2 Building symbol sense
Techno logy can be used in many ways to help students make
sense of symbols and
symbolic expressions. We give two brief examples.
Example. Unpacking symbolic expressions
One approach to making sense of the densely packed symbolic expressions students
encounter at the tertiary level is to use technology to "unpack" them and
investigate
their parts. (This is the essence of analysis.)
For the infinite harmonic series discussed above, for instance, the Maple
command
> s .= n -> evalf ( sum(1/k, k=1..n) ) .
defines the partial sum function s(n). Evaluating s(n) is
now easy for specific inputs n.
> s(10), s(20), s(30), s(40), s(50), s(60),
2.929, 3.598, 3.995, 4.279, 4.499, 4.680
The results show s(n) increasing, though slowly, with n.
That's a good start, but it leaves wide open the deeper question of convergence
or divergence. Further experimentation (and perhaps some hints) might eventually
suggest successively doubling inputs to s.
> s(10), s(20), s(40), s(80), s(160), s(320),
2.929, 3.598, 4.279, 4.965, 5.656, 6.347
The situation is now much clearer, successive doubling of
n causes essentially linear
increase in s(n) (by about 0.7) each time, and analogy with logarithmic growth
begins
to appear.
Example. Looking closely at squares
Another technology-aided approach to giving meaning to
symbols is to look very closely,
from several viewpoints, at apparently familiar symbolic objects. Almost every
American
college student "knows," for instance, that
a fact that, while undeniably true, is almost entirely
valueless without some deeper
sense of what the symbolized objects and operations really mean. Here, too,
students
might use technology to help de-crypt the symbols, perhaps by plotting
appropriate
functions, zooming in on graphs , or calculating related derivatives.
For variety, let me suggest another approach to looking much more "structurally"
than usual at the squaring function, this time beginning from a numerical
perspective .
What structure should a student see in the following list?
1 4 9 16 25 36 49 64 81 100 121 144 169 196 225 ...
The first answer is obvious-even the dullest student with
any recent memory of mental
or paper-and-pencil arithmetic sees the squares of successive integers.
So far so good, but let's keep looking. Taking successive differences in the
first list
reveals the simpler pattern of successive odd numbers.
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31 33 ...
Taking differences again gives an even simpler list.
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 ...
And so on. (Taking differences soon becomes less
interesting.)
Starting from these basic structural ideas, students can move in many possible
directions to explore-and perhaps solve-new but related structural questions.
What happens if our original list arises by sampling not
the basic quadratic function
f(n) = n2, but some other quadratic, say g(n) = n2 + 2n + 3? Are the
first differences still in arithmetic progression? Are the second differences
still
constant?
How do differences behave if the original list samples
the cubic function n3? Or
the exponential function 2n?
How do differences behave if the original list samples
the cubic function n3? Or
the exponential function 2n?
What happens if we move in the "opposite" direction,
finding successive sums
rather than differences? How does the "constant of summation" affect the
results?
Quite different structural questions could also be
explored. Students might notice,
for example, that successive squares alternate between exact multiples of 4 and
numbers
of the form 4k + 1. Or they might see pattern in the last decimal digits of
successive
squares.
0 1 4 9 6 5 6 9 4 1 0 1 4 9 6 5 6 9 4 1 0 1 4 ...
And so on, perhaps, into areas of modular arithmetic.
3 Beyond symbolics. exploring structures
We have argued that technology can help students build
better symbol sense for tertiary
mathematics. But why is symbol sense worth acquiring? Where does it lead?
We should acknowledge first that, in actual practice and despite the presence of
technology that could enable better things, a lot of tertiary mathematics still
boils
down to performing symbolic algorithms. As Ralston [2] says about college
calculus in
the USA.
. . . despite so-called calculus reform, the aim of most
college calculus courses
still seems to be to create a student-machine in which functions are fed to
its maw and derivatives and integrals emerge at the other end.
In mathematical reality, of course, tertiary mathematics
is about much more than
algorithm performance, and technology may help us refocus attention where it
belongs.
The calculus for instance, can be about mathematical objects and ideas-function,
limit, derivative, differential equation, integral, infinite series-not just
about formal
calculations with these objects.
In my opinion, the true Holy Grail at the tertiary level
is mathematical structure.
Some italics may be in order .
Understanding basic mathematics profoundly means
proficiency at detecting,
recognizing, and exploiting structure, and at drawing useful connections
among different structures.
The preceding example illustrates most of these points,
the basic structure of successive
squares, once recognized and slightly manipulated, leads naturally to simpler or
more
complex structures, and to new, deeper, and more interesting questions.
There is nothing new about this focus on structure,
mathematics is often described,
in varying language, as the science of pattern. What may need emphasis, though,
is
the special importance of mathematical structure in tertiary-level mathematics,
where
students meet new structures, and relations among them, in potentially
bewildering
variety, ranging from abelian groups to planar graphs.
Quadratic polynomials. symbols reinforcing structure
We close with a final illustration of a pedagogical
strategy-looking closely, perhaps
using technology, at familiar objects-that focuses attention both on symbolics
and on
structures.
Quadratic polynomials are an excellent source of
simple-but not trivial-examples,
students should know them intimately. In calculus, quadratic polynomials
illustrate several
important notions, including local linearity and "quadraticity", global
nonlinearity,
the meaning of the second derivatives, and geometric convexity. Quadratics also
illustrate
the possibility and the advantage of algebraic factoring, and more generally of
the value of having convenient algebra formulas. One sees, easily, for instance,
that the
vertex of a quadratic polynomial lies midway between its roots, and that one
root of a
quadratic polynomial with rational coefficients is quadratic if and only if the
other root
is rational.
Example. finding Pythagorean triples
The rational root property just mentioned has an
interesting and perhaps unexpected
"structural" consequence. there are infinitely many Pythagorean triples, and
they correspond
in a natural way to rational points on the unit circle.
The idea is as follows. Given a nontrivial Pythagorean
triple (a, b, c) of integers,
with a2 + b2 = c2, we divide both sides by c2. Renaming x = a/c and y = b/c
gives a
rational point (x, y) on the unit circle
Since the process can (essentially) be reversed, hunting
for Pythagorean triples amounts
to finding rational points on the unit circle. A few solutions are obvious , one
is the
point (0, 1).
An ingenious way of finding other (indeed, essentially
all) rational points is to find
intersections of the unit circle with lines through (0, 1) that have rational
coefficients.
Each such line that is not vertical has an equation of the form the line y = mx
+ 1,
where the slope m is a rational number. Such a line intersects the unit circle
at a
simultaneous solution of
A little algebraic work (by hand or even by head) now
produces the one-variable
quadratic equation
Because the coefficients and the root x = 0 are all
rational numbers, so is the other
root. Since every line through (0, 1) with rational slope cuts the unit circle
in a rational
point we see that infinitely many rational points exist. A little more work
shows,
moreover, that our recipe produces all rational points. Combining symbols ,
algebra,
and various mathematical structures, we have solved a modest but nontrivial
problem-
and suggested methods of attack on many others.
4 Conclusion
As modern technology handles more and more of the
algorithmic aspects of mathematics,
even at the tertiary level, the importance of higher level mathematical
thinking-
symbol sense and facility with mathematical structure-become relatively more
important.
Used properly, high-level computing technology can help tertiary students see
beyond the mechanics toward what matters most. mathematical structure.
