> Jonathan Groves wrote (in part): > > http://mathforum.org/kb/message.jspa?messageID=7234498 > > > 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.
I agree that teachers reducing such strategies to more than cheerleading of some sort is not a good idea. Instead, the best way the students can get any benefit from these strategies is that teachers use them to encourage students to think beyond the surface level, beyond what is obvious, beyond the sort of answers that don't reveal any true understanding. For instance, I don't like students giving me answers such as "the recipe becomes smaller" when they are asked how cutting a recipe to 1/3 batch affects the amounts of ingredients in this new partial batch. Yes, that is true but requires no understanding of how to calculate 1/3 of a quantity. This week's discussion question at Argosy University asks students to read and interpret a graph about mean global temperatures that reveal that global warming is taking place (the graph is hard to read and the information provided is misleading, but that is another issue I won't go into detail about in this post). Several students, when asked what the graph says, had said no more than "global warming is occurring." Yes, that is true, but something like that is clear and does not require that the student know how to read the fine details of a graph or how to spot other trends. In fact, those students answers' on the question asking them to give values for specific years had showed me that they are struggling to read the details of this graph. In particular, I have already seen multiple students not reading the scale correctly. The temperature scale ranges from -0.8 to 0.8 Celsius, and the temperatures are deviations from some fixed mean global temperature that is considered "normal," but multiple students are giving temperatures of -2, 6, 8, etc. That is, they are not seeing the decimal points in the numbers marked on the temperature scale and thus are seeing temperatures range from -8 to 8 Celsius. In short, encouraging students is essential, but encouragement without any guidance leads nowhere. Yet too much guidance does not leave the student anything for him or her to think about.
It is true that some educators are claiming that such ideas such as discovery learning are new. For some (maybe many?) teachers, these ideas are new to them, so maybe that is why these educators seem to continue getting away with that. Other times I believe that some accuse educators unjustly of doing so in that the accusers read that meaning into the educators' messages about alternate teaching and learning styles. But even in those cases it can be hard to say when the messages do not make such claims because we cannot read the minds of the educators. In fact, Johnson and Rising not only do not make such claims, but they explictly say that the ideas of discovery learning have been around since at least the time of Socrates. In short, sometimes it is hard to tell for sure if messages about discovery learning and other forms of learning are being pressed as something new, but other times it is clear that the educators are not doing so, especially when they explictily mention that these ideas are not new.
> > 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? > > Nice. > > > 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.
Such questions like these you have proposed are better. I do take it that Johnson and Rising expect teachers to expand upon these questions by asking more specific questions to guide the students' thinking. They offer these ideas as starters for teachers who might need such ideas for questions to prompt discovery learning.
> > 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).
That is true, but then again if such complaints were fully justified, then we mathematicians would have to complain about the entire K-12 curriculum not offering any rigorous proofs. We would also have to complain that repeating decimal numbers should be removed from the K-12 curriculum for the same reason.
Alain Schremmer had mentioned a few weeks ago the mathematician Saunders MacLane complaining that Alain's students were not learning about epsilons and deltas in beginning calculus and equating their lack of learning about epsilons and deltas as lacking mathematical salvation. I do like the standards of proof that mathematicians have, of course, but many such proofs are not appropriate for beginning math students. Teaching a beginning calculus class this level of rigor is just as painful and helpful as beating your head against the wall. I agree with Alain: Logic can be used in math courses at these levels and that logic that is appropriate is common-sense logic. He is baffled by how so many take an "all or nothing" approach: Either we offer rigorous proofs (according to mathematicians' standards of proof), or we offer no proofs at all.
> > d. What is the maximum number of pieces of pie if a > round > > pie is divided by seven cuts? > > Probably need to define "cuts" . . .
Perhaps so. It is useful for students to work on some problems that are not so well defined. Many would gawk at that idea, but many real world problems are not well defined. For instance, real world data analysis problems in statistics are not well defined. We often must draw conclusions from data and other evidence without specifically guided by others as to what we need to say. I have volunteered to review some books for the MAA's library list, and they are not asking me specifically what to include in the review though I can look at some previous reviews to get some ideas.
> > 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?"
I like your approach. We could clarify the question by including various forms of equations of a line. All the equations of a line are equivalent (ignoring horizontal and vertical lines, that is) but do have different forms.
> > 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.
Good point. I have seen a similar kind of problem elsewhere before, but I don't remember where. That problem worded it so that it is clear we are looking for the relationship between the area in square units and the perimeter in units of length.
In fact, one point I'm trying to emphasize in my arithmetic book is that we cannot compare sizes of measurements unless they are expressed in the same units.
> > 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"?
I would ask for a precise relationship between the number of elements in the set and the number of subsets. If I were to use this question, I would hope that students discover that a set with n elements has 2^n subsets.
As for the answer "finite," I would have to encourage the student to be more specific in that we would like an expression or formula that tells us easily how many subsets we get without having to list them all.
> > 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, ...)
When I consider this question more clearly, I now realize that I am not sure what this means. Whatever the meaning, I'm sure that it is intended that the students use whichever definition of geometric figure is appropriate. The question could also be asked so that it narrows this exploration down to a specific class of geometric figures.
> > 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). > > Dave L. Renfro
In short, I am sure that Johnson and Rising intend for teachers to either word these questions as appropriate for their students or to ask them additional questions to guide their thinking.
Again, I believe it is good for students to work on some questions that are not so sharply worded because many real world questions are not like that. How to draw conclusions from data or other evidence is not necessarily something we can do by hard-and-fast rules. And doing so helps students to see that analyses that are not developed according to a teacher's guidelines are not necessarily incorrect analyses. Finally, we can help students to see that many questions cannot be sharply worded unless we already know exactly what we are looking for or what big idea or ideas we should note. If the question is one for which the answer is not known to anyone, then we often cannot word the question very sharply.