Goals for high school Instruction in Mathematics
by Jerome Dancis
Dept. of Mathematics, University of Maryland, College Park, MD 20742-4015
Telephone: 301 405 5120 FAX: 301 314 0827 e-mail: firstname.lastname@example.org
In macho style, the president has declared that mathematics education in American will become the best in the world. But, education is not a horse race. Also, this is an amorphous goal. Our students and schools should not be bent out of shape so that our president, at international meetings, can brag: "My students are smarter than your students".
Doing mathematics has been an important human activity for millennium, mainly because it is so useful for explaining all sorts of things from the mundane, (how much is a 15% tip) to very abstract physics and all sorts of useful technical and non-technical things in between. This article will articulate important goals, with reasons, for reshaping high school mathematics instruction 1. These goals are not profound, but achieving them are crucial for fully viable education programs. We need high school mathematics instruction, which will enable all students to comfortably and competently use mathematics, in courses from engineering to high school chemistry to college sociology and nutrition. Students need to be trained to think their way through problems -- not happening today.
In Oct. 1998, the professional society of school mathematics teachers, the National Council of Teachers of Mathematics, (NCTM) released its Proposed Principles and Standards (for the next decade). The goals listed below are not included therein. At the Nov. 1999 meeting of the National Commission on Mathematics and Science Teaching for the 21 st Century (appointed by Secretary of Education Riley), Chairman (and ex-astronaut), John Glenn raised the question: What should be the goals of school education in mathematics and science? The goals listed below were not discussed. Thus, these goals are not even on the agenda.
Goal: #1. Significantly increase the number of students comfortable with the Algebra and arithmetic aspects of college freshman social science courses.
Tax dollars are being used to teach college students how to do this problem in the elementary nutrition course on my college campus. We have selective admissions; not open admissions.
Problem 1: It is a fact that fat has 9 calories per gram and protein has 4 calories per gram. If a piece of meat consists of 100 grams of protein and 10 grams of fat, how many calories does it have? Answer: 490 calories. This is not a trick question.
Class time, used for this instruction, reduces time available for teaching nutrition.
In elementary sociology classes on my campus, students struggle with "percents" in problems like:
Problem 2. A school has 400 girls and 1000 boys. Suppose that 25% of the girls and 15% of the boys have blond hair. How many students have blond hair? Answer: 250; not a trick question.
A 1999 Department of Education statistical analysis by C. Adelman, showed that success in algebra in high school, is the best predictor of a college student earning a college degree.
#2. Significantly increase the number of students comfortable with the Algebra and arithmetic aspects of high school chemistry class. It should become the rule, rather than an exception, that high school students study both chemistry and physics. Indeed, this is a technological age. Lack of fluency in algebra channels many students away from high school science classes.
"When Grant Scott, a biology teacher, had to teach his chemistry students at Howard High School how to change centimeters to meters, he just told them to move the decimal two places -- rather than illustrating the concept. ... 'Forty-five minutes later, only three of them got it.' ", 2 The new (1999) California Standards require that students learn this in Grade 4; after all 100 centimeters make a meter, just like 100 cents make a dollar and 236 centimeters = 2.36 meters, just like 236 cents = $2.36.
#3. Significantly reduce the number of students needing remediation in mathematics in college.
Knowledge and understanding of high-school algebra is both useful and important for understanding the simple mathematics and other quantitative concepts that arise in a wide variety of college freshmen courses and lack of such student knowledge puts an unfair burden on both instructors and students. (For examples, see Problem 1 and 2 above.) There are massive numbers of college students retaking high school level mathematics courses.
The numbers of students required to take remedial courses in mathematics, has increased from one in four in 1989 to one in two in 1998, in the California State University System (CSU). And CSU admission is restricted to the top 30% of California high school graduates!
Elementary Algebra I (Math 001) was the largest single mathematics course on my campus in the 1990s. Half the students in Math 001, had received a grade of B or A in high school Algebra II3. Having to repeat two years of mathematics, already studied in high school, is not limited to the weak students, it extends to many strong students also. Dr. Frances Gulick (a lecturer in mathematics at my college), noted that about one third of the students in her precalculus class had already "completed" calculus in high school.
The number of students who pass Algebra I and the number of students taking advanced math courses are two items on a list of accountability measures for my local school system. The number of students who learn Algebra is not on this accountability list.
For the Montgomery County 4 Public Schools' Algebra 1 test, some high schools required a student to score only 30 points (out of 100) to pass and only 65 points for an A. (All the middle schools required 60 points for passing). Students, who scored only 50% will likely need remediation in college, but that is not the high school's problem.
#4. Significantly reduce the number of students who cannot have their first and second choices for a career or college major because their high school mathematics instruction was weak. Dr. Dr. Lucy Sells had observed that arriving in college with a weak mathematics background has forced many college women into the traditional female fields of teaching, social work and the humanities. Large numbers of college students have to switch out of engineering programs because of low grades in calculus courses. A major reason for their difficulties/failure is lack of fluency in high school mathematics.
"Although large numbers of US students entering the universities say they are interested in majoring in technical areas, very few actually get such degrees today. The total number of technical degrees awarded to US citizens recently is approximately 28,000 yearly, while there are currently about 100,000 new jobs in these areas each year"5 .
#5. Significantly reduce the prevalence of mathematics phobias, especially word problem phobias. It's only in America that adults and children say "I am not good at math." This is not said by educated people in the rest of the world. Outside of mathematics classes, virtually all mathematics problems are word problems. Learning mathematics skills are useless if a student cannot correctly translate word problems into equations. (Correctly solving the wrong equation is counterproductive.) Many students develop mathematics anxiety, which results in the students employing mathematics and science course avoidance strategies, thereby greatly limiting their educational and career choices.
#6. Remove the pedantic from mathematics classes.
Students' self confidence and self esteem are lowered and their mathematics phobias are raised, very inappropriately, by pedantic practices. Some examples from each level of school:
In elementary school, a student is left bewildered after receiving a grade of zero after having calculated 10 exercises correctly, but placed the dollar sign on the wrong side: 58$ + 23$ = 81$.6
The subject was factoring; a child factored all numerators correctly, factored all denominators correctly, but wrote her mathematically correct answer as 2x(3 ab). This problem was marked wrong -- no credit; the teacher's answer was 6ab. Upon appeal, 5 of 25 points were awarded.
Proving the obvious, in long complicated ways, is a good way to turn students off. In Algebra 2, students were presented with a fourteen step proof that (1/a) x 1/b = 1/ab. 7
#7. Raise the average score on the Math SATs from about 50% (30 correct answers out of 60 questions) to 83% (50 out of 60).
Students inability to do Math SAT type problems was the first deficiency noted as a reason for poor success of students in college freshmen geography courses8. Being able to do Math SAT type problems is essential for all courses with some quantitative aspects.
The Math S.A.T. problems are just elementary one minute problems, requiring 15 seconds of thought, precise reading and knowledge of algebraic concepts. But thinking about how to solve a problem is missing from traditional mathematics instruction; mostly, the students are programmed to solve "textbook problems". This is a major reason for low Math SAT scores.
Fluency in Algebra 1, including how to use it, corresponds to being able to solve 83% (or 50 out of 60) of the Math S.A.T. problems. But this corresponds to a S.A.T. score of about 700 (the 97th percentile). It is absurd that only 3% of students can do this well; this demands better instruction. Of course students should learn how to solve more sophisticated problems, but it will be a significant improvement if students become proficient at solving these one minute problems.
#8. Students should become fluent in deductive reasoning.
Until the 1960's, students studied Euclidean geometry in high school. Starting with a small number of axioms, they proved and watched the teacher prove many theorems. In this way they were provided with extensive training in deductive reasoning. Students learned that a statement in plane geometry was true because they had seen a proof of its validity. Since then, 100 theorems have been renamed axioms and their proofs (now being redundant) have disappeared from the textbooks. Now a statement is true in plane geometry because the book/teacher says it is so (It is an axiom.). Deductive proofs have been exiled to the last quarter of the textbook; not enough for students to learn this topic. The teaching of deductive proofs in plane geometry is banned in some school systems. Students now arrive in college with little or no training in deductive reasoning, an educational handicap.
Prof. Barry Simon, Chairman of the Mathematics Department at California Institute of Technology in, "A Plea in Defense of Euclidean Geometry "9 , "mourned this loss of what was a core part of education for centuries." ... as he noted "what is really important is the exposure to clear and rigorous arguments. ... "They can more readily see through the faulty reasoning so often presented in the media and by politicians". Also, they would have less difficulty adjusting to and understanding college courses.
Fortunately, the new (1999) California Standards bring back deductive proofs as the basis of high school Geometry, at least for California students.
#9. Raise the amount of mathematics taught back to the level of the 1950s.
My children (in the fast academic track) were taught one third less mathematics in high school than I was (in the standard academic track) in the 1950s.10 The solid geometry course and half of the 1950s Trig course have disappeared. Serious training in deductive proofs and word problems has disappeared.
Prof. Barry Simon (again): "The dumbing down of high school education in the United States, especially in mathematics and science, is a crime that must be laid at the doorstep of the educational establishment".
The verbose, 700 page NCTM proposed standards do not even consider the question of raising the content of the mathematics curriculum back to the levels of the 1950s.
"There are other hidden, but measurable, costs. Laurence Steinberg, a psychology professor at Temple University, noted last year that his institution's requirement for two semesters of psychological statistics for majors is not a cause to celebrate high standards. Rather, it is an admission that it now takes two semesters to learn what used to be done in one".11
The goals listed here are not being met, or even attempted. This results in many students being prevented from pursuing the college major or career of their choice. High school and college courses are being dumbed down (educational deflation) to accommodate students' lack of fluency in basic mathematics.