Sample virtual Algebra Tiles from Amplify's Polypad

CRA in the Secondary Math Classroom

Devan Smith, Bayside High School

January 2025

What is CRA?

The Concrete-Representational-Abstract (CRA) instructional framework is a three-phase approach used in primary and secondary mathematics to build conceptual understanding and procedural fluency.

Concrete Phase - Students use physical, hands-on manipulatives (e.g., base-ten blocks, algebra tiles, counters, fraction circles, geometric shapes) to explore mathematical concepts.
Representational Phase - Students transition to visual representations, such as drawings, diagrams, or graphs, to model the concepts introduced in the concrete phase.
Abstract Phase - Students use symbols, equations, and mathematical notation to solve problems without relying on physical or visual aids.

This framework scaffolds learning by helping students progress from tangible experiences to abstract reasoning, improving comprehension and retention of mathematical concepts. It is particularly effective for secondary mathematics students, including those who struggle with abstract reasoning or have learning difficulties.

Source: Third Space Learning https://thirdspacelearning.com/us/blog/concrete-representational-abstract-math-cpa/

This graphic was created by me for my master's project on the use of Virtual and Physical Manipulatives in the HS Geometry classroom, and shows in which stage physical and virtual manipulatives belong in the CRA framework.

Why CRA?

The CRA framework builds conceptual understanding through helping students grasp the "why" behind mathematical rules before applying them procedurally. Not only that, but CRA supports differentiation as it meets the needs of diverse learners, including those with disabilities or gaps in prior knowledge.

CRA has been successful in facilitating retention as it encourages connections between concepts, improving memory and recall. Finally, CRA bridges concrete to abstract thinking, by design. Using CRA instruction prepares students for more complex and symbolic mathematics, such as algebra, geometry, and trigonometry.

Mathematical rigor stool showing the importance of procedural fluency and conceptual understanding. For more detail on what is Mathematical Rigor, see the our page on the VBCPS Sine Wave.

How to CRA?

According to Pennsylvania Training and Technical Assistance Network (PaTTAN), "When using CRA, the teacher should provide multiple opportunities for practice and demonstration to help students achieve mastery of the mathematical concept" (2024). Explicit instruction that involves the use of manipulatives should also include the presentation of the numerical problem (Miller, Stringfellow, Kaffar, Ferreira, & Mancl, 2011). After multiple teacher demonstrations, students are given opportunities to practice the mathematical model and use verbalization while using the CRA sequence.

Kennard explains that progression through the phases is not always linear and different students may progress at different times. Students should have access to the manipulatives at all times and should be comfortable with and confident in their use (Kennard, n.d.).

Some teachers choose to leave the representational stage (also known as semi-concrete) out, but it is key to ensuring that children can make the link between a concrete resource and abstract notation.

Algebra Tiles

See this comprehensive resource on a lesson sequence for using Algebra Tiles from Vicky Kennard from the Australian Mathematical Sciences Institute (AMSI). In the resource, she claims that Algebra Tiles should continue to be used beyond primary levels as "Algebra tiles are a manipulative that can help develop student’s understanding and confidence with algebra, at many different levels" and "Students often experience difficulty with Algebra and the notation it employs" (Kennard, n.d.).

This month’s newsletter also covers how Algebra Tiles can be used in Grades 6 - Algebra 2. Below are some snapshots of how lessons can use Algebra Tiles from Kennard.

Modeling Integers with Algebra Tiles

Solving Linear Equations with Algebra Tiles

Multiplying Polynomials with Algebra Tiles (Representational)

Exploragons

An exploration lesson using Exploragons for the sum of exterior angles of triangles theorem can be found here.

Exploragons were developed by EAI Education to provide students with a hands-on tool to explore geometric concepts. According to the company website, Exploragons can support basic Geometric vocabulary, shape attributes, proper shape nomenclature, angle relationships and measurement, informal arguments involving angle relationships of polygons, and parallelism (EAI, 2022). In essence, Exploragons are sticks that snap together to form figures like triangles or parallel lines that allow students to visualize and manipulate measures to identify characteristics or patterns. There are eight different sticks of various lengths that are further differentiated by color. VBCPS has incorporated these in middle school instruction as a resource.

Virtual Manipulatives

You can use virtual manipulatives if you do not have access to physical ones. Desmos activities and GeoGebra often have embedded manipulatives, but there are also resources online such as https://mathsbot.com/manipulatives/tiles, https://www.didax.com/apps/geoboard/, and https://toytheater.com/fraction-strips/ (see examples below).

Virtual Algebra Tiles from Mathsbot

Didax Virtual Geoboard

Toy Theater Angle Manipulative

Toy Theater Fraction Strips

References

Johnson, E. (2024, December 17). What is the concrete representational abstract (CRA) approach and how to use it in your elementary math classroom. Third Space Learning. https://thirdspacelearning.com/us/blog/concrete-representational-abstract-math-cpa/

Kennard, V. (n.d.). Algebra tiles learning sequence. AMSI Calculate. https://calculate.org.au/wp-content/uploads/sites/15/2019/03/lesson-sequence-for-algebra-tiles.pdf

Pennsylvania Training and Technical Assistance Network. (2024, October 29). Concrete-representational-abstract: Instructional sequence for mathematics. PaTTAN. https://www.pattan.net/getmedia/9059e5f0-7edc-4391-8c8e-ebaf8c3c95d6/CRA_Methods0117

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