seem to operate in the Ed. Department these days.

Imparting nitty gritty details is a bit too "direct", I would guess. And from what I gather from these folks is you will not learn it unless you "construct" it yourself. Bleh.

uxolo said...

Yes, problem solving can be taught and you can even use direct instruction to develop this repertoire.

http://chemeng.mcmaster.ca/pbl/PBL.HTM

There are Americans who do this work, but McMaster's is the superstar.

ebaer said...

George Polya wrote a very influential book on teaching problem solving - back in the 1950s - called "How to Solve It." It is widely used by professional college-level educators in Math, Physics and other problem-based areas. In this view of problem-solving, problems are non-trivial questions that "requiring some degree of independence, judgment, originality, [and] creativity." There are several steps based on his work, each of which has many techniques that can help and be taught. The steps are: 1) Understanding the problem, 2)Devise a plan, 3)Execute the plan, 4)Look back. I know this seems simple, but a teacher can teach steps/techniques to do each of these effectively. Sometimes one technique will work, sometimes another and more often the solver will need multiple techniques. I should warn folks, however that this work is often used by the proponents of the "discovery" methods of teaching - something many members of this community are hostile to. This is because in this view of teaching of problem-solving teaching rote techniques such such as multiplying fractions is not considered problem solving. My personal opinion is one needs both to be an effective problem-solver

MathTeacher42 said...

bird at 7/20/10 8:20 AM

your last paragraph nailed it - for me. I hope it did for others.

Sahila - the either / or battle in math is due to the reformists IMPOSING their pedagogy turned theology. The "training" you get to endure & waste your time on is determined by the reformists. Your annual review - your employment and your rent - can be based on how well you implement reformist edu-babble.

There were a lot of things wrong with math ed. 30++ years ago. They did well at drill and kill, they did well discouraging those NOT on a competitive abstract math, engineering or physics track.

The reform crowd, to their credit, tried to fix lots of stuff. They worked hard. They meant well. They failed, which is going to happen you try new things. Unfortunately, their failure has crippled millions of kids who lack all kinds of basic skills needed to just get by, never mind really change things. What is unforgivable is that the reformists have dug in their heels defending & redefining their failure, instead of trying to figure out what is going to work.

BM

Sahila said...

Well Bob, as an outsider looking in, I have to say I think both the reformers and the traditionalists have dug their heels in and the kids are paying for this adult power struggle...which really is quite shameful (analogies of messy separations/divorce spring to mind)

People often ask me how I label myself in the spiritual sense - they want a descriptor they can use when they advertise that I'm going to be a guest speaker at their church service or event... I call myself a synthesist.... I've studied many different spiritual traditions, taken what is common at the core of them all, what works for me and is congruent with my own metaphysical experiences and and moulded that into something that is my unique understanding of the world... the advantage of that is that I can navigate most of the different traditions and can speak/interact meaningfully with members of all those traditions, whereas someone who only accepts one 'truth' cant... and in only accepting one 'truth' one is depriving oneself of depth and breadth of knowledge and experience and is setting up an artificial isolation, which only adds to the problem/difficulty...

Sahila said...

and the final factor that is missing in ideas about problem solving is that there is a huge place in the process for 'intuition'... where you just 'now', feel, see in your mind's eye a new way of looking at, doing, creating, overcoming a challenge...

and traditional math doesnt make space for that factor to be part of the equation (pardon the very lame pun)... but I'd venture to suggest that most of creativity, invention, problemd solving happens just as much due to intuition as it does due to fact and process...

Sahila said...

oops... should be:

..."where you just 'know'..."

one cup of coffee just isnt enough some days!

as WV says... graci for reading!

Patrick said...

Certainly problem solving can be taught. Just like other skills. Not everybody can become a great original mathematician, but pretty much everybody can learn to solve the mathematical problems that come up in life. (If we need to finish our hike down the mountain by the time it gets dark, and we hike downhill twice as fast as we hike uphill, what time do we need to turn around?) You teach how to solve many kinds of problems, and the student will recognize that kind of problem when it comes up again even in slightly different form.

The student must also learn multiplication tables and how to do long division. Without that ability to do arithmetic, it doesn't matter how well you can set up the problem because you won't be able to carry it through to a solution.

I've been glad to see estimating as a skill that was taught in my daughter's class (just finished 3rd grade). Again, though, knowing the times tables is essential to making a one-significant-figure estimate.

Sahila said...

Patrick - as I wrote above, my daughter never, ever managed to get the times tables stuck in her memory, and she's a successful 30 year old, with a degree, had management level positions, has travelled the world and is just embarking on opening her own business...

Me on the other hand, love arithmetic, love the pattern of numbers, used to win mental arithmetic challenges in primary school, used to ask my mother to create pages and pages of multiplication and long division problems for me to solve on long car and train journeys, and I completely lost the plot on higher level math when I was about 14-15 because what was being taught, didnt make sense BECAUSE OF THE MANNER IT WAS BEING TAUGHT - there was no real life context that made sense to me...

I thought it was stupid to replace numbers with letters of the alphabet to do equations - it no longer seemed real... I didnt think negative numbers were real - my logic said how can you go past zero if zero means nothing? So what was the point of spending time 'playing' with what I considered to be 'made up' numbers, a fantasy...

I couldnt accept answers such as: "that's just the way it is" ... me being me, its the same reason I left the catholic church behind when I was about 14 and my teachers couldnt explain/answer my questions in any meaningful way - I was just supposed to accept because 'that's the way it is'... and that blind acceptance is not a capacity I carry in my being...

ParentofThree said...

Yes problems solving skills can be taught, and yes they can be taught in the subject of math.

However, what students need to solve any math problem are basic math skills and they need to have some of these skills committted to memory, such as the multiplicaton table.

Reform math focuses on problem solving and working in groups, which is great. Reform math however does not instill basic math skills, especially in the early years, to memory.

If the K-5 math adoption as gone as adverstised in SPS, students would have received 15 minutes a day working in the Singapore math books, learning basic math skills, and then spent the remain time learning problem solving skills.

But that is not what happened, and we are now starting to see the results of the reform math in our schools.

Maureen said...

I'm finding this whole discussion confusing. (I haven't read all the posts in detail-so am cetainly missing some points) I'm shocked that so many of you feel that problem solving cannot and is not taught. I feel like I was definitely taught problem solving skills all through school (and I attended the cr@ppiest little Catholic schools you can imagine.) We were taught how to interpret the language of math problems and translate them into math operations. We were taught how to break questions (math or not) into their component parts to make them solvable. Looking back, I realize we were taught basic logic that came in handy later with if then else statements. We learned that there is a standard structure to a paragraph and to a research paper. We knew how to form a hypothesis, collect data and report it back.

We weren't given a template for interacting with others in groups, but we did practice it and got guidance from teachers when it wasn't working well.

I don't believe that teachers don't do this anymore, what else is teaching if not showing others how to problem solve on their own? I believe (wrongly?) that my kids have been taught these things--I just feel like they don't get enough practice applying them so don't always internalize what they know.

mirmac1 said...

Charlie, back to the nugget of your question:

although I'm not a teacher, I AM the mom of a child with mild autism spectrum disorder. Learning "problem-solving," cause and effect, and logical reasoning is difficult for her, even though she get's strong grades in domain knowledge. The issue is her difficulty with "theory of mind": to step out of the concept of "self" (what she sees, knows by direct experience, thinks) and place herself in the position of, say, the character in the word problem, and/or consider really abstract (to her) concepts. Here, specially designed instruction is imperative so that she may keep up with her peers who have an innate sense of what others may be thinking.

There are many children like her in our school system.

Anonymous said...

After a cursory glance through these 33 comments the problem that still needs to be solved:

discovering the equation that puts a null value on Sundquist & Maier's influence on School Board Policy.

ken berry

reader said...
This comment has been removed by the author.
reader said...

I have interviewed many, many young people seeking hi-tech jobs. Their skills always amaze me, as does the increase in skills beyond those who interviewed in previous decades. They are more innovative, creative, and they have applied more of their knowledge. Group problem solving, is an incredibly important skill, important for anything related to any job... or even just getting along. Perhaps, it's the most important skill. It is hard to teach, and hard to assess. That doesn't mean we should avoid it. It's always these old timer math teachers who poo-poo that. One step outside the classroom, and the importance of collaborative problem solving would be obvious.

While MathTeacher42 may think I'm condescending... this attitude represents the height of condescension. "Well, I learned to problem solve because I'm just really smart. Nobody fostered my curiousity or problem solving predilection. Just really too bad for those dummies that weren't born with superior minds. They should just stick to basics. Let's keep doing those people doing basics until they're good it... oops too bad never got to anything meaningful." It's exactly the attitude that will keep bloggers from being elected to anything.

The way to learn to problem solve... is to try it often, on well thought out successively challenging problems. And, you teach persistence. The way to teach problem solving, is to model it, and teach students how to create models. Maureen also lists many other ways.

I don't know much about math pedagogy, or what is taught about it in math-ed, or what math teachers are made to sit through. I can imagine it could be awful, and full of worthless edu-babble. I wouldn't say I'm a proponent of "reform". Just that there's a reasonable balance. My kids have always gotten a good dose of "commit to memory", even with the most reform curricula. But, that isn't the main point.

TechyMom said...

I work in high tech, and also have done a fair bit of interviewing. High tech jobs do require getting along in groups, brainstorming, and group problem solving. However, many high tech jobs, particularly those that involve writing code, also involve concentrated, individual thought, deep understanding of algorithms, and juggling of multiple variables in one's head.

There are also a fair number of people who really love the algorithmic thinking involved in old-fashioned math. It was my favorite subject, followed closely by programming, and what I see in my daughter's books leaves out most of what I really enjoyed about math.

We need to teach both. Most of the private schools I toured fully understood this. Why can't SPS?

WenD said...

Problem solving can be taught, but it's an arena ripe for differentiation.

The next step, explaining how you solved a problem, is still totally subjective, unless it's algebra. And at that point, where's the value in bumbling around when algebra is in fact a language. Sure, you can get a group of developers together, get em' drunk, and they might give you some ninja secrets on code, but at the end of the day, code is code. Now, finding a solution to a desire that doesn't make money, and make it so essential that it makes money and helps millions of people? In general, that doesn't rise up from group think. Creativity can be encouraged, but I don't think you can teach it.

Why does problem solving have to be an integral part of any curriculum? Ask 10 people how they fixed a leaky faucet, and you will probably listen to 10 unique explanations. It's like the classic interview question: if you were the first person to create a window blind, what steps would you take to design and make them? This is where technical writers make their rent.

On the emphasis of group learning, which is a regular component of EDM: for my kid, this requirement only caused frustration. Let's sit as a group and be confused, while the one kid that gets it just wants to move on. Is this really helping the teacher? In her case, no, it created more work.

I've heard a fan of EDM defend it with this: yes, it requires that you think differently, and isn't that a good thing?

I don't think so, not when my kid goes through grades 2, 3, and 4 feeling frustrated, with shaky number sense, utterly convinced, along with many of her peers, that she's an idiot. I, along with other parents, would offer help, an alternative narrative with scratch paper and manipulatives as needed, and were uniformly told by our children: "You're not doing it right!!!" Math anxiety, anyone?

My daughter is a solitary learner. Once she knows it, she's happy to share, but the problem solving part is done solo. Asking her to learn with 5-20 other kids doesn't work for her. I've been told she's on the cusp of an ADHD diagnosis. Who can say. She knows what works for her and what doesn't. At least she has that.

The most beneficial group activity she had re: math was the daily facts drill. Her teacher couldn't rely on EDM, had lots of parent help, and doing times tables, sort of round robin style, was fun. Beyond that, asking 5 kids to calculate the area of their classroom, to order carpet (watch out for the toxic glue!), ended up requiring way more help. If at least one child was an expert, sure, they can show what they know, but this method still required more work for the teacher, rather than teaching the method, then letting the kids break out in groups or work solo.

I'd like to think that Arne means well, but who is he listening to?

kprugman said...

Balance continually crops up in most discussions and I would ask how do you strike a balance between good books and bad books? The issue over curriculum isn't about balance, its about selecting the best textbook for students that will satisfy what parents want their kids to know. Teachers want textbooks that are easy to use, that engage all their students, and prepare them for the next level. The worst part about the DOE's exemplary math programs is that what the kids are supposed to learn gets usually forgotten from lack of use. Successful students revert to the traditional algorithms because those algorithms are more robust. They are self-checking. Students that use the non-traditional methods have to rely on calculators, the teacher, or other children to check their answers.

reader said...

Techmom, most people can do the code (yes, some are better than others)... the collaboration is a lot harder. And really, there isn't a shortage of excellent coders.

EDM doesn't place a huge value, or huge amounts of time on group problem solving. I'm not sure its good at that. But, on principle, if we had something that really worked for group problem solving, it would be great. And as for communication, this is also something that we never learned back in the old days of math. People who could multiply, maybe couldn't communicate about it. That's another skill.

Right, Wend. Teachers have to do more now. More is expected. Kids have to do more too. And, it is more difficult when you've got a disability as you mention. There's problem solving, there's critical thinking, there's communication. And, we're getting more out of it.

Whenever we hear about "learning the algorithms"... it always turns out to be about the arithmetic. People do love the arithmetic, but they've got to do more than that.

What I think really is a problem, is that the assessment tools aren't that good. If somebody can get an answer, or be a good mathematical thinker, but isn't a good communicator, or is ELL... the assessments don't know the difference.

reader said...

kprugman, we aren't talking about "good books" vs "bad books". We're talking about reform/constructivism vs traditional, and the balance therein. Both are religious points of view. EDM are perfectly fine books, all my teachers like them, and can use them well. It isn't all that different from Singapore, but has a heavier empahsis on problem solving, and more interesting problem solving. It is easier to do "Singapore", because it actually does less.

kprugman said...

What is reader's standard for r-i-g-o-r?

"A. If math were a color, it would be –, because –.
B. If it were a food, it would be –, because –.
C. If it were weather, it would be –, because –." - EDM Silliness

What would reader's solution be? Look at the results reader? Which group of students does better on tests? Students educated with Singapore textbooks or Students educated with EDM? Blame their poor result on the teachers ... Is that your religion?

Class societies, like the US, contain ‘caste-like’
cultural practices that are rooted in racial origins.

The stratification of students into lower math classes is one of those 'caste-like' practices. The racial practice by the dominant culture of using textbooks that pretend to teach fact to the lower socio-economic classes is not without historical precedence. Where else would a child be taught that 'math is hard' or that some people are born 'smart', while others will never get it.

A non-white has a higher probability of going to college, if they are educated in a different country, than if they were educated here in the US.

As an educator, doing nothing is the same as being a racist. It might shock reader, but the majority of classrooms have students that cannot be engaged because they cannot understand the textbooks they've been given to read.

kprugman said...

Reader confuses the DOE's exemplary pedagogy with an even longer word called, constructivism.

Little does reader know that constructism applies to all pedagogies, not his religion alone. How about we all chant the 'success for all' greeting and lets throw in some 'vedic math' so our students can do math 1500% faster according to researchers at Maharishi University in Bloomington. More ridiculous claims and no facts to back them up. We might as well all go believe in the sasquatch (another research project discussed at an MAA regional meeting. This isn't just a bandwagon, its a parade.

kprugman said...

The professors who had a hand in reform and were paid millions also built the assessments that had to be aligned to the DOE's carefully-tested books, but to their surprise, any gains measured in test scores from using these books are at best noise while the long-term trend remains down and out. It has been ages since we heard from them and most are now retired. I still remember one old fart - 'What children need are smaller textbooks.'

The only science more dismal than economics is math education in the US, unless you are an ecstatic.

kprugman said...

Reader sounds like he could have written EDM - "EDM does more, while Singapore does less [problem-solving?]."

One shouldn't confuse number of words in text with problems to solve. Muddled could also be meaningless.

And I suppose "because energy makes it go? would be your other response." What vitamins are you taking? Are you sure they weren't approved by the DOE too?

SPS mom said...
This comment has been removed by the author.
kprugman said...

http://www.thenewatlantis.com/publications/how-we-measure-up

Where 44 percent of Singapore’s students reached the TIMSS “advanced international benchmark,” only 7 percent of U.S. students did. And, in general, the longer students had remained in the U.S. school system, the worse they performed relative to their peers abroad.


Neener, neener!

kprugman said...

Not only does the US have a larger racial disparity in achievement, but now the US has fewer high achievers. This is the worst possible scenario. Better think fast reader, public schools have only been losing ground for the past 25 years and its DOE foolishness, assisted by the Governors Roundtable and a few lumpheads, like Steen and Merlino, that got us here.

Anonymous said...

I do believe that people's problem solving skills can be built up with repeated practice and exposure to other people's problem solving methods.

The current SPS curriculum does seem to have lots of exposure to different ways of looking at problems, and demonstrates lots of the "tricks" that we had to figure out for ourselves in the past. It hasn't been effective for my kids because they haven't had the opportunity to master and use whatever one or two methods work best for them first, before learning other ways of solving the same kinds of problems. Having to learn several methods before any one method is mastered has created a lot of confusion.

Some of the reform math goals do seem valuable, but there really are many people who are not able to successfully learn math with the current curriculum. My middle schooler has needed hours and hours of extra help and extra programs (expensive!) to learn the basic arithmetic that he hasn't learned with EDM and CMP. Then there's our college sophomore who is taking a pre-calculus class this summer, after a basic math class last spring because she didn't score high enough on her school's math placement test to take calculus. She was in Spectrum, took AP classes in high school, passed the WASL in 10th grade without a problem, and passed regular SPS calculus with an "A". Something is not working.

P.S. Sorry this ia Anonymous...it's my first post and I'm not "blog-savvy" enough yet to get my id to work. I'm sure I can get my middle schooler to help!

ParentofThree said...

"Something is not working."

What is not working is the level of rigor in the K-12 Reform Math that SPS has adopted. Has nothing to do with problem solving. You can take Discovery Math Algebra I and II in highshool, earn all As..but will not cover all the material that a traditional Algebra math track covers.

Your students experience is EXACTLY what college professors have been talking about now for some time. Students coming to campus ill prepared to persue degrees requiring math skills that should have been learned in highschool.

TechyMom said...

"most people can do the code (yes, some are better than others)... the collaboration is a lot harder. And really, there isn't a shortage of excellent coders."

Maybe this varies by company, but my experience is just about the opposite. Pretty much any bright person with reasonable people skills can do the collaborative design work after a few years of on-the-job training. Most coders, on the other hand, are fairly mediocre, and the really good ones are very, very difficult to find. Top computer science schools require very strong algorthmic thinking skills. Very few of recent grads I've interviewed went to public school, where most had when I started interviewing in the 90s. Some of that is likely due to the economy and the cool-factor of the tech industry waning, but it is still a remarkable difference.

The corporate IT department might be fine with the mediocre coders (and the applications show it), but that doesn't cut it for large-scale commercial software production (Microsoft, Adobe, Google, etc.).

Again, I think problem solving, group problem solving, logic, and algorithm thinking, and mental arithmetic are all very different skills, and that all are necessary for success where I work.

Jan said...

My take on the "collaboration" and "problem solving" issues is this:
1. Many (but not all) of the skills used in problem solving can be taught. A few elements (creativity, etc.) may be more in the nature of "gifts," but can be nurtured -- with the understanding that like musical ability or artistic talent - we aren't all going to end up in the same place.
2. The problem with using discovery methods that rely a great deal on collaborative and group work is that "math skills and knowledge" and "ability to problem solve in groups" are very different knowledge sets -- and to ignore the learning styles of children who best learn math through "mastery" methods is a huge disservice to them. First, teach them math -- in whatever way works (and it is not the job of the child to show up with learning abilities that match however teachers want to teach it -- it is the adults' responsibility to figure out how to best help kids learn). THEN, if you want, pose problems that require the development of problem solving skills and the ability to work together well in groups. In my oldest child's high school, the science department would announce, each term, a "problem" that required lots of complex thinking skills -- and the entire campus had 4 to 5 weeks to work on it -- it would be discussed in various science classes, kids interested in it could be found outside of classes discussing how they thought it should be approached, and what data they had developed, and how valid it was. But they didn't hold substantive skills (like math) hostage to whether, and how well, the kids could collaborate and problem solve as a group.

MathTeacher42 said...

before this diary disappears ... and another few days and another week winds away ...

THANKS EVERYONE - it is interesting and useful to hear what parents are thinking and what parents are living with.

I was discussing this thread with a co-worker ... to really really respond to everyone and their different points would take the rest of the summer - except we couldn't, because there is a lot of stuff which happens in school which isn't for public consumption*! I've yet to figure out how to explain this job to those who haven't done it, and, I don't think anyone has figured out how to explain it.

THANKS EVERYONE

BM

*remember - high school kids are NOT legal cuz they are physiologically immature!
IF teachers were to openly discuss issues in public forums, people would figure out which kids they were talking about - and then those ADOLESCENT issues for some kid would be public knowledge forever. Aside from the fact that it is just wrong to hang stupid ADOLESCENT things around people's necks forever,

aside from the little itty bitty thing - how would you like some dumb thing(s) you did at 15 or 17 or 15-17 hung around your neck forever (do unto others as you would ...)

aside from the decent thing to do, it is usually illegal to discuss stuff that adolescents do!

Rose M said...

My child had a high school math teacher who used an unusual strategy (to me anyway) to teach team problem solving. It was in an advanced class so that may have contributed to the reason it was successful.

Every Friday he gave a group quiz. He assigned the groups & they changed every week. He graded only one of the papers from each group. So everyone in the group got the grade of that one student. At the beginning of the year, kids were very upset about it. Then they learned to talk to each other about the problem solving, debate and defend their reasoning & trouble shoot each other’s papers. They had to work together & agree on how all their papers would look. With different personalities every week, they could not make assumptions about how to work together or who had the correct answers. For my child, having to debate other kids helped with sorting through problem solving techniques and cemented good ones.

A larger part of their grade was the individual unit tests he gave, but I came to think the group tests were a valuable learning tool.

dan dempsey said...

Anonymous said:

"Some of the reform math goals do seem valuable, but there really are many people who are not able to successfully learn math with the current curriculum."

Goals without effective well designed plans to attain the goals are worthless. SPS math has been delivering close to worthless for so many students at so many grade levels for so long that MGJ's decision to Appeal Spector's High School math instructional materials decision just fit right in. MGJ was backed by the often clueless rubber-stampers Carr, Sundquist, Martin-Morris and Maier.

Without a sound background of basics most everything becomes a problem.

kprugman said...

If the best that NCTM can do is lie with statistics and make reform math look superior to all the rest and both camps reado;u concede that, yes, there is an racial achievement gap in testing - then what will become the solution to our problem.

How long must children be forced to endure another textbook that is as bad as Core Plus or Discovering Math or Everyday Math or ...

Stay tuned while we watch Mr Gates and Mr Broad attempt to make KIPP Schools more palatable for the public - Success for All!!

Reform math, the public's scorn, makes for good bearbait throughout all of education. Any politicians with lofty aspirations and looking for a reality check or a chance to swim in quicksand ought to endorse Everyday Math, the pariah of leperish textbooks.

dan dempsey said...

It seems to me that if we are talking about better skills in solving story problems yes these can certainly be taught.

The Singapore Model Method which is started in grade 3 is certainly testament to that fact.

Again with proper skills, what might be thought of as problems can become exercises for the skilled student.

kprugman said...

Its been my experience that a 'popular' textbook is by necessity written by skilled writers.

There are themes within a book that get incorporated throughout the entire series. Every problem is selected and tested and weeded out. During the development phase of Singapore, CPM, and Challenging Mathematics each problem was carefully scrutinized and written over and over for a particular spectrum of students.

Foremost each book is written in a style that can be understood by a student who's primary language is not English. That requires a very, sophisticated and talented writer.

These books were originally written for sheltered English classes and they were so popular with students and parents the textbooks were introduced into mainstream English classes. You have the same problems in the US as you do in Asia and Canada - how do you mainstream non-English speaking students.

The TIMSS results speak for themselves and the US publishers have fabricated the research - foremost by disallowing any research that compares Singapore with current state-adopted materials.

Its because billions have gone into their own research and they can't start over.

kprugman said...

The US textbooks are deeply flawed in many respects - principally, they were written, but not adequately tested or carefully edited.