It's Elementary: Introducing Algebraic Thinking Before High School
Sitting in Mrs. Peavey's Algebra I class, I experienced algebra much like millions of other Americans—as an intensive study of the last three letters of the alphabet. I failed to grasp the importance of algebra—how it provides support for almost all of mathematics or to understand its power as a tool for analytical thinking. It was a course I endured to get into college.
Algebra for All
Thirty years later, algebra is not just for those who plan to attend college, but for everyone. Robert Moses, founder of the Algebra Project, says that in today's technological society, algebra has become a gatekeeper for citizenship and economic access. As the world has become more technological, the reasoning and problem solving that algebra demands are required in a variety of workplace settings. We also see evidence of the growing importance of algebra in standards and assessments. National and state assessments include algebraic skills at the eighth-grade level and many high school exit exams now test algebraic proficiency. It seems the mantra "algebra for all" has been firmly established. Johnny Lott, president of the National Council of Teachers of Mathematics (NCTM), agrees. "I think most everybody recognizes the importance of algebra. It is a question of how they introduce it and when," he says.
James Kaput, a researcher from the University of Massachusetts, Dartmouth, believes that by "algebrafying" the K–12 curriculum, we can fulfill the promise of algebra for all and eliminate "the most pernicious curricular element of today's school mathematics — late, abrupt, isolated, and superficial high school algebra courses" (Kaput, 2000). The idea isn't new. Kaput, other researchers and educators, and the NCTM have been promoting algebra as a K–12 experience, integrating algebraic thinking and reasoning throughout the mathematics curriculum.
University of Wisconsin researcher Linda Levi, who has been working on a study called the Early Algebra Project for the past eight years, emphasizes, "We're not saying you should be teaching high school algebra to elementary school children." Instead, Levi and her colleagues in the Early Algebra Project, Thomas Carpenter and Megan Loef Franke, believe teachers should engage children in learning about the general principles of mathematics as they are learning arithmetic. They say that the learning of arithmetic is often isolated from other related mathematical ideas. This deprives students of powerful ways of thinking about mathematics and can make it more difficult for students to learn algebra later on. Many students studying high school algebra don't see the procedures they use to solve equations or simplify expressions as based on the same properties that they used in arithmetic computation (Carpenter, Franke, & Levi, 2003).
The Early Algebra Research Project
The Early Algebra Research Project began in 1996 under the direction of Thomas Carpenter, director of the National Center for Improving Student Learning and Achievement in Mathematics; Megan Loef Franke, an associate professor at the University of California, Los Angeles, and director of Center X: Where Research and Practice Intersect for Urban School Professionals; and Linda Levi, associate researcher at the Wisconsin Center for Education Research. It grew out of the Cognitively Guided Instruction research program begun in 1985.
The study, which initially began in Madison, Wisconsin, involved approximately 240 elementary school students and their teachers. It found that innovative professional development and refocused mathematics instruction paved the way for elementary school children to begin to reason algebraically.
The researchers are now conducting a large-scale experimental study in Los Angeles, involving about 5,000 elementary school students and their teachers. The study is examining the effects of the teacher professional development program on students' algebraic understandings.
Levi says the researchers have collected achievement data for the students involved and will complete their analysis in 2004.
Levi explains, "Kids come to school with a very rich understanding of numbers and operations. They may still make mistakes when counting but they solve many math problems. A lot of kindergartners come in knowing that when you add zero to a number, the number doesn't change. That is a big principle in mathematics. And they can talk about it. Maybe they can't write it down or can't read it if you write it down, but they can start talking about things that they know to always be true in math." Levi adds that teachers often don't realize how powerful the patterns or generalizations that their students express can be. These expressions should be seen as opportunities for class discussions so that all of the students have access to these ideas. "As teachers, it's really our job to understand how children think about mathematics when they come to school and build on this informal understanding," she says.
Fostering Students' Thinking
According to Blanton and Kaput (2003), teachers must find ways to support algebraic thinking and create a classroom culture that values "students modeling, exploring, arguing, predicting, conjecturing, and testing their ideas, as well as practicing computational skills." They suggest that teachers "algebrafy" current curriculum materials by using existing arithmetic activities and word problems, transforming them from problems with a single numerical answer to opportunities for discovering patterns and making conjectures or generalizations about mathematical facts and relationships and justifying them. This can be as simple as encouraging children to discuss why they believe a mathematical statement or solution to a problem is correct. Blanton and Kaput suggest teachers use the following prompts as ways to extend student thinking:
- Tell me what you were thinking.
- Did you solve this in a different way?
- How do you know this is true?
- Does this always work?
In their pilot study involving 240 students, Carpenter, Franke, and Levi found that teachers have good luck beginning discussions among students and eliciting generalizations from students using true-false and open-number sentences (see examples in the sidebar "Number Sentences Used to Elicit Generalizations"). For students in upper elementary school this can lead to discussion of what is required to justify a generalization.
Number Sentences Used to Elicit Generalizations
Below are examples of number sentences teachers used to help students articulate mathematical generalizations.
|E X A M P L E S||78 + 0 = 78; 23 + 7 = 23 *|
|"When you add zero to a number, you get the number you started with."|
|E X A M P L E S||96 - 96 = 0; 74 - ____ = 74|
|"When you subtract a number from itself, you get zero."|
|E X A M P L E S||96 x 0 = 0; 43 x 0 = 43*|
|"When you multiply a number times zero, you get zero."|
|E X A M P L E S||65 x 54 = 54 x 65; 94 x 71 = 71 x ____|
|"When multiplying two numbers, you can change the order of the numbers."|
*denotes a false number sentence
Source: National Center for Improving Student Learning & Achievement in Mathematics and Science. (2000). Building a Foundation for Learning Algebra in the Elementary Grades.
The Notion of Equality and Relational Thinking
One of the major concepts that Carpenter, Franke, Levi, and other researchers have written a lot about is getting children to understand that the equal sign represents a relationship. At the beginning of the Early Algebra Project, participating teachers presented the following problem to their students:
8 + 4 =____ + 5
Eighty-four percent of 145 sixth-grade students gave the solution to the problem as "12." Another 14 percent gave the solution as "17." It became clear through subsequent class discussions that to these students, the equal sign meant "carry out the operation." They had not learned that the equal sign expresses a relationship between the numbers on each side of the equal sign. Levi says, "We're advocating that when teachers begin using the equal sign with children, they use it in a way that encourages an understanding of a relationship between two quantities rather than just a signal to perform the operation. Number sentences such as 6 = 6 and 8 = 7 + 1 need to be included when teachers begin introducing the equal sign."
This type of relational thinking is crucial to students who are learning algebra but it also enhances computation skills. "If you look at algebra in a more general sense," says Levi, "what you are really looking for is the major unifying principles and properties of mathematics. As soon as kids start learning how to count, and then add, subtract, multiply, and divide, they are encountering these major principles. It makes computation a lot more efficient and accurate. For example, if kids understand the distributive property, their multiplication strategies are much more efficient and accurate than if they are trying to do repeated addition over and over again." Teachers can also provide opportunities for building computation skill in the context of finding and generalizing mathematical patterns and relationships.
How do teachers know if a student is using relational thinking? Levi explains, "We eventually want children to solve a problem like 397 + 248 = 396 + t without computing. Initially children will solve this problem by adding 397 and 248 getting 645 and then figuring out what they have to add to 396 to get 645. But by the end of elementary school, I want kids to look at the whole number sentence and realize that since 397 is 1 more than 396, t has to be one more than 248. There are relationships such as this one for subtraction, multiplication, and division as well. I want children to fully understand the operations with known quantities before they start a formal study of algebra where many of the quantities are variables or unknowns."
Why Understanding Equality Matters
Children must understand that equality is a relationship that expresses the idea that two mathematical expressions hold the same value. It is important for children to understand this idea for two reasons. First, children need this understanding to think about the relationships expressed by number sentences. For example, the number sentence 7 + 8 = 7 + 7 + 1 expresses a mathematical relationship that is central to arithmetic. When a child says, "I don't remember what 7 plus 8 is, but I do know that 7 plus 7 is 14 and then 1 more would make 15," he or she is explaining a very important relationship that is expressed by that number sentence. Children who understand equality will have a way of representing such arithmetic ideas; thus they will be able to communicate and further reflect on these ideas. A child who has many opportunities to express and reflect on such number sentences as 17 - 9 = 17 - 10 + 1 might be able to solve more difficult problems, such as 45-18, by expressing 45 - 18 = 45 - 20 + 2. This example shows the advantages of integrating the teaching of arithmetic with the teaching of algebra. By doing so, teachers can help children increase their understanding of arithmetic at the same time that they learn algebraic concepts.
A second reason that understanding equality as a relationship is important is that a lack of such understanding is one of the major stumbling blocks for students when they move from arithmetic to algebra (Kieran, 1981 & Matz, 1982). Consider, for example, the equation 4x + 27 = 87. Many would begin to solve this equation by subtracting 27 from both sides of the equal sign. Why may we do so? If the equal sign signifies a relationship between two expressions, it makes sense that if two quantities are equal, then 27 less of the first quantity must equal 27 less of the second quantity. What about children who think that the equal sign means that they should do something? What chance do they have of being able to understand the reason that subtracting 27 from both sides of an equation maintains the equality relationship? These students can only try to memorize a series of rules for solving equations. Because such rules are not embedded in understanding, students are highly likely to remember them incorrectly and not be able to apply them flexibly. For these reasons, children must understand that equality is a relationship rather than a signal to do something.
Source: Falkner, K. P., L. Levi, and T. P. Carpenter (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(1), p. 234. Reprinted with permission from the National Council of Teachers of Mathematics.
How Do We Get Teachers to Think Algebraically?
Elementary school teachers will need professional development to integrate algebraic thinking into their classrooms, as they typically have experienced algebra much like the majority of us—as Algebra I and II in high school and college. Blanton and Kaput (2003) write, "Elementary teachers need their own experiences with a richer and more connected algebra and an understanding of how to build these opportunities for their students."
A critical component of the Early Algebra Project has been its professional development for the teachers involved in the project. The project enabled teachers to spend time together discussing mathematics and their students' thinking. One of the principals in the Early Algebra Project requested that teachers bring in examples of their students' work and discuss with her what they were learning in the project. Such support can go a long way in encouraging teacher development.
In the Classroom
"Build a Foundation for Learning Algebra"
Here are a few ways to provide a foundation for learning algebra.
Ask questions that provide a window into children's understanding of important mathematical ideas. For example, students' responses to the number sentence
9 + 6 = __ + 8 tells a great deal about their understanding of the meaning of the equal sign. Probe students' reasons for their answers. Ask students why they answered as they did.
Provide students opportunities to discuss and resolve different conceptions of mathematical ideas. For example, different conceptions of the equal sign that emerge from students' solutions to the open number sentence 9 + 6 = __ + 8 can provide the basis for a productive discussion.
Provide students with equations that help them understand that the equal sign represents a relation between numbers, not a signal to carry out the preceding calculation. Examples include __= 8 + 9, 8 + 6 = 6 + , 9 + 6 = __ + 8, Vary the format of number sentences. Include sentences in which the answer does not come right after the equal sign.
Provide students with true and false number sentences that challenge their misconceptions about the equal sign (e.g., 8 = 5 + 3, 9 = 9, 7 - 4 = 7 - 4).
Provide students problems that encourage them to make generalizations about basic number properties (see "Number Sentences to Elicit Generalizations.") When they provide an answer to one of the problems, ask them how they know their answer is correct. That often will result in their stating a generalization such as "When you subtract a number from itself, you get zero." When they do state a generalization like this, ask for example, "Is that true for all numbers?"
Have students justify generalizations they or their peers propose. Justification of generalizations requires more than providing a lot of examples (e.g., 8 x 5 = 5 x 8). By expecting children to justify their claims, you can help them gain skills in presenting mathematical arguments and proofs. Use the questions "Will that be true for all numbers?" and "How do you know that is true for all numbers?" repeatedly to encourage students to recognize that they need to justify their claims in mathematics.
Reprinted from K–12 Mathematics & Science: Teaching Considerations (Fall 2000), published by the National Center for Improving Student Learning & Achievement in Mathematics and Science, Wisconsin Center for Education Research, Madison, Wisconsin.
References and Suggested Reading
- Blanton, M. L., & Kaput, J. J. (2003). Developing elementary teachers' "algebra eyes and ears." Teaching Children Mathematics, 10(2).
- Carpenter, T.C., et al. (1999). Children's Mathematics: Cognitively Guided Instruction (with two multimedia CDs). Portsmouth, NH: Heinemann.
- Carpenter, T. C., Franke, M. L., & Levi, L. (2003). Thinking Mathematically: Integrating Arithmetic and Algebra in Elementary School. Portsmouth, NH: Heinemann.
- Falkner, K. P., Levi, L., & Carpenter, T. P. (1999). Children's understanding of equality: A foundation for algebra. Teaching Children Mathematics, 6(1).
- Kaput, J. J. (2000). Transforming algebra from an engine of inequity to an engine of mathematical power by "algebrafying" the K–12 curriculum. Dartmouth, MA: National Center for Improving Student Learning and Achievement in Mathematics and Science. (ERIC Document Reproduction Service No. ED 441 664).
- Moses, R. P., & Cobb, C. E. (2001). Radical Equations: Math Literacy and Civil Rights. Boston: Beacon Press.
- National Center for Improving Student Learning & Achievement in Mathematics & Science. (2000). Building a foundation for learning algebra in the elementary grades. In Brief: K-12 Mathematics & Science Research Implications, 1(2).
- National Council of Teachers of Mathematics. (1999). Algebraic thinking: Grades K-12. Reston, VA: NCTM.
Eisenhower National Clearinghouse (ENC)
Most mathematics and science teachers are probably familiar with the ENC and its magazine, ENC Focus. The Web site contains all sorts of lesson plans, activities, and resources. Most materials are free and online. A search for "algebraic thinking" on www.enc.org yielded 300 suggestions.
National Council of Teachers of Mathematics
The Web site of the National Council of Teachers of Mathematics is geared to members of the organization, but includes a problem of the week for elementary, middle school, and high school levels as well as some lesson plans and activities that everyone may access. Also online are abstracts for recent issues of NCTM journals such as Teaching Children Mathematics and Mathematics Teaching in High School.
This is the NCTM Web site focused on the NCTM Principles and Standards for School Mathematics. It contains activities, resources, and lesson plans based on the standards and includes interactive and multimedia math investigations.
Figure This! is a Web site cosponsored by the National Council of Teachers of Mathematics, the National Action Committee for Minorities in Engineering, and Widemeyer Communications. It features mathematics challenges for families of middle school students and includes interesting problems and math facts. "Teacher's Corner" provides details on how to conduct a family math challenge at your school.
The National Center for Improving Student Learning and Achievement in Mathematics
Look under Teachers' Resources on this site for a section called "Building Students' Algebraic Reasoning." Here you will find articles, activities, and lesson plans to extend algebraic thinking. The Web site also includes research summaries, newsletters, and other publications.
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