Full Answer Section

• 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.