Foundations of Math

Foundations of Math

Should more of your learners with disabilities be proficient in math? This course, focused on mathematical content knowledge, allows general and special education teachers to examine mathematical procedures from a conceptual standpoint and increase sense making for learners who struggle with methods reliant on memorization and procedure. Participants will learn how making pedagogical shifts can increase student engagement and improve math achievement for every learner.

What are the Components of Number Sense?

Reading is not reading if all of the components of reading are not working together (such as fluency, comprehension, vocabulary, phonemic awareness, and phonics).  Math is not math if learners are not engaging in, and making connections between, multiple components of number sense.

The phrase “Number Sense” is commonly used, and means different things to different people.

In this training, number sense refers to the integration of the 8 components presented in the Components of Number Sense “wheel,” each component connected to every other component through language.  During the training, each component is defined, and participants learn classroom applications, diagnosis questions, and research related to the component.

components of number sense wheel
The Components of Number Sense Wheel with the eight components listed around the outside: Quantity and Magnitude, Numeration, Equality, Base Ten, Form of a Number, Proportional Reasoning, and Algebraic and Geometric Thinking. Copyright 2007, Cain, Doggett, Faulkner, Hale, and the North Carolina Department of Public Instruction.


What is the Prototype for Lesson Construction?

Math in the course is taught using the Prototype for Lesson Construction. Participants see it modeled, work through it themselves, and have time to adapt their own lessons to fit this framework.

The prototype is based in developmental and cognitive research about how the brain learns math.

Arrows in the diagram show the need to first build relationships between quantities and language.  Second, learners extend that knowledge to understand relationships between quantities, vocabulary, and discourse.  Two way arrows between the ovals indicate the need to be fluent between representations, and move among and between quantity, structure, language, and symbols.

prototype for lesson construction flowchart
The Prototype for Lesson Construction contains three ovals with text inside and headlines under the ovals. First oval: Quantity, Touchable Visual, Concrete display of concept. Second oval: Mathematical Structure, Discussion: Makes sense of concept, Discussion of the concrete. Third oval: Symbols, Learn to record these ideas, Simply record keeping! Copyright V. Faulkner and North Carolina Department of Public Instruction Task Force, adapted from Sharon Griffin, 2003.


Why Try Something New?

Learners of all ages should receive a cohesive and coherent mathematics education that allows them to make sense of their world, interact in it, and contribute to it.

Mathematics is often taught procedurally, focusing on memorization and using steps to solve problems. This approach can confuse learners, particularly those with disabilities. Memorizing and keeping track of when to apply certain procedures (and when not to) is difficult.

With greater mathematical content knowledge, educators can reduce efforts aimed at making learners better memorizers and step followers and increase robust mathematical instruction. This will allow learners to make connections between math content and apply knowledge to new math topics and real life situations.

Evidence Base

  • Mathematics in the 21st Century: What Mathematical Knowledge is Needed for Teaching Mathematics (Remarks made by Deborah L. Ball at the Secretary's Summit on Mathematics at the U.S. Department of Education in 2003.)
  • Teaching Number Sense
  • The Learning Trajectories
Foundations of Math: Teaching Students with Significant Disabilities

A Foundations of Math course was also developed for teachers who work with students with significant disabilities and complex communication needs. The course pedagogy and philosophy are identical to Foundations of Math. Additionally, this course incorporates augmentative and alternative communication (AAC) strategies, video examples of instruction and assessment, and research specific to this student population.

Foundations of Math Video Series

If you are interested in learning more, check out our full Foundations of Math video series on our YouTube channel.

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