U.S. mathematics students

Why Are U.S. Mathematics Students Falling Behind Their International Peers?
by: Kristy Vernille


Introduction
For more than 30 years, the United States has been concerned about the performance of their students in relation to their international peers.Since the appearance of Sputnik in the late 1950s, national leaders and educators have focused on the importance of helping U.S. students stay internationally competitive in mathematics.In 1964, the First International Mathematics Study was conducted, which gave us the first set of data as to how poorly the U.S. was performing in comparison to other leading industrialized countries.Almost twenty years later a second study was conducted, and about fifteen years after the second study came the third.All three of these international studies have shown the same results; the U.S. was not performing as well as their international competitors.The results of these studies have caused Americans to question their approach to the teaching of mathematics.
In this document I will be comparing the educational systems of the United States, France, and Japan.The reason for selecting these two countries in my comparison was that both countries out-performed the U.S. in all math subject areas on all three of the international studies conducted.I intend on identifying ways that the U.S. could improve their curriculum and procedures based on the information I have gathered on these two other countries.

Background
The First International Mathematics Study (FIMS) was conducted in 1964.This study included twelve countries, including the United States.It collected data on 13-year-olds and students in their final year of secondary school.The First Study was the International Association for the Evaluation of Educational Achievement’s(IEA) initial attempt to identify factors associated with differences in student achievement (Medrich & Griffith, 1992).
It wasn’t until about twenty years later, in 1981-1982, that the Second International Mathematics Study (SIMS) was performed.This Study was a comprehensive survey of the learning and teaching of mathematics in the schools of approximately twenty-four countries around the world.It was the results of this Study that “provoked considerable controversy when it revealed that American students were distinctly mediocre in mathematics when compared to their peers in most other countries”(Fowler & Poetter, 1999, p.1). Such controversy encouraged the writing of several documents addressing educational reform and the need for national standards.Some of these documents included A Nation At Risk: The Imperative for Educational Reform, which was written to help define problems troubling American education and provide solutions and recommendations for educational improvement, The Underachieving Curriculum:Assessing U.S. School Mathematics From An International Perspective, which highlighted the findings of SIMS, and the NCTM’s Curriculum and Evaluation Standards for School Mathematics (1989), which made an attempt to create a set of standard to guide the revision of the school mathematics curriculum.
In 1995, the Third International Mathematics and Science Study (TIMSS) was performed.The study mainly focused on the participation of eighth grade level students, but also gained participation at the fourth and twelfth grade levels.A total of 42 countries participated in the three grade levels and their achievement in Mathematics and science was examined.Two important documents that emerged a few years later as the result of SIMS and TIMSS, in which I would like to talk about more in depth, was the NCTM’s Principles and Standards for School Mathematics (PSSM) and the Mathematics Achieve Partnership’s Foundations for Success: Mathematics for the Middle Grades (FfS).

PPSM and FfS

From the results of the Second International Mathematics Study and also from the Third International Mathematics and Science Study, United States educators have seen a pattern among the top performing countries.The pattern is that most of these countries have nation-wide standards.Before the NCTM’s Standards of 1989 were released, the U.S. did not have such national standards.Hence, the production of the NCTM’s Principles and Standards for School Mathematicsand the Mathematics Achieve Partnership’s Foundations for Success: Mathematics for the Middle Grades.
Principles and Standards for School Mathematicswas intended to be a resource and guide for the decision makers who affect the mathematics education of students in PreK through grade 12 (NCTM, 2000).The document was produced by the National Council of Teachers of Mathematics (NCTM), an international professional organization committed to excellence in mathematics teaching and learning for all students.The NCTM’s three previous documents, Curriculum and Evaluation Standards for School Mathematics (1989), Professional Standards for Teaching Mathematics (1991), and Assessment Standards for School Mathematics (1995), were all important attempts by a professional organization to develop and articulate explicit and extensive goals for teachers and policymakers.The final document PSSM is a single resource that can be used to improve mathematics curricula, teaching and assessment.It is a hope of many that PSSM will give some direction for school curricula and that students will be challenged in a way comparable to international standards (NCTM, 2000).
In May 1999, Achieve and 10 reform-minded states announced the Mathematics Achievement Partnership (MAP), an extraordinary project designed to help states work together to raise expectations and measure results — using a common, internationally rigorous yardstick.MAP's work is grounded in TIMSS, which presents a sobering picture of how well U.S. students perform compared to their peers around the world.They found from the results that U.S. students do quite well in fourth grade, average in the middle grades, and fall to the bottom of the international rankings by the end of high school.The Foundations for Success: Mathematics for the Middles Grades document released its first draft in June of 2001.It attempts to provide the framework for what American students need to know to meet their potential for learning challenging mathematics in the middle grades.The expectations found in the FfS document are rigorous and will not be achieved right away, but the potential is possible for U.S. students who, by the fourth grade, already are performing among the top students internationally in mathematics (MAP, 2001).
Both the PSSMand FfS documents have good intensions for U.S. math students.With time, both MAP and NCTM believe that the standards set forth in their respective documents are attainable and will help students to rise above the international average when in competition with their peers.

Overview of Countries
Japan
Japan has a national curriculum that all elementary and secondary schools must follow.The schools are headed by the Ministry of Education, which is an administrative body responsible for school education.The Ministry supervises and finances local boards of education.They also prepare and distribute a course of study that forms a standard curriculum that all textbooks must follow (Miyake, Nagasaki, 1997).
The school year begins April 1st and ends March 31st.Most elementary and secondary schools are in session for 35 weeks or 190 days. Public school students attend school Monday through Friday and two to three Saturdays per month.Elementary school students attend 6 hours per day M-F, while secondary school students attend 7 hours per day M-F.All students receive four hours of instruction on the Saturdays that they attend (Miyake, Nagasaki, 1997).
Class size in Japan is rather large.In the elementary schools, class size is an average of 29 students.The lower secondary school, equivalent to our middle school, has an average of 34 students, while the upper secondary school has a class average of 40 students (Miyake, Nagasaki, 1997).
Teachers in Japan must have a four-year bachelor’s degree, with several courses in educational theory and pedagogy as part of their degree (Miyake, Nagasaki, 1997).
The Japanese education system is based on the idea that all children have the potential to learn, and therefore the curriculum in all subjects is virtually the same for all students through lower secondary school.Therefore, neither elementary nor lower secondary schools group students by ability level.Students that have difficulty keeping up hire private tutors and attend after-school and weekend courses (AFT & NCISE, 1997).
Japanese students have a very strong commitment to hard work and excellence.Families and the school system instill a strong work ethic.From the very early grades, the curriculum is designed to build a disciplined commitment to hard work.Japanese students know that this hard work will pay off and this attitude is nurtured and reinforced from a very young age (AFT & NCISE, 1997).
The high schools in Japan must be applied to and one must take a rigorous entrance exam.Admission is based on course grades and on test performance.This process is very competitive, so students are usually guided by their teachers to apply where they will most likely be accepted.For the lower achieving students, they usually attend schools with a more vocationally oriented curriculum (AFT & NCISE, 1997).
Japanese students who plan on attending a college or university must pass a college entrance exam.They see their entire schooling as preparation for the examination.Students understand that these tests are important to their future and understand that job opportunities and career advancement is better for graduates of the best colleges and universities (AFT & NCISE, 1997).

France


Schools in France are headed by the French National Ministry of Education.It is the Ministry that defines educational goals, programs, levels of diplomas, and appoints teachers.It mandates the number of hours elementary and secondary school students must spend in each subject at each grade level.For each subject, it outlines both general principles of instruction and specific concepts to be mastered.All students regardless of ability and achievement take the same national curriculum during their first five years of elementary school and the first two years of lower secondary school (the same as our grades 6 and 7).More than three –quarters of the students will continue through grade 8 and 9 with a common national curriculum (Servant, 1997).

The law states that the school year must be 36 weeks long.Primary school students have 26 hours of class per week, and lower secondary students have an average of 27 hours per week.At the secondary level, schools organize their time in various ways and the length of the day can also vary significantly during the week (Servant, 1997).
The average number of students in preprimary classes is about 27.Primary schools have an average class size of 23, lower secondary has an average of 25, and upper secondary has an average size of 22 in the public schools (Servant, 1997).
The basic requirement for primary and secondary teachers is a 3- year university degree.After obtaining their degree, preservice teachers must pass competitive examinations (Servant, 1997).
Students can earn the lower secondary diploma, brevet de college, at the completion of ninth grade.In order to earn this diploma, students must meet two requirements: complete four years of nationally prescribed common courses in the general track of lower secondary school, and sit for brevet exams in French, math, and history/geography.The diploma awarded is based in exam results and classroom performance in all subjects the last two years of lower secondary school.Students therefore have high motivation to take their coursework seriously since their course grades are of much importance in earning a diploma.Although the diploma itself is not a formal requirement, 90% of students take the exam.It appears that students must achieve a level of performance in their coursework equivalent to passing the exam if they wish to continue in the academic track at upper secondary school (AFT & NCISE, 1997).
Once in an academic track in upper secondary school, students have 3 more years to complete of schooling before going onto a college or university.In the first year of upper secondary school, students follow a common curriculum, which makes the course load quite heavy.Students have to be successful in this year in order to be promoted to the second year.This is not automatic, and approximately 78% move to the second year (AFT & NCISE, 1997).
If a student wants to go onto a university, they must earn the baccalaureat. This is France’s renowned secondary school diploma, which can only be earned by passing challenging national examinations at the end of 12th grade (AFT & NCISE, 1997).
United States


States have primary responsibility for education in the U.S.Historically, the federal government has played a limited role in areas such as curriculum and assessment.States and local districts have shared the majority of the responsibility in these areas, and each state has approached these responsibilities differently.Some states have defined a core curriculum and have developed statewide assessments.Others have entrusted local districts with these responsibilities.As a result of such a decentralized system, there has been a wide variation in curricula and performance expectations across the country (AFT & NCISE, 1997).

The U.S. does not have clear and rigorous standards.Therefore it is relatively easy for students to pass through the system.Students move from grade to grade without having to systematically demonstrate competency in any subject matter.Even though they take numerous standardized tests, their achievement on these exams does not usually affect their progress through the system (AFT & NCISE, 1997).
Most schools hold classes between the beginning of September and the end of May.The typical calendar year turns out to be 175-190 instructional days.The school week is Monday through Friday, and each day is approximately six hours long.In 1993-94 school year, the pupil-teacher ratios averaged 17 in elementary grades and 11 in secondary grades (Robeck, 1997).
Teachers may be certified in two different ways.The first way is to earn a four-year degree in education.The second certification can be obtained by earning an education certificate through one of two years’ study after a four-year degree in another area (Robeck, 1997).
The United States has the largest and most diverse postsecondary education in the world.Unlike Japan and France, U.S. students are not required to take common exams in order to gain admission into college.Since there is no single examination students must take, admissions officers use several tests in order to make their selections.These tests are the SAT I, SAT II, ACT, and AP (AFT & NCISE, 1997).

SIMS Findings
The Underachieving Curriculum: Assessing U.S. School Mathematics From an International Perspective report was intended to highlight the major findings of the Second International Mathematics Study.It is with this information, provided by this document, that I wish to compare the U.S. with Japan and France.

In Population A, which included seventh and eighth graders, Japan obtained the highest achievement scores of all countries in the study.The U.S. achieved slightly above the international average in computational arithmetic, and well below the international average in non-computational arithmetic.Achievement in geometry for the U.S. was among the bottom 25% of all countries.As for France, they scored well above the international average.
In Population B, which corresponded to the U.S. twelfth grades, Japan had the second highest achievement scores along with France achieving above the international average.As for the U.S. students, they scored below the international average.The achievement of the Calculus classes, which are the nation’s best students, was at or near the average achievement of the advanced secondary school mathematics students in other countries.The achievement of the U.S. precalculus students, which represent the majority of twelfth grade college-preparatory math students, was substantially below the international average.In some cases, the U.S. ranked with the lower ¼ of all countries in the Study and was the lowest of the advanced industrialized countries.
Other Research Findings


In all three of the International Studies of Mathematics, the results have been quite similar.The results from TIMSS show that in fourth grade, U.S. achievement is above the international average, in eighth grade, achievement is at or a little below the average, and by the last year of school, students’ achievement is well below the average.Anyone finding out this data would ask the same question, namely, why is there a continuous decline from elementary school through high school, and what can the United States do to better improve achievement?
In elementary school, the topics taught in mathematics are quite similar across the globe.Since U.S. students tend to get along well with the basics, when it comes to testing achievement, we usually perform above the international average (AFT, 1998).

The middle school years are where the research tends to show a sudden decrease in achievement.In eighth grade, our students are still studying topics that the rest of the world’s students have already mastered.“Mathematics instruction in the middle school years does not take previously taught content to more complex levels, nor does it introduce challenging material that prepares students for higher-level content in the eighth grade” (AFT, 1998, p.4).Therefore, when countries such as Japan and France are moving onto topics such as algebra and geometry, the U.S. is spending considerable time on whole-number computation and fractions and decimals (AFT, 1998).The TIMSS study concluded that the mathematical content, in textbooks and in actual practice, in the U.S. compared to other countries is less advanced.


How Are Other Countries Attaining Their High Achievement?


Japan
Over the years, American schools are constantly being compared to Japan’s schools.They are considered to be a “powerhouse” over us.So we ask the question, why?What is going on in their schools that is not occurring in ours?
Japan’s cultural emphasis on mathematics is a huge factor.Parents and society have great concern over high achievement, so they regard this subject as very important.The home is considered a powerful educational institution.Parents provide the motivation for their children to succeed at very young ages.They provide private tutoring when their children are not performing as well as others, or when they need help with passing entrance examinations into college or university (Dutton, 1977).
Another factor of Japan’s success is how much time they spend in the classroom.School is in session 5-6 days a week.Therefore, students in Japan average at least 8 more hours of schooling a month than U.S. students.(Miyake, Nagasaki, 1997).Research has also shown that Japanese students take more mathematics courses than American students do (Stigler, 1988).
Japanese superiority in math exists as early as kindergarten, and it is remarkable by the time the children reach fifth grade.The dominance of these students is not limited to basic computational skills but extends to nearly every math-related area that has been tested (Stigler, 1988).This supremacy can be credited to the amount of verbal explanation that occurs during a mathematics class.Japanese teachers constantly stop to discuss and explain the topic at hand.The teachers give, and ask students to give, lengthy verbal explanations of mathematical concepts and algorithms, opposed to American teachers who are more likely to stress participation in non-verbal activities or ask short-answer questions to lead students into a new topic.Japanese teachers not only explain more but also produce more complicated and abstract explanations than American teachers, especially in the first grade (Stigler, 1988).
Japanese classrooms appear to move at a move relaxed pace than American classrooms.Only teachers in Japan were ever observed to spend and entire forty-minute lesson on one or two problems.Japanese teachers seem not to rush through material but rather are constantly pausing to discuss and explain.They are well prepared, enthusiastic about helping children learn the topic at hand, and are continual in their efforts to secure pupil mastery (Dutton, 1977).The relaxed pace of learning in these classrooms, combined with the high level achievement, is a fact worthy of more reflection.Understanding takes time, and maybe spending that time in the early stages will lead to future benefits (Stigler, 1988).
The way Japanese schools evaluated students’ work is quite different from American schools.In Japan, if a student has produced an incorrect solution, they would be asked to present it to the entire class for discussion and correction.American teachers tend to evaluate work more privately than Japanese teachers.They were more likely to limit public evaluations simply to reporting how many problems were answered correctly (Stigler, 1988).
From Japanese observations, there were two important findings.The first one was that young children are competent of responding to, and understanding complex verbal explanations.The second finding was that it is possible to stress both concrete experiences and verbal explanations at the same time.In fact, it is possible that both are necessary to promote high levels of learning (Stigler, 1988).

France
France has also ranked above the United States in all three international studies.Researchers feel that the French example is potentially helpful to Americans for three reasons.The first is that they feel France is culturally closer to the U.S. than some other countries.Second, both France and the United States have very diverse populations.The third reason is that France has a relatively large gap between its upper and lower class, similar to the U.S. (Fowler & Poetter, 1999).
France has some practices and polices that benefit both advantaged and disadvantaged students.Children may enter preschool and attend an all day preschool program at the age of two.This is completely optional, however 90% are in attendance by the age of three.Even though the children play most of the day, the enrichment of language and the development of “rudimentary notions” on math are also part of the school day.The French believe that this highly developed educational program will help children from non-French speaking and poor homes enter first grade with a better chance for success in school than they would have had otherwise (Fowler & Poetter, 1999).
In French elementary schools, problem solving plays a central role in the teaching and learning of mathematics.“The French view the ability to understand and master math concepts in number and arithmetic, geometry, and measurement as keys for students to be able to solve problems that are new and about which students have little previous knowledge” (Fowler & Poetter, 1999, p. 12).In American schools, problem solving is typically viewed as doing “word problems” which challenge the student to pull out and isolate a math skill for operation.However, in France, they begin at early ages to advance mathematical thinking by teaching students to generate new methods for organizing and addressing a problem and for creating alternative solutions.Math skills are rarely taught in isolation from their relationship and application to genuine, life-like situations (Fowler & Poetter, 1999).

One way teachers evaluate student learning is through student notebooks.Each student has a separate notebook for each subject they have in elementary school.In their math notebook, students are expected to keep precise notes on math and document their math thinking about complex, well-situated problems from their own environment.The teachers, in turn, keep an ongoing record and dialogue with the students about math in their individual notebooks (Fowler & Poetter, 1999).
In French classrooms, students are expected to be attentive, conscientious, productive, thorough, and receptive.Observation of these classrooms have shown that these rules are held and taught in school and that this “regulative” discourse is strongly framed.Students, however, are also encouraged to be creative and interactive (Fowler & Poetter, 1999).
French teachers almost always use whole class instruction and expect every child to participate and to attempt to solve problems.They also organize each math lesson as a tightly structured “sequence” of activities that grow out of an initial, problematic situations.Most of the class time is devoted to whole group discussion and analysis of problems.The pace is very rapid and the students play an active role.When the time comes to working on individual notebooks, the students understand that they are to produce a solution to the problems and that, not only should they try to solve the problems correctly but also that their solution must be neatly presented and must correspond to a particular format (Fowler & Poetter, 1999).
French students spend most of their time in math class participating with the teacher in the development of concepts.When the students work individually, they usually “apply the learned procedures to new situations rather than either practicing routine procedures or inventing new ones” (Fowler & Poetter, 1999, p. 22).
Teacher recruitment and selection processes in France guarantee that knowledgeable professionals teach elementary mathematics.This is because the French school system that great mathematical learning takes place at young ages, and therefore, teachers have a very important role.The teaching is said to be strongly paced, and the teacher determines what the students will study and how rapidly they will progress while following the national curriculum.Historically, this strong pacing has been the major cause of a high retention rate in the elementary grades and into the middle grades.
Another reason why it is so important to have knowledgeable teachers is because teachers try to develop mathematical concepts rather than simply stating them.The French do not use a skill and drill approach to teaching mathematics.Their role includes acting as a discussion leader, selecting the sequence of problems to be studied and providing feedback to students (Fowler & Poetter, 1999).

Conclusion
Both Japan and France have national curriculums and nationwide assessments.President Clinton, in response to the poor showing of U.S. eighth graders on TIMSS, proposed that there be a voluntary national test of mathematics achievement of eighth grade students benchmarked to international standards (AFT, 1998).This proposal would at least attempt to encourage the U.S. to implement a national assessment as in the other two countries.
American students also need to have common curriculums.Both the PSSM and FfS documents are trying to accomplish just this.Research has found that the U.S. curriculum is dramatically differentiated at the eighth grade level.Four programs were identified in a research study, and each had extremely different math content.The U.S. curriculum is characterized by a great deal of repetition and review, with the result that topics are covered with little intensity (IEA, 1987).
American students need to become better problem solvers.Items on the international tests are open response and require that students show how they solve problems.U.S. tests are predominantly multiple-choice items that require little intellectual demand associated when determining an answer.
The French have a great deal to teach Americans about effective teaching and learning in schools.With the support of a focused national curriculum, appropriate pedagogy and assessment practices, and extremely knowledgeable teachers, French students develop math skills and problem solving abilities early in their educational careers.It is believed that as a result of their rigorous preparation in the early grades that French students consistently score higher on standardized tests (Fowler & Poetter, 1999).
The French example suggests that “one of the most effective ways to improve the teaching of mathematics in American elementary schools would be focus on teacher selection and educational processes.If the American states adopted policies which guaranteed that only people who are competent in mathematics and comfortable discussing it, raising questions about it, and helping children explore it taught in American schools, we would probably see a remarkable improvement in the mathematics achievement of American children within a generation” (Fowler & Poetter, 1999, p. 35)

A Math Paradox:

A Math Paradox:

The Widening Gap Between High School and College MathThree surprising reasons why

By Joseph Ganem, Ph.D.

We are in the midst of paradox in math education. As more states strive to improve math curricula and raise standardized test scores, more students show up to college unprepared for college-level math. The failure of pre-college math education has profound implications for the future of physics programs in the United States. A recent article in my local paper, the Baltimore Sun: "A Failing Grade for Maryland Math," highlighted this problem that I believe is not unique to Maryland. It prompted me to reflect on the causes.

The newspaper article explained that the math taught in Maryland high schools is deemed insufficient by many colleges. According to the article 49% of high school graduates in Maryland take non-credit remedial math courses in college before they can take math courses for credit. In many cases incoming college students cannot do basic arithmetic even after passing all the high school math tests. The problem appears to be worsening and students are unaware of their lack of math understanding. The article reported that students are actually shocked when they are placed into remedial math.

The article did not shock me. It described my observations exactly. In recent years I've witnessed first hand the disconnect between the high school and college math curricula. As a parent of three children with current ages 14, 17, and 20, I've done my share of tutoring for middle school and high school math and I know how little understanding is conveyed in those math classes. Ironically much of the problem arises from a blind focus on raising math standards.

For example, the problems assigned to my children have become progressively more difficult through the years to the point of being bizarre. My wife keeps shaking her head at how parents without my level of math expertise assist their children. My eighth-grade daughter asked me one evening how to perform matrix inversions. I teach matrix inversion in my sophomore-level mathematical methods course for physics majors. It is difficult for me to do matrix inversions off the top of my head. I needed to refresh my memory by pulling Boas' book: Mathematical Methods in the Physical Sciences off my shelf. Not exactly eighth grade reading material.

On another night my eighth-grader brought home a word problem that read: If John can complete the same work in 2 hours and that it takes Mary 5 hours to complete, how much time will it take to complete the work if John and Mary work together? That's an easy problem if you know about rate equations. Add the reciprocals of 2 and 5 and reciprocate back to get the total time. However it took me a lot of thought to arrive at an explanation of my method comprehensible to an eighth-grader.

My other daughter struggled through a high-school trigonometry course filled with problems that I might assign to my upper-class physics majors. I certainly wouldn't assign problems at such a high level to college freshmen. I kept asking her how she was taught to do the problems. I wondered if the teacher knew special techniques unknown to me that made solving them much easier. Alas no such techniques ever materialized. The problems were as difficult as I judged. At least I could solve the problems, a feat the teacher couldn't manage in a number of cases.

For example one problem involved proving a complicated trigonometric identity. My daughter brought it to me saying she had tried but couldn't find a solution. I saw immediately that the textbook had an error that rendered the problem meaningless. One side of the problem had a combination of trigonometric functions with odd symmetry and for the other side the symmetry was clearly even. I told her it was not an identity and that fact could be proven with a simple numerical substitution on each side. If it is an identity the equality condition must hold for all values of the angle. A single numerical counter example proves that it is not an identity. It only took one try to find a counter example.

The next day she reported to me that the teacher couldn't solve the problem.

"Did you tell him that it is impossible?" I asked.

"I told him it was not an identity and if he put numbers in he would find that out. He didn't believe me. He just said 'We'll see'."

The teacher never talked about that problem again. He did teach the class about the symmetry properties of trigonometric functions but evidently he didn't understand the usefulness of that knowledge.

At the same time I work the summer orientation sessions at Loyola College registering incoming freshmen for classes. Time and again students cannot pass the placement exam for college calculus. Many students cannot pass the exam for pre-calculus and that saddles them with a non-credit remedial math course--the problem described in the newspaper article. Without the ability to take college-level math the choices students have for majors are severely limited. No college-level math course means not majoring in any of the sciences, engineering, computer, business, or social science programs.
A colleague in the engineering department who also works summer orientation complained to me that many students who wanted to major in engineering could not place into calculus. The engineering program is structured so that no calculus means no physics freshman year and no physics means no engineering courses until it's too late to complete the program in four years. For all practical purposes readiness for calculus as an entering freshman determines choice of major and career. The math placement test given to incoming freshmen at orientation has much higher stakes than any test given in high school. But, the placement test has no course grade or teacher evaluation associated with it. No one but the student has any responsibility for or stake in its outcome.

Through the years I've found it discouraging as a faculty member to see so many high aspirations dashed at orientation before classes even begin. I tell students with poor math placement scores to go home, review high school math over the summer and take the test again. But, few take my advice. Most students with poor placement scores switch to majors that do not have significant math requirements.

So if eighth graders are taught math at the level of a college sophomore why are graduating seniors struggling? How can students who have studied college level math for years need remedial math when they finally arrive at college? From my knowledge of both curricula I see three problems.

1. Confusing difficulty with rigor. It appears to me that the creators of the grade school math curricula believe that "rigor" means pushing students to do ever more difficult problems at a younger age. It's like teaching difficult concerti to novice musicians before they master the basics of their instruments. Rigor-defined by the dictionary in the context of mathematics as a "scrupulous or inflexible accuracy"-is best obtained by learning age-appropriate concepts and techniques. Attempting difficult problems without the proper foundation is actually an impediment to developing rigor.

Rigor is critical to math and science because it allows practitioners to navigate novel problems and still arrive at a correct answer. But if the novel problems are so difficult that a higher authority must always be consulted, rigorous thinking will never develop. The student will see mathematical reasoning as a mysterious process that only experts with advanced degrees consulting books filled with incomprehensible hieroglyphics can fathom. Students need to be challenged but in such a way that they learn independent thinking. Pushing problems that are always beyond their ability to comprehend teaches dependence-the opposite of what is needed to develop rigor.

2. Mistaking process for understanding. Just because a student can perform a technique that solves a difficult problem doesn't mean that he or she understands the problem. There is a delightful story recounted by Richard Feynman in his book: Surely You're Joking, Mr. Feynman!: Adventures of a Curious Character, that recounts an arithmetic competition between him and an abacus salesman. (The incident happened in the 1950's before the invention of calculators.)

The salesman came into a bar and wanted to demonstrate the superiority of his device to the proprietors through a timed competition on various kinds of arithmetic problems. Feynman was asked to do the pencil and paper arithmetic so that the salesman could demonstrate that his method was much faster. Feynman lost when the problems were simple addition. But he was very competitive at multiplication and won easily at the apparently impossible task of finding a cubed root. The salesman was totally bewildered by the outcome and left completely discouraged. How could Feynman have a comparative advantage at hard problems when he lagged far behind at the easy ones?

Months later the salesman met Feynman at a different bar and asked him how he could do the cubed root so quickly. But when Feynman tried to explain his reasoning he discovered the salesman had no understanding of arithmetic. All he did was move beads on an abacus. It was not possible for Feynman to teach the salesman additional mathematics because despite appearances he understood absolutely nothing. The salesman left even more discouraged than before.

This is the problem with teaching eighth-graders techniques such as matrix inversion. The arithmetic steps can be memorized but it will be a long time, if ever, before the concept and motivation for the process is understood. That raises the question of what exactly is being accomplished with such a curricula? Learning techniques without understanding them does no good in preparing students for college. At the college level emphasis is on understanding, not memorization and computational prowess.

3. Teaching concepts that are developmentally inappropriate. Teaching advanced algebra in middle school pushes concepts on students that are beyond normal development at that age. Walking is not taught to six-month olds and reading is not taught to two-year olds because children are not developmentally ready at those ages for those skills. When it comes to math, all teachers dream of arriving at a crystal clear explanation of a concept that will cause an immediate "aha" moment for the student. But those flashes of insight cannot happen until the student is developmentally ready. Because math involves knowledge and understanding of symbolic representations for abstract concepts it is extremely difficult to short cut development.

When I tutored my other daughter in seventh grade algebra, in her words she "found it creepy" that I knew how to do every single problem in her rather large textbook. When I related the remark to a fellow physicist he said: "But its algebra. There are only three or four things you have to know." Yes, but it took me years of development before I understood there were only a few things you had to know to do algebra. I can't tell my seventh grader or anyone else without the proper developmental background the few things you have to know for algebra and send them off to do every problem in the book.

All three of these problems are the result of the adult obsession with testing and the need to show year-to-year improvement in test scores. Age-appropriate development and understanding of mathematical concepts does not advance at a rate fast enough to please test-obsessed lawmakers. But adults using test scores to reward or punish other adults are doing a disservice to the children they claim to be helping.

It does not matter the exact age that you learned to walk. What matters is that you learned to walk at a developmentally appropriate time. To do my job as a physicist I need to know matrix inversion. It didn't hurt my career that I learned that technique in college rather than in eighth grade. What mattered was that I understood enough about math when I got to college that I could take calculus. Memorizing a long list of advanced techniques to appease test scorers does not constitute an understanding.

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Joseph Ganem is a professor of physics at Loyola University Maryland and author of the award-winning book: The Two Headed Quarter: How to See Through Deceptive Numbers to Save Money on Everything You Buy, that teaches quantitative reasoning applied to financial decisions.

Previously Published by The Daily Riff 4/15/10Originally published by APS Physics.org


Related post on The Daily Riff:Why Other Countries Do Better in Math: Should Parents "Race to the Tutor?"

Why mathematics?

Mathematics is perhaps the purest of all the pure mental endeavors of humankind. During the times of Euclid and Pythagoras, mathematics was seen as a pure mental exercise that could deliver truth and nothing but truth. Today, this view is subject to a minor modification: the ``truth'' is related to a set of starting propositions (called axioms). An axiom can be, in some cases, completely lacking intuitive content and beyond intuitive or empirical verification. In most of the cases, axioms are, however, propositions which we take as self-evident. From this view, the relevance of mathematics to our topic can be seen the following two ways:

The (apparent?) sense of absoluteness and universality of mathematics on its own and its relationship to thoughts;

The efficacy of pure mathematical argument on physical reality (by way of sophisticated theoretic physics).

For one thing, mathematics is seen by many as an exact deductive science which has its own reality. But unlike other disciplines in natural sciences, they think, a theorem is absolutely and universally true. As long as a theorem is proved by a mathematician, all mathematicians should be able to prove (at least to verify) the theorem as well and the theorem is considered simply proved. The strong belief that mathematics forms a consistent unity may justify our calling it mathematical realism. In this sense, mathematical objects (such as numbers, theorems, proofs, etc.) exist on their own and have objective existence independent of the minds of mathematicians. We may call them ``mathematical reality.'' According to this position, the job of mathematicians, exactly as their colleagues in physics, is to discover the hidden reality, so that the truth can ``fall into place.''

It is indeed this fascinating belief that has raised an interesting question: what exactly are the rules of mathematical reasoning and why don't the outcomes contradict each other? This is a topic of mathematical logic. Many questions are answered positively in this domain – mathematically Interestingly, as by-products of this discipline, different ``logics'' have been discovered (or developed). For example, the first order intuitionist logic that turns down the law of double-negation can be still shown to be compact and complete. Nevertheless, there are also many puzzling and pessimistic results, for example Gödel's Incompleteness Theorem .

It turns out that the development of mathematical logic has in many ways also aided the growth of modern computer science -- formal language, automata theory, proof theory, and recursion theory, to name a few area strongly influenced by mathematical logic. Moreover, it was the ambition of a branch of computer science -- artificial intelligence (AI) that again brought to light profound problems about the definition of mind. This, no doubt, will have significant impact on natural language understanding and/or processing. In fact, it is because of our customary way of treating logic (indeed, classical first order logic) as a better way of reasoning (for some, it is the perfect way) and taking other everyday reasoning (non-monotonic, modal, context-sensitive) as frictional or impure forms thereof that has led to many difficulties in AI (see Chapter 1 for examples).

We have to see that mathematics plays a crucial role in our contemporary understanding of physical reality. In a sense, this role is active and somewhat tyrannical. For one thing, mathematics is not just a crucial tool for describing experiments or observation. Rather, the description and prediction power of mathematics is attributed to Nature's agreeing with mathematics. Einstein, for example, spent more than eight years of his lifetime devoted to the development of the General Theory of Relativity without the slightest clue from physical experiments and observations. The ultra-high agreement of the General Theory of Relativity to observed data in some areas  certainly suggests that it is not merely a matter of luck. There must have been something in Einstein's mind that held the key to the mystery of the universe.

Indeed, many important discoveries of today's physics are guided by mathematical theories rather than the other way around (Gedanken experiments with pencil and paper alone are in principle mathematical exercises). The role of experiments is to confirm or refute an existing mathematical theory. The job of experiment is therefore passive in this sense. An experimentalist physicist will not be surprised to see outcomes predicted by a mathematical theory. On the contrary, she is surprised when the phenomenon predicted by the theory is not there.

An observation of the power of logic/mathematics renders a naive sub-symbolic approach highly implausible. For one thing, the sub-symbolic school is an alternative view seeing frictionless reasoning as an idealized version of a more subtle classical physical activity and attacking the difficulties of AI from the bottom up. In light of the efficacy of mathematics and logic, it is hardly imaginable that a mental framework emerging from this classical substrate may give rise to a highly abstract understanding of multidimensional geometry, for example.

(Author unknown)