Although there are subtle differences in the ways that algebra is defined in mathematics education

  1. According to Briahier, “Although there are subtle differences in the ways that algebra is defined in mathematics education, it is important to think of it as a language and a content area, rather than a course that one takes in secondary or middle school” (p. 340). What do you think he means by algebra as a language? What would that conception of algebra mean for teaching?

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Briahier’s statement that algebra should be thought of as a language and a content area, rather than just a course, is a powerful one. Thinking of algebra as a language implies that it’s not just a set of rules and procedures to memorize, but a way of thinking and communicating mathematical ideas. It’s a system of symbols and syntax that allows us to express relationships between quantities, generalize patterns, and solve problems.

Here’s a breakdown of what “algebra as a language” might mean:

  • Symbols and Syntax: Just like any language, algebra has its own set of symbols (variables, constants, operators) and rules for how they can be combined (order of operations, properties of equality). Learning algebra is like learning the vocabulary and grammar of this language.

 

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  • Expressing Relationships: Algebra allows us to express relationships between quantities in a concise and general way. For example, the formula A = lw expresses the relationship between the area of a rectangle, its length, and its width, and it applies to all rectangles.
  • Generalization: Algebra enables us to generalize patterns and make statements that are true for a whole class of cases. For example, the formula (a + b)² = a² + 2ab + b² is true for all real numbers a and b.
  • Problem Solving: Algebra provides tools for solving problems by translating them into symbolic form, manipulating the symbols according to the rules of algebra, and then interpreting the results.

What this conception of algebra means for teaching:

If we view algebra as a language, it has significant implications for how we teach it:

  • Emphasis on Meaning: Instead of just memorizing rules, students need to understand the meaning behind the symbols and operations. Why does the order of operations matter? What does it mean to solve for x? We need to help students develop a deep conceptual understanding.
  • Focus on Communication: Students need opportunities to express mathematical ideas in algebraic language and to interpret the algebraic expressions of others. This means reading, writing, and speaking algebra.
  • Connections to Other Representations: Algebra shouldn’t be taught in isolation. We need to connect algebraic representations to other representations, such as graphs, tables, and diagrams. This helps students see the same mathematical ideas from different perspectives.
  • Problem Solving as Translation: Problem solving should be seen as a process of translating real-world situations or mathematical problems into algebraic language, manipulating the symbols to find a solution, and then translating the solution back into the original context.
  • Developing Fluency: Just like learning a spoken language, becoming fluent in algebra takes time and practice. Students need opportunities to use the language in a variety of contexts to develop fluency and automaticity.
  • Focus on Reasoning: Algebra is not just about getting the right answer; it’s also about understanding why the answer is right. Students need to develop their reasoning skills and be able to justify their solutions.

In short, teaching algebra as a language means moving away from a purely procedural approach and towards a more conceptual and communicative approach. It means helping students develop not just the skills to manipulate symbols, but also the understanding and fluency to use algebra as a tool for thinking and communicating mathematically. It means creating a learning environment where students are actively engaged in making sense of algebraic ideas and using them to solve problems.

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