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OCR for page 57

TEA CHINO
Hi
There is little we do in America that is more important
than teaching. Elective teaching of mathematics requires
appropriate pedagogical and mathematical foundations, but
thrives only in an environment of trust which encourages
leadership and innovation. In short, teaching must become
more professional.
Under current conditions, most school teachers face nearly
insurmountable obstacles. Lacking freedom to establish fully
either ends or means, teachers rarely have the opportunity
to exercise truly professional responsibilities. What emerges,
often, are foreshortened ideals and shattered dreams.
Evidence from many sources shows that the least e~ec-
tive mode for mathematics learning is the one that prevails
in most of America's classrooms: lecturing and listening.
Despite daily homework, for most students and most teach-
ers mathematics continues to be primarily a passive activity:
teachers prescribe; students transcribe. Students simply do
not retain for long what they learn by imitation from lec-
tures, worksheets, or routine homework. Presentation and
repetition help students do well on standardized tests and
lower-order skills, but they are generally ineffective as teach-
ing strategies for long-term learning, for higher-order think-
ing, and for versatile problem-solving.
Teachers, however, almost always present mathematics as
an established doctrine to be learned just as it was taught.
This "broadcast" metaphor for learning leads students to
expect that mathematics is about right answers rather than
about clear creative thinking. In the early grades, arithmetic
becomes the stalking horse for this authoritarian mode! of
learning, sowing seeds of expectation that dominate student
attitudes all the way through college.
- , ,
_ _
. . ~. . .
Understanding Mathematics
Many students master the formalisms of mathematics
without, at first, any real understanding. Some go on
to achieve a retrospective understanding after they have
reached a more advanced vantage point. Surprisingly, many
mathematicians and scientists recall that their own educa
...learning through involvement
Myth: Learning mathematics
means mastering an immutable
set of basic skills.
Reality: Skills are to mathe-
matics what scales are to mu-
sic or spelling is to writing.
The objective of learning is to
write, to play music, or to solve
problems not just to master
skills. Practice with skills is just
one of many strategies used by
good teachers to help students
achieve the broader goals of
learning.
57

OCR for page 57

Teaching
Back to School
A table of data gives information
on stopping distances for several
cars in terms of speed, weight,
and types of brakes (drum, disc,
antilock). The information in-
cludes both reaction time dis-
tance as well as braking distance.
Develop a model for these data
in terms of graphs, equations, or
computer programs that would
enable one to predict how other
cars would handle under similar
circumstances. Compare what the
model predicts with the advice
. . . . .
given In driver ec ~ucat~on courses.
58
tion fits this model; rarely is anything learned well until it is
revisited from a more advanced perspective.
The vast majority of students never move beyond for-
mal knowledge since they do not persist in subsequent work
to reach the point where the veil of confusion is lifted.
(Those who do persist are likely to be the ones who sub-
sequently go on to careers in science.) Present educational
practice in the United States offers students only one path
to understanding a long, dimly lit journey through a moun-
tain of meaningless manipulations, with the reward of power
and understanding available only to those who complete the
ourney.
P· · · · · . . . . · . . .
resent educational practice offers
mathematics students only a dim light at the
end of a very long tunnel.
Most students do not find the light at the end of the tunnel
sufficient to illuminate their journey through the mathemat-
ics curriculum. Far too many abandon their effort before
receiving any benefit from the power of retrospective un-
derstanding. To improve mathematics education for all stu
dents. we need to exnancl teaching r~rnctin~s that Name ~nr1
. ~ . . ~ . .. . ~ . ~ . . ~ .
motivate students as they struggle with their own 1earnlng.
In addition to beckoning with the light of future understand-
ing at the end of the tunnel, we need even more to increase
illumination in the interior of the tunnel.
:Learning Mathematics
In reality, no one can teach mathematics. Effective teach-
ers are those who can stimulate students to learn mathemat-
ics. Educational research offers compelling evidence that stu-
dents learn mathematics well only when they construct their
own mathematical understanding. To understand what they

OCR for page 57

...learning through involvement
learn, they must enact for themselves verbs that permeate the
mathematics curriculum: "examine," "represent," "trans-
f 0 r m , " " s 0 ~ v e , " " a p p ~ y , " " p r 0 v e , " " c 0 m m u n i c a t e . " T h i s h a p -
pens most readily when students work in groups, engage
in discussion, make presentations, and in other ways take
charge of their own learning.
All students engage in a great deal of invention as they
learn mathematics; they impose their own interpretation on
what is presented to create a theory that makes sense to them.
Students do not learn simply a subset of what they have been
shown. Instead, they use new information to modify their
prior beliefs. As a consequence, each student's knowledge of
mathematics is uniquely personal.
_P tudents retain best the mathematics that
they learn by processes of internal construction
and experience.
Evidence that students construct a hierarchy of under-
standing through processes of assimilation and accommo-
dation with prior belief is not new; hints can be found in
the work of Piaget over fifty years ago. Insights from con-
temporary cognitive science help confirm these earlier ob-
servations by establishing a theoretical framework based on
evidence from many fields of study.
Engaging Students
No teaching can be effective if it does not respond to stu-
dents' prior ideas. Teachers need to listen as much as they
need to speak. They need to resist the temptation to con-
tro! classroom ideas so that students can gain a sense of
ownership over what they are learning. Doing this requires
genuine give-and-take in the mathematics classroom, both
among students and between students and teachers. The
Myth: Students learn by remem-
bering what they are taught.
Reality: Students construct mean-
ing as they learn mathematics.
They use what they are taught to
modify their prior beliefs and be-
havior, not simply to record and
store what they are told. It is stu-
dents' acts of construction and
invention that build their math-
ematical power and enable them
to solve problems they have never
seen before.
59

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Teaching
Myth: The best way to learn
how to solve complex problems
is to decompose them into a se-
quence of basic skills which can
then be mastered one at a time.
Reality: Research in learning as
well as instructional practice of
other countries offer little sup-
port for this strategy of teach-
ing. There is abundant evidence
that mastery of necessary skills
is rarely sufficient for solving
complex problems. Moreover,
many other countries introduce
students to complex problems
well before they have studied all
of the prerequisite skills. Those
students often invent effective ap-
proaches to the problem, thereby
gaining valuable experience in
higher-order thinking.
Robust Arithmetic
How do you add up long lists
of numbers? A dozen different
people do it in a dozen differ-
ent ways-top down, bottom up,
grouping by tens, bunching, and
various mixtures. There is no sin-
gle correct method.
60
best way to develop effective logical thinking is to encourage
open discussion and honest criticism of ideas.
Clear presentations by themselves are inadequate to re-
place existing misconceptions with correct ideas. What stu-
dents have constructed for themselves, however inadequate
it may be, is often too deeply ingrained to be dislodged with
a lecture followed by a few exercises. To change beliefs,
students need to have a stake in the outcome.
Honest questions by teachers are rare in mathematics
classrooms. Most teachers ask rhetorical questions because
they are not so much interested in what students really think
as in whether they know the right answer. Soon, students are
plaguing teachers with their own rhetorical question: "Can't
you just tell me the answer?"
·~ rid
~. ~
when students explore mathematics on their own, they
construct strategies that bear little resemblance to the canon-
ical examples presented in standard textbooks. Just as chil-
oren need the opportunity to learn from mistakes, so stu-
dents need an environment for learning mathematics that
provides generous room for trial and error. In the long
run, it is not the memorization of mathematical skills that
is particularly important without constant use, skills fade
rapidly but the confidence that one knows how to find and
use mathematical tools whenever they become necessary.
There is no way to build this confidence except through the
process of creating, constructing, and discovering mathemat
~cs.
·
M ...............................
athematics teachers must involve
students in their own learning.
Classes in which students are told how to solve a quadratic
equation and then assigned a dozen homework problems to
learn the approved method will rarely stimulate much lasting
mathematical knowledge. Far better is an approach in which

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...learning through involvemer't
students encounter such equations in a natural context; ex-
plore several approaches to solutions, including estimation,
graphs, computers, and algebra; then compare various ap-
proaches and argue about their merits. Of course, classes
such as this where active learning is a way of life require
more time and energy on the part of both teachers and stu-
dents than either is accustomed to giving under present con
· ~
c Tons.
Teachers' roles should include those of consultant, mod-
erator, and interlocutor, not just presenter and authority.
Classroom activities must encourage students to express their
approaches, both orally and in writing. Students must en-
gage mathematics as a human activity; they must learn to
work cooperatively in small teams to solve problems as well
as to argue convincingly for their approach amid conflicting
ideas and strategies.
There is a price to pay for less directive strategies of teach-
ing. In many cases, greater instructional effort may be re-
quired. in those parts of the curriculum where mathematics
directly serapes another discipline (for example, engineering),
students may not march through the required curriculum at
the expected rate. In the long run, however, less teaching
will yield more learning. As students begin to take responsi-
bility for their own work, they will learn how to learn as well
as what to learn.
%,, ~
Impact of Computers
Calculators and computers compel reexamination of prior-
ities for mathematics education. How many adults, whether
store clerks or bookkeepers, still do long division (or even
long multiplication) by paper and pencil? How many scien
.. . . ~ . ~ . ~
tlStS or engineers use paper-and-pencil methods to carry out
their scientific calculations? Who would trust a bank that
kept its records in ledgerbooks?
Those who use mathematics in the workplace-account-
ants, engineers, scientists rarely use paper-and-pencil pro-
cedures any more, certainly not for significant or complex
analyses. Electronic spreadsheets, numerical analysis pack-
ages, symbolic computer systems, and sophisticated com
Calculators vs. Computers
Polls show that the public gener-
ally thinks that, in mathematics
education, calculators are bad
while computers are good. Peo-
ple believe that calculators will
prevent children from mastering
arithmetic, an important burden
which their parents remember
bearing with courage and pride.
Computers, on the other hand,
are not perceived as shortcuts to
undermine school traditions, but
as new tools necessary to soci-
ety that children who understand
mathematics must learn to use.
What the public fails to recog-
nize is that both calculators and
computers are equally essential to
mathematics education and have
equal potential for wise use or for
abuse.
61

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Teaching
Tomorrow's Computers
Even as teachers struggle to adapt
yesterday's curriculum to today's
computers, industrial leaders
are designing tomorrow's tech-
nology. Multimegabyte memory
and gigabyte storage support un-
precedented graphics, unleash-
ing potential for interactive text-
books, remote classrooms, and
integrated learning environments.
As today's computer visions be-
come tomorrow's verities, they
will revolutionize the way mathe-
matics is practiced and the way it
is learned.
62
puter graphics have become the power tools of mathematics
in industry. Even research mathematicians now use comput-
ers to aid exploration, conjecture, and proof.
In spite of the intimate intellectual link between mathe
matics and computing, school mathematics has responded
hardly at all to curricular changes implied by the computer
revolution. Curricula, texts, tests, and teaching habits but
not the students are all products of the precomputer age.
Little could be worse for mathematics education than an en-
vironment in which schools hold students back from learning
what they find natural.
It is true, as many say, that we are not sure how best to
teach mathematics with computers. Nevertheless, despite
risks of venturing into unfamiliar territory, society has much
to gain from the increasing role of calculators and computers
in mathematics education:
· School mathematics can become more like the mathemat-
ics people actually use, both on the job and in scientific
applications. By using machines to expedite calculations,
students can experience mathematics as it really is as a
tentative exploratory discipline in which risks and failures
yield clues to success.
· Weakness in algebraic skills need no longer prevent stu-
dents from understanding ideas in more advanced math-
ematics. Just as computerized spelling checkers permit
writers to express ideas without the psychological block of
terrible spelling, so will the new calculators enable moti-
vated students who are weak in algebra or trigonometry to
persevere in calculus or statistics. Calculators in the cIass-
room can help make higher mathematics more accessible.
· Mathematics learning can become more active and dy-
namic, hence more elective. By carrying much of the
computational burden of mathematics homework, calcu-
lators and computers enable students to explore a wider va-
riety of examples; to witness the dynamic nature of math
ematical processes; to engage realistic applications using
typical not oversimplified data; and to focus on impor-
tant concepts rather than routine calculation.
· Students can explore mathematics on their own, to ask and
answer countless "what if'' questions. Although calculators

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...learning through ir~volrement
and computers will not necessarily cause students to think
for themselves, they can provide an environment in which
student-generated mathematical ideas can thrive.
Time invested in mathematics study can build long-lasting
intuition and insight, not just short-lived strategies for cal-
culation. Innovative instruction based on a new symbio-
sis of machine calculation and human thinking can shift
the balance of learning toward understanding, insight, and
mathematical intuition.
.
Ten years ago, arithmetic fell to the power of inexpensive
hand calculators; five years ago, scientific calculators offered
at the touch of a button more sophisticated numerical math-
ematics than most students knew anything about. Today's
calculators can do a large fraction of all techniques taught
in the first two years of college mathematics. Tomorrow's
calculators will do what computers do today.
Priorities for mathematics education must
change to reflect the way computers are used in
mathematics.
The ready availability of versatile calculators and com-
puters establishes new ground rules for mathematics edu-
cation. Template exercises and mimicry mathematics- the
staple diet of today's texts will diminish under the assault
of machines that specialize in mimicry. instructors will be
forced to change their approach and their assignments. it
will no longer do for teachers to teach as they were taught in
the paper-and-pencil era.
Education of Teachers
Mathematics is taught in every grade throughout the entire
thirteen years of school, K-12; consistent growth in skills,
Myth: The way to improve stu-
dents' mathematical performance
is to stress the basics.
Reality: Basics from the past, es-
pecially manual arithmetic, are of
less value today than yesterday-
except to score well on tests of
basic skills. Today's students
need to learn when to use math-
ematics as much as they need to
learn how to use it. Basic skills
for the twenty-first century in-
clude more than just manual
mathematics.
63

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Teaching
Back to School
Record the height and weight of
everyone in your class and see
what relationship you can deter-
mine. Do age or sex help make
the relationship clear?
64
maturity, and understanding is essential as students move
from one level to the next. As a chain that breaks at its
weakest link, mathematics instruction is especially vuIner-
able to weakness in any grade or course. For this reason,
the preparation of mathematics teachers is a crucial factor
in revitalizing curricular practice.
Too often, elementary teachers take only one course in
mathematics, approaching it with trepidation and leaving it
with relief. Such experiences leave many elementary teach-
ers totally unprepared to inspire children with confidence
in their own mathematical abilities. What is worse, experi-
enced elementary teachers often move up to middle grades
(because of imbalance in enrolIments) without learning any
more mathematics.
Those who would teach mathematics need to learn con
temporary mathematics appropriate to the grades they will
teach, in a style consistent with the way in which they will
be expected to teach. They also need to learn how students
learn what we know from research (not much, but impor-
tant), and what we do not know (a great deal). They need
to learn science (including technology, business, and social
science) so that they can teach mathematics in the contexts
where it arises most naturally in measurement, graphs, pre-
diction, decisions, and data analysis. And they need to learn
the history of mathematics and its impact on society, for
it is only through history that teachers will come to know
that mathematics changes and to see the differences between
contemporary and ancient mathematics.
The United States is one of the few countries in the world
that continues to pretend-despite substantial evidence to
the contrary that elementary school teachers are able to
teach all subjects equally well. it is time that we identify a
cadre of teachers with special interests in mathematics and
science who would be well prepared to teach young children
both mathematics and science in an integrated, discovery-
based environment.

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...learning through involvement
T· -
he United States must create a tradition
of elementary school specialists to teach
mathematics and science.
· · -
Many models for mathematics specialists are possible,
most of which are in place in different school districts today.
Implementation can range from paired classes-one teacher
for language arts, the other for mathematics and science to
certified specialists who lead curricular development and as-
sist regular classroom teachers. Many teachers already have
the interest, experience, and enthusiasm for such positions;
others could qualify through special summer institutes.
To encourage more widespread adoption of diverse pat-
terns for mathematics specialists, states must alter certifica-
tion requirements to encourage these new models. Then uni-
versities must implement new courses with open construc-
tive instructors so that prospective school teachers can grow
in confidence as a result of their university study of mathe-
matics. The content of the special mathematics courses for
prospective elementary and middle school teachers those
who do not undertake a standard mathematics major must
be infused by examples of mathematics in the world that the
child sees (sports, architecture, house, and home), examples
that illustrate change, quantity, shape, chance, and dimen-
sion.
Teachers themselves need experience in doing mathema-
t~cs In exploring, guessing, testing, estimating, arguing, and
proving in order to develop confidence that they can re-
spond constructively to unexpected conjectures that emerge
as students follow their own paths in approaching math-
ematical problems. Too often, mathematics teachers are
afraid that someone will ask a question that they cannot
answer. Insecurity breeds rigidity, the antithesis of mathe-
matical power.
Since teachers teach much as they were taught, university
courses for prospective teachers must exemplify the highest
standards for instruction. However, most mathematics that
Back to School
You have 10 items in your gro-
cery cart. Six people are wait-
ing in the express lane (10 items
or less); one person is waiting in
lane 1 and two people are wait-
ing in lane 3. The other lanes are
closed. What additional infor-
mation do you need to know in
order to determine which lane to
join?
65

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Teaching
66
teachers have studied has been presented only in the author-
itarian framework of Moses coming down from Mt. Sinai.
Very few teachers have had the experience of constructing
for themselves any of the mathematics that they are asked
to teach, of listening to students who are developing their
own mathematical understandings, or of guiding students to
their own discovery of mathematical insights.
P· · . . . . . . · . . . . . . . . . . . . . . . . . . . . . . . . .
respective teachers should learn
mathematics in a manner that encourages active
engagement with mathematical ideas.
All students, and especially prospective teachers, should
learn mathematics as a process of constructing and inter-
preting patterns, of discovering strategies for solving prob-
lems, and of exploring the beauty and applications of math-
ematics. Above all, courses taken by prospective teachers
must create in these teachers confidence in their own abil-
ities to help students discover richness and excitement in
mathematics.
Resources
Textbooks and their ancillary products (worksheets, home-
work exercises, testbanks) dominate mathematics teaching at
all levels, from primary school through college. In no other
subject do students operate so close to a single prescribed
text; neither library work nor laboratory work, neither term
papers nor special projects are common parts of mathemat-
ics instruction. Classroom mathematics is the study of set
texts and set problems that rarely have any parallel either in
the world of work or in the many disciplines that depend on
mathematics as a tool.
Quite apart from the limitations imposed on classroom
practice by excessive reliance on textbooks, the very impor

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...learning through involvement
lance of the text as the primary medium of instruction poses
unique challenges and opportunities:
How can major school textbook series or course texts adapt
quickly to changes in curricular goals or emerging technol-
ogy?
· Are we on the verge of a post-Gutenberg revolution in
which computer communications can deliver flexible in-
teractive texts more readily and efficiently than the print-
ing press?
How can texts and software act as incentives rather than as
brakes for the newly emerging standards for school math-
ematics?
Even while educators work to reduce the dominance of
text-based learning in mathematics classrooms, publishers
and teachers need to explore new modes of publication that
will enable good innovative ideas to enter expeditiously into
typical classroom practice. Texts, software, computer net-
works, and databases will blend in coming years into a new
hybrid educational and information resource. It is already
true that the most common application of school mathemat-
ics is to program formulas into computer spreadsheets. As
texts evolve and software matures, both must be synchro-
nized with forward-Iooking curricular and classroom objec-
tives of mathematics education.
Assessment
Governors and political leaders in all fifty states are advo-
cating assessment in order to raise expectations and evaluate
programs. Tests serve many important purposes. They allow
students to recognize personal success; they enable teach-
ers to judge students' progress; they provide administrators
means to measure the effectiveness of instruction; and they
afford the public accountability for the use of public funds.
When designed and used properly, tests and other assess-
ment instruments provide feedback that is essential for any
system to maintain steady progress toward its objectives.
"According to virtually
all studies of the matter,
textbooks have become
the de facto curriculum
of the public schools . .
It is therefore critical
that textbooks stimulate
rather than deaden stu-
d~ents' curiosity, and that
teacher manuals encour-
age rather than squelch
teachers' initiative and
flexibility."
Harriet Tyson-Bernstein
· · ~
67

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Teaching
Myth: Minimal competency ex
. . . .
am~nat~ons raise stuc tent accom-
plishment.
Reality: The typical effect of a
. . .
minima competency examlna-
tion is to reallocate limited in-
structional resources from the
average students who have no
trouble passing such a test to the
weakest students who need spe-
cial help. Since the effort to bring
all students up to the level set by
the exam is so great, the progress
of the majority is often slowed
down while that of a few is im-
proved. Many weak students
fall further behind; often, the
floor is too low for all. On the
whole, more students lose than
gain from such rigid structures.
68
Unfortunately, tests in mathematics education are rarely
used in a manner appropriate to their design. Tests designed
for diagnostic purposes are often used for evaluating pro-
grams; scores from self-selected populations (for example,
takers of Scholastic Aptitude Tests) are used to compare
districts and states; and commonly used achievement tests
stress simple skills rather than sophisticated tasks, not be-
cause such skills are more important, but because they are
easier to measure.
Tests are dear to the public; they produce winners and
losers, as do sports playoffs, primary elections, and lotter-
ies. Tests also symbolize commitment to things we value
to facts and information that we once learned and that we
believe all children should still learn. In America (but not
in other countries), objective, multiple-choice tests are the
norm; they are efficient, economical, and seemingly fair.
l V I ..................
1 ~ Mathematical assessment in America
relies excessively on misleading multiple-choice
tests.
Nonetheless, multiple-choice tests as used in America lead
to widespread abuses, which the public rarely recognizes:
· Tests become ends in themselves, not means to assess ed-
ucational objectives. Knowing this, teachers often teach
to the tests, not to the curriculum or to the children.
Tests stress lower- rather than higher-order thinking, em-
phasizing student responses to test items rather than orig-
inal thinking and expression.
Test scores are sensitive to special coaching, which aggra-
vates existing inequities in opportunities to learn.
Tests reinforce in students, teachers, and the public the
narrow image of mathematics as a subject with unique
correct answers.
· Timed tests stressing speed inhibit learning for many stu-
dents.

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.
· Normed tests ignore the vast differences in rates at which
children learn.
· Tests provide snapshots of performance under the most
stressful environment for students rather than continu-
ous information about performance in a supportive atmo-
sphere.
· Poor test scores lead students to poor self-images, destroy-
ing rather than building confidence.
Similar problems arise when detailed learner outcomes
rather than teacher judgments define the objectives of
courses. Like items on objective tests, specific learner out-
comes bias teacher effort and constrain student learning. The
most important goals for mathematical learning cannot be
atomized into tiny morsels of knowledge.
Too often, good intentions in testing can lead to very bad
results. Minimal competency testing often leads to minimal
performance, where the floor becomes a ceiling. In stress-
ing the importance of basic skills, such tests fad! to encour-
age able students to progress as far as they can. As political
pressure for state assessment begins to encompass higher ed-
ucation, where assessment is even more complex, it is vitally
important that the mathematical community agree on proper
standards for assessment.
W ...............................
hat is tested is what gets taught. Tests
must measure what is most important.
Assessment should be an integral part of teaching. it is the
mechanism whereby teachers can learn how students think
about mathematics as well as what students are able to ac-
complish. But tests also are used to compare classes and
schools, to evaluate teachers, and to place students in future
courses or careers. Because assessment is so pervasive and
has such powerful impact on the lives of both students and
teachers, it is very important that assessment practice align
properly both with the purpose of the test and with curricular
objectives.
..learning through involvement
Myth: Only objective tests yield
reliable results.
Reality: Experience in evaluat-
ing student writing shows that
trained readers judging whole es-
says produce results more aligned
to the goal of high-quality writ-
ing than do objective exams of
grammar and vocabulary. Sim-
ilar experiences show that one
can reliably judge scientific un-
derstanding by observing student
teams in a laboratory. Effective
means of assessing operational
knowledge of mathematics must
be similarly broad, reflecting the
full environment in which em-
ployees and citizens will need to
use their mathematical power.
69

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Teaching
Voice of Experience
"We weren't happy with per-
formance on conceptual and
problem-solving parts of national
standardized tests. So we threw
out the texts ant! are teaching ele-
mentary school mathematics with
concrete materials and teacher-
made worksheets. Now our kids
are doing much better on na-
tional and state tests and when
we hear from parents, it is to fine!
out how to get their kids into our
program.
70
To assess development of a student's mathematical power,
a teacher needs to use a mixture of means: essays, home-
work, projects, short answers, quizzes, blackboard work,
journals, oral interviews, and group projects. Only broad-
based assessment can reflect fairly the important, higher-
order objectives of mathematics curricula.
As we need standards for curricula, so we need standards
for assessment. We must ensure that tests measure what
is of value, not just what is easy to test.
-Ray Whinnem
If we want stu-
dents to investigate, explore, and discover, assessment must
not measure just mimicry mathematics. By confusing means
and ends, by making testing more important than learning,
present practice holds today's students hostage to yesterday's
mistakes.

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OCR for page 57

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