Math Forum Discussions

Jonathan Groves

Posts: 2,068
From: Kaplan University, Argosy University, Florida Institute of Technology
Registered: 8/18/05

Re: Mathematician
Posted: Oct 10, 2010 11:32 AM

Plain Text

On 10/8/2010 at 5:02 pm, Dave L. Renfro wrote:

> 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.

Dave,

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.

Jonathan Groves