Foundations of Math

Foundations of Math was developed through a federal grant at the North Carolina Department of Public Instruction Exceptional Children’s Division. The course was designed by a team of two math and two special education educators. It is based on the research of Deborah Ball concluding teacher content knowledge is linked to student math achievement.

Often special education personnel are responsible for the math education of a range of students (e.g., age, grade level, disability, educational setting) yet have little or no math training from a university program or through ongoing professional learning. Furthermore, general education math teachers may feel under-equipped to meet the needs of students with learning disabilities in their classrooms.

This course will develop the mathematical knowledge of special and general educators who work with students with disabilities or students who struggle in math. The focus of the course is on the Prototype for Lesson Construction and the Components of Number Sense. The course is designed to deepen the teachers’ content knowledge and connect procedural knowledge to conceptual understanding. An emphasis is placed on formatting lesson construction to develop strong mathematical foundations in students.

Evidence Base

This course draws from research related to teacher content knowledge, how children learn math, and the connections between teacher content knowledge and student math achievement.

Mathematics in the 21st Century: What Mathematical Knowledge is Needed for Teaching Mathematics? - This course is predicated on research that identified a connection between teacher content knowledge and student achievement. 

Teaching Number Sense - Research on how students cognitively develop mathematical understanding from conceptual to procedural knowledge informs the Prototype for Lesson Construction used in this course.

The Learning Trajectories - The trajectories define natural developmental progressions that children go through as they develop mathematical understanding. This course shows how guiding students through these levels of thinking strengthens foundational mathematical understanding.

The Components of Number Sense - The Components of Number Sense provides a framework for teachers to utilize when planning and delivering math lessons. It allows for guidance and scaffolding of math concepts for both the teacher and student to deepen their content knowledge.

How to Receive This Training

You can choose to partner with Alt+Shift or attend a scheduled Statewide Event. Read more about each option below. If you’d like to be notified when the next statewide event becomes available, you can request to be notified.

Partner with Alt+Shift

Training is provided to ISD staff as part of an ISD partnership. Training is typically provided to the entire district, building, or program staff. This depends on the specific ISD's implementation plan. Training is one piece of the partnership. Strategic planning, implementation support, and capacity building are also addressed through the partnership.

Attend a Statewide Event

Statewide events are opportunities to receive training, but with limited opportunities for follow up support. Participants can expect to gain ideas and strategies that would be usable immediately in their practice, and to gain a better understanding of the nature of the training as part of an exploration process for sites considering a partnership with Alt+Shift.

There are no upcoming events for this training.

Implementation Stories

Alt+Shift asked partners around the state to share their implementation experiences and the impact on adults and students where they work.

Heidie Lehmkuhle

6-8 grade Interventionist

In what ways have you implemented Foundations of Math?

I am using many components from the training to better meet the needs of my students. Specifically, I use the Number Knowledge Test to collect data on students' gaps. Doing this has allowed me to gain insights as to where students lack understanding and to build their understanding in the areas they need the most support. My lessons follow the concrete, representational, and abstract model. The best part is all my students can access the curriculum because of what I am now doing in my classroom. Furthermore, they are being challenged at the level they need to be.

I now recognize that while my previous attempts to fill in students' gaps of understanding were with the best of intentions, I was missing the mark. The information provided in Foundations of Math is incredibly powerful because it allows me to deepen my instruction in ways that meets my students' needs. Additionally, I did not have to completely change my practices in order to implement what I learned at this training. My mentor teachers and administration have taught me there is not a single program, method, or strategy that can replace quality instruction. They have guided me to understand that good teaching is simply that, good teaching. Foundations of Math adds to my instructional strategies. It  does not replace them. What has changed is my perspective of students' learning experiences and how I can best meet their many critical needs.


What impact has Foundations of Math had on you and those you work with?

Foundations of Math is impacting the experiences my students have with math. They now can access the curriculum and deeply understand mathematical concepts. Students are making connections between a variety of areas and their everyday lives. Because of that, they are having positive experiences. Several of my students have had many negative experiences with math. They are several grade levels behind their peers and have developed a fixed mindset. Combining Foundations of Math with good teaching practices, and quality relationships, is paving the way to new opportunities for my students.Teaching students who have had many negative experiences with math can be difficult. Foundations of Math recognizes that education is not a 'one size fits all' program. It is a way of thinking and forming instruction that makes connections for students. It is not a prescribed formula that must be followed. It has made my instruction more authentic and genuine, and that is what my students deserve. While it can be difficult, it is worth it!


What is one challenge you encountered when implementing FoM and how did you overcome it?

One challenge I encountered when implementing Foundations of Math was proctoring the Number Knowledge Test for all my students. This test is similar to a verbal interview with students. It provides powerful and specific information about each student's understanding. With it I am getting more information than what questions students get right or wrong. I now understand what mental strategies students have in order to solve problems and what aspects of each problem they struggled with. Helping students build their understanding needs to be about more than simply helping them get the answer right. The Number Knowledge Test is helping me focus on a student’s process and strategies.


Giving the Number Knowledge Test was challenging because I had to give my attention to one student while still being responsible for the rest of my class. In order to overcome this challenge, I did several things. First, I collaborated with our district’s math instructional coach, who proctored some of the tests. I also asked my colleagues to let me use their paraprofessionals. I took time establishing classroom expectations for this test, and I communicated with our director of student achievement. Implementing this test for all my students was challenging, but the information I gained was invaluable.


Describe one implementation success you have had.

I first started to feel the powerful benefits of Foundations of Math when I heard one specific phrase said by one of my students. We were discussing the connections we found for a specific topic. Then one of my students who was initially convinced he was "not a math person" added an incredible contribution to our discussion. All the students were left in awe at the math that was in their minds. Then another student muttered six words that proved to me an impact was being made. He said, "wow, this class is so advanced." They all agreed as they were still processing the information before them. It was my turn to have the 'deer in the headlights' look. I realized for the first time in their lives many of these students truly believed they were 'advanced math' learners. 

Continue the conversation with Heidie Lehmkuhle

Julia Alder

Math and Science Curriculum Specialist

I utilize aspects of Foundations of Math in my job to collaborate with classroom teachers to implement instructional strategies that develop the components of number sense. I often use examples from the book Knowing and Teaching Elementary Mathematics by Liping Ma. I also use articles about designing challenging curriculum for all students. Making mathematics visible and accessible has been an area we are exploring and trying to make part of every math classroom in Troy.

children working together in a workbook

Foundations of Math has impacted how I work with teachers, principals, and support staff. I have been able to cite research and practices that are necessary to uncover what students know.


One challenge has been reaching every teacher. We have overcome part of the challenge by offering professional learning sessions in August. These allow teachers some choice in professional development. The district also provides grade level specific professional learning at least three times per year for one teacher per elementary building. We have also increased the opportunity for teachers to see and learn about math instructional practices through classroom visits.

Continue the conversation with Julia Alder

Michelle Marcus

Second Grade Teacher

The main focus for implementation of Foundations of Math in my teaching is utilizing the Concrete-Representational-Abstract (CRA) model in my planning and teaching each day. It allows all students an entry point into the content. I have also begun paying closer attention to vocabulary and the components of number sense in my lesson planning and instruction. My colleagues and I have come to realize that math is not just a page in a book. Manipulatives, vocabulary, discussion, modeling, and equations all hold an important place in every math lesson.

A challenge I encounter is knowing and understanding what misconceptions students bring with them to math class. To overcome this, I am working on deepening my own understanding of mathematics. By building upon my knowledge base, I hope to be better prepared to pre-plan for these instances.


My next step in implementing Foundations of Math is to continue to work with the CRA model in my lesson planning and instruction of our base ten system. We can build ideas and prove strategies work by using manipulatives.

Continue the conversation with Michelle Marcus

What Others are Saying

We asked our partnership sites to share their experience related to the training. 

Tonya Paige
Bernie Anderson

Additional Information

Foundations of Math: Teaching Students with Significant Disabilities (FoM: SD)

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.

Visit Foundations of Math: Teaching Students with Significant Disabilities

Foundations of Math Video Series

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

View the Foundations of Math video series

Foundations of Math Handbook

The Foundations of Math handbook provides a course overview, descriptions of key concepts from the course, and answers to frequently asked questions.  It was written to help our partnership sites answer questions they were getting from classroom staff, building and district administrators, and parents.

Get the Foundations of Math Handbook