#### What is CRA? How does it relate to teaching kids at home or in the classroom?

"Concrete Representational Abstract (CRA)" is a three step instructional approach that has been found to be highly effective in teaching math concepts.

The first step is called the concrete stage. It is known as the “doing” stage and involves physically manipulating objects to solve a math problem.

The second representational (semi-concrete) stage is the next step. It is known as the “seeing” stage and involves using images to represent objects to solve a math problem.

The final step in this approach is called the abstract stage. It is known as the “symbolic” stage and involves using only numbers and symbols to solve a math problem. CRA is a gradual systematic approach.

Each stage builds on to the previous stage and therefore must be taught in sequence. This approach is most commonly used in elementary grades, but can be found in some middle and high school classrooms.

#### Here are 10 Easy Steps for Teaching Your Students:

- 1Teach the math concept using manipulatives (concrete level).
- 2Allow ample opportunities for students to practice the concept using various manipulatives
- 3Make sure students understand the concept at the concrete level before moving on to the representational level.
- 4Introduce pictures to represent objects (representational level). Then model the concept.
- 5Provide plenty of time for students to practice the concept using drawn or virtual images.
- 6Check student understanding. Do not move to the abstract if students haven’t mastered the representational level.
- 7Teach students the math concept using only numbers and symbols (abstract level). Model the concept.
- 8Provide plenty of opportunities for students to practice using only numbers and symbols.
- 9Check student understanding. If students are struggling, go back to the concrete and representational levels.
- 10Once the concept is mastered at the abstract level, periodically bring back the concept for students to practice and keep their skills fresh.

Remember that modelling the concept and providing lots of opportunities to practice is extremely important at all three levels. Also, do not rush through the levels. Students need time to make connections and build on what they already know. Give them time to process the information before moving on to the next level.

#### To teach CRA effectively, you need concrete manipulatives

**Here are some examples of concrete manipulatives. This means you're taking the numbers off the page and making them hands-on for your student. It's a much more effective way of teaching and retaining information!**

**For Stage 1 (the concrete stage) you could use:**

- colored chips
- beans
- unifix cubes
- candy (e.g Skittles)
- popsicle sticks
- fraction blocks
- fraction pizzas/cakes

However, the favourite amongst teachers and parents alike is the Integer Block Kit. This complete set of manipulatives is used through each lesson of the Math-U-See program from Primary to Grade 12 maths, building on specific colours and research-proven methods to teach numbers concretely. Here's how the Math-U-See program works.

Here's what those blocks look like (kind of like Lego):

**For Stage 2 (the representational part) you could use:**

- tally marks
- dots
- circles
- pictures of objects

**Benefits of using both Concrete and Representational examples:**

- Provides students with a structured way to learn math concepts
- Students are able to build a better connection when moving through the levels of understanding from concrete to abstract
- Makes learning accessible to all learners (including those with math learning disabilities)
- Taught explicitly using a multi-sensory approach
- Follows Universal Design for Learning guidelines
- Research has proven that this method is effective
- Able to use across grade levels, from early elementary through high school
- Aligned with NCTM standards
- Helps students learn concepts before learning rules
- Can be used in small groups or entire class

**Negatives**

- Not commonly used past upper elementary grades (though it should be!)

#### Concrete and Representational examples are also used for students with different learning styles

Using these examples to teach maths (or any subject for that matter!) has profound benefits. Have you heard of the VARK method? It's about using all different styles of learning to teach a concept - visual, auditory, reading and kinaesthetic methods.

Here's why it works:

- Visual-Spatial: Helps students to think in pictures and create a mental image to retain concepts.
- Verbal-Linguistic: Helps students to organize words in math problems in a way that makes sense.
- Bodily-Kinesthetic: Students learn through hands-on activities.
- Logical-Mathematical: Students use logic to organize information, classify and categorize, make connections and build relationships.

#### You can use the same methods to teach any subject

**The importance of keeping in mind the CRA principles and using them with your student is that not only they learn and understand the concept better, but it's research proven that this method of teaching helps students retain the information for a much longer time!**

There's also more benefits for you and your student:

- Using CRA methodology provides multiple ways to teach math concepts
- Multiple means of representation offered through the use of various manipulative items, visual images, and technology (SmartBoard, computer games/software, video, etc.)
- Allows options for how students learn and express their understanding of a math concept (assessment example: Use SmartBoard clickers to ease student anxiety when having them give answers to math problems. This will in turn increase student engagement and participation.)
- Flexible methods for engaging students (able to incorporate student interests and use real life examples)
- Accessible to all students regardless of ability level
- Allows for accommodations to be made
- Learning is active, fun and engaging

Research shows that using the CRA approach is very effective for students who have a learning disability in math (Anstrom, n.d.). Students are more apt to gain and retain an understanding of math concepts when they are taught using CRA (Anstrom, n.d.)

According to the research cited by Terry Anstrom (n.d.), “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.”

So, now you have a whole heap of research and the important benefits of teaching through using CRA methodology with your students - at home and in the classroom. Thousands of teachers who use the Math-U-See program say it really does work!

If you have any questions, let us know

Warmly,

The Team at Maths Australia

**References:**

Anstrom, T. (n.d.). Supporting students in mathematics through the use of manipulatives. Washington, DC: Center for Implementing Technology in Education. Retrieved April 14, 2012.

Bender, W. (2009). Differentiating math instruction: Strategies that work for K-8 classrooms. Thousand Oaks: Corwin Press.

Sousa, D. (2008). How the brain learns mathematics. Thousand Oaks: Corwin Press.

The Access Center: Improving Outcomes for All Students K-8. (n.d.). Concrete-Representational-Abstract instructional approach. Retrieved April 14, 2012.

**There are also references to the article from "Making Education Fun"**