1. Conceptual Understanding: Students should develop a deep understanding of mathematical concepts, not just procedural fluency. They should be able to explain their reasoning and make connections between different mathematical ideas.
2. Procedural Fluency: Students should be able to perform calculations and carry out procedures efficiently and accurately. This doesn't mean rote memorization, but rather understanding *why* procedures work.
3. Strategic Competence: Students should be able to formulate, represent, and solve mathematical problems. This involves selecting appropriate strategies, applying them effectively, and checking the reasonableness of solutions.
4. Adaptive Reasoning: Students should be able to think logically, reflect on their thinking, and justify their conclusions. They should be able to analyze situations, make generalizations, and evaluate their own work and the work of others.
5. Productive Disposition: Students should develop a positive attitude towards mathematics, believing in their ability to do mathematics and valuing its importance. This includes persistence, curiosity, and a willingness to engage in challenging mathematical tasks.
It's crucial to remember these are interconnected and not independent goals. A student who possesses strong conceptual understanding is more likely to achieve procedural fluency and strategic competence, and a positive disposition supports all the other aspects. The NCTM emphasizes the holistic development of mathematical proficiency, rather than focusing on a rigid list of five separate goals.