## ABSTRACT

Mathematical symbolism generally-and symbolic algebra in
particular -is among mathematics'

most powerful intellectual and practical tools. Knowing mathematics well enough
to

use it effectively requires a degree of comfort and ease with basic symbolics.
Helping students

acquire symbolic fluency and intuition has traditionally been an important, and
sometimes

daunting, goal of mathematics education. Cheap, convenient, and widely available
technologies

can now handle a good share of the standard symbolic ope rations of undergraduate

mathematics. differentiation, integration, solution of certain DEs, factoring
and expansion

in many forms, and so on. Does it follow that teaching these topics, and even
some of the

techniques, is now a waste of time?

The short answer is "no." On the contrary, as machines do
more and more lower-level

symbolic operations, higher-level thinking and deeper understanding of what is
really happening

become more, not less, important. Numerical computing has not made numerical

viewpoints obsolete, neither will computer algebra render symbolic mathematics
obsolete.

The key question is how to help students develop that bred-in-the-bone "symbol
sense" that

all mathematicians seem to have. What really matters is that students use
mathematical

symbolism effectively to pose worthwhile problems in tractable forms. Once
properly posed,

such problems are well on the way to solution, very often with the help of
technology. The

longer answer, which I'll explore in the paper, concerns choosing mathematical
content and

pedagogical strategies wisely in light of today's technology.

## Introduction

What does it mean to know and do mathematics effectively
at the tertiary level? How

do the answers reflect the present and future, when mathematical technology,
including

symbol-manipulating technology, is already widely available, and will probably
soon be

ubiquitous.

What should college-level students in particular know and
what should they be able

to do, in order to be mathematically educated in a technology-rich environment?
How

can we teachers help bring students to this kind of knowing?

I approach these questions from a perspective that's
fairly common in the United

States. I'm a generalist mathematician who teaches reasonably pure mathematics
to

North American college students. About one-third of my students in an average
class

intend, with varying degrees of intellectual seriousness and interest, to
complete a 4-

year mathematics major. Only a small minority (no more than 10%) of students
plan

postgraduate study in mathematics. A more typical student plans to work after
graduation

in a technical but not university-level academic job, such as software
engineering,

database management, or high school teaching.

I am a practitioner, not an expert researcher in
mathematics education, and so will

not pre sume to offer advice on the education research agenda or how it should be
carried

out. What I do hope to contribute is a teacherly and mathematical perspective on
what

content, techniques, and ideas related to symbolic mathematics I think are
mathematically

most important to a modern tertiary student clientele, and how I think students

can be helped-often with technological assistance-to acquire these advantages.

## 1 The technology background

Disputes over educational uses of mathematical technology
have been around as long

as the technology itself. Years ago one heard the "desert island" argument from
opponents

of instructional technology. Students who use calculators for school arithmetic

might suffer disproportionately if later shipwrecked on low-tech islands. This
argument

is seldom heard anymore, it was killed either by the rising availability of
cheap calculators

or by the worldwide decline in passenger marine travel. In any event, there's no

doubt that many students can now keep readily and affordably at hand the
technology

needed to perform a huge share of the algorithms usually encountered even in
tertiary

mathematics. It's well known, for instance, that the TI-89 handles integrals,
derivatives,

partial fractions, and much more. But did you know that the TI-89 can also

handle many of the residue calculations given as exercises in complex analysis
texts?

With powerful computer algebra systems such as Maple and Mathematica also
becoming

more affordable and available to students, the technology background has shifted

markedly.

With the desert island argument no longer operable,
technology opponents resort to

other arguments. Technology takes too much time to learn, students can't think
in the

presence of machines, technology use is just a post-modern cover for dumbing
courses

down-another nail in the co n of civilization. I find these arguments unconvincing
at

best and dishonest at worst. For instance, how much do you think your students
really

struggle with technology as they pirate music MP3s from the Web? The dumbing-down

argument may be worst of all. it is simple "calumny" (as Tony Ralston put it in
[2])

to equate technology-based reform with lowered intellectual standards or
expectations.

This is not to deny, on the other hand, the existence of
good, important, and (in

my opinion) still open questions surrounding pedagogical uses of technology.
Owning a

calculator that "knows" how to expand rational functions in partial fractions
does not

necessarily obviate the need to understand something of the idea-and perhaps
even of

the process-by hand or by head.

At the school level, arguments over technology use often
touch on the role and

importance of paper and pencil arithmetic (PPA) in technology-rich environments.
At

one extreme are calculator abolitionists, asserting (with perhaps more vehemence
than

evidence) that calculator use is somehow inimical to reason-children can be
taught

to push buttons or to think, but not both. At the opposite end of the spectrum
are

other abolitionist, such as Tony Ralston, who advocate abolition not of
calculators but

of PPA as an explicit goal of K-12 mathematics education. (In his eloquent paper
[2],

Ralston also recommends greatly increased emphasis on mental arithmetic, and
perhaps

also on mental algebra.)

Coexisting with this clash of opinions is some basic
agreement on ultimate goals.

In the end, most of us care far more about whether students can pose and solve
novel

and challenging problems than about what technology might be used along the way.

What counts most is effective mathematical thinking, which comprises such
elements as

basic symbol sense and facility with mathematical structures. What is mainly at
issue,

I believe, is whether technology can help or must hurt the cause of teaching
students

to think well mathematically.

## Number sense and symbol sense

At the elementary level, what may really matter less than
PPA facility and speed is

number sense, that intuition for numbers that includes such things as an ability
to

estimate magnitudes, an eye for obviously wrong answers, and an instinct for
choosing

(rather than necessarily performing) the arithmetic operation needed to solve a
given

problem.

At the secondary and tertiary levels, the mathematical
symbols under study become

much more general than numerals, and the degree of abstraction tends to rise as
students

progress. The objects symbols stand for in more advanced mathematics might

be unknown numerical quantities, functions, operators, spaces of various sorts,
or even

more abstract objects. At these higher levels of study the analogue of number
sense is

"symbol sense," as defined by Arcavi[1] and others. Symbol sense is harder to
define

and delimit than number sense-appropriately enough, given the greater
mathematical

depth and breadth of, say, polynomial algebra as compared to integer arithmetic.

(Arcavi lists at least seven aspects of symbol sense-only one of which involves
actual

symbolic manipulation.) Arcavi links symbol sense closely to algebra in
particular-

acquisition of symbol sense is the proper goal of teaching algebra. For example,
a

student with good symbol sense should sense that something is amiss with an
"equation"

like

She should also know-without calculation-that of

one is right and one is wrong, and that equations of the
form

something and
something

can be arranged to hold, while

something

is unpromising.

In this paper I want to take very broad views of both "symbol sense" and
"algebra."

By symbol sense I mean a very general ability to extract mathematical meaning
and

structure from symbols, to encode meaning effciently in symbols, and to
manipulate

symbols effectively to discover new mathematical meaning and structure. By
"algebra"

I mean symbolic operations in general, including not only algebra in the
classical sense

but also such things as formal differentiation and expansion in power series.

Definitions may differ, but whatever one means exactly by
"symbol sense", it's clear

that tertiary-level mathematics takes a lot of it. Tertiary mathematics is a
symbolrich

domain, and doing mathematics successfully at this level requires considerable

comfort and sophistication with symbols. Above all, students need a clear sense
of the

things symbols represent, and how to extract meaning and structural information
from

symbolic expressions.

Perhaps this should all go without saying-who could doubt
that symbols ought

to mean something to students? In practice, however, we've all seen students
floating

freely in the symbolic ether, blithely manipulating symbols but seldom touching
any

concrete mathematical ground. For example, many students struggle to make sense
of

a symbolic expression such as

This is hardly surprising, after all, the statement's
truth or falsity is far from obvious

to a newcomer to infinite series. But a more basic source of difficulty, I
believe, is that

the expression's meaning-let alone its truth or falsity-is highly compressed in
the

symbolic representation. "Unpacking" the symbolism to reveal meaning and
structure

can be a daunting challenge in its own right, as we see when students confuse or
con ate

the terms and the partial sums of an infinite series.

This brings me to my main questions.

1. How can we use technology-and symbol-manipulating
technology in particular-

to help students acquire symbol sense in the broad sense discussed above?

2. Where does better symbol sense lead? How can students use better symbol sense

to understand mathematics more profoundly?