Response to Fourth Set of Questions
from NCTM Commission on the Future of the Standards
MAA President's Task Force on the NCTM Standards
Reported by Ken Ross
Question 1. In previous responses, ARGs have
highlighted the importance of mathematical reasoning, of deduction and
formal proof, and of mathematical disposition. These topics relate to what
might be called the "nature of mathematics." What should K-12 students
learn about the *nature of mathematics*? At what grade levels? You might
find it useful to consider Evaluation Standards 7 and 10 from the
Curriculum and Evaluation Standards, which follow.
Evaluation Standard 7: Reasoning
The assessment of students' ability to reason mathematically should provide
evidence that they can-
- use inductive reasoning to recognize patterns and form conjectures;
- use reasoning to develop plausible arguments for mathematical
- use proportional and spatial reasoning to solve problems;
- use deductive reasoning to verify conclusions, judge the validity of
arguments, and construct valid arguments;
- analyze situations to determine common properties and structures;
- appreciate the axiomatic nature of mathematics.
Evaluation Standard 10: Mathematical Disposition
The assessment of students' mathematical disposition should seek
information about their-
- confidence in using mathematics to solve problems, to communicate
ideas, and to reason;
- flexibility in exploring mathematical ideas and trying alternative
methods in solving problems;
- willingness to persevere in mathematical tasks;
- interest, curiosity, and inventiveness in doing mathematics;
- inclination to monitor and reflect on their own thinking and
- valuing of the application of mathematics to situations arising in
other disciplines and everyday experience;
- appreciation of the role of mathematics in our culture and its value
as a tool and as a language.
If, as seems to be the case, "mathematical disposition" is the assessment
of students' mathematical disposition to seek information about a variety
of things, we do not understand the inclusion of Evaluation Standard 10
here. Nevertheless, we want to concur most emphatically with the
importance of the second and third bullets.
The remainder of this question deals with the "essence" of mathematics
which we believe to be its abstraction. From it, the discipline derives
its great power for application to a great number of seemingly unrelated
areas. A singular feature of mathematics which gives it power is that it
approaches a problem by stripping it down to the essentials and throwing
away all irrelevant data. In this way, widely dissimilar problems are seen
to come under a single heading while a simple idea turns out to be
applicable to diverse situations. Often this process will transform what
seems to be a hard problem into an easier problem. Mathematics teachers at
all grade levels should be so well versed in this concept of the field that
they transmit a sense of excitement, interest, and joy in having students
learn an important fundamental discipline.
As students proceed through the grades, they should gain an understanding
and appreciation of the exactness, elegance, and beauty of the field. They
should be aware that precise logical thinking dominates in mathematics, so
that terms must be clear and well defined. Hypotheses, whether implicitly
understood or stated explicitly, must be taken into account in the solution
of any problem. Precise logical thinking is the only way to communicate
mathematics reliably and gain consensus about the workings of complex
phenomena, make dependable predictions, and achieve universal
applicability. We also stress precise logical thinking in mathematics,
since otherwise many students find it difficult because of the sense of
alienation they experience in trying to learn the subject. They perceive
that it is a game whose rules have been set randomly and arbitrarily in
ways that they cannot hope to understand. Consequently, most players
(students) view mathematics as something that does not concern them. Under
the circumstances, it would be a miracle if they could be persuaded to work
hard to achieve its mastery.
Studies from around the world have shown that students' serious problems
with mathematics begin when they study properties of fractions. We believe
that the introduction of fractions in the elementary grades is often so
lacking in logic and precision that the rules for adding, multiplying and
dividing fractions are presented in a random and arbitrary manner. As a
result, many students do not take what they learn seriously. It should be
no surprise that when asked to add fractions a/b and c/d, many students opt
to write (a+c)/(b+d). Similar difficulties occur with the rules for
working with signed numbers and the laws of exponents.
- Students should be invited to make mathematical conjectures as soon
as they have enough arithmetical knowledge to do so, and thereafter
students should realize when they are making a conjecture and when they are
firmly establishing a statement. For example, early in their elementary
school careers students can be asked to start with a number that is a
multiple of three, add together its digits, then add together the digits of
the resulting number if it has more than one digit, and continue until the
result has only one digit. The students can be invited to do this with a
collection of different numbers they select themselves that are all
multiples of three, and then make a conjecture based on the observed
outcomes. The teacher should congratulate students on correct conjectures,
but should stress right from the beginning that since the students have not
shown this for every integer that is a multiple of three, they cannot yet
be certain that this really is true for all such numbers. They can also
explore the inverse of the conjecture by trying the same process on numbers
that are not multiples of three. The advent of technology has made
the numerical checking of many simple conjectures fashionable and painless.
Unfortunately it may increase the likelihood of confusing--in the students'
minds--the distinction between checking a general statement for a few
special cases and knowing the reasoning that underlies that statement. We
believe NCTM2000 would do well to underscore this distinction.
- Students should be given a gentle introduction to proof early in the
elementary grades. For example, the teacher can ask students how many
digits there will be in the product of two two-digit numbers. After the
students experiment awhile and decide that there can be either three or
four, the teacher can ask them how they can be sure there are no other
possibilities. Perhaps spontaneously, and almost certainly with a little
prodding, the students should be able to come up with an acceptable
argument based on the two extremal cases. (While this depends on some
order properties that the students will almost certainly not have studied
rigorously, their proof will be appropriate and acceptably rigorous for
their grade level.) This would also give the teacher an opportunity to
point out that to answer a question like this, it is not enough for
students to give examples, because they could give lots of examples to
support the erroneous conclusion that the answer is always three.
While they are in high school, students' experiences of proof should
include other mathematics besides geometry. In elementary algebra classes
they can be asked to give counterexamples showing that commonly used, but
erroneous, formulas such as (a+b)2=a2+b2
are incorrect. Or they could be asked to find values of a and b for which
such a formula is true (the difficulty of which might help bolster a sense
for the general falseness of the formula). In grades 11 and 12, students
could be asked to verify basic divisibility properties of integers, such
as: if a, b, and c are integers and a is a factor of both b and c, then a
is a factor of b+c.
- Students should be given an introduction to the principles of
deductive reasoning, beginning in the early elementary grades with the
notion that the truth of a statement does not imply the truth of its
converse. They should also explore the truth and falsity of statements
involving the quantifiers "some," "all" and "no." As the students progress
through the grades, other items can be added to their store of knowledge
about the principles of deductive reasoning. Before they are ready to take
high school mathematics courses, they should know that one counterexample
disproves a statement, but no finite number of examples that illustrate a
general result is sufficient for a proof. By the time they reach the end
of their high school mathematical career they should have learned all of
the basic principles of deductive reasoning used in mathematical arguments,
including argument by contraposition. Given a concrete argument within
their scope, they should be able to identify the hypotheses and
conclusions. They should also be able to pick up glaring logical errors
and to identify simple reasoning from assumptions to conclusions. There
should be some evidence that some of this can be formalized. But
formalization should be on examples that are understood, not memorized, and
for which the need to generalize using symbolic notation is recognized.
- Students should be given enough exposure to deductive reasoning from
a system of axioms that they understand something of the axiomatic nature
of mathematics. This is important, not just for mathematics, but also to
make sure students understand that the end products of deductive reasoning
in any situation are usually based on the underlying system of axioms that
a person accepts as obviously true. It would seem that the appropriate
place for this is the traditional one, namely somewhere in the middle of
the high school years.
- The curriculum should avoid the inclusion of topics if they are not
actually needed at a certain grade level and if they cannot be
satisfactorily explained at that grade level. As a specific example, a
detailed investigation about the decimal expansions of rational and
irrational numbers should be deferred to grade 11 or 12. At that time,
teachers should discuss how to take a repeating decimal and find its
fractional representation and how it happens that the decimal expansion of
a fraction is bound to terminate or repeat. This is also the appropriate
time to prove results such as the irrationality of the square root of 2.
When a number such as pi comes up in the earlier grades, it can simply be
noted (to be discussed in greater detail later) that no matter how accurate
one's calculator, it will never display an exact decimal representation for
the number. Thus, when one substitutes a decimal version into an
expression, one will always have to round it to a certain number of decimal
places and use an approximately-equal sign. Practice working with such
numbers in grades 6-9 can turn a discussion of irrationality in the 11th or
12th grade into a meaningful experience.
Question 2. What are the four or five most
important geometric concepts or themes that should be included in the K-12
First, we would like to express our hope that this question, and answers to
it, will not limit the coverage of geometry to four or five concepts and
hence lead to the continued neglect of geometry in the K-12 curriculum.
Rather, we interpret the question as asking for help in identifying some
key themes that can be used to strengthen geometry in the K-12 curriculum.
We shall identify five possible themes below, but would first like to call
attention to the current failure to devote at least a full semester to
proofs in Euclidean geometry. We are concerned that many of the usual
presentations of Euclidean geometry, with an emphasis on the foundational
and obvious materials, do not appeal to most students. But by being
selective in presenting the foundations, and by adopting the method of
local axiomatics, students can be led quickly to beautiful and nontrivial
theorems such as the nine-point circle and the Simson line. *These* should
appeal to all students. We strongly advocate emphasizing Euclidean-
geometry-with-proofs in the curriculum because NCTM's call for developing
higher order thinking skills resonates with us. It seems to us that
Euclidean-geometry-with-proofs is the ideal part of the high school
curriculum to develop the ability to formulate conjectures (by simple
drawings) and to check their correctness, often using multi-step reasoning.
The visual appeal of the theorems provides additional incentive.
We strongly recommend that students receive one full year of mathematics in
grades 8-12 that requires them to deal directly with geometry and the
visualization of geometric objects, in both two and three dimensions. This
one-year coverage of geometry need not necessarily all take place in the
same grade; it could be spread over two or more grades or with portions of
other appropriate courses satisfying this requirement.
- Coordinate geometry, which reflects the relationship between
geometry and number that arises from the fact that real numbers can be
represented as points on a line.
- Similarity and its relationship to proportional reasoning.
Similarity is a geometric idea that has many applications, one of
them being a proof of the Pythagorean theorem.
- Measures of geometric quantities, such as length, area, volume, and
angle, and the relationships between them in specific situations. This
provides a connection between number and shape and includes the Pythagorean
theorem and the constant ratio between a circle's circumference and its
- Visualization of geometric objects, particularly three-dimensional
ones. (This is one place where computers can play a useful role.) In
particular, students should be able to visualize and abstract various plane
- Symmetry, in its broadest sense, is a very rich theme in geometry.
From it one can develop the ideas of congruence, and it provides a natural
way to introduce functions (transformations) into geometry.
Question 3. In Round 3 responses, ARGs
identified a number of areas within discrete mathematics that should be
considered in Standards 2000, including iteration and recursion, graph
theory, the binomial theorem, and combinatorics. Should there be a separate
standard addressing discrete mathematics across grades K-12, or should
these topics be dispersed among the other proposed Standards (number,
algebra and functions, geometry, measurement, and probability and
statistics)? You might find it useful to consider Standard 12 for grades
9-12 from the Curriculum and Evaluation Standards; which follows.
Standard 12: Discrete Math
In grades 9-12, the mathematics curriculum should include topics from
discrete mathematics so that all students can-
- represent problem situations using discrete structures such as
finite graphs, matrices, sequences, and recurrence relations;
- represent and analyze finite graphs using matrices;
- develop and analyze algorithms;
- solve enumeration and finite probability problems;
and so that, in addition, college-intending students can-
- represent and solve problems using linear programming and difference
- investigate problem situations that arise in connection with
computer validation and the applications of algorithms.
We favor dispersing these items among the other areas covered by the
proposed standards. While discrete mathematics is an important part of
mathematics, the K-12 curriculum is already crowded. We are concerned that
if a great deal of emphasis is put on discrete mathematics as a separate
topic, then it is likely to crowd out other things that may be more basic
to a student's success in later mathematical studies.
We suggest that, of the areas within discrete mathematics listed in
question 3, only the binomial theorem and basic combinatorics be included
in the "core" high school curriculum. Moreover, they should be included
among the topics for algebra II, though they will reappear in probability.
In particular, we recommend that students in algebra II use fundamental
counting principles to compute combinations and permutations; use
combinations and permutations to compute probabilities; and use the
binomial theorem to expand binomial expressions which are raised to
positive integer powers.
Finally, we note that algorithms should be a part of just about every
mathematics course. Sequences and recurrence relations would be
appropriate topics in an elective course. On the other hand, there is
probably no room for linear programming in the K-12 mathematics curriculum,
not even for college-intending students.
OTHER AREAS WE WOULD LIKE TO ADDRESS
4. The first question asked about the "nature of mathematics,"
which we have addressed without discussing statistics. However, statistics
is finding a place in the curriculum. We strongly support the continued
inclusion of statistics and probability in the K-12 curriculum. It should
not be an optional topic.
5. We want to reiterate the importance of teachers raising their
expectations of ALL students.
6. We have been advised that we should avoid discussing teacher
preparation, because a different committee will be considering that issue.
That committee will need to take account of our emphasis on the importance
of students developing a gradually increasing ability to reason logically
and deductively starting before high school. In particular, we are
suggesting that middle school teachers should actually prove certain things
for their students. We acknowledge here that we are not currently
adequately preparing teachers for these tasks and recommend that curricular
materials be written so that the mathematical facts and their rationales
are clearly and explicitly stated. In particular, this means that, if a
topic is introduced through a discovery activity, the students should
receive follow-up materials giving a clear summary of the main insights to
which they were supposed to be led by doing the activity.
7. The MAA Task Force on the NCTM Standards recommends that the
revised Standards include a brief glossary of those terms in the revised
Standards that have been the subject of misinterpretation or
misunderstanding or are likely to be given different interpretations by
different people. Such a glossary should be brief, only include terms
actually appearing in the revised Standards, and not include the
definitions of mathematical terms whose meanings are unambiguous across the
various mathematical communities. Inclusion of such a glossary would go
far to address one of the most frequently heard criticisms of the present
NCTM Standards, namely that its readers have interpreted terms used in
these Standards in entirely different ways. Giving clear definitions of
such terms would be a very important contribution to clarity in the revised
Here are examples of some terms which, if they appear in the Revised
Standards, should be considered for inclusion in the glossary.
- abstraction, generalization, ideal
- algorithm, animation, simulation, application, model, concrete
model, mathematical model, real-world
- big ideas, context, conceptual, intuitive
- exercise, problem, open-ended problem
- expression, symbol, function; variable, unknown, name
- integrated curriculum
- practice, drill
- proof, illustration, justification, verification, rigorous
- rote skills, themes, strands
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Last modified: Wed Jul 22 11:26:36 -0500 1998