Many students struggle with algebraic ideas because they have not developed the conceptual understanding (Hattie, 129 pag.)

Are you a “just the facts ma’am” mathematician, or do you have deep conceptual understanding of mathematics? How did Algebra I, Algebra II, and Calculus go for you? Did you love it, just survive it, or flat-out hate it?

What if we focus on depth of knowledge at an early age? How might we change the future for our young learners?

Imagine you are back in Algebra I, Algebra II, or Calculus working with polynomials. Do you have conceptual understanding, procedural fluency, or both?

Learning has to start with fundamental conceptual understanding, skills, and vocabulary. You have to know something before you can do something with it. Then, with appropriate instruction about how to relate and extend ideas, surface learning transforms into deep learning. Deep learning is an important foundation for students to then apply what they’ve learned in new and novel situations, which happens at the transfer phase. (Hattie, 35 pag)

What if, at the elementary school level, deep conceptual numeracy is developed, learned, and transferred?

Our brains are made up of ‘distributed networks’,and when we handle knowledge, different areas of the brain light up and communicate with each other. When we work on mathematics, in particular, brain activity is distributed between many different networks, which include two visual pathways: the ventral and dorsal visual pathways (see fig 1). Neuroimaging has shown that even when people work on a number calculation,such as 12 x 25, with symbolic digits (12 and 25) our mathematical thinking is grounded in visual processing. (Boaler, n pag.)

Using concreteness as a foundation for abstraction is not just good for mathematical instruction; it is a basic principle of understanding. (Heath and Heath, 106 pag.)`

A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success. (Boaler, n pag.)

Well, that’s worth repeating, huh?

*A number line representation of number quantity has been shown in cognitive studies to be particularly important for the development of numerical knowledge and a precursor of children’s academic success.*

Often, we rush to efficiency – to “just the facts ma’am” mathematics. Surface knowledge – memorized facts – is critical to success, but that is not the end goal of learning. The goal of all learning is transfer.

When we use number lines to support conceptual understanding of number quantity and operations, we deepen and strengthen mathematical foundation. Our young students are learning that multiplication is repeated addition, that 4 x 5 is 5 four times, which lays the foundation for being able to transfer to the following polynomials.

**a + a + a +a = 4a**

and

** a + 3b +a + 3b = 2a + 6b**

Abstraction demands some concrete foundation. Trying to teach an abstract principle without concrete foundations is like trying to start a house by building a roof in the air. (Heath and Heath, 106 pag.)

How might we focus on deep learning and transfer learning by studying and learning visually? What if we embrace seeing as understanding so that we learn to show what we know more than one way?

“Seeing as Understanding: The Importance of Visual Mathematics for Our Brain and Learning.” *Journal of Applied & Computational Mathematics* 05.05 (2016): n. pag. *Youcubed*. Standford University, 12 May. 2016. Web. 18 Mar. 2017.

Hattie, John A. (Allan); Fisher, Douglas B.; Frey, Nancy; Gojak, Linda M.; Moore, Sara Delano; Mellman, William L.. Visible Learning for Mathematics, Grades K-12: What Works Best to Optimize Student Learning (Corwin Mathematics Series) (p. 35). SAGE Publications. Kindle Edition.

Heath, Chip. Made to Stick: Why Some Ideas Survive and Others Die (p. 106). Random House Publishing Group. Kindle Edition.