[Prev][Next][Index][Thread]
(Radically) revamping the early grades curriculum
Okay, since you poor well-meaning people have all mailed me expressing
interest in this, I guess I have to start actually putting it together.
Here's at least a first approximation. This being only an approximation to
an approximation, I am sure that we will all treat is as such and point out
its strengths and weaknesses (those won't need to much pointing out) rather
than saying something obvious like "this won't work", which is almost
always true and almost always irrelevant about first approximtions.
Perhaps I should start out with some weaknesses I see:
1. Stakeholders (school board, parents, community and business leaders)
are going to rebel.
2. Many teachers are not going to be happy with it--it will be a lot
harder, in some sense than explaining a bunch of rote procedures.
3. Standardized tests are just going to have to change.
Many of you should probably read no further--if your blood is already
boiling at these statements, you'll have an ulcer by the end of this post.
Okay, now, assuming that the faint-hearted (read "sensible") have stopped
reading, here it comes.
(Previously referred to as "How I would change mathematics education (if
the schools were foolish enough to listen to me") This just deals with the
early grades, at this point.
The basic plan would be to restructure the early grades mathematics courses
completely. The motivating idea in these grades is going to be reasoning
abilities and problem solving. Kids are naturally perceptive, inquisitive,
and creative, and nearly every advance that has ever been made in
mathematics has been a result of applying these qualities (which, in the
current system, essentially have to be re-learned in graduate school).
Teaching meaningless (to them) procedures at this age at best completely
fails to make use of these attributes, and at worst stunts them.
I am not necessarily saying that arithmetic should be avoided, but that it
should only be taught where it comes up as a useful means of solving a
problem. Kids always want to know why they are learning what they are
learning, and there's usually not much to tell them when it comes to things
like long division or even subtraction. We often respond to this by coming
up with real world problems that use these skills, but even then we are
bending over backwards, since we came up with the application in order to
motivate the teaching of a procedure that we had other motivations for
teaching (like "It's on the standardized test.").
Instead of teaching unmotivated subjects, math could be an activity (and
reflection) time. The activities would be chosen from various areas of
mathematics, but motivated in ways that make sense to kids.
Here's an example from graph theory, due to Mike Fellows and Nancy Casey.
Draw a graph (network) on the floor in masking tape, and tell the kids that
the edges (connecting lines) of the graph are city streets and the vertices
(nodes, or intersection points) are street corners. Have them put down
large colored markers on the street corners to indicate the locations of
ice cream stands. Their goal is to do this in such a way that no matter
where you are in the city, you can get to an ice cream stand within one
block. (This is called finding a dominating set for the graph, but don't
tell them that.)
That's not hard to do--but once they have accomplished that, take one of
the ice cream stands away and see if they can rearrange the remaining ones
in such a way that the conditions are still satisfied. What _is_ hard to
do is to come up with a dominating set which is minimal, that is, for a
given graph, find the smallest number of ice cream stands necessary. There
are mathematicians and computer scientists working on this problem as we
type, and no one knows a fast way to do it that works in all cases.
However, it makes for a very challenging, yet fun and naturally
collaborative activity for early grades students (yes, it's been tried, and
they eat it up). In fact, kids have been able to rediscover, on their own,
algorithms originally proposed by research mathematicians.
Now, what's the point of all this? Well, there are lots of advantages to
this approach (at least in theory). For one, the questions we're asking
kids to think about are natural and interesting to them. (In my uninformed
opinion, that's pretty much enough all by itself...) Second, they are
learning about actual research questions that people have jobs trying to
solve. Third, they are learning about reasoning, developing algorithms
(after all, if you just want something that can perform algorithms, the
price has come way down on hand-held prgrammable machines), collaboration,
and proof, which will be invaluable to them in many aspects of their lives.
Now, a lot of arithmetic will probably come up naturally. Some graph
theoretic problems, for example, come down to whether or not there are an
even or odd number of streets leading from each street corner, for example
("Can you take a walk in the city in a way that you walk along all the
streets, but don't walk down any street twice?") I believe that if
arithmetic comes up in these contexts, kids will be a lot more willing to
learn about it (and at first they could just use lookup tables for addition
and multiplication--they'll learn a lot more from noticing patterns on
those tables than they will from memorizing them).
'Yeah, but when do they ever learn to [insert mathematical operation]?"
Well, there are a lot of possible answers to this question. One is,
"whenever there is a reason for them to" as above. It would be possible to
make a lot of these operations come up naturally. You may remember that
the people that discovered these operations did it because something they
were already doing motivated it (as opposed to something that they might
need to do in some contrived situation later in life).
Another possible answer is, "They can learn [insert operation] in two weeks
max a little later on in their educational careers.". Let me illustrate by
teaching you how to subtract a three digit number from a larger one.
Here's how it works:
825
-568
-----
take the bottom number, and rewrite it a digit at a time in
nine-complements. (That means write the difference between it and nine in
it's place.) Take away the minus sign while you're at it.
825
431
---
now add them
825
+431
----
1256
and take that pesky one off of the front of the sum and add it to the rest
256
+ 1
----
257
voila. 825 - 568 = 257. (You should be able to prove that this works (why
don't we get into groups and figure it out?), and generalize it to more
digits (hint--it's easy) and figure out what to do if the bottom number is
bigger.)
Now, if I can teach you how to subtract multidigit numbers in 15 minutes,
suppose, worst case, it takes you three weeks to show a fifth grader how to
do it. We've still saved a boatload of time on subtraction compared to the
current system.
Okay, well, I'm just going to send this as a rough rough draft, since my
ride is leaving. My basic point is that we could come up with a coherent
set of activities for young kids to do that would
a) make sense to them
b) be fun for them to do
c) teach them logic and reasoning skills
d) be naturally collaborative
e) address a host of learning styles
and possibly
f) fail to have them doing the basic four operations by fifth grade.
But, gee, I mean,
g) I'll bet you could make up the "lost time" on basic operations in a
fraction of the time we currently spend on it
My reasons for believeing this? You would be working with a bunch of kids
that liked math. They would be old enough to appreciate the usefulness of
being able to do these operations. You could build on all the activities
that they had been participating in so far--the basic operations are very
useful in most mathematical situations, so you'll have a whole bank of
things they've seen where the skills at hand could have been (or were)
helpful. They would have learned about logic and reason and how to "think
mathematically" and might be able to come up with their own algorithms.
Of course, their parents are going nuts at this point, seeing that their
kids haven't been forced down the same hellish regiment of addition of
fractions and multiplying decimals thay they had to go through. In this
respect, the plan is almost wholly unworkable. But if you could put
something together that was coherent and well thought out enough, and
demonstrated its usefulness with a district or group (perhaps a home
schooling group or a private school) that was willing to experiment a
little, perhaps some of those kinds of barriers could be overcome.
The truth is, we don't teach mathematics to kids. We teach arithmetic. I
think kids would like mathematics if they ever got to see it. It is a fact
that kids who see arithmetic taught for arithmetic's sake are often
intelligent enough to see the futility of the exercises, and thus at this
crucial and impressionable age we steal from them all that is fun,
wonderful, and joyous about math and replace it will a pale, lifeless
parody of the subject. It would seem that at one point we would wake up
and see that _the subject matter_ needs to change in a fundamental way.
Why do we persist in believing that what was prescribed in the 1800's as an
appropriate mathematics curriculum is appropriate today? (I have a few
possible answers to this, but let's save those for another time.)
More to come!
mike
Michael South
Shodor Education Foundation, Inc.
628 Gary St
Durham, NC 27703
(919) 688-2176 (voice)
(919) 688-2697 (fax)
http://www.shodor.org
Follow-Ups: