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).

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.

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."

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.

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.

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