> 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?"
This is a good example of why I tend to be indifferent at best to the cheerleading I come across for most math education ideas -- the ideas are NOT new to most anyone who has taught at least a few years and who isn't totally incompetent in subject matter knowledge. Also, many of the same ideas are recycled and offered as if they are somehow "new", despite the fact that in the case of the many, many similar previous phrase-driven strategies over the previous 100+ years one can find the same things being advocated.
Now I don't mean to disparage the suggestions themselves. Indeed, these are great suggestions and every teacher should strive to follow them. What I do have issue with is in giving these suggestions a name such as "discovery learning", with the idea of marketing the suggestions as something new and profound.
> 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?
I'd be more specific. Also, I'd include other versions, such as: Let m and n be positive integers. If m is even and n is even, is m^n always even, always odd, or sometimes even and sometimes odd? Repeat with the three other variations -- "m is odd and n is even", "m is even and n is odd", "m is odd and n is odd". The reason I'd include other things (in a book or article for teachers) is that "everyone" knows about sums and products of odd/even integers, so to be useful one would want to include something teachers might find novel.
> c. Why is 1.999....=2?
A good discussion idea, but I suspect many mathematicians would cringe at this being used (teachers ignoring subtle limiting issues involved).
> d. What is the maximum number of pieces of pie if a round > pie is divided by seven cuts?
Probably need to define "cuts" . . .
> e. How are the slope and y-intercept of a line related to > the equation of a line?
There is not a "the equation" of a line. Again (as in 'b'), why not introduce something slightly foreign to the average teacher, such as: "Discuss the coordinate-geometrical significance of 'a' and 'b' in the equation x/a + y/b = 1?"
> f. How is the perimeter of a right triangle related to > its area?
This needs to be worded a bit more carefully to avoid comparing apples with oranges. For example, 3 cm is neither greater than, less than, nor equal to 6 cm^2.
> g. What is the number of subsets of a finite set? (The book > does not say "finite" but should.)
This needs to be more precisely worded, since there is not a "the number" for the question asked. Also, what if someone answers "finite"?
> h. How can the formulas for areas of geometric figures > be related to the area of a rectangle?
Does "geometric figure" mean polygon? Is a Sierpinski triangle a geometric figure? Is a parabola a geometric figure? Is a line (or a line segment) a geometric figure?
Also, the question should be worded so that "formulas" doesn't appear. (Formulas aren't unique, formulas aren't the essential idea being investigated, ...)
> i. What equality properties apply to inequalities? For > those that do not, can you find conditions for which > these equality properties apply to inequalities?
How about some specific examples a teacher could use, such as discussing whether "if x \= y and y \= z, then x \= z" is always true.
> 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."
This is something I've brought up in classes since the first class I taught (back in 1983). I've not really encountered all that many students who believed they needed to find the LCD (but I have encountered some), but I have often heard teachers SAY you need to find the LCD, even when I'm sure they know you don't, but the teachers still say "need", much in the same way that students will write on paper things like 2x + 3 = 7 = 2x = 4 = x = 2 while knowing (if you ask them to carefully examine the equalities) that these equalities are inconsistent.
> 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.
This is certainly something I've had trouble with. It doesn't take reading books or having someone tell you about it (unless it's someone observing a class when you do it), it takes practicing not explaining too much and maybe even writing little notes to yourself to take with you to class (e.g. sticky notes with large capital-lettered phrases such as "DON'T TELL STUDENTS EVERYTHING!" stuck in the appropriate places in your handwritten class notes).