STATEMENT ON THE NATION’S REPORT CARD:
Algebra I and Geometry Curricula:
Results From the 2005 High School Transcript Mathematics Curriculum Study
Member, National Assessment Governing Board;
Teacher and Mathematics Department Chair,
Bartholomew Consolidated School Corporation (Columbus, Ind.)
Over the past two decades there has been a major effort throughout the country to have students take more rigorous mathematics courses in high school. Today’s report shows that rigorous content within the courses is more important than the course titles. Clearly, not all algebra I and geometry courses are equal.
The most striking example of a course delivering less than what its title indicates is honors algebra 1. According to a careful accounting of what was covered in the textbooks of the public high school graduates featured in this analysis, 73 percent of the students in courses with this title actually received a curriculum ranked as intermediate algebra 1. Only 18 percent received a rigorous curriculum, which was much less than the 34 percent receiving a rigorous curriculum in a “regular” algebra 1 class.
When the researchers looked at the content of the courses taken by students in different racial/ethnic groups, they found few differences. The only substantial variation was in courses labeled “honors geometry,” where fewer black and Hispanic students than white and Asian students were receiving rigorous content. In the other two versions of geometry, the content delivered to students in the different racial/ethnic groups was broadly the same.
Overall, as expected, when mathematics content was more rigorous, mathematics achievement was higher. But the gains were not uniform across all racial/ethnic groups, and the gaps between them did not disappear. In both algebra and geometry, whites had higher average National Assessment of Educational Progress (NAEP) scores than blacks or Hispanics, regardless of the course level actually taken.
This new NAEP study does not try to explain why this is happening. But other NAEP data give a strong clue—and some reason to expect that over time the differences may diminish. NAEP’s eighth-grade results continue to show substantial racial gaps. In 2011, the average NAEP mathematics score for white students in eighth grade was 31 points higher than the average for blacks and 23 points higher than the average for Hispanics. It seems pretty clear to me that when students enter a high school algebra or geometry class with different levels of preparation it is likely their achievement at the end of the course will be different no matter what textbook material their teachers try to cover.
However, the eighth-grade achievement gaps, while still far too wide, are being reduced. From 2000 to 2011, the average NAEP mathematics score of eighth-grade whites increased by 9 points, compared with an 18-point gain for blacks and a 17-point increase for Hispanics. If these trends continue, the gaps in high school achievement should be reduced too.
In my school we don’t have a course labeled “honors algebra 1.” We do, however, have “honors” courses in geometry, algebra 2, and pre-calculus. I think this study points out the importance of each school and school district ensuring that there is a significant difference in any course labeled “honors,” whether it is in the curriculum or the implementation.
Not too many years ago, my state, Indiana, required two years of mathematics for a high school diploma but with no specification of what those two years should include. We had a significant number of students who graduated with mathematics credits in general mathematics, consumer mathematics, or courses with some other name that really meant pre-algebra.
Now algebra 1 is the lowest-level mathematics course that can be offered in high school. All graduates must pass algebra 1 and geometry, and in fact without a signed waiver from their parents, they must also take a course in algebra 2. But the report shows clearly that just having higher standards in high school is not enough to produce higher achievement or reduce achievement gaps, and it will not be enough for meeting the Common Core State Standards unless achievement is raised before students enter high school.
The study also shows wide discrepancies in the proportion of students taking algebra 1 before high school: 30 percent of Asian/Pacific Islanders, 23 percent of whites, 10 percent of Hispanics, and only 8 percent of black students.
The challenge for teachers and schools is to make the algebra 1 curriculum accessible to all students without sacrificing rigor or “dumbing it down.” That’s difficult to do when student preparation prior to high school is so uneven, but I believe it can be done with careful attention to three things. One is an approach that balances conceptual understanding and skills. Too often we as mathematics educators have focused on skills without understanding. What might make sense to the teacher appears to many students as doing random things with strange mathematics symbols. One example of this is the different ways to teach multiplying and factoring polynomials. I have seen students successfully master this skill by connecting it to finding an area, while a more symbolic approach can become rote memorization with little chance of remembering it over the long term.
This also illustrates a second important component for student understanding and achievement. It is important that students see problems in mathematics not just as a set of rules for moving symbols around, but also as a way to work meaningfully with tables, graphs, diagrams, equations, and verbal descriptions of real situations. Students must see the connections among these different forms. In mathematics education, this is referred to as multiple representations and connections.
The final important component lies in the answer to the question, “Who is doing the mathematics?” I am a very strong believer that textbooks matter, but no matter which textbook a school is using, class time must be spent with the students doing the mathematics, not watching the teacher do the mathematics.
This report shows that simply covering the same material in a textbook will not produce the same results. What is achieved depends, in part, on what students bring to high school from their elementary and middle school mathematics preparation and also on what their teachers do to make new concepts accessible to all. The challenge is not just to high schools but also to all schools from kindergarten—and even pre-K—onward.
But, of course, that’s a tall order.