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Friday, March 16, 2007

What Algebra should we teach?

Algebra is fundamental to understanding mathematical thought. It encompasses an understanding of patterns and functions, ways of representing and analyzing mathematical structures and situations, models that represent quantitative ideas and relationships, and approaches that analyze change in a variety of situations.

Most people recognize that algebra is needed by scientists or engineers, but algebraic thinking and reasoning are also used in many other occupations, including health care providers, graphic designers, and home builders.

Algebra in the Early Grades

Algebra is not an abstract, difficult school subject intended for the few who choose to study advanced mathematics and science. It is fundamental to a basic education of all students, starting in the earliest grades. Young children's understanding of algebra builds on their understanding of number ideas and arithmetic concepts and properties. For example, to help students learn the basic facts of addition, students should work with missing addends, such as + 4 = 12. Here, the box represents a fixed unknown the students need to identify. This is one way to help young students understand the concept of variables.

Another example of building algebraic thinking is the zero property of addition. Students learn that adding "0" to any number produces a sum equal to that number, as in 0 + 5 = 5. In working with variables, this property may be represented in expressions such as 0 + = or + 0 = . Later, letters like "a" and "b" may be used as variables, resulting in statements similar to 0 + a = a or b + 0 = b. Important algebraic concepts like this can and should be taught in the early grades.

Middle-grades instruction must build on and continue to develop such concepts in preparation for formal instruction in algebra. For example, in middle school connections to data analysis and geometry play a role in helping children grasp patterns, functions, and ways of representing mathematical situations algebraically. A good illustration of this is helping a box manufacturing company determine the dimensions of a carton that must hold 35 cups stacked on top of each other. In order to find the dimensions of the carton, students make a table to record the number of cups and their stacked height. Using different types of cups reveals various patterns that can be represented on a coordinate graph. In this type of activity, algebraic reasoning about patterns and functions and the use of variables strengthens students' understanding of algebra and make a connection to real-life situations.

A Variety of Courses

All children must have access to algebra. It is critical when considering how best to prepare students for algebra to underscore the importance of an algebra content strand throughout the middle grades. However, the approaches described below clearly have benefits for many students.

A two-year algebra course. A two-year course in algebra spreads the content of a regular year-long algebra course over two years. It is designed for students for whom the content would be too challenging at its normal pace and gives them needed time to strengthen their understanding of patterns and functions, relationships among arithmetic operations, mathematical structures, and algebraic properties.

Exploratory algebra. An exploratory algebra course, also called hands-on algebra or algebra investigations, centers around students constructing their own understanding of algebra by uncovering facts and relationships. A course offering this approach requires careful management and monitoring by skilled and knowledgeable teachers to guarantee that students meet the goals and objectives of instruction.

Pre-algebra. A pre-algebra course, in preparation for regular algebra instruction, is designed for students who have adequately mastered arithmetic knowledge, skills, and procedures, but have not made the necessary connections between arithmetic and algebraic ways of thinking and reasoning. A pre-algebra course can give students the opportunity to bridge arithmetic and algebra with activities and experiences that deal with variables and functions, connections between arithmetic and formal mathematical structures, models of mathematical situations, and ideas about change.

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