Tom Loveless, Brookings Institution
Trends in Math Achievement: The Importance of Basic Skills
Today I will attempt to answer two questions. The first is: how are American students doing in mathematics? To answer this question, I will review trends on the two math tests of the National Assessment of Educational Progress (NAEP). We will see that although students have made progress in some areas of math, when it comes to basic arithmetic—in particular, the ability to compute—growth has been disappointing. In recent years, progress has ground to a halt. That leads to the second question. Why should we care? Some people argue that students no longer need to learn how to compute now that calculators are widely available. Are basic math skills still important? I will argue that they are. Unless every student receives a thorough grounding in arithmetic, the nation has little chance of achieving ambitious goals in mathematics.
Let's look at some data.
Main and Trend
The NAEP administers two math tests, the main and the longterm trend. The main NAEP test has been given since 1990. As shown in Figure 1, main NAEP scores have increased significantly—scores for 12th, 8th, and 4th grades are all up.
Figure 2 shows scores on the trend NAEP. Since 1990, scores are up, but not as much as on the main NAEP. One can see this better by converting the two tests' gains to comparable units. Table 1 does that, displaying gains in standard deviations, a unit of measure used by statisticians. Gains on the main NAEP are at least twice as large as those registered on the trend. The difference between the two tests is most apparent with the youngest students, nine year olds and fourth graders. They accomplished a huge gain on the main NAEP, .29 standard deviations, approximately equal to one year's worth of mathematics. But on the trend NAEP, the gain was miniscule, equal to about one month of learning.
Why are the two tests telling completely different stories? The most likely reason is that they assess different skills. The main NAEP devotes more items to measuring students' skills in areas such as geometry, problem solving, and data analysis. For assessing basic skills, and especially for measuring computation skills, the trend NAEP is the better test. As the NAEP framework states, the trend test is the one that can "provide insight into students' computational abilities." (Quotation from page 9, 2005 NAEP Mathematics Framework).
With that in mind, some Brookings colleagues and I identified computation items on the long term NAEP, grouped them by topic, and calculated gains for both the 1980s and 1990s. It is important to note that the findings are not conclusive. The analyses are based on a small number of NAEP items, and though this is the best available evidence on computation skills at the national level, the findings are merely suggestive. That said, the findings pinpoint important areas where problems may be developing.
Computation
Table 2 breaks down computation scores into ten separate areas of skill. The percentage of students answering items correctly is presented for 1982, 1990, and 1999. The last two columns compare gains or losses in the 1980s and 1990s. The better performing decade is shaded in yellow. You can see that, overall, students gained more in computation in the 1980s than in the 1990s. Eight of the ten clusters favor the 1980s. Two of the clusters favor the 1990s. Scanning down the 1990s column and examining the sign of each cell, you will notice that most cells are negative. Students lost ground in most computation skills during the 1990s. Only in computing percentages at ages 13 and 17 did students register gains.
Take a closer look at the scores for nine year olds. These skills comprise the basic arithmetic that all fourth graders are expected to master—addition, subtraction, multiplication, and division of whole numbers. All four areas reversed direction in the 1990s, turning solid gains that were made in the 1980s into losses. Not only that, but the declines came from levels that weren't very high at the beginning of the 1990s—certainly not at a level that is acceptable for such fundamental material.
I taught sixth grade for several years in California. My grading scale for tests and quizzes was pretty straightforward: 90's were "A's," 80's were "B's," 70's were "C's," 60's were "D's," and below 60% was an "F.". The 1999 NAEP scores for nine year olds would have received the following letter grades in my class: in addition, a "C," in subtraction, a"D," (with the benefit of rounding), in multiplication, an "F," and in division, an "F." Those are not good grades. Think of it this way. Youngsters who have not mastered whole number arithmetic by the end of 4th grade are at risk of later becoming remedial students in mathematics. Half of the nation's nine year olds missed the multiplication and division items on the trend NAEP the last time the test was given.
A similar concern can be raised about the performance of thirteen and seventeen year olds. Their level of proficiency on computation skills remains unacceptably low. Look closely at fractions. Proficiency with fractions is critical in preparation for algebra. In 1999, only about half of thirteen and seventeen year olds could compute accurately with fractions on the NAEP. Students who leave eighth grade not knowing how to compute with fractions enter high school as remedial math students. Students who leave high school lacking proficiency with fractions are inadequately prepared for college mathematics. On the most recent trend NAEP, both age groups were less proficient at computing with fractions than in 1982, twenty years ago.
Should we be worried if the evidence suggests that students are not learning how to compute?
We should—for the following reasons.
Why Important?
Basic skills serve equity. As shown in Table 3, the lack of progress in computation skills has disproportionately affected AfricanAmerican students. The blackwhite achievement gap expanded in every computation skill area in the 1990s. This is typical of what happens when basic skills are shortchanged. The students who pay the biggest price are those with the least to lose, those for whom the educational system has never worked very well. When basic skills are not taught, the least privileged in our society—those who cannot afford tutors, fancy computer programs, or academic summer camps—suffer the biggest losses.
Basic skills are necessary to advance in math. Insisting that students master computation skills is not to advocate that they stop at the basics. Basic skills are a floor, not a ceiling. Students must learn arithmetic so that they can move on to more demanding mathematics—algebra, geometry, calculus. An emphasis on the basics should never be used as an excuse to straightjacket students or to slow their progress in the math curriculum
Basic skills predict adult earnings. In recent years, a growing body of research has documented that the skills and knowledge students learn in school is correlated with success later in life. In their landmark study showing the impact of basic skills on adult earnings, Richard Murnane and Frank Levy conclude, "mastery of skills taught in American schools no later than the eighth grade is an increasingly important determinant of subsequent wages."
A final word on the controversy surrounding basic skills. I am mystified when some analysts refer to a concern for arithmetic and computation skills as advocating "back to basics." Well, as an old elementary teacher, I am very concerned about American fourth graders learning arithmetic. A 50% proficiency rate is unacceptable. But I don't want to go back to anything. I want to go forward on the basics. Back to basics implies there was a golden age when everyone learned essential skills. That age has never existed. To ensure that every fourth grader is proficient at whole number arithmetic means that we must go forward, not backward. We must go forward on basic skills if a more equitable school system is a national goal; we must go forward if American students are to be prepared for higher level mathematics; we must go forward if young people are to master the skills correlated with middle class employment as adults. Back to basics is a bad idea. There is nothing to go back to but mediocrity and failure. It is time to go forward as a nation on basic skills.
Today the President addresses a serious issue in American education—How well we as a nation do in educating our children in mathematics. Dr. Loveless has addressed this issue using data from our nation's report card—NAEP. I will address the same issue but from an international point of view.
The data are clear. Recent results from the Third International Mathematics and Science Study (TIMSS) show that US eighth and twelfth graders do not do well by international standards—ranking below average in both grades and, in fact, near the bottom of the international rankings on a mathematics literacy test at the end of high school. Even our best students in mathematics taking an advanced mathematics test do not fare well against their counterparts in the other countries. Those results were obtained in 1995 but even a retesting four years later in 1999 produced the same disappointing results. Put simply—there is no evidence to suggest that we as a nation are doing better, at least relative to other countries. New international results will soon become available as a third retesting will occur this spring, but I fear there is little reason to expect major improvement.
Why do I feel this way? When you add the fourth grade results where our students were above average to the picture, a pattern emerges of a steady decline in our international ranking from fourth to twelfth grade. This suggests that our students do not start out behind but increasingly fall behind during the middle and high school years. This suggests that the problem lies not with our children, but that it is the education system that is failing them. This, I believe, strongly supports the need for a major national initiative in mathematics education.
Make no mistake—the problems in mathematics education have implications that are real and not just academic. The implications are for individual children as they seek employment in an increasingly technological economy where, given the lowering of trade barriers, they are competing not only with each other, but with children from around the world. The TIMSS data suggest they will not fare well in such a competition. This has implications for our nation as a whole, as well. It is not clear how long we can make up for deficiencies in our own educational system by importing the needed talent from other countries. Such a brain chase has become more competitive as other nations also compete for the same individuals. As the New York Times recently reported, the nation may soon regret not being concerned with the proper education of its own native born.
To stop here only restates what many already know—we, as adults, are failing our own children. What else have we learned from the international studies that might help us to respond to this serious situation. Other nations' representatives often ask me why does our student achievement not improve especially given that we are constantly reforming mathematics education in the US. The short answer is that we often engage in reform that is not based on scientific evidence but rather on opinion and someone's ideology. TIMSS offers us a good opportunity to use scientifically collected data on some 50 countries to find a more promising answer to the question of what we should do to improve the mathematics education of all children so that we truly do not leave any of them behind.
TIMSS results suggest that the top achieving countries have coherent, focused and demanding mathematics curricula. What would a coherent curriculum look like? A coherent curriculum leads students through a sequence of topics and performances over the grades that reflects the logical and sequential nature of knowledge in mathematics. Such a curriculum helps students to move from particular knowledge and skills toward an understanding of deeper structures, more complex ideas and mathematical reasoning including problem solving. For example, students should be expected to master the basic concept of number and basic computational skills in the early grades before they tackle more difficult mathematics.
What does the US curriculum look like? The US curriculum as reflected in many of the states' standards and in our nation's textbooks tends to reflect an arbitrariness where topics appear somewhat haphazardly throughout the grades. For example, teachers are expected to introduce relatively advanced mathematics in the earliest grades before students have had an opportunity to master basic concepts and computational skills. Secondly, the curriculum continues to focus on basic computational skills through grade eight and perhaps beyond. I would argue that if the logic of mathematics is not transparent to students, then it becomes difficult for them to develop a deep understanding that would lead to higher achievement.
What does a focused and rigorous curriculum look like in the top achieving countries? The number of topics that children are expected to learn at a given grade level is relatively small, permitting a thorough and deep coverage of each topic. For example, nine topics are the average number intended in the second grade. The US by contrast expects second grade teachers to cover twice as many mathematics topics. The result is a characterization of the US curriculum as a mile wide and an inch deep.
Coherent standards move from the simple to the complex. By the middle grades the top achieving countries do not intend that children should continue to study basic computation skills but rather that they begin the transition to the study of algebra, including linear equations and functions, geometry and even in some cases, basic trigonometry. By the end of eighth grade in these countries children have mostly completed US high school courses in algebra I and geometry. By contrast, most US students are destined to mostly continue the study of arithmetic. In fact, we estimate that at the end of eighth grade US students are some two or more years behind their counterparts around the world.
All of this is related to what students learn. That is why schools matter. The major policy implication of all of this is if we are serious about providing all students with a challenging mathematics curriculum it must be coherent, focused and demanding not by our own sense of what this might mean, but by international standards. We expect this of our companies, why would we expect less for our children's education. This implies we must secure the advice of the research mathematics community in this process together with those who understand children and how they learn mathematics.
But, this will not be enough. It is necessary to change our curricular expectations, but it is not sufficient to increase the achievement of all of our students. Recent research involving the top achieving countries suggests that the preparation of middle school teachers in mathematics includes a demanding level of preparation in theoretical mathematics as well as preparation in topic specific pedagogy, i.e., how to teach particular mathematics topics to children of a certain age. The level of formal mathematics training is very demanding in these countries. This required level of knowledge for eighth grade teachers gives some insight as to how such a demanding curriculum can be required in other countries and not just for their elite, but for all children—a goal we only seek, but one that is realized in many European and Asian countries.
Secretary Paige and his department have identified the national problem that many mathematics teachers do not have a major or even a minor in mathematics. The problem, however, is even more severe. Data collected from a group of districts that are in many ways similar to the US indicate the severity of the problem. Over half of the sixth through eighth grade mathematics teachers have neither a major or a minor in mathematics. For those teachers only onefourth feel, by their own assessment, well prepared academically to teach a basic set of topics—most dealing with arithmetic only. We must address this issue of teacher quality and one important way is to begin by including mathematical knowledge as a key component in the definition of teacher quality.
We, in this nation, have set a goal to provide all children with a demanding mathematics curriculum that leads to greater learning. The goal is right, but the road there is demanding. Curriculum must be rigorous and coherent by international standards. It must be focused. It must require our middle schools to expect more of our students. It must be taught by teachers well prepared in mathematics and in instructional approaches that themselves are steeped in mathematics as well as cognitive theories of how children learn. And, it must be for all children.
Excellence in mathematics must be our highest national priority if we are to fulfill the true promise of America for all of our children. To do otherwise is unconscionable.


