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OCR for page 17
Redesign from ~ Technological
Perspective
_
=_  _
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_ . ~
Of the many forces at work
that are changing the way
mathematics is learned, the
impact of technology is both
most urgent and most contro
versial. In less than two
decades society has moved
from primitive electronic cal
culators to desktop worksta
tions that are as powerful as
the largest computers of only
a few years ago. The unprece
dented magnitude and
speed of change in technol
ogy has created a consider
able degree of professional confusion and public alarm about
mathematics education,
Mathematicians and parents are divided on the wisdom of
early and widespread use of calculators before children have
mastered arithmetic by traditional means. Calculators are
seen as doing to arithmetic what many believe television has
clone to reading. Concern about further deterioration in basic
skills fuels a general fear of change that has produced a sus
tained public debate about the wisdom of calculator use,
Many who view mathematics as an ideal instrument to filter
students into professional and educational tracks believe that
calculators make this filtering less effectiveby enabling too
many students to score well despite weakness in traditional
skills. The image of a calculator as an inappropriate intellectu
a~ crutch is so deeply ingrained in many adult's mindsespe
cially among mathematicians and scientiststhat most col
lege entrance exams do not permit calculator use.
Advocates of early and unrestricted use of calculators
including virtually all mathematics educators argue on the
basis of student motivation, classroom realism, and needs of
OCR for page 17
18
Reshaping School Mathematics
the workplace. Calculators are seen as an effective tool to
transform the typical arithmetic lesson from worksheet
drucigery into motivated exploration. Appropriate use of cal
culators can enhance opportunity for chilciren to learn higher
orcler thinking skills without first mastering standard computa
tional algorithms. Incleecl, early informal experience with multi
ple approaches to arithmetic problems~including calculators,
fingers, ancl other devicesprovides a secure base for subse
quent study of stanclarcl techniques. Calculators enable cur
ricula to move beyond emphasis on mechanics to experience
with icleas.
In contrast to calculators, use of computers in the schools is
rather widely supported by the general public. However,
among mathematicians and classroom teachers, their use is
just as controversial as calculatorsancl for essentially the
same reasons. Many mathematicians and teachers fear That
time spent learning to use computer programswhether it be
programming languages such as Logo or Basic or packages
such as Mathematicais time subtracted from what they
believe to be the central lessons of mathematics: solving prob
lems with paper, pencil, ancl pure thought. Electronic aicis like
computers should be used by professionals to implement
quickly anci accurately what they have aireacly learnecl, not
usecl in education as an alternative to traclitional techniques
for developing skill ancl unclerstanding.
Despite the controversy, most mathematics educators who
have studied the issues ancl the evidence have concluded
that the potential benefit to mathematics education is enor
mous, well worth the extra effort ancl increased risk associated
with venturing into uncharted territory (Wi~f, 1982; Fey, 1984;
Hansen, 1984; Smith et al., 1988~. Increased use of technology
in mathematics education is inevitable, but wise use is not
automatic. Technology has more to offer education than just
hightech flash carcis. Effective use of calculators ancl com
puters requires objectives for mathematics education that are
aligned with the mathematical neecis of the information age.
New Opportunities
From fourfunction calculators to clesktop workstations,
computer technology is poised to make an extraordinary
impact on the content and presentation of mathematics edu
cation. Computing crevices will:
· Decrease the value of many manual skills traditionally
taught in the school mathematics curriculum;
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19
A Philosophy and Framework
· Increase the importance of many areas of mathematics
that now are rarely taught;
· Focus attention as much on problem formulation as on
problem solving;
· Make possible tools for teaching and learning of a sophis
tication still largely undreamed of by most mathematics
eclucators.
More than any other empow
erment of technology, comput
er graphics will, in particular,
totally transform the way math
ematics is used. Because the
United States still leacis the world
in most aspects of computer
technology, we have a unique
opportunity to grasp the poten
tial of this technology and use it to make dramatic improve
ments in mathematics education.
A growing volume of research supports appropriate use of
calculators in any grade. It is now clear that an understanding
of arithmetic can be developed with a curriculum that uses
estimation, mental arithmetic, and calculators, with reduced
instruction in manual calculation. ~ncleecl, mental arithmetic
may replace written methocis as the basic skill of our computer
age.
Since few arithmetic calculations are done most efficiently
using paper and pencil, the level of arithmetic skill that is the
current goal in most elementary school classrooms is far in
excess of what is neeclec] for tomorrow's society. IncJeecl, there
is some evidence that overemphasis on manual skills hinders
the chilcl's learning of when and how to use them. Too often,
skill rather than meaning becomes the message.
Thus, any reform of school mathematics must entail a major
reduction in the time spent on teaching traditional arithmetic
skills. Technological developments suggest strongly that even
those aspects of the secondary school curriculum that are ori
entecl mainly to development of algebraic skills such as poly
nomial arithmetic no longer serve a compelling purpose. In a
computer age, facility in these skills is not an absolute prerequi
site either to the use of mathematics or to further study in
mathematically based fields.
Use a calculator to find three different numbers
whose product is 7429. How many different
answers can you find? Write a paragraph
explaining what you did, why you did it, and
how well it worked.
New Priorities
Reclucing priority on development of routine skills will allow a
variety of clesirab~e consequences. There will be more time to
Exploring
Numbers
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20
Reshaping School Mathematics
develop understanding of processes and reasoning that lie at
the heart of mathematical problem solving (Conference
Board of the Mathematical Sciences (CBMS), 19831. Indeed,
enabling students to solve a variety of problems is one of the
main purposes of school mathematics education.
By reducing emphasis on manual skills, it will be easier to
develop a curriculum that will allow all students some level of
mathematical accomplishment while retaining the interest
and enthusiasm of the more able students. The current
emphasis on manual arithmetic skills, which any observant stu
dent knows are seldom used outside school, contributes to dis
taste for mathematics in many able students. For slower stu
denTs who fail to achieve quick mastery of arithmetic skills,
there is no path to future success in mathematics. Less stress on
skills will make possible an elementary school mathematics
curriculum in which lack of success in one area will not neces
sarily preclude success in another.
In such a curriculum, it will be possible to emphasize approx
imation and estimation, topics that play essential roles in many
areas of mathematics (Schoen, 19861. Is it more important for
a student to be able to multiply 2507 x 4131 precisely or to be
able to say that the result is about 10 million? Often, the
approximate answer is not only sufficient, but it also provides
more insight than the exact answer. Moreover, the approxi
mate answer provides a quick check on the result of any
exact proceclure, whether by a hand moving a pencil or by
fingers pushing buttons on a calculator,
A broader curriculum stressing a variety of mathematical
strategies will make it possible to teach material to students in
each grade that will be useful to them no matter when they
end their mathematics education, At the same time, students
preparing for further study of mathematics will be stimulated
by early glimpses via the power of the computer~of what
lies aheacl.
Finally, computers and calculators have changed not only
what mathematics is important, but also how mathematics
should be taught (Zorn, 1987~:
· Computers and calculators change what is feasible and
what is important. They make the difficult easy and the
infeasible possible. For example, computers can display
and manipulate mathematical objects such as compli
catecl threeclimensional forms that cannot reasonably
be studied without computers. As a consequence, stu
clents can solve realistic problems that are relevant to
their everyday experiences and that have the potential
of stimulating continuing interest in mathematics.
· Computers free the teacher for those tasks that only a
teacher can do. For example, teacher and students can
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21
A Philosophy and Framework
together explore conjectures. Computers provide a
dynamic and graphic medium that offers many effective
ways to present mathematical ideas,
Technology makes mathematics realistic. Before the
advent of calculators and computers, even the most
able students could not perform calculations required for
most realistic problems. Nor, for that matter, could teach
ers do such calculations for students without spending far
too much time on the computations themselves. Now,
computation itself is no longer a barrier. If the problem
can be grasped by the stuclents, then it can be solved.
Real date from real experiments can be analyzed. Equa
tions that represent significant physical situations can be
solved, Many sophisticated concepts can be made more
intelligible with computers than through any other means.
The Year 2000
The eventual use of technology in the teaching and learn
ing of mathematics can be seen, at best, only dimly today,
Few classrooms today are equipped to make the use of com
puters convenient and inviting for teachers. Software, even
when of high quality, is open relevant only to narrow curricular
objectives. Too often it is neither teacherfriendly nor student
friendly. Rarely is it coordinated with textbooks or curriculum.
Despite these current problems, which are legacies of old
technology, workstations of the 1990s will offer powerful, flexi
ble environments that will make possible a much improved
symbiosis between teacher and technology.
Developments in computer technology, both hardware and
software, are notoriously hard to predict. Still, enough is known
now to be able to make some reasonable predictions about
what is desirable and feasible for computers in schools in the
year 2000:
· All students should have available handheld calculators
with a functionality appropriate to their grade level. Cal
culators suitable for secondary school will by then have
symbolic and graphics capability sufficient for all high
school level mathematics.
· All mathematics classrooms should have permanently
installed contemporary computers with display units con
venienfly visible by all students. New schools may be
equipped with desks that include builtin computers.
· There should be sufficient computer facilities available for
laboratory and outofclassroom neecis of all students. In
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22
Reshaping School Mathematics
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Facilities like these are beyond the
budgets of most school districts toclay.
However, by 2000 their relative cost
should be much less than it is toclay.
Districts need to plan now to ensure
calculator and computer facilities ade
quate for all classes and all stuclents.
School boards and administrators must
plan school budgets to ensure full
access to the tools of learning, espe
cially for districts with limited resources.
This is an area in which government, business, and industry can
effectively enhance education by cooperating on a plan to
ensure full technological support in every classroom in Ameri
ca. Curricular blueprints developed today must be based on
the technological reality of tomorrow's schools.
Research Findings
Of all the influences that shape mathematics education,
technology stancis out as the one with greatest potential for
revolutionary impact. it is also the area of greatest public con
cern, since it is so new, Without a rich base of experience on
which to draw, it is very difficult to say just how technology
can be most effectively used in mathematics eclucation. For
tunately, sufficient research has been clone by pioneers in this
field to suggest general trencis and likely results.
The effects of calculators in school mathematics have been
studied in over 100 formal investigations during the past 15
years. These studies have tested the impact of a variety of
kinds of calculator usefrom limited access in carefully select
ed situations to access for all aspects of mathematics instruc
tion and testing. There have been two major summaries of
research on calculator usage (Hembree and Dessart, 1986;
Suydam, 1986~. In almost every reported stucly, the perfor
mance of groups using calculators equaled or exceeded that
of control groups denied calculator use.
The recent Hembree and Dessart mesaanalysis of 79 calcu
lator studies sorted out the effects of calculator use on six
dimensions of attitude toward mathematics as well as on the
OCR for page 17
23
A Philosophy and Framework
acquisition, retention, and transfer of computational skill, con
ceptual understanding, and problemsolving ability. The analy
sis led to this conclusion (Hembree and Dessert, 1986~:
Students who use calculators in concert with tradi
tional instruction maintain their paperandpencil
skills without apparent harm. Indeecl, use of calcuia
tors can improve the average student's basic skills
with paper and pencil, both in basic operations and
in problem solving.
Research suggests that access to calculators in a well
plannecl program of instruction is not likely to obstruct
achievement of skill in traditional arithmetic proceclures. More
optimistically, it appears that when students have access to
calculators for learning and achievement testing, they per
form at significantly higher levels on both computation and
problem solving. In particular, stuclents using calculators seem
better able to focus on correct analysis of problem situations,
The earliest educational use of computers was focused on
computerassisted instruction (GAI), often based on pro
grammed learning, most frequently for drill on rote skills. Sever
al reviews of research on fine effectiveness of CAI (e.g.,
BangertDrowns et al., 1985) have concluded that it is general
ly very effective, giving better achievement in shorter time
than traditional instruction.
Lately, principles of artificial intelligence have been applied
to design of sophisticated tutors for algebra, geometry, and
calculus. The designers suggest that the use of such tutors can
yield ciramatic increases in student achievement. However, no
data are yet available about the use of such tutors in realistic
classroom settings.
There are several kinds of computerbased systems that give
students powerful new tools for learning in an exploratory envi
ronment. Best known is Logo; its turtle graphics teach students
concepts of geometry, algebra, and higherorder thinking
(Papert, 1980~. Although research findings have failed to con
firm the strongest claims that Logo clevelops a high level of
general reasoning, a variety of studies have found positive
effects on more specific instructional goals (Campbell, 1989~.
Moreover, thousands of classroom teachers have been con
vinced by firsthand experience that Logo is a powerful
instructional tool.
A different sort of computerbased exploratory tool is pro
videcl by the Geometric Supposer (Schwartz and Yerushalmy,
1987) and Geodraw (Bell, 1987~. Each provides students with
open but guided environments for exploring the results of geo
metric constructions. Green Globs (Dugdale, 1982) provides a
comparable setting for algebraic exploration. Although there
is Title formal research describing the effects of these learning
OCR for page 17
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Reshaping School Mathematics
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and teaching tools, there are
some suggestions (Yerushalmy
et al,, 1987) that students may
perform as well or better than
control students on traditional
criteria while at the same time
learning other objectives.
Some investigators have
studied the effects of computer
graphics on student under
standing of mathematical con.
cepts like function (Rhoads,
1986; Schoenfelcl, 1988a) or
statistics and data analysis
(Swift, 1984~. In each case, the
computer seems clearly to
enhance student interest and
understanding of important
ideas.
Although most studies have
focused, one way or another,
on finding better ways to reach traditional goals, there have
been some daring departures from conventional curriculum
priorities. Both Lesh (1987) and Heid and Kunkle (1988) tested
the effects of experimental algebra instruction in which stu
dents used symbol manipulation software to perform routine
tasks like solving equations. Each found that students who
were freed from the traditional symbolic procedural aspects of
problem solving became much more adept at problem for
mulation and interpretation.
In two similar studies of computeraided calculus, Heid
(1988) and Palmiter (1986) found that students who learned
calculus with the aid of computer software developed a
much deeper understanding of fundamental concepts than
did students in traditional skilloriented courses. Heid also found
that her students picked up needed procedural knowledge in
a short time period following careful instruction in conceptual
background, and Palmiter found that her students acquired
their understanding much more quickly than students in con
ventional courses.
Open Questions
Most current research addresses fundamental questions of
technology applied to the mathematics curriculum: What are
the essential interactions among conceptual development,
procedural knowledge, and problem solving? This research
OCR for page 17
25
A Philosophy and Framework
indicates that access to computers and calculators need not
hinder attainment of traditional curricular objectives, and that
it may substantially advance it. Unfortunately, there is no con
sensus on how to investigate new effects such as the improve
ment of higherorder thinking skills. A series of articles in Ecluca
tional Researcher (Becker, 1987; Papert, 1987; Pea, 1987b;
Walker, 1987) illustrate the wide diversity of opinion on this
topic, A key concern is the extent to which the development
of mathematical power can be inferred from written test per
formance or within the limited time spans of most research
studies,
It has sometimes been proposed that the availability of
computers would, more or less in itself, produce significant
improvements in mathematical thinking, From the few
attempts that have been made to measure changes in rea
soning power, it is possible to conclude that such advance
ments cannot come from trivial technological fixes, Repeated
attempts to document such change has yet to reveal a lasting
effect for example, studies of the effect of Logo on planning,
of the impact of Pascal on understanding of algebraic syntax,
and of the cognitive impact of learning metaprinciples of pro
gramming in Basic, While these results do not necessarily imply
that computers will not improve mathematical thinking, they
do suggest that simplistic approaches are not likely to pro
duce measurable improvements.
Rapid changes in the objectives and strategies of mathe
matics education have outpaced the evidence of effective
ness provided by educational research (NRC, 1985~. Introduc
tion of calculators and computers, especially, opens up many
new issues that need careful study:
· Organization for Learning. Changes in curriculum, in
teaching practice, and in the educational role of com
puters and calculators provide both opportunity and
compelling need for new research on the effectiveness
of different strategies. Computers virtually compel
reordering and new combinations of traditional topics.
What orders yield optimal learning?
· Levels of Learning. Technology makes possible earlier
introduction of certain topics (e.g., decimals). What is the
relation between the stage of introduction and ultimate
understanding?
· Modes of Learning. As instruction recognizes an active
role for students in constructing their own knowledge, we
need to monitor the longterm impact of this approach
on stuclents' abilities to learn and to use mathematical
concepts throughout their lives,
· Manipulative Skills. Powerful calculators compel reexami
nation of traditional priorities for arithmetic and algebraic
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26
Reshaping School Mathematics
skills Which skills best support mathematical power, and
when must they be taught?
· Procedural and Conceptual Knowledge. What level of
manipulative skill is necessary in order to be able to
understandand thus usemathematics in a problem
solving context?
· Transfer of Knowledge. How can school instruction pro
vicle students with a background that will enable them to
apply what they have learned in outofschool contexts?
· Instructional Uses of Technology. Technological research
has just begun to create tools with the power to alter sig
nificantly the traditional process of instruction. What kinds
of mathematical comprehension can these new tools fess
ter?