Article | How to Make Math Concrete & Visible

Does this word problem seem familiar? Amy has 46 crayons. Roy borrowed some crayons and now she has only 17 left with her. How many crayons did Roy take from Amy? Math has always been perceived as an abstract subject that is filled with numbers, variables, equations, and word problems. It’s therefore not surprising that many students often find the subject uninteresting and difficult to grasp.

Children of all ages and even adults can learn better if they are able to visualize things and have concrete and real experiences. And this has always been a challenge with teaching math.

Mathematics instruction is a complex process that attempts to make abstract concepts tangible, difficult ideas understandable, and multifaceted problems solvable. Visual representations bring research-based options, tools, and alternatives to bear in meeting the instructional challenge of mathematics education (Gersten et al., 2008).
Source: Evidence for Education, Volume III. Issue I. 2008

In his research, psychologist Jean Piaget stated that, children are active learners, and they can master concepts by progressing through three levels of knowledge–concrete, pictorial, and abstract. The Concrete-Representational-Abstract (CRA) technique is a widely acceptable and proven technique for teaching math. The CRA technique introduces mathematical concepts in three-steps.

  1. Concrete –  The teacher uses concrete materials (such as colored blocks, beans or cubes) and models the mathematical concept to be learned.
  2. Representational – Then, she draws pictures or diagrams (also known as thinking blocks) to helps students visualize the concept.
  3. Abstract – Finally, she presents the concept in mathematical terms (word problem, equation).

The transition from one step to the next may not be immediate and often varies. Simple concepts of addition and subtraction are easy to understand in a single session, but concepts like multiplication and division often require multiple sessions before students get confident with them. Teachers, should allow students to spend sufficient amount of time in each phase, and only after ensuring that students have grasped the concept in a phase should they move to the next one.

In the following video, Dr. Yeap Ban Har talks about how the Model Method which is based on the principles of CRA.

Model Method for Math

Number bonds is another basic technique used in Singapore to introduce numbers in primary grades. In the following video, Dr. Yeap Ban Har explains the concept of Number Bonds.

Number Bonds for Math

And, in the following video you can see how a teacher introduces the concept of Number Bonds in class.

Using Number Bonds in Grade 1

The CRA technique is effective because it brings math to life. Words and numbers take the form of objects that students can touch and interact with, and manipulate in different ways to create different meanings. It also engages students and promotes active learning. Seeing and making combinations helps them better understand concepts and retain them.

CRA is based on using manipulatives to teach math. These maninpulatives can be easily made from day-to-day material and go a long way in enhancing the learning experience. Do read our Micro-innovation Monday series article – MI Monday | Low-cost, high impact classroom resources, by STIR Education to see how Devanik, a teacher in a government school developed a set of sustainable and low-cost classroom resources to teach math, and improved student performance by 72%.

This week’s articles and resources will feature innovative practices for teaching Mathematics in class. If you have a practice or product that has worked effectively for your students and would like to share it with our community, then Contact Us and we will be happy to feature you.


Kanchan Shine

Kanchan Shine

Passionate about everything related to education. I believe that the best kind of learning happens through play, experiments and fun! I love watching how children learn and love to implement play-based, hands-on teaching approaches. I get my thrill by planning activities for my children (6 yo girl & 3 yo boy) and watching them learn while having fun!

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