Mathematics - American Education
In education, the science of numbers and all subtopics thereof, including arithmetic, geometry, algebra, trigonometry and calculus. Often used inaccurately as a synonym for arithmetic, mathematics is a far broader field that includes arithmetic as but one of its basic elements. The reach of mathematics education stretches from preschool to university graduate schools. Ideally, formal mathematics education begins in kindergarten with woodblocks and other manipulatives, which students learn to sort by size, color, shape and other characteristics. Kindergarten students also learn to count from one to 20, to tell time, to measure comparatively (bigger than or shorter than, for example) and to add and subtract single-digit numbers.
First grade math education extends counting to 100, calculation to double digits, measurement to a variety of media such as money, time, capacity, weight and temperature, and also introduces simple geometry and the concepts of estimating. Second grade extends numeration to 1,000, calculation into multiplication, division and the realm of algebra. Geometry is extended to three dimensions. Third grade introduces multiplication tables through 12 and multiplication and division of two- and three-digit numbers. Measurement of length, area, volume and weight is added to geometry. Students also learn the basics of probability and how to read graphs, along with the use of calculators and computers. use of calculators and computers.
In fourth and fifth grades, students progress to more advanced work, mastering all basic arithmetic functions, along with place values, ratios, rounding and approximation, decimals, fractions, graphing, the banking process, means, medians, modes and so on. Story problems introduce them to algebra, while the use of protractors and graph paper carries them into advanced geometry, and more complex computer software enables them to model two- and three-dimensional shapes. Sixth grade ends elementary school mathematics education, with training in the associative, commutative and distributive properties of numerical expressions; exponents, square roots and cube roots; percentages; formulas for calculating length, area and volume of geometric shapes and figures; the Pythagorean theorem and angle-sum theorem.
In middle school, seventh grade students study prealgebra and the use of computers for problem solving, while eighth graders study first-year algebra. Public high schools require only two or three years of mathematics for graduation in the academic track: Algebra I in ninth grade; Euclidean plane geometry in tenth grade; and Algebra II and trigonometry in eleventh grade. When offered, twelfth grade mathematics generally covers advanced algebra, trigonometry and precalculus. Academically advanced secondary schools compress the entire mathematics curriculum to permit the study of calculus in twelfth grade.
Although the above math curricula do not differ substantially from those in the rest of the industrialized world, their administration in American public schools had been far poorer than in educational systems elsewhere, and it fell short of the standards set by the National Council of Teachers of Mathematics. In 1988, for example, American 13-year-olds ranked last among students from Canada, Ireland, Korea, Spain and the United Kingdom. Nine-year-old Americans ranked ninth among students from 10 nations participating in the International Assessment of Educational Progress in 1991, scoring an average of 58% on a mathematics test. The other participating nations and their scores were Korea (75), Hungary (68), Taiwan (68), former Soviet Union (66), Israel (64), Spain (62), Ireland (60), Canada (60) and Slovenia (56). By 2000, the picture had not changed much, despite vast increases in federal, state and local spending on educational reforms. Although the United States spent an average of $8,855 per student on primary and secondary school education—highest in the world but for Switzerland, which spent $9,780— American 15-year-olds ranked only 18th in the world in reading literacy, 28th in mathematics literacy and 14th in science literacy. On the 2005 National Assessment of Educational Progress test in math, only 30% of American fourth and eighth graders scored at the “proficient” level, meaning they could perform multistep problems and showed familiarity with algebra. Nationally, colleges identified 22% of incoming freshmen as needing remedial math. Minority students present an even more dismal scoring record, with black high school seniors scoring 10% below their white classmates and Hispanic students scoring 7% lower than whites.
In standardized tests given to a representative sample of 250,000 American fourth, eighth and twelfth graders as part of the U.S. Department of Education’s National Assessment of Educational Progress, 75% scored below standards set by the National Council of Teachers of Mathematics in 2003. Based on three levels of achievement—“basic,” “proficient” and “advanced”—the tests found that 76% of American students met basic standards, but only 31% were proficient and 4% advanced. “Basic” was defined as partial mastery of mathematics ideas expected at specific ages; “proficient” meant mastery of all mathematics ideas expected at specific ages; and “advanced” performance represented superior work above minimum expectations. The results have been a significant—often dramatic—rise in the level of what American public schools deem proficiency in mathematics as measured by American government tests, with average proficiency for all students climbing 5.5% in the decade ending in 2000. The percentage of nine-yearolds (fourth grade) exhibiting proficiency jumped 60%; the percentage of 13-year-olds (eighth grade) 29%, and the percentage of 17- year-olds (twelfth grade) nearly 18%. Unfortunately, whatever American public schools and the designers of their mathematics tests defined as proficiency in mathematics still left American students woefully behind students in the rest of the industrialized world.
Critics of mathematics instruction in the United States contend it fails to teach students reasoning skills. Indeed, in another nationwide test administered to 250,000 students, the Department of Education’s National Center for Education Statistics found that only 16% of fourth graders, 8% of eighth graders and 9% of twelfth graders correctly answered mathematics questions requiring problem-solving skills. The test covered algebra, geometry, measurements, statistics and data analysis. Fourth graders were asked, for example, to use words and pictures to show how a boy who eats half a pizza may eat more pizza than a girl who eats half of a different pizza. Minority students fared far worse than white students. Among twelfth graders, 10% of white students did well, compared to 4% of blacks and Hispanics. Students in poor urban areas did far worse than those in more affluent areas, and private school students had substantially higher scores than public school students. Contrary to the myth that boys are innately more proficient in math than girls, there were no differences between male and female scores in any of the various age groups.
The demand for improvement in student problem-solving skills led to changes in the training of mathematics teachers and an adaptation of the way the math curriculum is presented. Students are encouraged to use calculators and computers at an early age to eliminate the drudgery of pencil and paper. The new approach relies less on memorization than on instruction in the basic concepts of mathematics. In simplest terms, third grade problems such as this: A boy has three apples and eats one. How many apples does he have left? have been replaced with problems such as this: Three students have two apples. How do we share? “Traditional math instruction for third- and fourth-grade students,” explains the National Council of Teachers of Mathematics, “emphasizes memorization, drill and practice.” Problems like the one below ask children to focus on finding the total, not on understanding what the numbers mean.
Susie has two quarters.
John has three dimes.
Joey has a nickel.
How much money do Susie, John and Joey have together?
New teaching methods use math as a way to reason, communicate and solve real problems. Problems like the ones below emphasize hands-on activity and real-life applications.
I have six coins worth 42 cents; what coins do you think I have? Is there more than one answer?
I have some pennies, nickels and dimes in my pocket. I put three of the coins in my hand. How much money do you think I have? Can you list all the possible amounts I have when I pick three coins?
Critics of the new approach to mathematics teaching have no argument against increased instruction in problem solving, but they disagree with the widespread use of calculators and computers in teaching basic mathematics functions they believe should be committed to memory. Moreover, they point out that poor school districts cannot afford to provide students with computers, while poor families can ill afford to provide their children with sophisticated calculators. Nevertheless, mathematics instruction is making a dramatic shift from a narrow focus on routine skills to development of broad mathematical skills such as performing mental calculations and estimates and recognizing which mathematical methods are appropriate for solving various types of problems.