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by Kathy Checkley
Table of Contents
Mathematics, in its widest significance, is the development of all types of formal, necessary, deductive reasoning.
—Alfred North Whitehead, A Treatise on Universal Algebra
It wasn't easy to be a pioneering U.S. math teacher in the days before the National Council of Teachers of Mathematics (NCTM) released the Curriculum and Evaluation Standards for School Mathematics. Just ask Mark Saul, an award-winning teacher who spent more than 30 years working with a wide range of students from the 3rd through the 12th grades. Pushing the envelope was discouraged, he recalls; innovators, rebuked. If a teacher taught math in a traditional way and it “didn't work”—if students didn't understand the content—colleagues were sympathetic. “They would say, ‘Oh, you had a bad lot,’” Saul explains. If a teacher got the same result after trying something extra or different, however, colleagues were less generous. The response then became, “It's your fault.”
Fortunately, with the release of the NCTM standards in 1989, what was once radical became standard, says Saul, who also served as a program officer at the National Science Foundation. The standards-based approach to mathematics education legitimized his routinely going outside the box to help students see how great mathematics could be.
Nearly two decades and one revision later, the standards (now entitled Principles and Standards for School Mathematics) continue to support and inspire the practice of radical educators, like Saul. Their staying power underscores the need for a shared, national understanding of the math content that should be emphasized, pre-K through 12th grade, along with the processes and attitudes children should attain.
Articulation across the grades is a big concern in mathematics education today, Saul affirms. The NCTM standards, he notes, outline the broad areas of mathematical knowledge that students should, ideally, build over time and grade levels. Achievement in algebra in high school, therefore, depends on students learning to think algebraically in earlier grades. Likewise, U.S. students will fare better on international exams that emphasize problem solving if they learn, even as early as kindergarten, how to pull problems apart and identify the essence of those problems.
“Nearly all other countries, especially those that outperform the United States in mathematics, have a continuous math program from elementary through secondary school,” observed Cathy Seeley, president of NCTM, during an online chat with educators from around the world (Seeley, 2005, para. 18). “I like the idea of introducing algebraic ideas in the elementary grades and in middle school,” she continued (2005, para. 42).
Algebra is something that you [can] start as soon as you enter school.
—Cathy Seeley Algebra, K–12
That Seeley likes the idea of introducing algebra in the early grades is somewhat an understatement. Emphasizing algebraic thinking in the early grades was the professional development focus for NCTM in 2004–05. The rationale for the emphasis was clear, according to Seeley: too many U.S. high school students struggle with algebra. For a long time, she says, there was a “very strong numerical focus in K–8.” Students then had to make a giant leap into algebra, but it doesn't have to be that way, Seeley asserts. “Algebra is something that you [can] start as soon as you enter school.”
By Cathy Seeley
Developing algebraic thinking is a process, not an event. Algebraic thinking includes recognizing and analyzing patterns, studying and representing relationships, making generalizations, and analyzing how things change.
At the earliest grades, young children work with patterns. At an early age, children have a natural love of mathematics, and their curiosity is a strong motivator as they try to describe and extend patterns of shapes, colors, sounds, and, eventually, letters and numbers. And at a young age, children can begin to make generalizations about patterns that seem to be the same or different. This kind of categorizing and generalizing is an important developmental step on the journey toward algebraic thinking.
Throughout the elementary grades, patterns are not only an object of study but a tool as well. As students develop their understanding of numbers, they can use patterns in arrays of dots or objects to help them recognize what 6 is or whether 2 is larger than 3. As they explore and understand addition, subtraction, multiplication, and division, they can look for patterns that help them learn procedures and facts. Patterns in rows and columns of objects help students get a sense of multiplication and see that facts make sense. Patterns within the multiplication table itself are interesting to children and help them both learn their facts and understand relationships among those facts. The process of noticing and exploring patterns sets the stage for looking at more complex relationships, including proportionality, in later grades.
As students move into the middle grades, their mathematics experience can focus on connecting their work with numbers and operations to more symbolic work with equations and expressions. At this level, the focus of the mathematics program should be on proportionality—perhaps the most important connecting idea in the entire preK–12 mathematics curriculum. This concept should take students well beyond the study of ratios, proportions, and percent. A real understanding of proportionality allows students to connect their experience with numbers and operations to ideas that they have studied in geometry, measurement, and data analysis. They begin to get a sense of how two quantities can be related proportionally, as seen on maps, scale drawings, and similar figures, or in calculating sales tax or commissions.
A solid understanding of proportionality sets the stage for students to succeed in the more formal study of algebra. From this base, notions of linearity and linear functions emerge naturally. As students explore how to use linear functions to solve problems, the bigger world of functions that may not be linear begins to open for them. Looking at what is the same and what is different among functions lies at the heart of understanding algebraic skills and processes.
Source: Adapted with permission from “A Journey in Algebraic Thinking,” by C. Seeley, 2004. Retrieved May 16, 2005, from www.nctm.org/news/president/2004_09president.htm
It's not teaching algebra, per se, notes Thomas P. Carpenter, emeritus professor of curriculum and instruction at the University of Wisconsin–Madison. “We're not saying ‘Let's go down and teach kids how to solve equations,’” he says. Instead, it involves helping children recognize patterns and teaching arithmetic in ways that are more consistent with how it is used in formal algebra.
For example, teachers may ask a child to solve this equation: 10 + 2 = __ by asking, “What is 10 plus 2?” If the teacher never points out that the equal sign means the same as, students may mistakenly come to think that the equal sign merely signals that they have to perform some kind of operation. A slight shift in language, “What is 10 plus 2 the same as?” can help students build a deeper understanding of equivalence (Carpenter & Romberg, 2004, p. 39; see Resource Review: All About the Equal Sign, p. 20).
A slight shift in language can help students deepen understanding of equivalence.
Helping students develop a deep understanding of what “equal” means is the subject of a featured lesson in Powerful Practices in Mathematics and Science, a multimedia resource based on the research of the National Center for Improving Student Learning and Achievement in Mathematics and Science (NCISLA).
In the lesson for grade 4, which was videotaped for one of the two CD-ROMs included in the resource, the teacher writes this number sentence on the board:
8 + 4 = __ + 5
Students initially believe that the number 12 will make the number sentence true.
Recognizing that her students hold a common misconception that the equal sign symbols an operation, the teacher helps students come to see why 12 can't be right. She doesn't immediately correct the students; instead, she uses the incorrect response as an entry point into a lesson about equivalence.
First, the teacher poses number sentences in nonstandard forms:
7 = 3 + 4 and 6 = 6 + 0
Then, the teacher challenges students to determine whether these equations are true or false. Working together to discuss the problems, children come to see that the equal sign signifies balance. As the teacher works with groups of students, she gradually introduces the language that will guide students in their discussions. For example, she uses phrases like “is the same as” to convey the correct meaning of the equal sign.
When all the students are ready, the teacher asks them to share their strategies and explain their reasoning. She does not tell students if an answer is wrong but gives other examples and asks questions that enable students to ultimately figure out what works mathematically.
The lesson on the equal sign is one example of many included in Powerful Practices that helps illustrate an NCISLA-approved approach to teaching mathematics and science. Thomas P. Carpenter, emeritus professor at the University of Wisconsin and a coauthor of the materials, says the program is designed to focus on how mathematicians and scientists approach unknowns, to highlight and “demystify” the practices they follow to “come up with explanations for the underlying causes of things.”
The three practices that mathematicians and scientists engage in—and that students can, too—are constructing models, making generalizations, and justifying those generalizations. “These are sense-making activities,” says Carpenter. “Mathematics and science becomes easier to learn when it's done this way. The content sticks.”
Source: Adapted from Powerful Practices in Mathematics and Science (pp. 1, 39), by P. T. Carpenter and T. A. Romberg, 2004, Madison, WI: The Board of Regents, University of Wisconsin System. The program is distributed by Learning Point Associates. Order copies of Powerful Practices online at mscproducts@contact.learningpt.org.
When armed with the correct meaning of the equal sign, students are better equipped to see how numbers and symbols can be interchangeable in mathematical equations. Students can start substituting variables for numbers much earlier than traditionally thought, Carpenter says. And, if students learn these fundamental ideas early, “there's not so great a transition when algebra [the subject] comes.”
If introducing algebraic concepts earlier in a child's schooling helps make the move to the subject more seamless, then it's reasonable to expect that more children will achieve in algebra. And that, says Robert P. Moses, is important for two reasons. One reason is equity. “Algebra is a gatekeeper subject,” he told Educational Leadership. “Too many poor children and children of color are denied access to upper-level math classes—to full citizenship, really—because they don't know algebra,” states Moses (Checkley, 2001, p. 6), who founded the Algebra Project, a national mathematics literacy effort aimed at helping low-income students and students of color achieve higher-level mathematical skills.
A second reason to introduce algebra early is because of technology and the careers it spawns. For students to get a job and support a family in a world “driven by technology,” students need a new literacy, Moses asserted. “Computers are run by symbolic systems. To understand the language of computers, we must have an understanding of a mathematics that encodes quantitative data and creates symbolic representations. The place in the curriculum where students are introduced to such a language is algebra” (Checkley, 2001, p. 6).
The attention to strengthening students' algebraic thinking in the early grades is matched by an effort to help students hone their problem-solving skills—an area that's ripe for improvement, given how U.S. students fared on the Program for International Students Assessment (PISA) in 2004.
Reform-minded educators want to be sure that the mathematics curricula allow students to grapple with the myriad of problems they will eventually encounter, says Barbara J. Reys, distinguished professor of mathematics at the University of Missouri. Children must learn the skills of mathematics, but they must also learn to use those skills to “reduce a complex situation into something they can represent and sort out,” she says. Children are “presented with lots of numerical information,” observes Reys. “The more comfortable they are in analyzing that data, the better the decisions they'll make.”
The ability to problem solve is a tremendous life skill that has a much broader application than school mathematics, says NCTM's Seeley. In learning to problem solve, students think about cause and effect, about actions and consequences, she asserts. Seeley concedes that the PISA results were an appropriate indicator that problem-solving opportunities need to be better integrated into lessons. “We need to give more than lip service to problem solving,” she notes, and one way to do that is to create challenging problems that reflect the world that students know.
For example, many students may someday want to convince their parents to take them to an amusement park, such as Disneyland or Disney World. Through an 8-unit lesson for grades 3–5, teams of students learn where to find data to make an informed pitch: they study maps, consider different routes, visit Web sites to get airline schedules and costs, and analyze their data. Each team then presents its vacation plan to the class and discusses the best features of each plan (NCTM, 2005).
Another example: There will be a day when the children who are now in elementary school will have to buy cell phones. Will they “just go with the first plan they come to?” asks Reys. Or will they use their mathematical tools to determine the best plan for them? If students have been taught how to approach a problem and organize data in different ways—to put it into equations or a graph or spreadsheet—they can compare the costs and benefits of each plan and make an informed decision, Reys asserts.
Will students just go with the first plan they come to? Or will they use mathematical tools to determine which is the best plan?
That experience will also prepare them for the next issue they'll need to address, she adds. Teachers can ask students to think about the problem-solving process and determine how to use what they learned to make sense of another puzzler. “It's not just about solving this problem,” says Reys. “It's also about developing some confidence in their ability to solve other problems.”
Another math confidence booster involves communication. When students can explain how they solved a particular problem, when they can discuss their strategies, they solidify their understanding, say educators.
It's not just about solving this problem. It's also about developing some confidence in their ability to solve other problems.
“In order to make connections, we have to have conversations about math,” states Gail Underwood, a 2nd grade teacher at Grant Elementary School in Columbia, Missouri. Sometimes a discussion is held before an activity, Underwood explains. She asks the students—or they ask each other: Do you have a plan? If you don't have a plan, how can you approach the problem?
The students continue the conversation as they work in pairs or groups (or alone, if preferred) to solve the problem. The group then reconvenes for what Underwood calls a Math Congress. In this forum, students share their strategies and Underwood may introduce math concepts she thinks are important are to discuss. “It's a thoughtful process. There is a purpose,” Underwood states.
That purpose includes clarifying thinking and helping students see that there are many approaches to solving a problem, she says. The Math Congress also helps build a sense of community in the classroom. Underwood, who received a 2004 Presidential Award for Excellence in Mathematics and Science Teaching, notes that it takes time, but by October of any given year, her students are usually willing to say to their peers: “This is what I'm thinking,” or “I've got two answers, but I'm not sure which one is right, so I'm bringing it to the community.”
In doing so, says Underwood, students see themselves as people who make mistakes but who, through discussion, can find a strategy that will lead them to the right response. It's an ongoing life lesson, she observes, because students learn how to ask for help, and, if they see a mistake, they'll know how to correct the result in polite and respectful way.
Emphasizing communication includes giving students more opportunities to write about their mathematical understandings, according to those who crafted the NCTM standards documents. When asked to explain their problem-solving processes or to discuss how the math they learned might be used in the real world, students deepen their understanding of concepts and clarify their thinking (Goldsby & Cozza, 2002; Sjoberg, Slavit, & Coon, 2004).
Students' writing about their thinking also gives teachers alternative methods of assessment and better prepares students for high-stakes tests. In Missouri, for example, a huge portion of the state proficiency test requires that students communicate their thinking, says Underwood. Students have to be able to explain how they came to their conclusions, she states.
Underwood's students receive a mathematics education that is vastly different from the one she received. “I don't remember having those discussions about where to start with problem solving,” she says. What she does recall is the rote memorization of math facts and worksheets filled with problems that she dutifully completed and turned in, which were corrected (in red ink) and returned—no discussion necessary.
Many of Underwood's colleagues—as well as the parents of her students—had similar experiences. This is one reason that the pace of reform has been sluggish: prudence impedes progress. “How do you help people see that how we did things 20 years ago may not be the right way?” she asks, noting that her experience with school-level instructional change has convinced her that “we have to be thoughtful and careful in how we bring about reform.”
One way to reassure those who are anxious is to point out that computation is not ignored when K–6 teachers emphasize algebraic thinking, problem solving, and communication. In the context of solving a problem, for example, a teacher can ask students to recite their times tables and discuss how this knowledge better equips them to solve the problem. It's also appropriate for children to practice computation as a separate activity, so long as “the practice does not become the major activity in mathematics classes,” writes Michael T. Battista in “Research and Reform in Mathematics Education” (2001, p. 75). Otherwise, students will come to think that mathematics “involves nothing more than memorizing the rule you must follow to get the right answer,” he states (p. 50).
“It's important to acknowledge that computation is an important part of a balanced math program,” Seeley agrees. “Even in an era of calculators, we still want kids to do basic arithmetic. They have to have so much more than that, however. They have to know what to do with the arithmetic they learn,” she states.
Just as students need to know how to apply what they learn, educators need to know how to analyze the results of reforms they implement—and alter the course, if necessary, say experts. Communication between those “who do research in math and those who teach children,” must be enhanced, says Saul, who served on the RAND Mathematics Study Panel, which was convened in 1999 to study the lackluster performance in mathematics by U.S. students, determine some of the causes for poor achievement, and identify solutions.
The panel's work culminated in Mathematical Proficiency for All Students: Toward a Strategic Research and Development Program in Mathematics Education, a report released to the U.S. Department of Education in 2002. The study suggests that a program of research and development can lead to an increase in students' mathematics proficiency (see A Cycle of Research, Development, and Improvement, p. 28). Such a program could help determine
As pointed out in the report, teaching and learning will improve because such a program institutes a cycle of continuous improvement: there is initial research, development, and improved knowledge and practice, followed by evaluation, which leads to new research, new development, and improved knowledge and practice (RAND Mathematics Study Panel, 2003).
The RAND panel's work coincided with the NCTM's effort to update their original standards. Released in 2000, the new Principles and Standards identified priorities that corresponded with the RAND study's conclusions. The new NCTM standards, for example, state that students must develop computational fluency. That ability is included in the description of mathematical proficiency in the RAND report. The NCTM standards also stipulate that by the end of 8th grade, students should have a strong foundation in algebra and geometry. Emphasizing algebra K–12 is a key research focus recommended in the RAND study.
That mathematics researchers and mathematics practitioners should reach similar conclusions about priorities in practice bodes well for improving mathematics education, suggests Seeley, who stresses a need for real collaborations between mathematicians, researchers, and mathematics educators. That way, she notes, “people are working together and learning from each other.”
Saul agrees. Such collaboration reinforces what he believes is the essence of standards-based reform: improved student achievement. Conversations between teachers and researchers, Saul says, should be “based on what we want students to learn.”
Fortunately, Seeley observes, the underlying theme of the work completed by the RAND panel and NCTM, is that “we all want students to do math better.”
As many educators point out, the trend toward emphasizing algebraic thinking, problem solving, and communication in K–6 school mathematics is not a new trend. Still, as NCTM's Cathy Seeley acknowledges, “Making that change is pretty challenging.” It's helpful, therefore, to keep in mind that
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