Dear Colleagues,
a few years ago Tony Gardiner and I prepared a sample list of archetypal mathematical problems, very elementary, very natural; these problems are not artificial "brainteasers" (I hate "brainteasers" as pollutants of mathematical education). Have a look at
http://manchester.academia.edu/AlexandreBorovik/Papers/370441/A_dozen_problems
When I hear about generic "problem solving", I wonder: will it help to solve any of these problems? For me it is a basic criterion of efficiency of "problem solving" methods.
Seasons Greetings  Alexandre Borovik
On 23/12/2010 22:47, Jonathan Groves wrote:
> Anne and others,
>
> Another problem is that many mathematics classes reduce "problem
> solving" to mere "exercise solving" by reducing everything to
> recipes, by assigning as homework only those questions that are
> similar to worked examples in the book (mimic the solution just
> by changing the numbers), and giving only these kinds of
> questions on exams. Thus, according to traditional ways of
> assessing students' learning of mathematics, everything is just
> okiedokie if students can work questions similar to the workedout
> examples in a textbook. It does not matter if they have no
> earthly clue about how to approach a problem they do not know
> immediately how to solve as long as this problem is not one
> of the "standard" ones.
>
> Yet another problem is that only the "final answer" matters
> on many mathematics exams. As long as the student can get
> the correct number for the answer, everything is okiedokie.
> Their work does not matter. Their ability to write clear,
> correct solutions using correct mathematical language and
> notation also does not matter. I blame this attitude in
> mathematics teaching for one major reason why many of my
> students are very sloppy about writing their mathematical
> solutions and why they complain when I insist in discussions
> that they show their work and explanations; what they want
> is to get away with writing just a number down for the
> answer. Many of the statistics projects I review for
> the Kaplan Math Center are also very sloppily written.
> When only the "final answer" matters, much of the excitement
> and lessons we can learn through problem solving are lost.
> Problem solving motivates mathematics not merely because
> we can find "the answer" but because solutions to problems
> reveal further questions and techniques that we can use on
> other problems. There is also strong pleasure and excitement
> from finding a particularly elegant solution to a problem.
> A beautiful solution to a problem is a work of art.
> Yet these lessons cannot be taught and the role of problem
> solving in mathematics cannot be seen if students hide their
> solutions on exams and other assignments and treat only the
> "final answer" as what is important, not "the stuff
> in between."
>
>
>
>
> Jonathan Groves
>
>
>
>
> On 12/20/2010 at 3:01 am EST, Anne Watson wrote:
>
>> I am entering this discussion rather late I expect.
>>
>> One of the problems around this issue is the language
>> itself. There are indeed many teachers who make the
>> goals of problemsolving the problemsolving methods
>> themselves, and not the mathematics, and also not the
>> specific mathematical contributions to
>> problemsolving. The truncated problemsolving
>> algorithm of 'understand the problem; make a plan;
>> carry out a plan; evaluate the plan' is far too vague
>> and generic. What does it take to understand a
>> problem mathematically? You have to coordinate a
>> search for variables with your knowledge of how the
>> situation works, or is likely to work; you have to
>> get a sense of covariation, constraints, the domain
>> of enquiry in terms of those variables. You have to
>> set up some representation and  if you are really
>> good make that representation minimal in terms of
>> the numkber of variables and amenable to variation
>>  etc. etc. Yet many mathematics lessons orientate
>> themselves around trial and error approaches,
>> arithmetical or engineering type testing, until the
>> 'problem' is 'solved'. Once a problem is solved 
>> who is interested in generalisation or
>> mathematisation for other similar problems?
>>
>> I agree with Jonathan that there is too little
>> written about these particularly mathematical aspects
>> of mathematical problemsolving  as a field we are
>> inarticulate about what it is like to gaze and gaze
>> at a situation, trying this, trying that, until
>> suddenly we see a familiar structure that we can
>> symbolise and manipulate in order to gain insight.
>> This isn't quite like 'reducing the problem to one
>> e you know how to solve' which in Polya's hands is a
>> mathematising technique but in less skilled hands is
>> a mishmash.
>>
>> The missing ingredient is what the Freudenthal
>> Institute call vertical mathematisation  to work on
>> several problems that have some structrure in common,
>> and then to extract from that family of situations
>> the core mathematical structure that can become a
>> tool. Generic PS algorithms just talk about
>> evaluating the answer; mathematicians ask 'what other
>> problems have I worked on that have similarities to
>> this?' Until this stage is included in problembased
>> mathematics teaching  as the main part of the
>> process instead of as an optional addon  it will be
>> quite legitimate to be criticised. I speak as someone
>> who used extended exploratory tasks throughout my own
>> school teaching  with some success  and know well
>> the pitfalls. Studnets will not learn to be
>> mathematical if the end point of problemsolving is
>> the answer. They will also not learn to be
>> mathematical if the end point of anything is an
>> answer, so this applies to socalled traditional
>> methods too.
>>
>>
>>
>> Anne Watson
>> Professor of Mathematics Education, Department of
>> Education, University of Oxford (Linacre College)
>> ________________________________________
>> From: Postcalculus mathematics education
>> [[log in to unmask]] On Behalf Of Jonathan Groves
>> [[log in to unmask]]
>> Sent: 19 December 2010 15:01
>> To: [log in to unmask]
>> Subject: Re: Latest Doceamus against inquirybased
>> learning
>>
>> Richard,
>>
>> Thanks for mentioning this article in the AMS and for
>> posting this
>> information on the reactions of others on the RUME
>> list.
>>
>> It is clear that the authors John Sweller, Richard
>> Clark, and Paul
>> Kirschner misunderstand what many advocates of
>> teaching problem solving
>> say about this. Are there some who want to try to
>> teach problem solving
>> without really teaching content? Are there some who
>> want to try to
>> teach problem solving with "minimal guidance"? I
>> would not be surprised
>> if this is so. Perhaps such people exist in large
>> numbers. But not all
>> advocates of a problem solving approach to
>> mathematics do this or
>> suggest that we take such an extreme approach to
>> problem solving. This
>> is yet another classic example modeled after the
>> fights in the Math
>> Wars: Assume that all opponents take extreme
>> positions, refute those
>> extreme positions, and then conclude that all your
>> opponents are wrong
>> and hence that your views are the only sane ones to
>> take.
>>
>> Yes, there is value in refuting extreme positions,
>> but all such refutals
>> do (as long as they focus on just those extreme
>> positions) is show that
>> such an extreme view will not worknot that all
>> views are unworkable.
>> Hence, we cannot prove that a problem solving
>> approach to teaching
>> mathematics cannot work simply by refuting extreme
>> views on this
>> approach even if it were true that all advocates take
>> such an
>> extreme view.
>>
>> General problem solving skills greatly enhance the
>> teaching and learning
>> of problem solving in mathematics because general
>> problem solving skills
>> teach us to think. Many students fail to learn
>> simply because they
>> were never taught and never encouraged to think,
>> simply to regurgitate
>> or simply to do the workno matter how shoddy the
>> work isor just
>> show up for class for an A. And many approaches to
>> problem solving
>> in mathematics are applications of general problem
>> solving: Reduce
>> the problem to one you know how to solve, consider
>> special cases,
>> do the required research if you need to know
>> something to solve
>> the problem that you do not know, and so on. Other
>> approaches
>> are mathematicsspecific such as proofs, but even
>> much of our
>> thinking in discovering a proof is similar to the
>> kind of thinking
>> we do for general problem solving.
>>
>> It is also clear that the authors have ignored
>> mountains of evidence
>> that the teaching of mathematics via studying worked
>> examples does
>> not in itself teach students much of anything,
>> especially when
>> all the students are asked to do on homework and
>> exams is to
>> replicate the solutions to similar questions.
>> Studying worked
>> examples can definitely help students in their
>> learning of
>> mathematics, but they need to study the underlying
>> logic behind
>> the mathematics being used. They should also study
>> what
>> problem solving techniques and mathematical ideas are
>> being
>> illustrated and used in these solutions. And they
>> should seek
>> understanding from these solutions, not memorization.
>> We know that mathematicians have learned a lot of
>> mathematics
>> and problem solving by studying others' solutions to
>> problems.
>> But they did not just sit there and try to memorize
>> the
>> solutions. That is, they study these solutions in
>> ways that
>> most students do not. Finally, students will not get
>> far if the
>> only questions they end up working for homework and
>> quizzes and
>> exams and other assignments are similar to the
>> workedout examples;
>> they must work on problems that they are not expected
>> to know
>> immediately how to solve. In short, studying the
>> work of others
>> in mathematics can contribute greatly to our learning
>> of
>> mathematics only when we ask ourselves the right
>> questions
>> while studying the work. And few students ask
>> themselves
>> the right questions, which is why they gain little
>> value
>> from studying the work of others. Why is none of
>> this crucial
>> information stated? Without it, all we end up doing
>> is
>> continuing to teach mathematics mindlessly via
>> approaches
>> that contradict how people actually learn
>> mathematics.
>>
>>
>>
>>
>> Jonathan Groves
>>
>>
>>
>>
>>
>> On 12/9/2010 at 2:30 pm, Richard Hake wrote:
>>
>>> Some subscribers to MathEdu might be interested in
>> a
>>> recent post
>>> "Latest Doceamus against inquirybased learning"
>>> [Hake (2010)]. The
>>> abstract reads:
>>>
>>> ********************************************
>>> ABSTRACT: A recent thread "Latest Doceamus against
>>> inquirybased
>>> learning" on the "Research in Undergraduate
>>> Mathematics Education"
>>> (RUME) list has discussed an article by Sweller,
>>> Clark,& Kirschner
>>> (SCK) (2010) titled "Teaching General
>> ProblemSolving
>>> Skills is Not a
>>> Substitute for, or a Viable Addition to, Teaching
>>> Mathematics"
>>> <http://bit.ly/i7wori> and published in the
>> November
>>> 2010 issue of
>>> "Notices of the AMS."
>>>
>>> Mathproblem guru Alan Schoenfeld wrote: "Sweller
>> and
>>> colleagues set
>>> up a straw man, the notion of 'general problem
>>> solving' as a
>>> counterpoint to mathematical knowledge [but] there
>>> are techniques of
>>> mathematical problem solving, and there's plenty of
>>> evidence that
>>> students can learn them, so the opposition Sweller
>>> and colleagues use
>>> to frame their paper is nonsensical. . . . ."
>>>
>>> Sweller and colleagues have previously set up straw
>>> men  the title
>>> of their tract "Why Minimal Guidance During
>>> Instruction Does Not
>>> Work: An Analysis of the Failure of Constructivist,
>>> Discovery,
>>> ProblemBased, Experiential, and InquiryBased
>>> Teaching"
>>> <http://bit.ly/duJVG4> invokes the straw men
>>> "Minimal Guidance" and
>>> "Failure" but critics of Sweller et al. point out
>>> that these methods
>>> are generally neither: (a) "unguided or minimally
>>> guided," nor (b)
>>> "failures."
>>> ******************************************
>>>
>>> To access the complete 23 kB post please click on
>>> <http://bit.ly/etl0dc>.
>>>
>>>
>>> Richard Hake, Emeritus Professor of Physics,
>> Indiana
>>> University
>>> Honorary Member, Curmudgeon Lodge of Deventer, The
>>> Netherlands
>>> President, PEdants for Definitive Academic
>> References
>>> which Recognize the
>>> Invention of the Internet (PEDARRII)
>>> <[log in to unmask]>
>>> <http://www.physics.indiana.edu/~hake>
>>> <http://www.physics.indiana.edu/~sdi>
>>> <http://HakesEdStuff.blogspot.com>
>>> <http://iub.academia.edu/RichardHake>
>>>
>>> REFERENCES [URL shortened by<http://bit.ly/> and
>>> accessed on 09
>>> December 2010.]
>>> Hake, R.R. 2010. "Latest Doceamus against
>>> inquirybased learning,"
>>> online on the OPEN! AERAL archives at
>>> <http://bit.ly/etl0dc>. Post
>>> of 8 Dec 2010 19:58:520800 to AERAL and NetGold.
>>> The abstract and
>>> link to the complete post are being distributed to
>>> various discussion
>>> lists and are also on my blog "Hake'sEdStuff" at
>>> <http://bit.ly/fFzJn8> with a provision for
>> comments.
Professor Alexandre Borovik * http://www.maths.manchester.ac.uk/~avb/
Blog: http://micromath.wordpress.com/
