As adults, we are probably very familiar with math “procedures”—stacking two numbers on top of each other to add, “carrying the one,” and recording the numbers below the equal bar. Procedures like this have become second nature to us, and many of us might not make the connection that “carrying the one” is the act of “regrouping” to make a 10. We understand how to solve the problem (i.e., procedural understanding), but not the why behind the answer or the concepts being employed (i.e., conceptual understanding). Ultimately, though, students need both procedural and conceptual understanding. Conceptual understanding should be built in the early years by using a variety of “active math” activities, including talking, play, movement, and interaction with physical objects.
Why Does Conceptual Understanding Matter?
Memorizing math procedures might not seem like such a big deal, but memorization without understanding may cause confusion and conflict with how a student is correctly thinking about how to solve a math problem. Overall, we’ve come very far in how we instruct mathematics to include more flexibility in problem solving and move away from strict, rote memorization.
When we look at the joint National Association of Teachers of Mathematics/National Association for the Education of Young Children (NAEYC) position statement for early childhood mathematics, we see that the upper-level computation procedures are built on a solid foundation of mathematical understanding through meaningful mathematical experiences that provide connections for number sense. This ability to be very flexible with numbers is a key component in being able to efficiently and accurately arrive at answers in the National Council of Teachers of Mathematics (NCTM) position statement for fluency.
How Does Conceptual Understanding Connect to Procedural Understanding?
The math steps or procedures used arise from the “conceptual understanding,” which is the underlying math that allows those steps to work. These quick procedures are just an additional representation of the underlying mathematics. Effective teaching practices provide opportunities to help students connect the underlying concepts with the steps and procedures so they can completely understand why those procedures work versus just memorizing them (NCTM, 2014). An example of this would be to add 15 + 16 with the “stacking” method. We usually put the 1 down and “carry” a one, which represents a group of 10. When the student knows that 5 and 6 make 11, the student could also add this way: 10 + 10 + 10 +1. The math didn’t change, but the student can use flexible thinking from the understanding of the place value.
Effective mathematics teaching and curriculum in the early years also include experiences that develop mathematical reasoning and connection of concepts through a range of methods and strategies, including talking, play, movement, and interaction with their surroundings (NAEYC, 2002). Engaging, encouraging, and positive mathematics experiences are important in the first six years of life to nurture the development of curiosity, flexibility, and inventiveness (NCTM, 2013).
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How Do Students Develop Conceptual Understanding When They’re Very Young?
Experiences at the pre-K level are providing the foundation for “decomposing,” or breaking apart, numbers. Composing and decomposing numbers involves combining and separating them to make parts and wholes (NCTM, 2011). For example, within a quantity of 5 there is a quantity of 3 and a quantity of 2 (students can show this with their physical objects).
This idea of “breaking apart” a quantity is what creates flexibility in number sense. In developing an understanding of the counting numbers, students learn that a number can be decomposed as a sum of two parts in many ways. This knowledge of “breaking apart” numbers, along with the commutative and associative properties, is foundational to many addition and subtraction fact strategies (NCTM, 2011).
In our example of 15 + 16, when students first learn to add two-digit numbers, they could break apart the 15 into a group of 10 and 5 cubes. When they add 16, they make groups of 10 cubes and 6 cubes. They can rearrange the numbers, or cubes, and add the two 10s first to get 20 and then add the 5 and the 6 to make 10 and 1. They can show the total as 20 + 10 + 1. This is the same as adding 5 + 6 to make 11, “carry” the group of 10 to get three groups of 10, for a total of 31.
- Developing number sense in students is a multi-sensory endeavor that serves them well as they progress through their math journey.
- Students use sight, sound, touch, and physical movement to understand quantities.
- Pairing numbers with concrete objects strengthens students’ number understanding.
- Using language to describe quantities, materials to represent quantities, and meaningful activities to explore quantities will build a strong math foundation (Gurganus, 2004 in Sousa, 2008).
- The concrete, hands-on action helps students understand more abstract procedures. This deep understanding will allow for more “stickiness” of procedural fluency.
Carpenter, T., Franke, M., Johnson, N., Turrou, A., & Wager, A. (2017). Young children’s mathematics: Cognitively guided instruction in early childhood education. Heinemann: Portsmouth, NH.
National Association for the Education of Young Children. (2002). Early childhood mathematics: Promoting good beginnings. National Association for the Education of Young Children.
National Council of Teachers of Mathematics. (2011). Developing essential understanding of addition and subtraction in pre-K–Grade 2. NCTM: Reston, VA.
NCTM (2013). Mathematics in early childhood learning: A position of the National Council of Teachers of Mathematics. NCTM. https://www.nctm.org/Standards-and-Positions/Position-Statements/Mathematics-in-Early-Childhood-Learning/
NCTM. (2014). Procedural fluency in mathematics: A position of the National Council of Teachers of Mathematics. NCTM. https://www.nctm.org/Standards-and-Positions/Position-Statements/Procedural-Fluency-in-Mathematics/
Sousa, D. (2008). How the brain learns mathematics. Corwin Press: Thousand Oaks, CA.
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