Transforming Math Learning: CRA and Multi-Sensory Strategies for Neurodiverse Students
Understanding the Concrete-Representational-Abstract (CRA) Instructional Approach
The CRA approach is a well-established instructional method designed to help students develop a robust understanding of mathematical concepts by moving through three distinct stages: Concrete, Representational, and Abstract.
Chips for Equivalent Fractions
Using two colored chips to show the concrete, drawing the chips for representation, and the abstraction of the fractions 2/3 and ? / 9. Note: The two-color chips might be different colors, such as red/yellow. Image Source
Concrete Stage
The concrete stage is the foundation of the CRA approach. At this level, students engage with physical objects to explore mathematical concepts. For instance, when learning about addition, a student might use counters, blocks, or beads to combine groups of objects physically. This hands-on experience is crucial because it allows students to interact directly with the concept they are learning, making the abstract idea of addition more tangible and understandable.
Example:
Imagine a student learning about fractions for the first time. At the concrete stage, they might use fraction tiles or even pieces of a chocolate bar to represent different fractions. By physically manipulating these objects, the student can see how fractions work, such as how two 1/4 tiles can be combined to make 1/2.Representational Stage:
Once students are comfortable with the concrete stage, they move on to the representational stage. Here, they draw pictures or use diagrams to represent the math concepts. This stage acts as a bridge between the concrete objects and abstract symbols. Instead of using physical counters, a student might draw circles or tallies to represent quantities. This visual representation helps students transition from concrete manipulation to understanding mathematical symbols.
Example:
Continuing with the fractions example, in the representational stage, the student might now draw pictures of the fraction tiles or chocolate bars on paper. (This is one reason I avoid using circles as the predominant representation of fractions–they’re very hard to draw and subdivide appropriately!) They could shade in parts of a circle to represent 1/4 or 1/2, helping them visualize the concept without needing physical objects.Abstract Stage:
In the abstract stage, students use mathematical symbols and numbers without relying on physical objects or drawings. This is where they begin to solve equations and perform calculations using numbers alone. The abstract stage is critical because it’s where students demonstrate their ability to understand and manipulate mathematical concepts mentally.
Example:
The student working with fractions could now add fractions like 1/4 + 1/4 = 1/2 using only numbers without drawing or manipulating objects.
Base Ten Blocks for Place Value
Showing place value using base 10 blocks for concrete, a drawing for representational, and then the expanded form of 300 + 4 + 2 and standard form of 342. Note that this is a picture; concrete would use traditional base 10 blocks. Image Source
Additional Considerations in the CRA Approach
Addressing Learning Difficulties
One of the key strengths of the CRA approach is its flexibility. If a student struggles at any stage, it’s often because they moved on before fully mastering the previous level. For instance, if a student finds abstraction challenging, it might be beneficial to revisit the representational stage, allowing them to draw diagrams to reinforce their understanding. Similarly, if a student struggles at the representational level, returning to the concrete stage with physical objects can help solidify their grasp of the concept.
Representation vs. Visualization
Some educators distinguish between representation and visualization in the CRA progression, adding a step between representation and abstraction. Before moving to abstraction, students should be able to visualize the representational stage mentally without having to draw it out. While this step isn’t widely adopted, it can be helpful for students who struggle to transition from representation to abstraction.
Example: A student working on solving equations might first draw out the problem, then practice visualizing the steps in their mind before finally solving the equation abstractly. This additional step can help solidify their understanding and make the abstract stage less intimidating.
While the CRA approach offers a structured pathway for developing a deep understanding of math concepts, integrating multi-sensory instruction can further enhance this process by engaging students’ different senses, making learning more dynamic and effective.
Exploring Multi-Sensory Math Instruction
Multi-sensory instruction is an approach that engages multiple senses—visual, auditory, and kinesthetic—to enhance learning. This method recognizes that students learn differently and that combining sensory inputs can significantly boost comprehension and retention. Unlike traditional approaches that may rely heavily on visual or auditory instruction alone, multi-sensory instruction integrates various sensory modalities, offering students multiple pathways to understand and remember concepts.
Visual: Visual learners benefit from seeing information. In math, this might include using images, diagrams, and visual aids to represent concepts. Visual aids can range from simple drawings on a whiteboard to more complex visual models like graphs or geometric shapes.
Example:
When teaching geometry, a teacher might use a combination of diagrams and physical models to help students visualize shapes and their properties. Seeing the shapes in different contexts can help students better understand the relationships between them.
Auditory: Auditory learners benefit from hearing information. In math instruction, this might involve verbal explanations, discussions, and even using songs or rhythms to explain mathematical ideas. Incorporating auditory elements helps these learners process information more effectively.
Example:
A teacher might use rhythmic patterns or songs to help students remember multiplication tables. By associating the numbers with a rhythm or melody, students can more easily recall the information during tests or homework.
Kinesthetic: Kinesthetic learners benefit from moving and doing. This might involve hands-on activities, like manipulating objects, using gestures, or even incorporating physical movement into learning math concepts. For kinesthetic learners, the act of physically engaging with the material helps cement their understanding. Note: Using objects is the hallmark of the Concrete part of the CRA approach discussed above.
Example:
In a lesson on measurement, students might use rulers, tape measures, or even their own bodies to measure objects around the classroom. By physically engaging with the concept of measurement, they gain a more intuitive understanding of it.
Why It Works
Multi-sensory instruction is particularly effective for students who struggle with traditional methods because it taps into multiple learning pathways. For instance, a student might better grasp the concept of fractions by physically cutting up a chocolate bar (kinesthetic), discussing what they’re doing (auditory), and seeing the pieces (visual). This approach reinforces the concept and caters to different learning styles, making it more likely that the student will retain the information.
Practical Applications and Benefits: For Parents
You don’t need to be a math expert to support your child’s learning at home. The CRA approach and multi-sensory instruction can be easily adapted for home use with simple activities and tools. Here are some practical tips:
Implementing the CRA Approach at Home
Start with concrete objects like toys, food, or household objects to teach basic math concepts. For example, you can use buttons or coins to practice counting, addition, or subtraction. Once your child is comfortable, draw pictures or use diagrams to represent the math concepts. Finally, encourage them to solve problems using numbers alone.
Example:
If your child is learning multiplication, start with concrete objects like small toys. Have them group the toys to visually and physically represent the multiplication process. Then, move to drawing groups of objects on paper, and finally, work on solving multiplication problems using numbers alone.
Incorporating Multi-Sensory Activities
Engage your child’s senses to make math learning more interactive and enjoyable. Use visual aids, like colorful charts or apps that animate math problems. Incorporate auditory elements by discussing math problems out loud or using songs to reinforce math facts. Encourage kinesthetic learning by involving movement, such as jumping several times to represent addition or using their hands to physically measure objects.
Example:
For a kinesthetic activity, you could create a “math obstacle course” where your child has to solve math problems at different stations. For each correct answer, they move on to the next station, combining physical movement with math practice.
Practical Applications and Benefits: For Educators
Incorporating CRA and multi-sensory strategies in the classroom can transform math instruction. Here’s how educators can effectively apply these methods:
Implementing the CRA Approach in the Classroom
Integrate the CRA approach into your lessons by starting with manipulatives or hands-on activities. For example, use fraction bars or circles that students can physically manipulate when teaching fractions. Gradually transition to having students draw their own representations of the fractions and, eventually, solve fraction problems abstractly with numbers alone. (Again, circles are hard to draw! Use rectangles, too, so students can move to the R part more seamlessly!)
Example:
In a unit on fractions, begin with students using physical fraction tiles to explore different fractional parts. As they become more comfortable, ask them to draw the fractions they created using the tiles. Finally, move to solving fraction problems on paper without any visual aids.
Multi-Sensory Techniques in the Classroom
Design lessons that incorporate visual, auditory, and kinesthetic elements. For visual learners, use diagrams, videos, or interactive whiteboards. For auditory learners, include discussions, oral explanations, and rhythmic activities. For kinesthetic learners, incorporate movement and hands-on activities into your lessons.
Example:
During a lesson on geometry, have students use string to physically create different shapes. They can measure angles with protractors, draw the shapes on paper, and discuss the properties of each shape in small groups. This multi-sensory approach helps cater to all types of learners.
Suitable For All, Essential for Some
The CRA approach and multi-sensory methods are not just beneficial for neurodiverse students—they’re valuable for all learners. These strategies align closely with the principles of Universal Design for Learning (UDL), which advocate for flexible teaching methods that cater to the diverse needs of all students. By incorporating CRA and multi-sensory instruction, educators can create a more inclusive learning environment where every student has the opportunity to succeed. While these approaches enhance learning for everyone, they are particularly essential for students with learning differences, ensuring they can engage with and master math concepts.
Example:
In a UDL-aligned classroom, a teacher might design lessons that offer multiple ways for students to engage with a math concept—through hands-on activities, visual aids, and oral explanations—ensuring that each student can access the material in the way that works best for them. Additionally, when planning these lessons, the teacher should consider how each activity can be adapted to meet the needs of all learners. For instance, while a multi-sensory lesson might include visual aids and hands-on activities, providing additional scaffolding, such as extra time or alternative ways for students to demonstrate their understanding to support those who need it, is essential.
Let’s dive into some specific types of neurodiversity and why these approaches are so critical.
Dyscalculia: Students with dyscalculia often struggle with understanding numbers and symbols. The CRA approach is particularly beneficial because it allows these students to build a solid foundational understanding through concrete and representational stages before moving to abstract concepts. Multi-sensory instruction further supports these learners by offering various ways to experience and understand math, catering to their unique needs.
Example:
A student with dyscalculia might begin learning multiplication by physically grouping objects and counting them. As they progress, they might draw pictures of the groups before finally moving to solving multiplication problems using numbers. This gradual progression helps them build confidence and understanding at each stage.
Dyslexia: Dyslexic students often benefit from multi-sensory instruction, which bridges the gap between verbal and visual representations of math concepts. Linking spoken instructions with visual aids and hands-on activities can help dyslexic students process and retain information more effectively.
Example:
When teaching word problems, a teacher might read the problem aloud, have the student draw a picture of the scenario, and then solve the problem using manipulatives. This multi-sensory approach helps the student connect the words to the math concepts in a meaningful way.ADHD: Students with ADHD often struggle to focus on traditional math instruction. Multi-sensory methods, which incorporate movement and interactive activities, can keep them engaged and help them channel their energy into learning. These strategies can be particularly effective in maintaining their attention and helping them grasp complex concepts.
Example:
A student with ADHD might benefit from a lesson that allows them to move around the room, using different stations to solve parts of a larger math problem. Each station could engage a different sense, keeping the student active and focused throughout the lesson.
A Brief Mention of Theoretical Underpinnings
These strategies are grounded in cognitive theories emphasizing how the brain processes and retains information. The CRA approach aligns with how learners naturally progress from concrete experiences to abstract thinking, making it a developmentally appropriate math teaching method. Multi-sensory instruction taps into the brain’s ability to create and strengthen neural connections through varied sensory inputs, which enhances understanding and retention.
Research in cognitive psychology supports the idea that engaging multiple senses can improve memory and learning outcomes. By providing students with various ways to interact with math concepts, educators can help them build stronger, more lasting connections in their minds.
Conclusion
The CRA approach and multi-sensory instruction offer powerful tools for making math accessible and engaging for all students, particularly those with learning differences. By incorporating these strategies at home or in the classroom, parents and educators can help students build confidence, reduce anxiety, and achieve success in math. Whether you’re a parent helping your child with homework or a teacher planning a lesson, these methods can transform how students experience and understand math. Give them a try and see the difference they can make in your child’s or student’s math journey.
