Michael Dougherty posted Oct 8, 2010 8:51 PM: > To me, math is almost automatically "discovery > learning." Maybe all subjects are but I could argue > math is more so. It's just a matter of how much of > it you want them to discover on their own. If I > teach trigonometric substitution or partial fractions > decompositions, they will still have to "discover" > the logic of it as they go, even if I completely > explain the logic to them in my always brilliant > lectures. I suppose if I want to give these two > topics a month instead of a week (collectively), they > might be able to "discover" it from something closer > to first principles, but no matter how much I explain > things they still have to work the problems to > "discover" what works and what does not. > > As they say, give a man a fish and you feed him for a > day; teach him to fish and you feed him for a > lifetime. > > OK, but now they're asking us to let him discover for > himself how to fish, perhaps out of desperation? > He'll know some aspects of fishing better than if > f you teach him, but he'll miss out on a lot of > details you could have taught him. And a lot of time > will be consumed where it did not have to be. > > As he gets older, it's good if we can teach him how > to find the resources to "teach himself," but in the > beginning it's better to present a logical context > and let them work through it. > > But I submit it's still "discovery," that they will > make working through problems we give them. The rest > is arguing about what level we want them to start, > and how much guidance to give them. Also, how much > of it we want to be a group activity. > > So when I hear "discovery learning," I'm hearing that > they think we explain too much. In math, that's > almost impossible, if at some point you make them > work problems on their own. > > - --Mike D. > I find myself in considerable agreement with Michael Dougherty's arguments above that practically ALL math learning is in fact 'discovery learning'.
However, in regard to the 'formal teaching of math', I feel that - while the 'discovery method' should have a pretty prominent place in the teacher's 'armoury-of-tools' - the teacher does have to be very sensitive to the student's complementary needs for discovering things on his/her own and for receiving needed guidance from the teacher before he/she gets seriously stuck. It's not easy at all to do this.
It becomes a whole lot easier through the modelling tools contained in the problem solving aid called the 'One Page Management System' (OPMS). In this process (for the specific case of 'math teaching+learning'), what happens is the following:
A: The teacher constructs an ongoing model fo the Mission: "to take Student ABC most effectively through all topics of the math syllabus";
B: The student (assuming he/she is around high-school level) constructs an ongoing model for the Mission: "To understand all topics in my math syllabus effectively and THEREBY to do well in all my math exams, tests and quizzes".
The ongoing development of these separate but complementary models would in the great majority of cases ensure that the teacher AND the student would accomplish those Missions. (The only fundamental requirement is that both teacher AND student should be genuinely interested enough in those Missions to spend around 10-15 minutes each day on developing them).
True, the teacher does have 25 or more students, agreed, but I believe it really should take no more than say half-hour per day to keep an integrated model for all the students up to date along with specific sub-models for the specific needs, quirks, etc of the various students. (As I'm not a 'teacher of math', I've never done this for an entire class - but I did succeed very well with helping one student overcome his severe weakness and fear of math). What was remarkable was that I most stringently did no 'math tuition' at all for that student (he got whatever 'math-help' he needed from his peers and his college math teachers) - I merely showed him:
- -- how to 'generate' appropriate responses to the various 'trigger questions' asked in the OPMS process;
- -- how to construct his own models from his elements generated; (In those days the software to do the modelling did not exist at all - it was still possible to do all the needed modelling needed on both sides in considerably less than 1 hour per day. Now the software is available, and it really takes no longer than minutes each day for an individual on any Mission).
- -- what to do at each stage of the process.
I had spent about 1 hour per day with that student for about 30 days, after which I had to leave Bangalore on a long-term assignment in Bombay. I left that student with a sizeable amount of 'individual, ongoing OPMS homework' to do on his own; there was no Internet those days; just about 8 months later that student wrote to me telling me that he had systematically done all his 'homework' and he was now able to understand everything he needed in Math and was now regularly getting over 75% in all his math exams etc. (Earlier, right through his school career, that student had never gotten above 45% in his math exams).
I observe that all the OPMS tool really did for that student was to give him the confidence that he did in fact possess all the needed resources of 'mind-and-heart' to conquer the 'math disabilities' that had previously held him back, nothing more: it was really all his own work that enabled him to overcome his math weaknesses.
To revert to Michael Dougherty's theme of 'discovery in math', I observe that this was actually an instance of 'discovery of self'. In the context of this specific example, the saying quoted by Mr Dougherty is most apt indeed: "As they say, give a man a fish and you feed him for a day; teach him to fish and you feed him for a lifetime".
I completely agree with the objections that Haim, Wayne Bishop and fellow travellers will undoubtedly raise that this one instance does not 'prove' my case. However, it does provide some strong evidence in favour.
Below high-school level, I've only had one experience: with my granddaughter, aged about 13 a year or so ago, who made the complaint to me that "Oh, math is SO boring!" In her case, I constructed a model to try to demonstrate to her that math was NOT boring at all - and succeeded to a fair extent in accomplishing just that. I've not yet exposed her to creating a model for herself, but shall do so in a year or two when she is at a maturity-level that would understand and appreciate such 'representations' of abstract 'mental models'.
The process has seen some remarkable successes (outside of 'math teaching+learning', a few of which are listed at the attachment titled "OPMS - in Outline" with the message at the thread "Social Promotion", pointed to above.
To provide some contrary examples of total failures I've experienced with the process:
1) I've never yet succeeded in getting Haim, Wayne Bishop, Robert Hansen to even take a serious look at the process - they have only been interested in 'cutting it down' as best they can;
2) I had some financiers who got themselves involved with my project - and took themselves out when the dot.com bust happened; one other financier, on learning from someone that this could be a very valuable tool, tried to grab ownership of my entire project for himself - at which point I walked out on him;
3) Shortly after 2003 when GW Bush launched his extremely foolish attack on Iraq, I tried to get US citizens interested in a project that I called "BUSHOUT - IMPEACH THE SCOUNDREL!" Of all the failures I've had (and I've had some) this was the one that disappointed me most.
Now, however, there are some financiers coming out from the woodworks once again; if things now move forward with the terms and conditions I am putting up, then we may see some significant development of OPMS pretty soon - one major application at that point will be to develop a specific spinoff from the generic OPMS tool to aid with the 'teaching+learning of math'.
> > Wayne, > > > > Certainly mandating discovery learning in these > ways > > is a bad idea > > because it is not essential to good teaching and > > because it is > > easy to botch in the hands of inexperienced > teachers. > > Teachers who > > do not feel comfortable using discovery learning > > should think > > twice before trying to use it. In short, discovery > > learning can > > be a useful approach to teaching and can be highly > > beneficial to > > students, but it is not essential to good teaching. > > And, like > > any approach to teaching, discovery learning is > best > > seen as > > something that can augment teaching and learning > and > > does not > > have to be seen as an "all or nothing" approach. > > Johnson and > > Rising's book "Guidelines for Teaching Mathematics" > > does not > > mention much about discovery learning, but they do > > point out that > > discovery learning is not appropriate in certain > > cases. It would > > be good if they had mentioned more specifics such > as > > discovery > > learning is not appropriate for those ideas that > > would require > > a mathematical genius or near genius to discover > with > > little or > > no assistance from the teacher or from others who > > already know > > those ideas. Perhaps the authors felt that such > > comments are > > not necessary. But they are necessary for those > who > > want to > > try to push discovery learning too far. Any > teaching > > method, > > whether discovery learning or anything else, used > to > > extremes > > leads to problems. > > > > > > > > Jonathan Groves