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
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 > okie-dokie if students can work questions similar to the worked-out > 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 okie-dokie. > 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 > 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 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 >> 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 >> e 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 >> Su >> 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 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 probl >> 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 >> 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 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 com >>> <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) >>> <firstname.lastname@example.org> >>> <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.