Introducing algebra in the 8th grade and earlier is only one of several changes in the U.S. math curricula that are needed to improve the math performance of the nation’s students, write the authors of a study included in Lessons Learned, What International Assessments Tell Us about Math Achievement, recently published by the Brookings Institution Press.
In other countries, algebra and geometry have been an important part of the middle school curriculum, not only for the elite, but for all students, the researchers write.
“Positioned for more than a century in the ninth grade, Algebra I recently began to move into the eighth grade, partly as a consequence of research showing that other countries were offering more algebra earlier than the United States was,” say researchers Jeremy Kilpatrick, Vilma Mesa and Finbarr Sloan. The researchers analyzed U.S. student performance on individual items in the Third International Mathematics and Science Study (TIMSS}.
Only in the U.S. are algebra and geometry taught separately in the typical college prep high-school curriculum, an approach that helps to account for disappointing U.S. student performance in TIMSS in many areas, they say.
“The integration is left up to the student, a task many find difficult or fail to complete on their own,” write a team of researchers “In other countries, students are taught algebra and geometry (and other areas of mathematics) simultaneously, in either integrated mathematics courses or parallel strands.”
U.S. school mathematics curricula also tend to treat algebra as generalized arithmetic rather than taking the function-oriented approach common in other countries and recently advocated in the United States, they write.
“Recent developments in the U.S. school mathematics curricula suggest than an approach to algebra as the study and use of functions rather than as simply equation solving and manipulation of expressions may be gaining ground,” they write. Functions are introduced as rules for connecting one number to another, they explain.
To identify strengths and weaknesses in U.S. students’ math performance in TIMSS, researchers identified test items that 25% or fewer of U.S. students answered correctly and those items which 75% or more of U.S. students answered correctly; they performed the analysis for Grade 4 and 8 students only for tests from 1995, 1999 and 2003.Each item was classified according to what it asked the student to do and to its cognitive demand (rich vs. lean content and open vs. constrained).
The items were classified as follows:
- informal algebra
- algebraic reasoning
- algebraic manipulation
Eighth-grade students performed better than expected considering few of them had any exposure to algebra by that age, they note. Based on their analysis, the researchers say U.S. 8th graders demonstrate relatively good understanding of:
- the notation for exponents,
- ability to interpret simple algebraic expressions,
- reasoning about sequences, and
- the notation for exponents.
In contrast, their performance is relatively weak in:
- interpreting symbols in an equation,
- completing a table showing a relation between variables,
- finding the sum of series expressed by verbal rules,
- identifying a pattern, manipulating a simple algebraic expression,
- extending sequences of geometric figures to find the pattern,
- solving word problems involving relations between quantities,
- translating from words into algebraic symbols, and,
- completing a geometric pattern.
The function approach to mathematics makes heavy use of technology in order to capitalize on its ability to manipulate linked tables, symbols and graphs representing functions, the authors write.
“This modeling approach to algebra, although growing, is still very much a minor theme in U.S. school mathematics,” they write.
While U.S. students say that their teachers spend half their lesson time relating mathematics to daily life, more than teachers in other countries, the researchers say U.S. students do relatively poorly in setting up an equation to model a real situation. The reason may be that teachers do not use high-complexity problems in their classes, they write. U.S. 8th-grade teachers also spend 30% of their lessons reviewing topics.
“U.S. Algebra Performance in an International Context” by Jeremy Kirkpatrick et al. Lessons Learned What International Assessments Tell Us about Math Achievement, Brookings Institution Press, 2007.
Published in ERN February 2008, Volume 21, Number 2