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

This Series (Arithmetic to Algebra through Pre-calculus) from MTE was designed to maximize student understanding and reduce the dependence on rote memory.

The Series, which emphasizes, A Critical Thinking Approach to Learning Mathematics includes the following:

	Competency Arithmetic
	Geometric & Measurement Topics
	Beginning Algebra	(1st year algebra)
	Intermediate Algebra	(2nd year algebra)
	Pre-Calculus	(Advanced Algebra & Trigonometry)

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The Texts and Materials of M.T.E. Ltd publishers are designed to Foster Reading, Understanding, and Computation -- not memory.

The Company motto is
"Striving for Student Understanding"
.
Student understanding is defined to be:
"The ability of a student to correctly state in their own words
mathematical concept learned".

The Problems of Traditional Mathematics Courses

Very few high school students enjoy learning math and even fewer of them are excelling at mathematics. Our colleges and universities have special programs designed to re-teach high school math, but students are still finding that it difficult to master math concepts under traditional approaches. Working harder under the wrong approach just doesn't always guarantee success.

So what's wrong - why isn't math fun, and why aren't students able to master mathematics?

The problem in traditional courses is created by using memorization as the predominate method for students to learn mathematics. Traditionally, instructors have relied on presenting examples, and upon hoping their students will memorize enough algorithmic approaches to pass an exam. Obviously, this doesn't work.

Traditional teaching has a second problem; it does not teach the language of math. Piaget stated that to understand a subject you must know the language of that subject. Memorization is a long way from understanding, and without understanding students will forget. Since most of the information that is memorized is stored in a person's "short term memory", students need a method to reconstruct concepts needed at a later date. The ability to read a math resource provides the possibility for this reconstruction. It is the forgetting of math concepts that has required colleges and universities to set up high school math refresher courses. Traditional approaches have relied upon review and then more review, but without understanding and the ability to reconstruct, there is often very little value in undergoing such reviews.

Traditional teaching has a third problem; students are not taught how to use critical thinking. Critical thinking analysis is needed to develop better problem solving skills. Students need help to develop critical thinking skills, it does not happen naturally; these skills are required to comprehend and develop a problem solving approach.

Following is a Report prepared by the
American Association for the Advancement of Science
(AAAS) April 26, 2002
About the problems with Traditional Math Textbooks

Entitled "Algebra for All - Not with Today's Textbooks"

Note all the textbooks reviewed in this report
were traditional

Washington, D.C.-The majority of textbooks used for algebra- considered the gateway to higher mathematics-have some potential to help students learn, but they also have serious weaknesses, according to a rigorous academic analysis by Project 2061, the long-term mathematics and science education reform initiative of (AAAS).

Seven of the 12 textbooks evaluated by the AAAS Project 2061 were considered adequate, however not one was rated highly. Five textbooks, including three that are widely used in American classrooms, were rated so inadequate that they lack potential for student learning.

"We need top-quality textbooks-not books that are barely acceptable- if we are to help every child understand algebra and eventually become competent in math and science," stated George Nelson, director of Project 2061. Many school districts now require algebra of all students and many of those taking algebra in high school are not able to pass college entrance exams causing them to retake this material over in college remedial courses - - wasting time and money.

"We do see solid improvement in some of the newer materials, but each book had major shortcomings," stated Nelson. "Unfortunately, the areas where most books are the weakest are those that are most critical in helping all students achieve, such as building on the knowledge that students may already have and dealing with their misconceptions."

This evaluation is the first to use Principles and Standards for School Mathematics, the recently-revised guidelines for what students should learn released earlier by the National Council of Teachers of Mathematics (NCTM). The standards are closely aligned with the learning goals of Project 2061's landmark Benchmarks for Science Literacy.

Highlights of AAAS Evaluation:

All of the textbooks present algebra using a variety of contexts and give students appropriate firsthand experiences with the concepts and skills; however, texts contained more material than can be presented in the allotted time and still ensure student understanding. Most traditional texts contain large sets of repetitive type exercises that have little educational value. This excess of material forces students and instructors to use memorization as the main method of learning and instruction.

Most of the textbooks do only an acceptable job of developing student ideas about algebra through representing ideas, demonstrating content, and providing appropriate practice. However, almost none have material that is designed to foster critical thinking or problem solving.

No textbook does a satisfactory job of providing assessments to help teachers make instructional decisions based specifically on what their students have-or have not-learned.

No textbook does a satisfactory job of building on students' existing ideas about algebra or helping them overcome their misconceptions or missing prerequisite knowledge.

The complete AAAS report is available on the internet.

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The Research Conducted by the Authors (CSBM).

The Critical Thinking Approach to Learning Mathematics Series are based on the results of nearly 20 years of research concerned with determining why students in high schools and college memorize math, but fail to understand most of the concepts. 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.

There seems to be two main factors that could be corrected in our teaching of math courses that would go a long way toward eliminating or reducing the current mathematics problem. These two factors are memorization and acceleration.

Memorization:

Students are trying to memorize all necessary facts that are needed to pass the course. Unfortunately, this is not possible for most students and such a course of action has no long-term benefits; hence causing the basic problem in mathematics.

Acceleration:

We are attempting to teach too much too fast in math courses. This, in turn, causes students to rely on memorization as their best hope to pass the course. We do not give them the proper amount of time to understand concepts, which reduces the need for memory.

SATISFACTON IS GUARANTEED on all M.T.E. products. If you are not satisfied with a M.T.E. product you have purchased, contact the publisher, in writing, within 6 weeks of the purchase date.

M.T.E. Ltd.
3095 South Trenton Street
Denver, Colorado 80231 - 4164
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In your letter, state the full name of the product, the MTE code number, and the date of purchase. You will receive a permission to return this product as soon as the information is verified (usually by return mail). You must pay the return shipping cost to M.T.E. If the product is in excellent condition, you will receive credit on your credit card or a check for the full refund. If the product is not in excellent condition, it will be returned to the sender.

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The Research Conducted by the Authors (CSBM)
Indicated Changes that Need to be Made
in the Traditional Teaching of Mathematics.
.

TEACHERS PREPARING TO TEACH A LESSON FROM
CRITICAL THINKING APPROACH TO INTERMEDIATE ALGEBRA

There is far too much content at each grade level, and every few years the amount of content is increased. Under these conditions, students are urged to memorize concepts without understanding in order to maintain an acceptable grade level on tests. Research has determined that, for most math students, the memorization of concepts without understanding is stored in the short-term memory and soon forgotten. A secret to student success in math is to reduce the amount of content at each level so that there is time to learn and understand the concepts. Unfortunately, depending on memory as the main method for learning math is having a very negative effect on students; increasingly, a large number of parents and teachers have found that students who used to like and enjoy math now hate and despise the subject. This is not good for a student or the future of our country."

The MTE research directors have been depicted as trying to stop a runaway roller coaster. The predominate guideline for mathematics curriculum the past few decades has been acceleration and this has resulted in all math courses having too much content. This means that most students must memorize math to pass the course. Unfortunately memorizing is short-term and not related to understanding mathematics. Yes, somehow we must stop this runaway roller coaster.

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Critical Thinking Approach to Correcting Problem
With The Curriculum.

The Curriculum of Critical Thinking Approach to Mathematics Series

Need to restructure and reduce curriculum

Over the years traditional texts have constantly added more and more content to each course causing instructors to "jam" too much content in too short a time. Traditional courses present an impossible number of concepts to be learned in one term, forcing students to memorize rather than understand concepts; this, in turn, forces much of the following term to be spent in review.

As an example of added more and more content to each course, consider the following:

My Father's Pre-calculus Text (Published in the late 1800's) is a text that has pages that are 5 inches by 7 inches and is 205 pages long. It was taught in a 16 week semester, 5 days a week, for one hour each day. This meant that the Instructor had 80 days of instruction and had to average about 2.5 pages per day.

One present day Pre-calculus Text has pages that are 8.5 inches by 11 inches and is 810 pages long. It is taught in a 14 week semester, 3 days a week, for one hour each day. This means that the Instructor had 42 days of instruction and must average about 19.3 pages per day.

Usually when I present this fact, people wonder what content has been added to the present day text. The early text contains almost every topic needed for pre-calculus course, but the truth is that some content has been added. In the late 1800's the course of Trigonometry was taught as a separate course because calculators had not been invented and most of the calculations at that time were done by logarithms and slide rulers. When the topics of logarithms and slide rulers are removed, the remaining content in Trigonometry is reduced to 2 Chapters. In the Critical Thinking Approach to Pre-Calculus Text the 2 Chapters of Trigonometry amounts to 64 pages with a page size of 7.5 by 10 inches.

In order to get a fair comparison, we will add the new trig content to my father's text, My father's text had 205 pages of 5 by 7 inches for an area of 7175 square inches and we add 4800 more square inches for Trig and get the sum of 11,975 square inches. Compare this to the total of 75,735 square inches in a present day text, A difference of 63,760 square inches more in the present day text.

The Critical Thinking Approach to Pre-Calculus has 354 pages as compared to the 810 pages of the modern traditional text. Other smaller modern traditional texts have 630 and 652 pages.

While this is certainly not a good scientific comparison, one can easily see that present day texts contain considerably more content than their predecessors or the Critical Thinking Approach to Pre-Calculus .

What then is this extra content?

Mostly review of previous courses and more examples and exercises.

The Curriculum of Critical Thinking Approach to Mathematics Series

The Critical Thinking Approach to Mathematics series of texts contain 70% or less that the amount of content in traditional texts. However, the Critical Thinking Approach to Learning Mathematics Series of M.T.E. contains all the necessary mathematical topics for each course, but eliminates the unnecessary and "frill and review" concepts in that course. The Units (Chapters) were determined by a survey of college and university professors. The professors were asked to determine which topics that incoming students should know, from Arithmetic to Pre-Calculus, to prepare these students sufficiently to enroll and pass Calculus or another university level mathematics course.

Units were subdivided into Sections, and then Objectives for each Section were developed. These texts were written using the Objectives as a guide.

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Critical Thinking Approach to Correcting Problem
of Reading Mathematics

Need to have more emphasis in reading math

TEACHING READING OF MATH FROM
CRITICAL THINKING APPROACH TO BEGINNING ALGEBRA

Students who have difficulty in math are often those who are not able to read their texts with understanding. These students cannot reconstruct math concepts after reading a math text and are handicapped by the inability to ask meaningful questions in class or be able to successfully review for upcoming tests.

Exercises for Vocabulary were designed and inserted in the Critical Thinking Approach to Mathematics materials at the end of each Section to help students learn terminology, both words and symbols, to be able to read and use reference material, and to be able to reconstruct concepts as needed.

Examples from the Critical Thinking Approach to Beginning Algebra Unit 9 page 27. In the Examples below, the answers are shown as italicized in parenthesis.

EXERCISES FOR VOCABULARY

Section D1

3. (positive integer) times a (negative integer) = (negative integer)

Section D2

4. Since the order in which two numbers are multiplied does not change the product, a positive integer times a negative integer is the same as a (negative integer)times a (positive integer)

5. To compute –6 times 3 we can (reverse)the order of the factors and determine the product of 3 times –6.

Section D3

7. Negative (integer) times a positive integer is another way to represent the opposite of a positive integer.

8. It is logical to assume that –1 times a negative integer represents the (opposite) of that negative integer.

9. One type of difficulty hat have with mathematics is that of not thinking od more than (one way) to interpret symbols.

Example 3 is an example of a question where students must extend the knowledge of their reading to answer the question. That is, while the concept is explained in the text, the question is not a direct quote of a sentence from the text. Students must use their knowledge of the concept to create an answer.

Example 8 is an example of a question involving symbols.

The Texts in this series are written at an appropriate level with understandable explanations.

For a more detailed paper concerning the importance of reading mathematics; consult Dr. Delano Wegener's paper on that topic: http://delwegmath.blogspot.com/.

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Need more emphasis on critical thinking and problem solving skills

Critical Thinking Approach to Correcting Problem
of Critical Thinking and Problem Solving

Some students who have difficulty in math because they cannot use critical thinking to improve their reasoning. They usually need help to improve their problem solving skills. The ability to solve problems and think critically is a valuable asset in seeking employment.

Exercises for Understanding containing discussion, explanation, and discovery exercises were designed and inserted in the Critical Thinking Approach to Mathematics materials at the end of each Section to improve critical thinking skills. If one is able to think critically, then problem solving is not a mystique.

There are three types exercises that foster critical thinking: discussion, explanation, and discovery, Some examples for understanding follow:

EXAMPLES OF DISCUSSION EXERCISES FOUND IN THE
EXERCISES FOR UNDERSTANDING

Unit 9

Section B6

51. On a series of 10 downs a football team made the following gains and losses: gained 7 yards, lost 2 yards, gained 5 yards, gained 12 yards, lost 6 yards, gained 4 yards, gained 16 yards, lost 17 yards, lost 6yards, and gained 12 yards. If they started on their own 20-yard line where are they at the end of these 10 plays? If you were the coach,what play would you call next?

Section F2

75. Discuss if and when the following statement is true: |a| = |–a|.

76. Discuss if and when the following statement is true: |b – a| = |a – b|.

***********

EXAMPLES OF EXPLANATION EXERCISES FOUND IN THE
EXERCISES FOR UNDERSTANDING

Unit 9

Section B5

53. If x represents an integer, explain why it would be correct to change"x + 0" to "x + (9 + – 9)"

Section F1

71. Explain the value of the expression |0| and why.

Section F2

74. Explain how –m may represent a positive number.

***********

EXAMPLE OF A DISCOVERY EXERCISE FOUND IN THE
EXERCISES FOR UNDERSTANDING

Unit 12

Section A2

78. Using as many as you need of the four basic operations, parenthesis and brackets, and exactly five digit fours; create as many notations as you can for the numbers 1 to 20.

Such as, a notation for the number seven is:

4 + [(4 + 4 + 4) ÷ 4].

***********

HOW TO EMPHASE PROBLEM SOLVING

If critical thinking skills are taught, because of the close relationship between critical thinking and problem solving, then presenting a few problems and guiding the students through problem solving techniques will suffice. In the Critical Thinking Approach to Mathematics Problem Solving is defined as:

Problem Solving is applied Critical Thinking.

In addition, a set of Problems for Problem Solving is included in the Instructor's Manual. This collection of problems, gathered from science and industry, is designed to give students practice in working with "real life" problems and provide an opportunity to use newly learned math concepts in a problem solving situation. This set of problems is designed to foster Critical Thinking skills by including an unfamiliar element as suggested by Dr. Polya of Stanford University. In the Critical Thinking Approach to Mathematics the problem is often completed in a series of "Challenges" where students are encouraged to verbalize and discuss their thinking processes.

MAKING DECISIONS IN BUSINESS
ABOUT EMPLOYEE MATH EDUCATION

***********

EXAMPLE OF PROBLEM FOR PROBLEM SOLVING FOUND
AT THE END OF THE INSTRUCTOR'S MANUAL

Unit 13

Section F

7. A builder makes and sales wooden platforms of 12feet, 18 feet, 24 feet, and 30 feet. The builder can save a considerable amount of money if the lumber used to build the platforms is all the same length so that there is no need to cut boards. What is the length of the longest board that can be purchased so that no board will have to be cut?

***********

CALCULATORS ARE SUGESSTED FOR SOME PROBLEMS
AN EXAMPLE OF THIS TYPE IS SHOWN BELOW

Unit 14

Section C

4. You receive a base salary of $20,000 a year. If your employer offers you the option of a raise of $1000 at the end of one year and $1000 per year after that, or $300 at the end of six months and $300 each six months after that, which should you choose?

Challenge 1 Guess which option will give you more money.

Challenge 2 Construct a 3-year table showing the 6-month amount of each option.

Challenge 3 Construct a second table showing the yearly of the two options..

Challenge 4 Determine any trends in the second table between the two options..

Challenge 5 Estimate the 5th, 6th, and 7th year amount for each option..

Challenge 6 Write a general equation for the nth year amount for each option.

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ADDITIONAL CONSIDERTIONS FOR
CRITICAL THINKING APPROACH
TO MATHEMATICS.

Need to continue emphasis on drill and practice

It is often said in the history of education that the "best" education took place in the one-room schoolhouse. If this is true, then it is because in a one-room schoolhouse education often came about as a result of older students teaching younger ones; both students gain from this experience. When a student is able to explain in his or her own words a math concept, that student understands the concept. The second secret to student success in learning math is to have students understand math concepts and demonstrate their understanding the concept by correctly explaining it in their own words to someone else. Homeschooling is a perfect environment for this to happen when a student can explain concepts to the parent instructor; in schools, this means a change in classroom procedure from an individualistic student approach to one of a small group student approach.

Committee Adopting Critical Approach to Mathematics Texts.

COMMITTEE ADOPTING CRITICAL THINKING APPROACH TO MATHEMATICS TEXTS

ABOUT THE INSTRUCTOR'S MANUAL

This Manual is designed to help Instructors, Teachers, and Tutors with the information they need to conduct a successful course. The Manual contains an Introduction with helpful teaching hints. A Unit by Unit commentary which includes: the purpose of each section, the Objectives of each Subsection, new vocabulary, answers to the Exercises for Understanding, the Solutions for the Problems for Problem Solving, a Complete set of answers to all exercises, and Student Copies for the Problems for Problem Solving.

ABOUT THE TEST KIT

University of Chicago Section of the Keller Plan played a major role in the initial development of the MTE test package. Since there were more than 1000 students enrolled each semester in the math program using the MTE Texts and because they all took a CRT Unit Test on the same day at the end of each Unit; it was necessary to be able to give and evaluate these tests over a weekend. In order to do this in such a short time period; it was necessary to use the multiple-choice format. On the other hand, we did not want the test to be "multiple-guess." It was in this connection that the University of Chicago staff were helpful enabling the development of valid, objective, efficient, and reliable tests.

Each Test Kit contains a Placement Test **, Criteria Reference Tests for each Unit, and a Final Test. The Placement Test can be used to determine if the student is properly placed in the series.

The Unit Criteria Referenced Tests will determine if the student has mastered the necessary content. The Final Test should be used as one part of a more comprehensive test with the other parts prepared by the instructor. The Test Kits have general instructions, a student copy for each test, and an answer sheet with other data.

The tests for the MTE Texts were developed and improved over a seven-year period. During this period over 11,000 students took various forms of the tests and a complete computer analyses of all student test results were made to increase their validity and reliability.

For a more detailed paper concerning the use and construction of tests; consult Dr. Delano Wegener's paper on that topic: http://www.delweg.com/dpwessay/tests.htm

FREE MATH HELP

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COMENTS FROM THE PUBLISHER ABOUT
CRITICAL THINKING APPROACH TO MATHEMATICS.

From the Publisher

This team of authors has taken the time and spent a great amount of effort to develop a series of texts with successful student learning as the main goal. Using an organizational scheme based on the objectives allows instructors and teachers to concentrate on any math concepts that students need to learn. The authors have incorporated understandable explanations; illustrations that clearly present concepts, and have included a variety of exercises that not only provide the necessary drill, but challenge students to think about math concepts. This series of texts is setting a new direction, marching counter to that of traditional texts which contain too much content; forcing students to memorize (not understand) as the only way to pass. This new direction in education taken by this team of authors is summed-up in their logo below.

THE M.T.E. LOGO

M.T.E. Ltd.
MATERIALS & TECHNOLOGY for EDUCATION
An Educational Math Publishing and Consulting Company

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	Changes that were Indicated by CSBM Research.		Critical Thinking Approach to Correcting Problem 1 The Curriculum.

	Critical Thinking Approach to Correcting Problem 2 Reading Mathematics.		Critical Thinking Approach to Correcting Problem 3 Critical Thinking and Problem Solving.

	What is Critical Thinking?		Additional Considerations for Critical Thinking Approach to Mathematics Texts Critical Thinking and Problem Solving.


	The Problems with Traditional Mathematics Courses		The Research Conducted by the Authors (CSBM)


	Publisher's Thoughts on the Critical Thinking Approach to Mathematics Texts

4.	Since the order in which two numbers are multiplied does not change the product, a positive integer times a negative integer is the same as a (negative integer)times a (positive integer)
5.	*To compute –6 times 3 we can (reverse)the order of the factors and determine the product of 3 times –6.*

7.	*Negative (integer)* times a positive integer is another way to represent the opposite of a positive integer.**
8.	*It is logical to assume that –1 times a negative integer represents the (opposite)* of that negative integer.**
9.	*One type of difficulty hat have with mathematics is that of not thinking od more than (one way)* to interpret symbols.**

75.	Discuss if and when the following statement is true: \|a\| = \|–a\|.
76.	Discuss if and when the following statement is true: \|b – a\| = \|a – b\|.

Challenge 1	Guess which option will give you more money.
Challenge 2	Construct a 3-year table showing the 6-month amount of each option.
Challenge 3	Construct a second table showing the yearly of the two options..
Challenge 4	Determine any trends in the second table between the two options..
Challenge 5	Estimate the 5th, 6th, and 7th year amount for each option..
Challenge 6	*Write a general equation for the nth* year amount for each option.**