Posted March 15, 2022 in Articles
Engaging more than just the eyes and ears, multisensory activities combined with a Concrete-Representational-Abstract instructional sequence helps students become competent mathematicians who understand the why behind concepts.
For many students with learning differences, math can be a major source of frustration. This is sometimes due to a disorder known as dyscalculia that impacts their understanding of numbers. Difficulties can also stem from low working memory, commonly associated with ADHD, which makes it hard to retain information (like a string of digits) while performing multi-step computations. Even students with dyslexia, who struggle primarily with reading, can get lost in the language aspect of a math lesson—finding great difficulty with word problems or tough terminology like polynomial or equation.
At Lawrence, we recognize memorizing formulas and algorithms is not enough to spark success for our learners. There is a deep body of evidence that shows students with learning differences become competent mathematicians if they are taught to understand the why behind concepts.
This is why, whether we’re helping our youngest learners count or conveying the advanced concepts of Calculus, we always present new mathematical information using a Concrete, Representational, Abstract sequence; reinforced with sight, sound, touch, and movement activities.
To really understand how this multiplies a learners’ mathematical understanding, let’s break it down:
The Concrete Representational Abstract Approach (CRA):
Concrete: The teacher introduces a new concept by modeling with tangible materials. This is where many of the hands-on activities and manipulatives come into play. Depending on the age of the learner, teachers might have students move beads along a string to convey basic addition and subtraction, stand up and move around the room to figure the percentage of students in a class, or use Algebra tiles to teach multiplying polynomials or factoring.
Representational: In this stage, the class draws a representation of the concept using symbols. This drawing might include pictures, circles, dots, or tally marks. Students are able to illustrate the concept with dry-erase markers, writing directly on their desk. The goal of this stage is to visualize the concrete model from the previous step on a flat surface. This becomes their portable memory.
Abstract: In this stage, the teacher helps students model the concept using only numbers, notation, and traditional operation symbols (%, ÷, ×, +, =).
A class of middle schoolers studying fractions might begin with the concrete stage by constructing a tower out of large wooden blocks. They’ll divide the blocks evenly into sections so they can visually see the whole, as well as the fractions within (i.e. ⅕, ⅖, ¾, ⅘). The class will then move onto the representational stage by drawing the tower, looking for different fraction patterns within the structure, and labeling them. And finally, once the concept is understood both concretely and representationally, students will interact with the traditional number problems, applying the understanding they gained in the two previous stages.
Other hallmarks of math instruction at Lawrence, include:
No matter the instructional level, teachers kick off every math class with a warm-up to help the class get in the groove and reinforce number fluency. Lower Schoolers might toss around a ball, while skip counting by 5s, while older students might start class with a math fact showdown using dominoes, playing cards, or the popular app Hit the Button.
Students adopt math language by actively thinking about definitions, even when defining numbers instead of words. For example, 23.6 is not described orally as twenty-three point six. Instead, it’s twenty-three and sixth-tenths. This helps learners better visualize and understand what the number represents.
Students are taught to recognize opportunities to distill the figures in any given equation. Take, for example, 72-69. Rather than borrowing from the larger number, students are taught to view number values in a way that is easier to parse down, such as: (60+12) - (60+9) = 12 - 9 = 3. Learning this approach makes mental math more automatic.
You might think this sounds like a lot to accomplish, but math class at Lawrence is longer than what is typical (75 minutes at Lower, 60 in Middle, and 45 in High School). This extended time frame allows our faculty to teach to mastery—but just as important, it creates a positive math mindset that follows students through college and their careers.
No matter your child’s age, math doesn’t have to be an unsolvable problem! Here are a few ways parents can make their home math-friendly and take some of the nerves out of numbers:
Student success increases when parents are involved in their children’s education, so ask about class, assignments, or what they learned that day.
Set goals and encourage your child to reach for them. Studies show that effort trumps ability when learning math, and high expectations can spark a growth mindset.
No negativity here—unless they’re numbers in an equation. Your view of math directly influences your child’s. Avoid saying things like I’ve never been good at math, or I’m not a math person, or This isn’t how I learned it.
Don’t play into outdated gender stereotypes. Everyone can be a math superstar, including your child! Foster confidence by highlighting achievements made by both male and female scientists and mathematicians.
Show how people use math every day, whether it’s while cooking and baking, playing music, or during the Browns game.
No one is a master from the beginning, and mistakes happen. Learning from and correcting them is an essential part of math.