Conceptual understanding – comprehension of mathematical concepts, operations, and relations This strand is all about knowing more than isolated facts. Students should be linking lessons together and understanding the connections in math. They should also be able to correlate new ideas to previously learned lessons. Students also should be trying to represent mathematical situations in different ways and be forming deep, rich connections.

“Conceptual understanding frequently results in students having less to learn because they can see the deeper similarities between superficially unrelated situations. “

Procedural fluency – skill in carrying out procedures flexibly, accurately, efficiently, and appropriately. Students should be efficient and accurate when performing basic computations without needing aids or tables. Students need to see that procedures can be developed that will help solve more problems. Without sufficient procedural fluency students cannot deepen their understandings and can then start disconnecting from math. If students learn without understanding they could separate what happens in school from what is outside.

“In the domain of number, procedural fluency is especially needed to support conceptual understanding of place value and the meanings of rational numbers.

Strategic competence – ability to formulate, represent, and solve mathematical problems. Students are often presented with clearly specified problems to solve, outside of school they encounter situations in which part of the difficulty is to figure out exactly what the problem is. Students should also know a variety of solution strategies and should know what each one will be most useful in certain situations. It is also important for everyone to start by building a mental image of the essential components.

“Not only do students need to be able to build representations of individual situations, but they also need to see that some representations share common mathematical structures.”

Adaptive reasoning – capacity for logical thought, reflection, explanation, and justification. This is the glue that holds everything together and that guides learners. Students who are disagreeing about an answer need to not questions anyone or anything except for their own reasoning, and whether it is valid or not. Some people say that kids do not start deep reasoning until 12 years old, but four and five-years old can explain their reasoning of how they solved the problem. A skill in this strand would to be able to justify your work. Some teachers truly believe that there is only one way to do something and being able to explain how you reached an answer can help everyone learn something new.

“Students need to be able to justify and explain ideas in order to make their reasoning clear, hone their reasoning skills, and improve their conceptual understanding.”

Productive disposition – habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with belief in diligence and one’s own efficacy. This skill can be built through frequent opportunities to make sense of mathematics, to see the benefits of determination and to experience the joy of math making sense. When students see themselves as capable of learning and using mathematics, they can begin to develop more layers of their other strand skills. These students will usually be confident in their knowledge and ability.

“The more mathematical concepts they understand, the more sensible mathematics becomes.”

https://www.nap.edu/read/9822/chapter/6#131