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.