Many students in U.S. schools have trouble understanding fractions and Algebra II, the one difficultly occurring at the end of elementary school, the other in high school. One reason is that schools generally focus on one aspect of mathematics—calculation—and often fail to address the second aspect—interpretation. Also responsible is the trajectory of school mathematics, which moves from the concrete and functional in lower grades to the abstract and apparently nonfunctional in high school. To improve instruction in math, teachers need to focus on communication, connections, and context. Students need to be able to explain in writing the meaning of data, tables, graphs, and formulas; they need to see other content-area teachers regularly using quantitative or logical arguments to tie evidence to conclusions; and they need a good diet of sophisticated, realistic, and meaningful problems to solve.

Through her work as a consultant, Burns has found that a handful of students in all classes lack an adequate foundation in basic math concepts and lag far behind their peers in both understanding and skills. Students who lack a foundation on which to build new learning are generally not well served even by well-planned, differentiated instruction; they require supplemental interventions to help them catch up. Burns shares nine strategies she has found essential to creating effective intervention lessons in math: (1) build scaffolding into lesson planning, (2) pace lessons carefully, (3) build in a routine of support, (4) foster student interaction, (5) make connections explicit, (6) encourage mental calculations, (7) use written calculations to track thinking, (8) provide practice, and (9) build in vocabulary instruction.

This article describes a mathematics lesson taught by a 5th grade teacher who engages her class in an in-depth examination of one student's incorrect solution to a problem. Because the teacher consistently asks her students to devise alternative calculation strategies and explain how those strategies work, the students have come to expect that mathematics makes sense and that they can solve problems through reasoning. The article also describes how a videotape of the lesson was used in a professional development seminar. Teachers in the seminar conducted their own study of the student's incorrect solution, and thus constructed for themselves more powerful understandings of mathematics teaching and learning. The author states that this kind of professional development is essential to help teachers reconceptualize mathematics as an interconnected body of ideas to be explored, rather than a set of facts, definitions, and procedures to be memorized and used.

The authors explain how math textbooks in Singapore—a country whose students consistently come in first on international math exams—use the bar model tool to help elementary school students master complex math concepts. Sample textbook problems illustrate how Singapore Math students, beginning in 3rd grade, learn to apply this tool to simple addition and subtraction problems, gradually progressing to more complex problems and laying the foundations for algebra. Some schools and many homeschoolers in the United States are now using Singapore Math, whose trademark strategy is simple explanations for hard concepts.

Singapore Math has had great success because it focuses on five essential elements. Its guiding framework presents a balanced and integrated vision that places problem solving at the center. Each element of the system—the framework, a common set of national standards, texts, tests, and teacher preparation programs— is carefully aligned to clear and common goals. With smaller and more targeted textbooks, Singapore has a clearer and more coherent mathematical focus at each grade level. Moreover, its textbooks include multiple representations and “think bubbles,” which clarify main concepts; also, they consistently include bar models to pictorially represent concepts. Finally, students are given rich multistep problems to complete, a practice that supports strong mathematical development.

The author, director of the University of Chicago School Mathematics Project, tackles the following question: Should the United States have national standards with teeth, that is, a single set of standards tied to assessments and agreed to by the states? Proponents advance five main arguments for implementing such a standard. In his rebuttal, the author points out that a single set of national standards in mathematics assures neither high student performance nor a healthy economy, that mathematics curriculums are fairly prescribed and uniform across states, that a national curriculum won't resolve the equity issue or speed up change in schools, and that the politicization of education in the United States has resulted in a loss of expert guidance in mathematics at the national level.

International research has shown that mathematics lessons in the United States often focus on procedure instead of engaging students in exploring math concepts. To help U.S. math teachers move beyond these ineffective teaching routines, the author recommends braiding together mathematics, language, and cognition. Both reading comprehension and mathematics are based on cognitive strategies, including making connections, asking questions, visualizing, inferring, predicting, determining importance, and synthesizing. The author describes a problem-solving model that he and his colleagues developed, which integrates these cognitive strategies into mathematical problem solving.

Black and Latino students are still underepresented in upper-level math classes in the United States, a fact which has serious implications for their academic achievement and futures. Walker provides six suggestions for how educators can encourage more black and Latino students to successfully take higher level math courses: (1) Expand our thinking about who can do mathematics; too often educators assume that minority students don't have the ability or interest to do higher level math. (2) Build on underrepresented students' existing academic communities. Walker's research reveals that minority students doing well in math often draw on networks of family and peers that support this achievement. (3) Learn from institutions that promote math excellence, such as historically black colleges and universities that graduate many minorities with math-related degrees. (4) Expand the options in school math courses. (5) Expand enrichment opportunities by providing more out-of-classroom mathematics experiences. (6) Make minority students less isolated in advanced mathematics courses.

Because of the complex interaction between economic class and learning, improving mathematics education for all while narrowing achievement gaps is no simple task. As a young teacher, the author conducted action research to determine how low-SES and high-SES 7th graders responded to a problem-centered mathematics curriculum. To her dismay, she discovered that her low-SES students seemed resistant to learning mathematics through problem-solving and discussion. In this article, she discusses the possible reasons for this difference in learning preferences. She endorses mathematics instruction centered on meaning and rich, complex math problems for all students, but concludes that low-SES students, as a group, need targeted support to benefit from such instruction. In addition, she urges schools to protect the interests of low-SES students by taking the time to carefully advise them (and their parents) about the mathematics course options available to them.

When a group of 8th graders at the Jakarta, Indonesia, International School came to class with extremely varied math skills, Suarez realized that no single learning goal was appropriate for the whole group. So Suarez and his colleagues developed a system of tiered instruction that enables students to study the same content at different levels of challenge. They organized their units of study into themes, with specific skill outcomes for each theme. Teachers then identified what tasks would represent foundational, intermediate, or advanced levels of understanding for each set of skills, and designated each level by a color. For each thematic unit, students choose whether to do green-level tasks and assessments (foundational level), blue-level (intermediate), or black-level (advanced). Suarez shares both test results and student and parent comments that show how tiered instruction has increased students' insights into their learning, their achievement, and especially their motivation in math.

Leanne R. Ketterlin-Geller, Kathleen Jungjohann, David J. Chard and Scott Baker

Much of the difficulty that students encounter in the transition from arithmetic to algebra stems from their early learning and understanding of arithmetic. Too often, students learn about the whole number system and the operations that govern that system as a set of procedures to solve addition, subtraction, multiplication, and division problems. Teachers may introduce number properties as “truths” or axioms without developing students' deep conceptual understanding. To provide rich and explicit instruction to students in early algebraic thinking, teachers should clearly model what they want students to be able to do. Students must understand variables and constants, decomposing and representing word problems algebraically, symbol manipulation, and functions to develop algebraic thinking.

Lesa M. Covington Clarkson, Gay Fawcett, Elaine Shannon-Smith and Nancy T. Goldman

Students, parents, and sometimes even teachers often believe that complex mathematics is too difficult for certain learners. But the programs described in this article demonstrate that anyone can learn, understand, and use complex math. Lesa M. Covington-Clarkson boosted the confidence of black 6th graders by teaching them to use graphing calculators. By exposing them to a complex tool often reserved for high school students, she hoped to encourage them to see higher mathematics as an option for future study.

Gay Fawcett and Elaine Shannon-Smith tell of a partnership between schools and community organizations to bring math-related activities to the wider community. When students and parents visited the zoo, the ball park, and other sites, they received cards or handouts with fun activities that reinforced concepts in the Ohio Mathematics Benchmarks.

Realizing that the students in her mathematics pedagogy class did not understand the meaning behind various mathematics procedures, Nancy T. Goldman engaged these teachers in training in activities designed to make math more meaningful. Her students would then be better prepared to help their students develop deeper mathematics understanding.

Michael Jellinek, Jeff Q. Bostic and Steven C. Schlozman

Specific guidelines can help a school meaningfully respond to an event that profoundly affects the school and community—a student's death. A crisis team should craft the initial announcement and decide how to disseminate information. School administrators need to monitor students and staff to decide how rapidly the school should resume a regular routine and how much time students and staff need to collectively grieve the loss. The various causes of death—an accident, a suicide, or a chronic illness—all present specific coping challenges for both students and staff. Initial and long-term memorials can productively channel energy and communicate key values.

When teachers at Sheridan Elementary School in Spokane, Washington, saw that only 46 percent of their students were performing at or above grade level in math, they decided to make a concerted effort to improve instruction. Teachers regularly met in grade-level teams with Gurule, the school's instructional coach, and developed strategies together. The 3rd, 4th, and 5th grade teams focused on improving students' problem-solving skills by having them work in small groups with an adult leader. They also established a process for problem solving that required students to read problems more carefully, encouraged in-class discussions of math problems, and gave students a rubric for self-assessment. After one year, the percentage of students at or above grade level leapt to 56 percent.

Carla Thompson presents research on computer use among teens, as well as math fear and math avoidance among adolescent females. The author asserts that Internet math activities are apt to engage female students by drawing on their interests in shopping, chatting, surfing, and community-building. Thompson suggests 19 Internet math activities for teachers to use with female students avoiding math.