Timothy W. Young

I WORKED FOR THIRTY-EIGHT YEARS IN THE PENNSYLVANIA PUBLIC EDUCATION SYSTEM AND RETIRED AFTER SERVING AS A HIGH SCHOOL PRINCIPAL FOR TWO SCHOOLS – CHARTIERS VALLEY HIGH SCHOOL, BRIDGEVILLE, PA AND GLENDALE JUNIOR/SENIOR HIGH SCHOOL, FLINTON, PA. I WAS ALSO A SECONDARY MATHEMATICS TEACHER – TYRONE AREA HIGH SCHOOL, TYRONE, PA AND CHARTIERS VALLEY HIGH SCHOOL, BRIDGEVILLE, PA.

Late in my public education career and after I retired, I became obsessed with mathematics education in the United States and changes which need to be made in order to help all students become successful in arithmetic and mathematics, starting in grade school through high school. Some of the methods used to teach arithmetic and mathematics in the United States must change, based on the "TIMSS" and "PISA" test scores for high school students.  The "TIMSS" and "PISA" tests are International Math Tests - and students in the United States frequently (always) score in the lowest quartile for high school students.  (Change needs to occur at both the university/college level when students are preparing to become elementary arithmetic teachers and secondary mathematics teachers, and change must also occur with in-service/professional development programs for teachers who are already teaching students.)  If this pattern continues the United States will fall further behind other industrialized countries with respect to technology, unless we import mathematicians and scientists, which is becoming increasingly more difficult because countries whose students score high on these tests are less willing to allow their best math and science students to leave their home country.  Therefore, the consequences for the United States in the future could be devastating. 

Shortly after I retired from Public Education, I became friends with Dr. Robert Siegler, who at that time was a Professor at Carnegie Mellon University. Dr. Siegler has devoted over 48 years of his life researching how students learn arithmetic and mathematics throughout the world.  After retiring from CMU he accepted a similar position at the Teachers College, Columbia University. He has continued his research and presently he has authored 9 books, edited 6 others and authored or co-authored over 250 articles and research papers.  For the past 25 plus years, his focus has concentrated on fractions (rational numbers) in arithmetic and the long-term effects students experience in advanced mathematics if they do not master concepts and procedures involving fractions.  For six years after I retired, I was an invited guest of Dr. Siegler’s to attend his weekly math research discussion meetings at CMU and other various meetings and speaker presentations at CMU. While attending these meetings and working directly with Dr. Siegler, I realized how much students struggle with all types of fraction, decimal, and percent problems in the United States compared to students in many other countries. I continually ask myself the question, “As an experienced math educator in the United States, "How would I teach fractions and arithmetic concepts starting with third and fourth grade students in arithmetic and then use the same rules and procedures to teach math to students in high school?" The goal for all arithmetic and mathematics teachers must be, "To have all students understand concepts involving all arithmetic and mathematics problems so the concepts discussed and presented help all students determine the correct rules and procedures to solve basic problems." Students must then master the rules and procedures used to solve basic problems, in order to be successful solving all types of problems, simple to the most difficult, in arithmetic and advanced math classes." Understanding arithmetic and mathematics concepts is an important first step, but if the rules and procedures required to solve problems are never "Learned and Remembered" then students are in trouble when they enter a higher-level math class.  Dr. Siegler’s research and the research of many other individuals around the world inspired me to write my textbook which is published and available to purchase on Amazon, titled “ONE THE MOST POWERFUL NUMBER”.   I suggest you google "Dr. Robert Siegler at the Teachers College Columbia University" and read some of his research papers and a book or two in order to understand challenges which teachers and students encounter every day in the classroom as it relates to arithmetic.  

The two main focus groups for my textbook, "ONE THE MOST POWERFUL NUMBER" are:  1.  Students in college planning to be an elementary teacher and 2.  Students in college preparing to be a secondary math teacher.  I have asked the following question to many elementary arithmetic teachers and secondary mathematics teachers, "Did you ever take a course in college which focused on preparing you to teach rules and procedures for the arithmetic or mathematics topics which are required in your classes?"  Teachers often respond that they took a course in college with a title similar to the following: "How to teach arithmetic or math", by focusing on teaching CONCEPTS, not how to teach RULES AND PROCEDURES.  The teaching point I am trying to emphasize is, "Every college student who is preparing to be an elementary arithmetic teacher or a secondary math teacher is required to take a course which will help them explain math concepts when they are in their own classroom, but very few students ever took a course which helped them prepare to teach students rules and procedures."  WOW, the last comment may be something everyone should start to think about.

With this thought in mind, it explains why most young teachers emphasize concepts, which is an extremely important first step when teaching arithmetic and mathematics, but it is not the final step in the teaching process if students are going to be successful in future arithmetic and mathematics classes.

The textbook, "ONE THE MOST POWERFUL NUMBER" fills the void between teaching concepts and teaching rules and procedures.  The textbook is a "How to teach arithmetic and math topics textbook focused on teaching rules and procedures."  After concepts have been repeatedly illustrated and the students understand why, then it is time to teach the next important step in the learning process which must be learned, if students are going to be successful in future arithmetic and math classes next year and beyond.  Teaching rules and procedures is not as fun as teaching concepts, and it requires special teaching skills which are not normally taught or learned in college. 

If you have questions, please contact me through my primary email, "twypitts1@gmail (obviously .com)".  There is a second book which is also available to purchase on Amazon titled "THE MOST POWERFUL NUMBER 1".  This book is a professional development book for experienced teachers, which covers the same topics, but the book is written in a slightly different style and there are problems for teachers to discuss in groups.  The college textbook has around 1000 problems which college students should solve and write detailed explanations.

When you enter the classroom the first time as the arithmetic or math teacher, and you illustrate the concepts for the topics required for your class then you must rely on your past experiences to teach the rules and procedures to solve the problems in the book.  Your past experience is usually based on how you learned and how your teachers taught you.  Hopefully, with experience, you develop you own approach to teaching rules and procedures for the topics you teach.  One important teaching point to keep in mind when you are teaching rules and procedures in any arithmetic or math class is, "Make sure the rules and procedures you are teaching are the same rules and procedures students will work with when they are in a higher-level math class next year or in 6 years."  If you are teaching tricks to solve simple problems in arithmetic or math class, "Will your students be able to use your tricks to solve more challenging problems?"

My textbook introduces two new theorems and uses these two theorems to help teachers teach the rules and procedures required to solve/simplify all types of fraction problems and conversions problems which students initially encounter as early as 4th grade but usually in 5th grade and then use the same rules and procedures to solve rational expression problems in algebra, trigonometric identities, calculus limit and derivative problems, etc.  The rules learned in arithmetic are the basis for the procedures required when students are in higher level math classes. The arithmetic rules never change the problems just become more exciting.

If college professors use my textbook as a template to help future elementary teachers and secondary math teachers to teach rules and procedures before they go out and do their student-teaching, they will have some new ideas about teaching rules and procedures which they may not have experienced as student when they were in elementary school, middle school, or high school.  The textbook is full of illustrations and written explanations about how to solve problems along with multiple student assignment questions.  

The course would be a semester course, and not only would I suggest that the students study the illustrations and solve all the problems in the assignments, but they should also read some of the research papers indicated throughout the textbook.  I learned a great deal about student problems when I read these research papers and only wish I had read more research math articles when I was a classroom teacher. 

The college textbook is a black covered book, and the professional development book has a white cover.

I have talked to over 50 high school math teachers who have taught Algebra 2, Precalculus, and/or Calculus, and asked them if they felt that many of their students frequently struggle when they start to solve/simplify rational expression problems.   Over 90% of these teachers agreed that this topic is one of the more challenging topics for students to master.  Keep in mind that there are 3 or 4 other topics which teachers identify which creates teaching challenges starting in Algebra 2 such as the Quadratic Formula, Logarithms, Conic Sections, etc. but we all also agree that mastering the procedures and rules to work with rational expressions is a requirement if students are going to continue and be successful in higher-level math courses. 

IMPORTANT COMMENT: 

Usually when I talk to math teachers, we end up agreeing that the rules taught in a normal elementary arithmetic class to solve and simplify fraction problems ARE NOT the same rules and procedures which students must use to work with rational expressions in algebra and higher-level math classes.  In arithmetic the students are expected to do a lot of work in their head, based on their knowledge of the multiplication and division facts, such as finding a common denominator then convert the given fractions into equivalent fractions with the new common denominator, when the students are required to add, subtract or compare fractions.  When students are required to reduce, multiply, or divide fractions they again will use their mental math skills to determine if there is a common number which will divide into both a number in the numerator and a number in the denominator, in order to make the numbers in both situations smaller prior to actually multiplying the numerators and denominators separately.  The final step in all arithmetic fraction situations is to determine if the end fraction can be written as a mixed number.

In algebra and higher-level math courses when students work with rational expression or identities or limits, etc.  The first step which is normally required when adding and subtracting fractions is to FACTOR all the denominators and then use the factors to determine the Least Common Denominator.  Next, find equivalent fractions by multiplying by a factor or factors based on a new set of rules which are not emphasized in arithmetic.  When reducing, multiplying or dividing challenging rational expressions the first step is to factor all the numerators and denominators in order to determine inf there are identical factors in both the numerators and denominators.  Wow, students never learn how to solve fraction arithmetic problems by focusing on factor8ng first.  This is a brand-new way to work with fractions and the fractions not only involve numbers, but the algebra fractions will involve numbers, variables, expressions, exponents, etc. Wow, no wonder students, even good students frequently struggle when they work with rational expressions the first time.

If you are an experienced secondary math teacher, you know what I am talking about.          "TWYPITTS1@GMAIL.COM"      (Small letters are fine.)

THIS IS WHY I WROTE THE SECOND BOOK, SO EXPERIENCED TEACHERS CAN READ AND STUDY THE PROCEDURES ILLUSTRATED THROUGH-OUT THE BOOK TO SOLVE CHALLENGING ARITHMETIC FRACTION PROBLEMS WHICH EMPHASIZE THE RULES REQUIRED TO WORK WITH RATIONAL EXPRESSIONS IN ALGEBRA AND THINK ABOUT HOW TO INCORPORATE THESE CHALLENGING ARITHMETIC PROBLEMS INTO YOUR REGULAR CLASSROOM LESSONS.

The fraction problems and other arithmetic problems illustrated in the book will prove to be a list of excellent topics for several in-service days for both elementary arithmetic teachers and secondary math teachers.  Superintendents, Principals, and all other school Administrators need to read and study the professional development book so they will have a solid understanding of the problems discussed in the book and the problems teachers encounter on a daily basis.  The book is written using basic terminology so anyone who wishes to learn how to help students learn to work with a wide variety of math problems will acquire some knowledge and teaching strategies which will help you as a teacher help your students be successful.

THE ULTIMATE GOAL IS TO MAKE SURE ALL YOUR STUDENTS "LEARN AND REMEMBER" HOW TO SOLVE A WIDE VARIETY OF ARITHMETIC AND MATH PROBLEMS THEN SCORE WELL ON ALL TEACHER TESTS AND EVEN THOSE NASTY TESTS REQURIED BY MANY STATES SOMETIME DURING THE SCHOOL YEAR.  

If you follow the teaching suggestions constantly illustrated and discussed throughout the book, you will not have to worry when your students take a standardized state math test, your students will amaze you and they will be proud of their accomplishments.  Which is the ultimate great feeling for any teacher.

WITHOUT CONCEPTS IT IS IMPOSSIBLE TO DEVELOP ARITHMETIC RULES AND PROCEDURES AND WITHOUT ARITHMETIC RULES AND PROCEDURES IT IS IMPOSSIBLE TO DEVELOP RULES IN ALGEBRA AND ADVANCED MATHEMATICS CLASSES.  "WITHOUT ARITHMETIC RULES AND PROCEDURES IT IS IMPOSSIBLE TO SOLVE ADVANCED MATHEMATICS PROBLEMS.  ARITHMETIC RULES AND PROCDURES ARE THE FOUNDATION TO UNDERSTANDING ADVANCED MATHEMATICS RULES AND PROCEDURES."

Based on the last sentence the arithmetic teacher is the most important person in a child's life if they are going to be successful in advanced mathematics classes.