Students' experiences with physical objects form the basis for learning at the abstract level (Skemp, 1987). According to cognitive psychologist Jerome Bruner, there are three stages of representation through which students demonstrate their understanding: enactive (use of concrete objects), iconic, and symbolic (1960, 1986). Teaching math in this order of stages will ensure that students grasp the true meaning of math concepts.
Concrete Representation
McNeil and Jarvin define manipulatives as concrete objects used to help students understand abstract concepts in the domain of mathematics (2007). These objects range from real items to symbolic representations. Presenting a math problem as a number story using students' names gives them a chance to act out mathematical concepts while using real or simulated objects. For example, Marie went to the store with her mom and bought 2 green apples and 2 red apples. I wonder how many apples she had in all? Marie would put real or plastic apples into a basket to "act out" the story, while the teacher guides students in counting the apples in the basket.
For subtraction, follow the same steps but change the question. Collins went to the grocery store with his mom. They bought 4 red apples but ate 2 of them. I wonder how many apples are left? Have Collins place 4 real or plastic apples into a basket and then take 2 apples out and place them out of sight. Show the students how to count the remaining apples. Once students have a grasp on using the actual objects, they can begin using manipulatives such as connecting cubes to represent the apples in the number story. They can also use virtual manipulatives, which are images to toggle on a screen via mouse or finger swipe. As McLennan (2014) states, "Early math is not about the rote learning of discrete facts like how much 5+7 equals. Rather, it is about children actively making sense of the world around them."
Drawing Representations
Once students can add and subtract using manipulatives, they can begin to make picture representations to help them. Continue to use real-life situations when modeling lessons, but begin to introduce equations along with the number story. Example: Introduce the equation 3 + 2 = __ while creating a real life situation. Taylor picked 3 flowers. Molly picked 2 flowers. How many flowers did they pick in all? We can show this number story as 3 + 2 = __. Model how to draw 3 circles to represent Taylor's flowers. Then draw 2 more circles to represent Molly's flowers. Encourage students to recount the circles as they draw to make sure they represent the correct number. Students can then count how many circles there are in all to find the sum.
When subtracting, show how to draw circles to represent the number given. Place an X on circles to show how numbers are taken away. Example: Taylor had 3 balloons but 1 of the balloons popped. How many balloons does she have left? We can show this number story as 3 − 1 = __. Draw three circles and place an X over one of the circles. Count the remaining circles to find the difference.
Thinking Abstractly
When students understand how to draw pictures to represent mathematical equations, they can begin to think abstractly by using numbers. Example: Let's find the sum for 3 + 3 = __. Students may count up from 3 to solve, or they could use their fingers as representations of the numbers. If students are subtracting, they should practice counting down to find the difference.
Not all students will be able to solve addition or subtraction problems without using some strategy. When used correctly, manipulatives can help students connect concrete representations to abstract situations. Far from toys, manipulatives are "powerful learning tools which build conceptual understanding of mathematics" (National Council of Supervisors of Mathematics Improving Student Achievement Series, 2013). By connecting math to real-world situations and using a progressive method such as Bruner suggested, we can ensure that students are truly understanding math rather than just producing an answer.