Some people who critique the traditional approach to algebra instruction cite papers like this one, a “research brief” by Toni M. Smith, published last year by the American Institutes for Research and entitled "Instructional Practices to Support Student Success in Algebra I."
Typically, instruction in algebra focuses on symbolic manipulation and algebraic procedures, with little attention to the connections between these procedures and the underlying mathematical concepts (Chazan and Yerushalmy, 2003). When students are expected to memorize and operate with a set of rules that are seemingly meaningless, they may become frustrated and eventually fail Algebra I.Smith is thus suggesting that your typical algebra class neglects to teach the meaning of the mathematical symbols. But times, she suggests, have changed:
For decades, the heart of algebra education has been learning to manipulate algebraic symbols to solve equations. Now, students are expected to know and be able to do more than solve equations. They are expected to demonstrate fluency in working with algebraic procedures, understand the associated mathematical concepts behind these procedures, and be able to articulate the connections between and among them.If "articulating connections" means doing so verbally, that's clearly new. But I hadn't realized that it's only now that, for the first time, math classes are (a) expecting students to work through algebraic procedures fluently and (b) expecting them to understand the underlying math concepts.
More specifically, as far as "articulating connections" goes:
Not only should students be able to solve an equation such as 3x + 5 = 20 for x by performing one or more steps, but also they should be able to articulate the mathematical principles (e.g., order of operations, properties of equality) that support the procedure they used and critique a different approach for solving that same equation.True, it's possible to stumble on the correct solution to this equation without understanding order of operations and properties of equality. But how about assigning a bunch more problems of similar and greater difficulty? Is it really possible to mindlessly follow the order of operations and perform the same operations on both sides of the "=" sign without understanding order of operations and the meaning of "="?
As far as understanding the underlying math concepts goes, now, for the first time, students are expected, when solving 3x + 5 = 20, to
understand that the solution they find represents the value for x in the function f(x) = 3x + 5 when f(x) is 21 [sic]. Not only should they be able to graph the function f(x) = 3x + 5 but also they should (a) understand why it would look like a line, (b) be able to identify the solution to the equation they solved on that line, and (c) know what kind of real-world relationship such a function would model.All of this, apparently, is completely absent from traditional algebra texts and traditional algebra classrooms, which, apparently, are "focusing solely on skill in manipulating algebraic symbols" without attaching "meaning to the associated algebraic symbols." This is too bad, because, as Smith explains to us:
Research indicates that when instruction attaches meaning to the associated algebraic symbols, students develop procedural fluency as well as conceptual understanding. This can be done in a number of ways, including through the use of technology.Luckily, though, technology is not essential:
One approach to assigning meaning to algebraic symbols involves algebraic expressions (e.g., 3x + 4). Students are given a list of algebraic expressions and asked to predict whether or not any of the algebraic expressions are equivalent. Some expressions are equivalent but look different (e.g., 2(x + y) and 2x + 2y) and others look similar but are not equivalent (e.g., 2(x – 3) and 2x – 3). After they make their predictions, students are asked to test them by substituting several numerical values for the variables in those expressions. This step emphasizes the meaning of variable. Once they have their results, students are then asked to provide a justification for what they found. Students who experienced this form of instruction on algebraic expressions produced higher pretest-posttest gains on measures of symbolic manipulation and understanding of variable than did students who received conventional, skills-based instruction (Graham and Thomas, 2000).Presumably the control group was never properly taught the meaning of parentheticals, or they wouldn't have seen 2(x + y) as more similar to 2x + y than to 2x + 2y.
Of course, the idea that any textbook or half-way competent teacher would teach symbols without teaching their associated meanings—whether in math class, music class, or Chinese class—is ludicrous. Equally ludicrous is that the only way to for students to learn the underlying mathematical concepts is to make predictions, test them out by plugging in numbers, and then provide a verbal justification for what they found.
How about instead doing what traditional math textbooks used to do, and have students represent real-world situations as algebraic expressions, translating words into mathematical symbols? For some nice examples of problems of this variety, rare in today's Reform Math texts, but presented to students at the very beginning of a traditional algebra text, see here.