“What’s four times four?”, my son asks.

“Sixteen,” I respond and then add, “How do you know that’s true?”

“Hrmmm. I know four times four is the same as four plus four plus four plus four and that’s the same as eight plus eight, which is sixteen.”

During my career, I’ve found that mathematical ideas include procedures, concepts, habits of mind, and declarative knowledge. Even in the brief exchange above, I claim that all four of these types of mathematical ideas are being used.

I define procedures as a sequence of steps intended to be used to solve specific mathematical problems, concepts are ideas that can be used with different procedures, habits of mind are general problem-solving strategies, and declarative knowledge is that which is known to be true without reference to other ideas.

  1. Procedures: My son knows that if he wants to add 4 numbers, he first adds two numbers together, then the last two numbers together, and then these two results together.
  2. Concepts: My son knows that one definition of multiplication is repeated addition and uses this idea to transform 4 times 4 into 4 plus 4 plus 4 plus 4.
  3. Habits of mind: My son knows that if one is not sure how to solve a problem that one can often change it into a different problem that one can solve. In this case, my son decides to change the multiplication problem into an addition problem.
  4. Declarative knowledge: My son knows that four plus four is eight and eight plus eight is 16 without reference to other ideas. It is often the case that things that are currently declarative knowledge are based on procedures and conceptual knowledge learned earlier.

This is why I find arguments about whether we should teach children procedures or teach them conceptually confusing — it’s not possible to do one or the other, students are always learning some mixture of all four types of mathematical knowledge.

“Knowledge is not tiny bits that we can count and represent by numbers, but a network of logically interconnected ideas, beliefs, and generalizations structured so it can be searched and used to work out and evaluate new ideas.”
– Graham Nuthall, The Hidden Lives of Learners.