The Matrix of Mathematical Big Ideas is a visualization that allows educators of elementary and middle-school-grades learners to make sense of the ways in which underlying mathematical ideas regarding number and operations can aid in developing deep, conceptual understanding and thereby increase the abilities of learners to actually reason mathematically.  

As indicated in its title, the organization of this visualization is in the form of a matrix. The operations are at the vertical axis, and the rational number forms - namely whole, standard decimal, common fraction, and integers - are at the horizontal axis. At the intersection of these axes are cells that display the various ways - pictorially and/or algorithmically - in which learners are able to conceive of the operations as a concept when the underlying properties of the operations as well as understandings of number are prioritized in instruction. As educators study the matrix, they will be able to envision the variations in the ways in which the properties and understandings of number will allow learners to take a critical approach to their work, increasing their ability to think flexibly and offer mathematically sound justifications of their thinking. Furthermore, the matrix provides educators a means by which to observe how the concepts of operations persists regardless of the rational number form used in a specific operation. Because the concept persists, educators, again, understand that conceptualizing the operations is of greater value to learners rather than providing instruction of algorithms that are limited to a certain number form and operation.