Your problem is a common one, establishing nonexistent strawmen at least at the precollegiate level. Few if any of us recommend nor do teachers use "pure 'lecture mode'", nor did we when I was a new high school teacher at the height of the New Math era, while pretending that "the 'discovery mode'" is some idealized teaching/learning environment that "does NOT mean that we have to do away with 'lectures' entirely". That is exactly what it means. Well constructed activities with minimal teacher - but judicious - direction can be a powerful teaching tool but, too often, it is mostly a waste of time. Many students don't "get it" - I have observed classrooms where even the "teacher" didn't - and, even when small learning groups do discover some solutions to a particular proposed problem, consummating the process with making sure that students aren't encouraged to believe that "all solutions are created equal"is frowned upon. Exposure to well-known efficient solution or solutions that are expected to be learned and known thereafter is an important - actually, essential - step in developing effective mathematical problem solvers.
Polya's great little books on problem solving are often misinterpreted as being directed toward our heterogeneous classrooms with the blind leading the blind. Most of the time - maybe all the time - he was focused on developing the natural talent of naturally talented students - often at the graduate level at Stanford, no less. When Yale's Begle held the first organizational meetings for establishing the SMSG (School Mathematics Study Group) initiating the New Math at Stanford, Polya was encouraged to participate, did attend some of their meetings, and then spoke forcefully against what they were preparing to do and then did. Although great for a few with some general improvements in curriculum and instruction for the many, it was a dismal failure in general. That was a good thing, Polya was right and the New Math died before it ever took hold. Regrettably, the New New Math of recent decades refuses to die in spite of its demonstrated and profound failure.
At 12:22 AM 2/7/2013, GS Chandy wrote: >With reference to "Polya's Ten Commandments for Teachers" (see Jerry >Becker's post pasted below), it strikes me that several of the >'commandments' are in fact a strong endorsement of teaching via the >'discovery mode' rather than via the pure 'lecture mode'. > >This does NOT mean that we have to do away with 'lectures' entirely >- merely that there is much more to effective learning by students >than simply 'lecturing' them in the hope that one is >'teaching'. Lectures should be recognized as constituting only a >minor part of the 'teaching+learning' dyad. > >GSC >Jerry Becker posted Feb 6, 2013 4:20 AM: > > *********************************** > > From Mathematics for Teaching website. See > > http://math4teaching.com/2012/07/28/g-polyas-ten-comma > > ndments-for-teachers/ > > *********************************** > > George Polya's Ten Commandments for Teachers > > > > [Posted by Erlina Ronda] > > > > This is George Polya's 10 commandments for teachers: > > > > 1. Be interested in your subject. > > > > 2. Know your subject.B > > 3. Know about the ways of learning: The best way to > > learn anything is > > to discover it by yourself. > > > > 4. Try to read the faces of your students, try to see > > their > > expectations and difficulties, put yourself in their > > place. > > > > 5. Give them not only information, but "know-how", > > attitudes of mind, > > the habit of methodical work. > > > > 6. Let them learn guessing. > > > > 7. Let them learn proving. > > > > 8. Look out for such features of the problem at hand > > as may be useful > > in solving the problems to come - try to disclose the > > general pattern > > that lies behind the present concrete situation. > > > > 9. Do not give away your whole secret at once-let the > > students guess > > before you tell it-let them find out by themselves as > > much as > > feasible. > > > > 10. Suggest it, do not force it down your throats. > > > > > > I got this from the plenary talk of Bernard Hodgson > > titled Whither > > the mathematics/didactics interconnection? at ICME > > 12, Korea, where > > he highlighted the important contribution of George > > Polya in making > > stronger the interconnection between mathematics and > > didactics and > > between mathematicians and mathematics educators. > > > > If it's too hard to commit the 10 commandments to > > memory then just > > remember the two rules below which is also from > > Polya. Combine it > > with Four Freedoms in the Classroom and you are all > > set. > > > > The first rule of teaching is to know what you are > > supposed to teach. > > > > The second rule of teaching is to know a little more > > than you are > > supposed to teach. > > > > ------------------- > > > > The Four Freedoms in the Classroom > > [http://math4teaching.com/2012/02/01/the-four-freedoms > > -in-the-classroom/] > > > > [Posted by Erlina Ronda] > > > > You will find that by providing the following > > freedoms in your > > classroom an improved learning environment will be > > created. > > > > The Freedom to Make Mistakes > > > > Help your students to approach the acquisition of > > knowledge with > > confidence. We all learn through our mistakes. Listen > > to and observe > > your students and encourage them to explain or > > demonstrate why they > > THINK what they do. Support them whenever they > > genuinely participate > > in the learning process. If your class is afraid to > > make mistakes > > they will never reach their potential. > > > > The Freedom to Ask Questions > > > > Remember that the questions students ask not only > > help us to assess > > where they are, but assist us to evaluate our own > > ability to foster > > learning. A student, having made an honest effort, > > must be encouraged > > to seek help. (There is no value in each of us > > re-inventing the > > wheel!). The strategy we adopt then should depend > > upon the student > > and the question but should never make the child feel > > that the > > question should never have been asked. > > > > The Freedom to Think for Oneself > > > > Encourage your class to reach their own solutions. Do > > not stifle > > thought by providing polished algorithms before > > allowing each student > > the opportunity of experiencing the rewarding > > satisfaction of > > achieving a solution, unaided. Once, we know that we > > can achieve, we > > may also appreciate seeing how others reached the > > same goal. SET THE > > CHILDREN FREE TO THINK. > > > > The Freedom to Choose their Own Method of Solution > > > > Allow each student to select his own path and you > > will be helping her > > to realize the importance of thinking about the > > subject rather than > > trying to remember. > > > > ******************************************* > > -- > > Jerry P. Becker > > Dept. of Curriculum & Instruction > > Southern Illinois University > > 625 Wham Drive > > Mail Code 4610 > > Carbondale, IL 62901-4610 > > Phone: (618) 453-4241 [O] > > (618) 457-8903 [H] > > Fax: (618) 453-4244 > > E-mail: firstname.lastname@example.org