Secondary School Mathematics Curriculum:
A New Teachers View
by
Nique Page
In order to implement the proposed NCTM Standards for secondary school
mathematics curriculum, it seems important to address topics that should
be included or excluded from the curriculum that has been implemented. My
belief is that the basic curriculum is good, however, the methods that are
used to teach the topics are in need of reform. However, there are a few
topics that need to be addressed. Due to advances in technology, the concern
for problem solving, and utility, the mathematics curriculum needs adjustments.
Below is a list of topics that I feel are problem areas in the curriculum.
Some of the topics that I have addressed deal with changing methods rather
than curriculum.
General Topics for All Secondary Courses
TOPIC ATTENTION NEEDED
Probability Increase or Add to current Curriculum
Statistics Increase or Add to current Curriculum
"Simple" Word Problems Decrease in current curriculum
Real Applications Increase in current curriculum
Problem Solving as "Algorithm" Remove and change approach
Oral & Written Expression Increase in Curriculum
Estimation Increase or Add to current Curriculum
Using Calculators and Computers Increase or Add to current Curriculum
Mathematics History Increase or add to current curriculum
In most curriculum guides and textbooks that I have seen, probability and
statistics are not even topics, or they are at the end which makes them
"optional". I feel that these two topics should be at the beginning
of every secondary mathematics class. Data analysis gives students a problem
solving tool that can be used throughout the course. Statistical data can
be used to form problem situations and apply new algebraic concepts. Probability
makes wonderful use of estimation, predicting, and can be used to develop
a sense of variable, and can allow students to create problem situations.
If these topics are addressed early in the course, students can apply algebraic
concepts to statistical data and to probability experiments.
Most courses present Problem solving as an isolated topic and present it
in an algorithmic way. Students are given "typical" word problems
that have a set algorithm solution. Problem solving should be an ongoing
process in any mathematics course. Students should be learning new problem
solving approaches and techniques everyday, and learn how to decide which
approaches are best for a particular situation. Students should be given
real world applications of the mathematics and be encouraged to explore
different ways to approach solutions.
One area that has been missing in mathematics curriculum is the teaching
of reading and writing. We encourage students to read in other disciplines,
but mathematics has traditionally been manipulation of symbols. Students
get "an answer" never knowing or explaining how or why. I think
that students should be encouraged to read mathematics articles, write journals,
and keep portfolios of their work. Communication is the link to everything.
If we cannot communicate what we know and learn, it is not very useful.
In mathematics, students are usually engaged in finding the "one correct
answer" and teachers want "exact" answers. I think that students
need more chances to use estimation skills, so that when they find a solution
to a problem, they can decide whether it is reasonable. Estimation is an
important everyday life skill. Driving, shopping, cooking, and other everyday
activities involve estimation. Students would be well served to see more
of it in school.
Students have a curiosity for "where things come from". I think
that it could be motivational for students to study historical topics in
mathematics. Study of ancient number systems can give students new arithmetic
tools and a deeper understanding of numbers. Study of historical events
that led to the development of the common mathematics that we use could
reveal to students the usefulness of mathematics. They would learn that
math evolved from a need to solve a problem.
Algebra I
TOPIC ATTENTION NEEDED
Simplification of Radicals Remove from course
Factoring to solve equations Decrease Attention
Paper & Pencil Graphing Decrease attention in some areas
Operations on Rational Expressions Decrease Attention
Geometric applications Increase or Add to current Curriculum
Algebra students are often engaged in simplifying radical expressions for
the sake of making them "look nice". Since calculators have become
common household/classroom items, it is not necessary for students to simplify
radical expressions. If radicals are involved in the solution of a problem,
students should be encouraged to use approximations because an approximation
is more useful in a real life application. It is more important to understand
the concept of square roots and what it means to find the square root of
a number.
Factoring is a skill that technology has made almost obsolete. I am still
undecided about how much factoring of polynomials should remain in the curriculum.
With current technology, students can estimate roots of polynomials by graphing.
Factoring in the past was the best tool students had for finding solutions.
However, with graphing calculators, computer graphers, and the like, students
have a much more powerful, and better yet, visual tool.
Paper and pencil graphing is another task that technology has replaced.
I think however, that some paper and pencil graphing is still important.
Students need to know what the calculator or computer is actually doing.
As long as the paper and pencil graphing are used to increase understanding
beyond what the electronic graphers can convey, I think it should be done.
Often students see Algebra as a collection of numbers and symbols to be
manipulated, and never see how it is useful. For this reason, I believe
that it is important for students to see Algebra as it is applied to Geometry.
Many of the problem situations that students encounter will be geometric
in nature.
Geometry
TOPIC ATTENTION NEEDED
Two column proof Decrease and move to end of course
Three dimensional geometry Increase and move to beginning of course
Transformations Increase
Paragraph proofs Increase
Often Geometry courses begin the year with abstract two column proofs
of geometric theorems. Although I feel it is important that student learn
to use the deductive reasoning involved in two column proofs, I believe
that they are not prepared for it at the beginning of a Geometry course.
Proof should come from asking "Why?". Students should explore
geometric figures and discover for themselves how and why things happen.
Technology is a wonderful tool for geometric exploration. Proof should come
from these explorations. Students should first do "informal proofs"
by explaining their findings in their own words. They should discuss their
intuitive reasons for why things are true. This should naturally develop
into more structured paragraph style proofs as students become more adept
at expressing their ideas. Finally students could be exposed to the formality
of a two column proof. However, I think that student should always be encouraged
to express proofs in their own words and style. I think two column proofs
discourage creativity, and encourage "tunneled", "one-answer"
thinking. Therefore, a paragraph type proof would serve the same purpose
without stifling the discovery.
Three-dimensional geometry is often brushed over. This seems absurd in light
of the fact that we live in a three dimensional world. Students need to
use three dimensional figures and explore them as much as figures in "flatland".
Three dimensional figures provide more real world applications for students
to explore. Students are likely to encounter three dimensional problems
in places like art, architecture, engineering and many other places, and
should be comfortable with them.
Transformations are an excellent place to connect geometry to Algebra. Students
can represent point as ordered pairs, and transformations lend themselves
to good problem solving activities. Activities like tesselations can be
used to help students visualize and use transformations. Representations
of motion can be done here as well.
Algebra II
TOPIC ATTENTION NEEDED
Geometric applications Increase or Add to current Curriculum
Simplification of Radicals Remove from course
Factoring to solve equations Decrease Attention
Paper & Pencil Graphing Decrease attention in some areas
Operations on Rational Expressions Decrease Attention
Matrices Increase attention on uses
Abstract Algebra Increase attention
Logarithm Tables/Interpolation Remove from course
Trigonometry Tables/Interpolation Remove from Course
Solving Systems of Equations Change approach
Many of the topics that I addressed in the Algebra I course apply to
the Algebra II course as well. Geometric application, simplification of
radicals, factoring, paper & pencil graphing, and simplification of
rational expressions are all in need of the same attention as in the Algebra
I course.
In addition, Algebra II students should spend time learning about matrices,
operations with matrices, and applications involving matrices. They should
be able to represent graphs using matrices. The computer age has made discrete
mathematics necessary for information processing.
Algebra II students should also do some elementary study of abstract algebra.
The study of our number system and its development and properties is very
important to the study of higher mathematics. I think that students could
explore what happens to our current number system when one or more of the
essential properties are eliminated. The study of different systems as they
compare to the real number system gives students a feel for the structure.
This is also a good place for students to be involved in more complex deductive
thought processes. With the proofs involved, students can gain experience
proving abstract ideas. This topic lends itself to developing thinking skills,
and reasoning abilities.
The time of log tables and trig tables is (or should be) over. Calculators
make trig and log tables obsolete. Why should students have to flip pages
of a table, and do complicated operations to interpolate a value, when a
calculator gives a better approximation in less time?
This topic has no usefulness other than to obtain an approximation that
a calculator can give.
Solving systems of equations has traditionally be taught by manipulating
the equations around to obtain an answer. Students should spend most of
their time graphing to find solutions than manipulating the equations. I
think that they should still use algebraic methods, but they should not
be tunneled into using methods that are cumbersome, or that they are not
comfortable using. For example, one student might be more comfortable using
substitution, and one might like to use linear combination, most will not
enjoy determinants. Students should have options of different methods. They
will soon discover which method works best for them. What is more important
is students understanding that they are finding a graphical intersection
(or lack thereof) of the two (or more) equations.
Summary
Secondary mathematics curriculum has been through incredible scrutiny in
recent years. I think this is partially a result of low test scores, and
a national concern for competing internationally. Because the curriculum
is constantly changing, I think that we, as teachers, need to stay in tune
with changes in ideas, technology, and curriculum reform. Part of this process
includes taking the time to ask ourselves why we teach what we teach, and
how can I teach it better. This is an ongoing process that teachers should
engage in daily. If more teachers would involve themselves in changing and
growing with society, I think curriculum changes could be more rapidly implemented
in our schools.