Arithmetic-Geometry-Algebra-Pre-Calculus; Math Texts

CHOOSING A TEXT

AND

VIEWS ABOUT TEACHING

CRITICAL THINKING APPROACH

ON WHAT IS IMPORTANT

ABOUT THIS PAPER

This Paper contains three Parts.

The purpose of Part 1, written by Dr. Mel Poage, is to provide some guidelines for those trying to decide on which math text to choose. It is a practical guide on "How to select a math text". This paper is based on the researches that lead to the development of the Critical Thinking Approach to Learning Mathematics Series with results gathered over 20 years. These researches were concerned with determining why students in high schools and college memorize math, but fail to understand most of the concepts. Since textbooks provide the basis for the curriculum in most mathematical classes it is important to pick a text that not only fits a department's ideas about the curriculum they want to present, but maximizes the potential for most students to learn and understand math. It is the ideas that maximize the potential for most students to learn math that will be the focus of this paper. The results of this research indicated that there are five major areas that are related to textbooks that improving the learning of mathematics at the high school and college levels. In no particular order the five categories are: Understanding Math Concepts, Buttonology, Vocabulary and Reading Math, Length of the Text, and Computation.

Part 2, written by Dr. Mel Poage,is meant to focus some attention on the direction of mathematics education in the United States. With the very low US ranking on international math tests, such as the TIMSS test, it is time to set a new direction for mathematics education. It is suggested that more attention be given to the success of students in the next course as a measure of achievement.

Part 3, written by Dr. Delano Wegener, provides an insight to the commonly held views about the teaching math and the present day textbooks of mathematics. This "tongue in cheek" satire points out how far adrift the United States is today from the views held many years ago in this country and those held today in Europe, Russia, and China. There are reasons that the students in our country are not learning and understanding mathematics. Teaching that has only the mission to enable students to pass state required tests does not equate with learning and understanding math.

To go directly to Part 2

To go directly to Part 3

PART 1

UNDERSTANDING

Unfortunately memorized concepts are stored in the student's short-term memory and are soon forgotten. This circumstance causes students to fail or get low grades in math on college or university entrance exams; causing them, in many cases, to have to take an entrance level mathematics review course repeating most of high school math. This in turn, usually eliminates any chance of taking a university level math course needed for a degree that leads to higher paying positions. Students cannot be good at problem solving unless they possess more than a short term "rote memory knowledge" of mathematics; they must have an understanding of the concepts involved. In recent times the best positions in business and industry have been offered to the graduates with the highest potential as problem solvers. Studying mathematics in general, and math materials in particular, will help develop better problem solving skills for most students.

Understanding is the key to retention and mathematics is the gateway to many occupations. This gateway is ever widening in the number of occupations requiring some competency in mathematics. Polya describes solving problems as "the most characteristically human activity"; that is,"the specific achievement of intelligence, and intelligence is the specific gift of mankind ...". The importance of problem solving is not debatable, but what constitutes a "problem to be solved" in mathematics, has caused debate. Most authorities now agree that the correlation to determining the answer to a traditional word problem and solving a "real world" problem is limited. Traditional word problems seldom represent a "real world" problem because they only require a student to select and apply an algorithm recently presented in the lecture. Traditional word problems do not contain any unfamiliar element, a critical part of a "real world" problem. The problems should contain an unfamiliar element and be designed to provide groups of students experience discussing data, possible methods of solution, discussions of trade-offs, possible decisions and possible ways of presenting solutions.

BUTTONOLOGY

The second point is now called Buttonology (not mine); certainly our new and exciting technology cannot be ignored in the teaching of mathematics, but there is a major point being missed by many. If a student understands a math concept, then learns how to process this concept on a calculator or computer it is a win-win situation; however, if the reverse is used only button pushers are produced and the employment opportunities are small and the understanding of math is nil, a lose-lose situation. To teach Buttonology will take additional time, but just where this time should come from has not yet been satisfactorily answered. Perhaps, as in the case of computers, a totally new course of study should be offered,
"Graphic Calculator Science".

LENGTH OF TEXT

Often over looked when choosing a text is its length. To illustrate this I need to go out side the curriculum of most high schools because I was able to only find the all the evidence needed to show this problem at the Pre-Calculus level. My father became an outstanding electrical engineer went to university during world war 1; I went to university just after world war 2; in the table below note how the size of Pre-Calculus texts grew over the last century:

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My father's text	205 pages	16 week semester
class 5 days a week	80 days of instruction	2.5 pages a day

My text	310 pages	2 - 12 week quarters
class 5 days a week	120 days of instruction	2.5 pages a day

Today's text	810 pages	14 week semester
class 3 days a week	42 days of instruction	19 pages a day

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VOCABULARY and READING

Research, particularly that done by Piaget, has shown that in order to understand mathematics, or any other subject, a person must be able to read and write the language. A human learns most by reading; whether it is from a text or a computer screen. There are four levels in learning a language, only two of these can be addressed in a math text. The first level, introduction of terms, and the second level, pointing out correct use of terms in context, should be incorporated in the text materials. To introduce the language of mathematics all special terms should be put in boldface where they first occur and be summarized along with special symbols at the end of each Section or Chapter. To insure the correct use of terms, the text should include some type of Exercises to emphasize the Vocabulary. In addition to these Exercises, the student should be encouraged to read their mathematics text to obtain understanding of concepts and answers to other exercises.

If you are doubtful about the necessity of reading to learn mathematics you should read the paper "College-Algebra.Com __Mathematics Without Definitions" by Dr. David Chard. According to Dr. David Chard, "Vocabulary knowledge is as essential to learning mathematics as it is to learning how to read."

Many parents have pointed out that they never used their math after they left high school; true for many of todays students . Now the gateway is ever widening in the number of occupations requiring some competency in mathematics. What happens to the person who finds out, that due to this new technology, they now need some of their math. Many return to college without a lot of success. A better solution is to teach students how to read math texts; research has found out that a student who understands a concept and knows how to read a text can quickly re-teach themselves the missing concept. Unfortunately a student who has never been taught how to read a math text will never be able to do so.

COMPUTATION

Computation has been the main focus of texts in the past, and while not part of the research this topic must be addressed. Texts must continue this attention as an important aspect of the learning of mathematics. While the use of modern calculators is not discouraged, restraint of their use should not allow buttonology to take over. People are seldom hired because they can push a button, the knowledge behind the need for that button is far more valuable. Research has determined that it is possible for many students to use a calculator to get an answer without any understanding of the math used in the concept. Exercises in traditional texts, for the most part, are numerical problems which provide the necessary drill and practice for reinforcement of algorithms. This necessary because some students must know the "how" before they can understand the "why".

Some texts are sold because they contain a large number of exercises. There is no need to "drown" students with extra long homework assignments. Several exercises on different aspects of the concept being presented is sufficient.

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PART 2

THE DIRECTION OF MATHEMATICS
EDUCATION

About 50 years ago the United States was first in mathematics education among the 30 or so industrialized nations . Since that time our standing has been falling. At this time we now stand at or near the bottom of the list. The data for this is provided by the TIMSS (International Mathematics and Science Study) the most recent was done in 2007. Students from the final year of high school (12th grade in the US) were divided into two categories: regular mathematics and advanced mathematics. This division was based on whether or not students had completed at least a pre-calculus course. There were 21 countries that competed in the regular math division and the United States placed 19th. In the advanced division there were 16 countries that competed and the United States was 15th. Despite all the effort that has been made to change direction of mathematics education, there has been very little accomplished.

The past few white house administrations have made special efforts to improve education, in particular, mathematics education. Some of these efforts have caused nation wide testing programs and while doing some good this effort has resulted in most teachers teaching how to pass tests instead of teaching mathematics. I first met this phenomenon very early in my teaching career; I was hired to teach high school geometry in New York where they had the "regency tests". Being from Colorado I was very confused when all the other geometry teachers began to teach old versions of the regency geometry test starting in the beginning of October. Well not being too smart and without many of the teaching skills I would develop later, I taught geometry all year long. When spring came and all geometry classes took the regency test and I just knew my classes would do a lot better than the others; unfortunately no, but they did a little bit better. As I thought about it over the years I have decided my geometry students at least knew more about geometry and its logic.

I heard on TV an idea that I think has great merit, one that could change the direction of mathematics education; one with which I have had some experience. This idea involves tracking students to see how well they perform in subsequent mathematics classes. Teachers can then be rewarded on how well their former students do; this reward would be in addition to their regular salary. This idea would concentrate the focus of teachers on making mathematics more understandable to students. As mentioned in various places in this web-site I was very fortunate to be involved in the CSBM (Continuous Sequence in Basic Mathematics) research. My experience with the tracking idea came as part of this research.

The student population for the CSBM research consisted of students who did not pass the University's Math entrance exam. Most of these students were pursuing nonmathematical or nonscientific degrees; those who took only the required mathematics courses. Students who passed the entrance exam were exempt from the CSBM program. I believe of one of the best indicators of the success of the Critical Thinking Approach to Mathematics used in the CSBM research was a research done by the Mathematical Association of America (MAA). During the time that the CSBM research was conducted it was the practice of the MAA to send teams to evaluate the strengths of Math Departments that requested this service. The MAA team that evaluated our university was unaware of the existence of the CSBM research project. After evaluating the success of some CSBM students the MAA team decided to do a longer more extensive study involving tracking students into math courses after CSBM. They decided to compare the final grades of CSBM students with non-CSBM students taking Business Calculus courses for the following reasons: it was easy to track the students, no CSBM professors were teaching these courses, and the study could be conducted by some of the universities own professors. The anticipation was that the non-CSBM students should do better because they did better in high school math in general than the CSBM students.

COMPARISON OF FINAL GRADES
IN BUSINESS CALCULUS

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	First Year	Second Year	Third Year
non-CSBM	816	532	983
Mean GPA	2.31	2.43	2.55

	First Year	Second Year	Third Year
CSBM	44	139	146
Mean GPA	3.33	2.88	2.95

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All were pleasantly surprised at the results of the MAA study.

Again the CSBM research project concentrated on the non mathematical student population (not the top 20%, but the bottom 80% of college students). The CSBM project had in its last eight years a success rate of 85% - 90% of the students who entered the program completed the program; the program required a passing mark of 87.5%. The student population for this period was almost 3000 students a semester. More than half of the graduates of CSBM did, by their own choice, enroll and successfully complete a calculus course. The University of Ohio was contacted and ask to prepare and grade a Locus of Control survey to measure the CBSM student attitudes. The results of this survey indicated that the CSBM graduates had a favorable (75% - 80%) attitude toward mathematics and the CSBM project.

PART 3

Doing a Different Equation

12 NOV 2009

This morning a cable news show reported that the number of deaths in the US due to H1N1 flu was 4000 rather than 1000 as had been reported earlier. The reporter went on to explain that the 1000 figure was the result of counting laboratory confirmed cases of H1N1 which resulted in death whereas the number 4000 is reached when all cases of H1N1 as identified by health professionals are counted.

AT that point the anchor took over and felt compelled to clarify and summarize the story with the following statement.

There aren’t more H1N1 deaths than before, it’s just that
we are doing a different equation.

The anchor’s statement is illustrative of broadly held misunderstandings and general ignorance of mathematics.

1. Counting is mathematics — mathematics is counting.

The anchor immediately thought of mathematics (equations, variables, and all that stuff) as soon as the subject of counting came up. Why on earth should the subject of counting conjure up visions of equations? Only ignorance of mathematics can be responsible. It is not explicit in his statement, but thirty years of experience prompts me to guess that he also thinks mathematics is about numbers only.

2. We do mathematics.

Listen to the words when students, teachers, parents, and the general public speak about mathematics. Compare those words and that language to the words and language used when other subjects are discussed.

Students study history, social studies, language arts, etc. They read literature, history or other subjects. They neither study nor read mathematics, but rather they do mathematics.

Teachers encourage their students to do their math every day to avoid falling behind. Parents check to insure their children have studied their spelling, read their literature, studied their history, and done their math.

Students and ultimately all of our society learn from this language and come to view mathematics as something we do. We think of mathematics more in terms of a motor skill rather than an intellectual activity. In fact, the impression that mathematics is a motor skill is so prevalent, that most people and indeed many teachers believe one learns mathematics by working hundreds of insipid problems listed in textbooks.

3. Everything mathematical is an equation.

A common response when shown any mathematical expression is to attempt to "solve it". It is a commonly held belief that mathematics consists entirely of equations. Such simplistic and completely wrong impressions of mathematics overlook the more important aspects of mathematics:

a) Mathematics as an intellectual activity,

b) Mathematics as an application of deductive reasoning,

c) Mathematics as a study of structures,

d) Mathematics as a precise, descriptive, detached language.

4. Equations always elicit the same response — do it.

It is a commonly held belief that mathematics consists of equations and it is believed that what one does with an equation is "find x".

When the question "What is the degree of the polynomial in the equation 3x³ + 2x² – 7x = 0?" is presented to a College Algebra class, sixty to seventy percent of the students will attempt to solve the equation.

Mathematics as an application of deductive reasoning,

There is little appreciation that in an equation such as 3x + 2 = 8, the variable x may be any number, but that some of those numbers make the statement true and some make the statement false.

In mathematics classes there is no knowledge nor interest in the relation between an equation such as 3x + 2 = 8 an its twin siblings 3x + 2 < 8 and 3x + 2 > 8 whose existence is guaranteed by the Law of Trichotomy and whose solution sets are related and delineated by that same Law of Trichotomy. For the majority of students and hence the majority of society they are three unrelated problems. I suspect that is because they are usually presented in totally unrelated and widely separated chapters in most algebra books.

Similar comments can be made with respect to virtually every topic in mathematics.

The fault does not lie with the students, the fault lies with teachers and what is promoted as mathematics education.

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After reading Dr. Del’s paper; Dr. Mel had this after thought.

Of course we DO mathematics!!!! We never "READ" math, or "THINK ABOUT" math, where ever did you get such silly notions. We also never teach math students: to DISCUSS math as one does in social studies or political science, "to "PRACTICE" math as one does music or sports, to use HANDS ON in math as one does in shop, home economics, chemistry, physics, or biology, or to "CREATE" math as one does in art and writing.

But; maybe we should.

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