Conrete-Representational-Abstract (CRA or CPA)
Teaching mathematics to elementary students can be both rewarding and challenging. To effectively engage young learners, educators often rely on a variety of teaching models that range from tangible to conceptual. Among these models, the concrete, pictorial, and abstract (CPA) approach is particularly effective in helping students solidify their understanding of mathematical concepts. This approach is present in many math programs including Math U See, Singapore Math, Right Start Math, and the Orton Gillingham approach.
Many students with dyscalculia need lots more time with concrete models than their peers. Don't be afraid to keep using manipulatives until the concept is fully mastered!
Concrete Models: Hands-On Learning
Concrete models utilize physical objects to help students grasp mathematical concepts. These manipulatives provide a tactile experience that can make abstract ideas more relatable. For instance, counting blocks, beads, or even everyday items like buttons can effectively illustrate basic operations such as addition and subtraction. By using concrete models, students can physically manipulate the objects to see how numbers interact in a visual and tangible way.
Some good examples of concrete models include:
Base ten blocks: These blocks can represent units, tens, and hundreds, making place value clear and understandable.
Math U See Manipulatives or Cuisinere Rods: This helps students visually see the pattern of numbers adding to 10.
Counters: Simple counters can visually demonstrate counting and even simple multiplication and division.
Rekenrek/Abacus: An age-old counting method with beads on a wire line. The absolute best models include a colour break between 5s and 6s and 50 and 60.
Pictorial Models: Visual Representation
Once students have a good understanding of a concept through concrete examples, the next step is the pictorial representation. Pictorial models involve using drawings, diagrams, or other visual aids that represent the same mathematical ideas. For example, after using blocks to understand addition problems, students can draw pictures of the blocks to represent the same addition problem visually.
This stage is vital as it bridges the gap between concrete models and abstract thinking. By providing a visual representation, students can enhance their understanding before moving on to more complex abstract concepts. Tools such as number lines, bar models, and charts are effective ways to present pictorial models in the classroom.
Abstract Models: Conceptual Thinking
The last model, the abstract approach, involves using symbols and numbers exclusively without physical or pictorial representation. This level focuses on numerical computations and equations, which may seem daunting at first for elementary students. To effectively transition students to this level, it's crucial to ensure they have grasped the concrete and pictorial stages.
For example, after using blocks to solve the problem 3 + 2, and drawing it out, the student would then write the equation using numbers and symbols. At this point, students can begin to understand mathematical operations abstractly, contributing to their overall mathematical proficiency in higher-level math.
Conclusion
Utilizing concrete, pictorial, and abstract models creates a comprehensive framework for teaching math to elementary students. Each model builds upon the previous one, allowing for a deeper and more intuitive understanding of mathematics. By integrating hands-on experiences, visual representations, and symbolic reasoning, educators can foster an environment where students not only learn math concepts but also gain confidence in their abilities.
Surrey, British Columbia, Canada
