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 problem-solving the problem-solving methods themselves, and not the mathematics, and also not the specific mathematical contributions to problem-solving. The truncated problem-solving 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 problem-solving - 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 you know how to solve' which in Polya's hands is a mathematising technique but in less skilled hands is a mish-mash.
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 problem-based mathematics teaching - as the main part of the process instead of as an optional add-on - 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 problem-solving is the answer. They will also not learn to be mathematical if the end point of anything is an answer, so this applies to so-called traditional methods too.
Anne Watson Professor of Mathematics Education, Department of Education, University of Oxford (Linacre College) ________________________________________ From: Post-calculus mathematics education [MATHEDU@JISCMAIL.AC.UK] On Behalf Of Jonathan Groves [JGroves@KAPLAN.EDU] Sent: 19 December 2010 15:01 To: MATHEDU@JISCMAIL.AC.UK Subject: Re: Latest Doceamus against inquiry-based learning
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 work--not 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 work--no matter how shoddy the work is--or 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 mathematics-specific 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 worked-out 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.
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 inquiry-based learning" > [Hake (2010)]. The > abstract reads: > > ******************************************** > ABSTRACT: A recent thread "Latest Doceamus against > inquiry-based > learning" on the "Research in Undergraduate > Mathematics Education" > (RUME) list has discussed an article by Sweller, > Clark, & Kirschner > (SCK) (2010) titled "Teaching General Problem-Solving > 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." > > Math-problem 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, > Problem-Based, Experiential, and Inquiry-Based > 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) "un-guided 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) > <email@example.com> > <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 > inquiry-based learning," > online on the OPEN! AERA-L archives at > <http://bit.ly/etl0dc>. Post > of 8 Dec 2010 19:58:52-0800 to AERA-L and Net-Gold. > 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.