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November 2007 | Volume **65** | Number **3**

**Making Math Count**
Pages 54-59

Sarah Theule Lubienski

To reach the goal of mathematics achievement for all, we must understand and address the obstacles faced by economically disadvantaged students.

“Mathematical power for all” has been the rallying cry of the National Council of Teachers of Mathematics (NCTM) for almost two decades. Through its various standards documents (see NCTM, 1989, 2000), the council has called for more engaging, meaningful mathematics instruction for all students. Such reforms are particularly important for students of low socioeconomic status (SES) and those in racial/ethnic minority groups, who have received more than their share of low-level, rote instruction (Knapp, Shields, & Turnbull, 1995).

Since 1990, educators have developed new curriculum frameworks and revised assessments to include more problem solving and more balance among the five mathematics strands identified by NCTM: number/operations, statistics/probability, measurement, geometry, and algebra/functions. Although there is encouraging news about the effects of these reforms on U.S. students in general, immense SES- and race-related achievement disparities persist. Addressing these achievement gaps will take concerted national effort.

As an eager young graduate student in the early 1990s, I was enthusiastic about the potential for the NCTM reforms to promote equity. Because I was a first-generation college student who had experienced firsthand how mathematics can serve as a ladder of socioeconomic mobility, I was especially interested in finding ways for other low-SES students to learn mathematics more effectively. After helping to develop a new problem-centered mathematics curriculum, I researched how low- and high-SES 7th graders reacted when I piloted those materials with them (Lubienski, 2000).

To my dismay, my low-SES students seemed resistant to learning mathematics through problem solving and discussion. Their higher-SES peers had the confidence to make sense of the mathematics for themselves; in contrast, more lower-SES students would often ask me to just “explain how to do it” or “tell the answer.” My higher-SES students found our class discussions about conflicting mathematical ideas interesting and informative; many lower-SES students complained that they became confused because they were not sure which ideas were right or wrong. Further, whereas the high-SES students usually noticed that the same mathematical ideas and procedures were repeated in different story contexts, the low-SES students would often become engaged with the real-world aspects of the problems, missing the mathematical point intended by the problems' authors.

For example, one of my students, Rose, took an unexpected approach to a problem that asked which of three popcorn containers sold at the movies was the “best buy.” Using the dimensions provided in the textbook, this bright, working-class student had no trouble finding the volumes of the various containers, which were shaped as a cone, a cylinder, and a rectangular prism. Although the point of the problem was for students to find the container with the lowest unit price (cost per cubic inch of popcorn), Rose used solid common sense to argue that “It depends on how much popcorn you want.”

Although Rose's approach was more sensible in the context of the problem than the unit-price approach, Rose missed the intended experience of comparing ratios. As another example, in a fractions problem that involved sharing a pizza, my lower-SES students were more likely than their higher-SES peers to consider whether each student would arrive at the pizza parlor on time and whether some students might be hungrier than others.

In general, my lower-SES 7th graders were more likely to use common-sense reasoning in mathematics class. This reasoning put them at a disadvantage when they missed the mathematical point of the problem. A similar pattern was noted in England, where researchers found that low-SES students tend to take the contexts of national exam questions more seriously than the test makers intended (Cooper & Dunne, 2000). We have a strange habit in mathematics education of wanting students to take “real-world” problems seriously (so that the students become engaged), but not too seriously (so that the students don't become distracted from the intended, abstract mathematics). Apparently, only some students know where to draw that line.

Curious to understand the differences in my students' approaches to learning, I turned to past research on cultural differences between social classes. Scholars have noted that whereas working-class jobs tend to require conformity to externally imposed routines, middle-class occupations allow for more autonomy and intellectual work. Play and work are more intermixed for professionals (for instance, many readers of
*Educational Leadership* are likely reading this article outside their regular work day). Those in working-class occupations typically experience a more stark separation between play and work. Similarly, studies on child rearing have found that working-class parents tend to be more overtly directive with their children, often showing or telling the children how to do things. In contrast, middle-class parents use more questioning and playfulness, guiding their children's problem solving through questions that focus attention on the general structure of problems (Bruner, 1975; Heath, 1983).

Although this research is sometimes criticized for promoting simplistic stereotypes, avoiding discussions of class differences is detrimental for low-SES students, whose strengths and needs we might then ignore. I am not suggesting that the distinctions above are true for every family or that children should necessarily receive instruction that matches their home environment. Indeed, one could argue that low-SES students are *most* in need of mathematics instruction that emphasizes questioning and problem solving. The point here is that mathematics teachers should pay attention to the particular orientation toward learning with which children have been raised, particularly when trying to implement instructional reforms.

Mathematics achievement is particularly important to our efforts to promote equity because it serves as a gatekeeper to high-status occupations and can provide a powerful ladder of mobility for low-SES students. And despite the obstacles faced by low-SES students, we have reason to be encouraged about the potential for mathematics instruction to make a difference.

Unlike scores in other subjects, such as reading, scores on the National Assessment of Educational Progress (NAEP) in mathematics have risen substantially since 1990 for both low- and high-SES students (Perie, Grigg, & Dion, 2005). These gains may be due at least in part to the fact that the NAEP mathematics test became aligned with the NCTM standards in 1990, incorporating items from NCTM's five mathematics strands as well as more conceptual and open-ended items. At the same time, mathematics instruction in many schools also became more aligned with the NCTM standards. Thus, the marked improvement in NAEP mathematics scores confirms that instructional changes can, indeed, improve students' achievement.

The more difficult problem remains, however: General instructional shifts that boost achievement for all students seem unlikely to seriously narrow achievement gaps without additional interventions. Indeed, despite monumental changes in mathematics education since 1989, we continue to have massive achievement gaps by SES and race/ethnicity, roughly equivalent to a difference of several grade levels (Lubienski, 2002). Because of the complex interaction between economic class and learning, improving mathematics education for all while narrowing achievement gaps is no simple task.

How can schools reduce the gaps in their mathematics test scores? Current research suggests several avenues for teachers and administrators to pursue.

Research shows that students tend to forget content over the summer, and this summer loss is greatest for low-SES children (Entwisle & Alexander, 1992). Students who approach mathematics by merely memorizing rules are less likely to retain what they have learned than are students who have deep understandings of mathematical concepts and relationships (and can then reconstruct any formulas they have forgotten). Unfortunately, according to NAEP survey data, low-SES and minority students are more likely than their more advantaged peers to agree with the statement, “Learning mathematics is mostly memorizing facts.” And this belief is strongly negatively correlated with achievement, even after controlling for SES and race (Lubienski, 2006).

To increase the mathematical power of low-SES and minority students, we must help them move beyond the belief that math is simply memorization. Consistent with NCTM's recommendations, we must teach these students with an emphasis on the meaning of concepts and procedures. Basic number facts are important—all students must master single-digit (0–9) addition, subtraction, multiplication, and division—but the meaning of the operation should come first. It makes no sense to assign dozens of division problems to students if they do not understand what division means. (For example, 24 ÷ 8 means “How many groups of 8 are in 24?” or “If I split 24 into 8 groups, how many will be in each group?”) Students can learn many basic facts while solving engaging problems that help them understand the meaning of operations and their applications.

This meaning-oriented approach is prevalent among the dozen elementary, middle school, and high school mathematics curriculums developed with funding from the National Science Foundation (see www2.edc.org/mcc/PDF/CurricSum8.pdf). Current data on these curriculums indicate that students who use them tend to be as strong at basic computation—and better at reasoning and problem solving—than students in more traditional curriculums (Senk & Thompson, 2003).

One of the main elements of the National Science Foundation–funded curriculums (and the current NCTM reform movement in general) is teaching mathematics through problem solving. A well-crafted mathematics problem can be a wonderful instructional tool, prompting students to grapple with important concepts and skills.

However, as I found with my low-SES 7th graders, teachers need to take extra care to ensure that their students—especially low-SES students—learn what is intended from such problems. This care can involve a series of focusing questions, a whole-class summarizing discussion, and journal prompts requiring students to record key definitions and formulas. (Many of the NSF-funded mathematics curriculums include these features in some form.)

For example, one Minneapolis high school teacher established structures to support her low-SES students' use of their textbook (Lubienski & Stilwell, 2003). She began each class by asking a student to recap the main mathematical ideas learned the previous day (which especially helped students who had been absent), and she concluded each day by eliciting and writing a class summary of the main mathematics terms and ideas the class had learned, which students then recorded in their math journals.

Some students might resist learning through problem solving and discussion, preferring to apply rules given by the teacher or text. Teachers might need to wean students away from their dependence on explicit instructions and to clearly explain the students' role in a problem-centered classroom. Yet, giving students the ability to make sense of mathematical ideas for themselves and to apply mathematics to important problems is a goal worth striving for—one that has been restricted to only high-SES students for too long.

One positive aspect of No Child Left Behind is that schools can no longer sweep their SES- and race-related achievement gaps under the rug. To use achievement data to address gaps, however, schools must go further than simply looking at means for various groups. Educators must identify which mathematical topics are most problematic for their underserved students. A detailed analysis of student standardized test data often yields this information, particularly if the testing company provides a spreadsheet containing item-level data by student, which the school can merge with demographic data that it has collected on each student.

If schools do not have access to such detailed data on their students' performance, or if teachers want more immediate feedback on the ways in which their current students compare with one another and with students nationwide, they can draw from NAEP's Web-based question tool (http://nces.ed.gov/nationsreportcard/itmrls), which offers a variety of assessment tasks within each mathematics strand. Teachers can generate a pool of items selected by mathematical strand, difficulty level, and format (multiple choice, short answer, or extended response) and administer the items to their students. They can then compare the results with detailed U.S. 4th, 8th, and 12th grade student performance data for each task (including differences by poverty level and race/ethnicity).

Schools might find that their disparities parallel patterns at the national level. According to NAEP data, the mathematics strand that tends to have the largest achievement disparities is measurement (Lubienski & Crockett, 2007). For example, one NAEP measurement question with large SES- and race-related disparities asked 4th graders to identify which of four items would most likely be measured in feet: a coin, a paper clip, a car, or the distance from New York City to Chicago. Whereas 76 percent of nonpoor students (those ineligible for free or reduced-price lunch) correctly selected *car*, only 57 percent of poor students did so.

Another pattern evident in NAEP data is that low-SES and minority students tend to perform worse on nonroutine problems. For example, the vast majority of 4th grade and 8th grade students from all race and class groups correctly answered basic computation problems, such as 238 + 462 = __. In contrast, there were large disparities on computation problems with extraneous information or multiple steps, such as this one:

Carl has 3 empty egg cartons and 34 eggs. If each carton holds 12 eggs, how many more eggs are needed to fill all 3 cartons?More than 53 percent of white, Asian, and nonpoor 4th graders answered this problem correctly, compared with only about one-third of black, Latino, and poor 4th graders.

I am not suggesting that when schools identify topic-specific disparities in student achievement, they should address these gaps by teaching to the test. Such a response might produce small gains for a few particular test items but would not produce meaningful learning. Before educators try to remediate gaps on the tests their students take, they should ask themselves whether the test content in question is worth teaching. If the answer is yes, then the school should critically examine its curriculum and instruction pertaining to that content and consider how it could be improved.

High-SES parents tend to look out for the interests of their own children, making sure that they have access to the best teachers and most challenging courses. Within mathematics education, parental influence on students' course placements can be especially powerful (Wayne & Youngs, 2003).

Overall, low-SES parents are less likely than high-SES parents to know the unwritten rules governing how they can lobby for their children's interests within schools. Instead of helping these parents, school policies often exacerbate the situation. For instance, I have seen many schools mask the fact that they initiate math tracks in elementary school, leaving only the most school-savvy parents aware that key decisions are being made about their children's futures.

To achieve equity, schools must ensure that low-SES and minority students get more than just the teachers and courses that are left over after others' requests have been fulfilled. For example, schools might assign low-SES students to classes first and then place high-SES students in the remaining spaces.

Schools should also take the time to carefully advise low-SES students and their parents about their options. Providing such advice is time well spent, because the choices students make about mathematics courses are likely to substantially affect their futures. Low-SES parents, in particular, need to be clearly informed about the differences between college-preparatory and vocational tracks, and how mathematics placement decisions they make early on can affect their children's future career options. (See www.sedl.org/pubs/quick-takes/qt_decisions.pdf for more information.) More fundamentally, schools should carefully study whether their teacher assignments and tracking practices are helping or hindering equity and ensure that no students are placed on dead-end, low-level tracks.

If we are truly committed to equitable outcomes, then we must commit *more*
resources to those students who most need them. To close achievement gaps in mathematics, we need to ensure that low-SES and minority students get the best teachers, the richest mathematics curriculums, the smallest class sizes, and the most careful guidance. Although we might strive to achieve “mathematical power for all,” we will not reach this goal if we focus on all students generally instead of addressing the particular barriers that historically underserved students face in learning mathematics.

## My “Aha!” Moment
The first “aha” moment in math that I remember was when I was 5 years old—and it wasn't in math class at all. I had just started school, and the teacher was getting us to make paper models of ice cream cones by drawing a semicircle, rolling it into a cone, and gluing the edges. I remember being intrigued that a flat piece of paper could make a curved shape. After that I just gradually acquired more and more bits of mathematical know-how. I think math started to come to life for me when I realized that you can invent new math for yourself. Or fairly new math—things that were new to you, and which no one had told you about, even if later you discovered they'd already been done. I liked finding integer solutions to various equations, inventing strange functions and finding their Taylor series, or whatever. The message was clear, even if I didn't make it explicit: You can be I think different people respond to different things, though. Some need to be convinced that a skill is useful in things that already interest them before they can get interested in something new. Some just enjoy being able to calculate things. Some can be switched on by an “aha!” moment. The opposite—being switched off by a teacher who may have had good intentions but pressed the wrong buttons—is all too common. Teachers who really enjoy their subject are better at avoiding turning students off than most. |

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**Sarah Theule Lubienski** is Associate Professor, Department of Curriculum and Instruction, University of Illinois at Urbana–Champaign; 217-333-1564; stl@uiuc.edu.

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