5 x 3 is not the same as 3 x 5? What are those strange arrays? The Internet is ablaze with criticism about the Common Core’s “cruel and unusual” new methods for teaching math. Our children’s math homework has pictures of cookies in place of numbers and we worry about what this means for their future! Whether or not we support the use of Common Core approaches in today’s classrooms, it is important that we as parents come to understand where these methods are coming from and what they mean.
Common Core math starts with an important underlying assumption: Numbers can be used more flexibly when they are taken apart and put back together. Just like children learn to decode words by putting together the sounds of individual letters, children can also learn to decompose and compose numbers by taking them apart and putting them back together. This often involves finding “hidden” numbers inside of larger numbers. For instance, the number 63 actually contains a “hidden” 60 inside of it, so we know that 63 is the same as 60 + 3. Likewise, the position of digits in the number conveys meaning, such that 22 also can be represented as 20 + 2. While numbers can be decomposed in a variety of ways, it is familiar and sensible to decompose numbers into sets of 1’s, 10’s, and 100’s, thus building on place value concepts.
A second fundamental principle of Common Core math involves using physical manipulatives as well as pictures to help children visualize numbers and the relationships among numbers. Children begin to form mental models of quantitative relationships by first using concrete objects such as two cookies. They then bridge their understanding to paper by representing those objects through pictorial methods such as . Finally, they are ready to connect more meaningfully with abstract numerals such as the number 2.
These fundamental concepts of composing and decomposing numbers, along with the process of visualizing numbers, are taught to young children and carry through Common Core math during their school years. Below is one example of how these approaches build coherently across many years of math.
1.Introducing Numbers with Ten Frames
When introducing numbers, the Common Core encourages children to compose and decompose them in a variety of ways. For instance, they learn that 7 can be composed of 6 + 1, 5 + 2, 3 + 4, and so forth. Along with this understanding, they also are introduced to strategies for visualizing numbers. Using a tool called the tens frame, children can begin to visualize 7 in different ways.
Looking closely at the tens frame, children can visualize how 5 + 2 = 7. Using the same image, they might even be able to visualize how 10 – 3 = 7. Using the tens frame helps children anchor their understanding of number in relation to benchmarks numbers of 5 and 10.
2.Multiplication with the Array Model
Children later apply their ability to compose and decompose numbers to addition, subtraction, multiplication, and division. Here again, this skill is paired with strategies that help children to visualize numbers. One such approach is used for multiplication, in which children decompose factors into extended form, then use an array to visualize the operation.
For instance, given the multiplication problem:
- Children would first decompose the factors into (10 + 4)(10 + 3).
- They would then arrange the extended factors on a visual array (see Figure 1).
- The extended factors are then multiplied together to generate the partial products of 12 + 30 + 40 + 100, which are added to together to find the final product of 182.
3.Connecting Arithmetic to Algebra
While younger children are busy composing and decomposing numbers and visualizing those numbers in spatial forms, they are actually building the foundations for algebra. The ways that children understand whole numbers in the multiplication problem above parallel how they will later come to understand variables such as a and b in the following example.
Take the basic algebra problem (a+b)(a+b). Using the traditional method, students could find this equal to a2 + 2ab + b2. However, students can also solve the problem using the modelshown above, replacing whole numbers with the variables a and b (see Figure 2). Rather than simply memorizing how to solve this algebra problem, students come to understand the ways in which mathematical patterns and processes appear time and again in more complex forms.
Traditional approaches to the teaching and learning of math have relied heavily on memorization, step-by-step procedures, and “plug and chug” approaches to solving math problems. Completing tasks such as double digit by double digit multiplication involved little more than using a rote multiplication procedure to arrive at an answer. The goal of Common Core is instead to focus students’ thinking on the important mathematical concepts involved in such a problem. Through the examples we have provided, students build deeper understanding of how to compose and decompose numbers, develop place value concepts, illustrate operations, and construct visual models of the mathematics at hand. The very good news is that when younger students better comprehend these underlying concepts, they are prepared to apply them throughout their school years. And the more we, as parents, understand the concepts the more support we can offer our children.