I did look up what Johnson and Rising's book "Guidelines for Teaching Mathematics" (2nd edition) says about discovery learning, and the book says more about discovery learning than what I remembered. Here are some things the book does say about discovery learning. I will not list everything, but here are some of the big ideas I find that are worth mentioning.
1. Discovery learning is a difficult teaching method because it must be continually adapted to students' questions and comments and what progress they have made thus far. We cannot plan extensively for discovery learning just as we cannot plan extensively in advance for a discussion; we will not know how the discussion will go or where it will lead until we actually do it.
2. As I had mentioned earlier, discovery learning is not appropriate for all situations. One example they give here is trying to get students to discover a definition.
3. The idea of discovery learning is that it helps students find their own meaning in the mathematical concepts and their own connections of that concept with their previous knowledge and experiences. Previous experiences happen to be one reason why our thinking about a concept makes perfect sense to ourselves and other students but makes no sense to someone else; that baffled student might not have had those experiences to make that explanation meaningful to him or her.
4. Some ideas for prompting students to think more deeply (examples taken straight from this book):
"Give me another example." "Do you believe that, Bill?" "How do you know that?" "Can anyone find a case for which John's rule does not work?" "That seems to work. Will it always?" "Have we forgotten any cases?"
5. The book points out some cautions to discovery learning (as quoted from the book):
a. Be sure that the correct generalizations are the end result. b. Do not expect everybody to discover every generalization. c. Do not plan to discover all the ideas of your course. Discovery of some ideas is too inefficient. Sometimes students do not need an intuitive, emprical, discovery approach to understand an idea. d. Expect discoveries to take time. e. Do not expect the generalization to be verbalized as soon as it is discovered. f. Avoid overstructuring experiences. g. Avoid jumping to conclusions on the basis of too few samples. h. Do not be negative, critical, or unreceptive to unusual or off-beat questions or suggestions. However, incorrect responses must not be accepted as true; and disruptive, nonessential explorations must be eliminated. Students should know that their status is not threatened by incorrect answers. i. Keep the student aware of the progress he is making. j. If possible, have crucial ideas "discovered" repeatedly or by different methods. k. Finally, each student must recognize why his discoveries are significant and how the ideas are incorporated in the structure involved.
6. The book gives some examples of ideas that students can try to discover for themselves:
a. The difference between the prime numbers 5 and 2 is 3. Why do no other prime numbers have this property? Here is a related one I have thought of: 3, 5, 7 are three consecutive odd natural numbers that are also prime numbers. Are there any other examples of three consecutive odd natural numbers where all three are prime numbers? If not, then why is this example the only one possible?
b. What do we know about sums and products of odd integers?
c. Why is 1.999....=2?
d. What is the maximum number of pieces of pie if a round pie is divided by seven cuts?
e. How are the slope and y-intercept of a line related to the equation of a line?
f. How is the perimeter of a right triangle related to its area?
g. What is the number of subsets of a finite set? (The book does not say "finite" but should.)
h. How can the formulas for areas of geometric figures be related to the area of a rectangle?
i. What equality properties apply to inequalities? For those that do not, can you find conditions for which these equality properties apply to inequalities?
Here is one I thought of:
j. Must we use the LCD to add or subtract fractions? Or will any common denominator work? If any common denominator works, can you see why? I like this one because I have seen many students who believe that adding or subtracting fractions using a common denominator besides the least common one is wrong simply because "that's not how I was taught to do that."
A comment to Mike Dougherty: In some sense, asking students to discover the reasoning and logic behind mathematics is discovery teaching. Sometimes this term refers to getting students to discover ideas and the underlying logic for themselves such as discovering and proving a theorem, something similar to what a mathematician has to do when developing a theory. Other times it can refer to having students fill in the details of the reasoning after the teacher has presented the big ideas and some outline of the reasoning with the details omitted so that students can try to fill those in for themselves. In these cases, the student is not asked to discover theorems but is asked instead to discover the proofs of them. It is clear from the examples given above from the book I had cited that this book uses the word "discovery learning" or "discovery teaching" in both of these senses.
I don't know if we explain too much, but I often question if we, including myself, explain too much too quickly before giving the students chances to think about and see these ideas for themselves. That is, if we explain too much up front, then we don't give students many chances to think for themselves. We also give students the impression that it is okay to take our word for it, especially permanently rather than just temporarily for convenience, even if they haven't the foggiest idea of why that is true. I don't see a problem with a student who wants to take our word for it for the time being, especially if they need to use that result immediately, if the student is willing to try later to see why that is true. Of course, if the proof of the result is beyond the scope of the course, then that is a completely different matter.
On 10/8/2010 at 11:21 am, Michael Dougherty wrote:
> 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. > > > 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