Math with Mary!

Author: Mary Headley – Elementary Math Specialist

The introduction of new math concepts can be described using three stages:

I. Concrete (the “doing” stage) – This stage involves both teacher and student modeling.

II. Pictorial (the “seeing” stage) – This stage transitions the concrete model into a representational level such as  drawing pictures or using dots or tallies, etc.

III. Abstract (the “symbolic” stage) – This stage uses numbers and mathematical symbols.

Using concrete models is the first step in building the meaning behind mathematical concepts.  These models include a variety of math manipulatives, measuring tools, and other objects that students can handle during a lesson. Research-based studies show that students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas and better apply these ideas to life situations.  (Harrison & Harrison, 1986; Suydam & Higgins, 1977)

Pictorial representations help teachers provide the perfect bridge between concrete representations and abstract algorithms. Pictorial representations include drawings, diagrams, charts and graphs that are drawn by the student or provided for the students to read and interpret. Pictured relationships show visual representations of the concrete manipulatives and help students visualize the mathematical operations. It is imperative that teachers explain how the pictorial examples relate to the concrete examples.

“Up the Hill” Manipulatives

Connecting the dots between the concrete, pictorial, and abstract is the glue that cements the learning for students. This connection provides the understanding that students need to demonstrate a problem or operation using symbolic representations such as numbers. The meaning of symbols and numbers must be rooted in experiences with real objects (concrete) and pictorial representations. Otherwise the symbolic operations (abstract) become rote repetitions of memorized procedures with no understanding.

The gradual movement from concrete to pictorial to abstract benefits all students and helps to prevent the frustration that some students feel when instructed only with abstract processes and procedures.

Perhaps this article has caused you to think about exploring multiple ways to teach math.  Would you like to observe and experience the conceptual development of content? Do you want to give students multiple strategies for success? Would it help you to see how manipulatives can be used to build the meaning behind math concepts?

If the answer to these questions is yes, you may be interested in Math with Mary, an online resource tool that offers professional learning modules designed to build teacher content knowledge and teacher confidence with the use of manipulatives. These modules are hosted by Mary Headley, Education Specialist for K-5 Mathematics at Education Service Center Region XIII, and will walk participants through the use of a specific manipulative which will allow students to explore and develop a variety of math concepts. Using the strategies presented, students will be able to visualize the math while engaging in strategies that build conceptual understanding.

The first course module, Math with Mary: Multiplication with Base Ten Blocks (FA1224478), is appropriate for grades 3-6 and is currently available on E-Campus. This course lays the foundation for understanding multiplication of 2 digit numbers and beyond. Student expectations related to Number and Operations emphasize the use of concrete models and visual representation of numbers and operations. The Multiplication with Base Ten Blocks course supports student expectations outlined in the TEKS and will help teachers build the bridge between concrete models, pictorial representations and the abstract multiplication algorithm. (2 hours CE credit)



Harrison , M., & Harrison, B., “Developing Numeration Concepts and Skills,”  Arithmetic Teacher 33 (1986): 1–21.

Suydam, M. N.; & J. L. Higgins,  Activity-based Learning in Elementary School Mathematics: Recommendations from Research. Columbus, OH: ERIC Center for Science, Mathematics, and Environmental Education, 1977.

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Math With Mary

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