Given my generation’s math education, sometimes I am surprised that I became a math teacher. As a student I figured out methods and patterns, completed my work, did “well,” got the A and had no idea what I had just completed. The teacher demonstrated a (singular) procedure to solve a particular kind of problem, and then similar problems were introduced for the class to solve together. Afterwards, students practiced many similar problems independently.
Today we know that teaching mathematics this procedural way may actually hinder learning. Children can become dependent on tricks and rules that do not work in all situations, making it more difficult to apply their knowledge to new situations (McNeil and Alibali, 2005).
Here is an example of procedural thinking: When I ask adults to define “mean” or “average,” I consistently hear, “It’s when you add the values together and divide that sum by the number of values that you added.” While this response tells me the correct procedure for finding the mean, it demonstrates absolutely no understanding of its significance. When I ask respondents to explain differently, they cannot. The procedure, while helpful, does not explain the concept which in this example is a redistribution of the values so that one common amount is representative of all values.
Teaching conceptual math
Please be patient with today’s math teachers. Chances are that we are teaching differently than the way that you were taught. We are trying to get it right by teaching students conceptual understandings, the meanings and relationships that eventually are performed efficiently through procedures or algorithms. Concepts are ideas, connections, meanings, relationships… We teach students the significance of the number 10 in our Base 10 number system. We use pictures as the beginning representation of multi-digit multiplication. We round numbers first by using a number line and noting distances to benchmark numbers. We teach repeated subtraction and then partial quotient long division before we teach students traditional long division; now forgetting “Divide Multiply Subtract Compare Bring down Repeat” will not lead to frustration or disaster. We want students to understand the significances that lead to procedures.
Understanding vs. memorizing
Research abounds that supports the efficacy of understanding rather than memorizing and the longer term retention of math ideas that are understood rather than memorized. (Borasi & Rose, 1989). When we teach concepts and connections, rather than facts and procedures in isolation, students are more likely to relate novel problems to familiar ones, leading to increased likelihood of transfer of information (Fuchs et al, 2004). Our goals are for your children to understand math and to be comfortable working with new math ideas while persevering and exploring. Hopeful by-products will be enjoyment and appreciation of the beauty of math.
Learn more about best practices in teaching math:
- Borasi, R. & Rose, B. (1989) “Journal Writing and Mathematics Instruction.” Educational Studies in Mathematics, Vol. 20, 347-365.
- Ferguson, Kyle, "Inquiry Based Mathematics Instruction Versus Traditional Mathematics Instruction: The Effect on Student Understanding and Comprehension in an Eighth Grade Pre-Algebra Classroom" (2010). Master of Education Thesis & Projects. Paper 26.
- Fuchs, L. S., Fuchs, D., Finelli, R., Courey, S. J., & Hamlett, C. L. (2004). Expanding schema-based transfer instruction to help third graders solve real-life mathematical problems. American Educational Research Journal, 41(2), 419-445.
- McNeil, N.M. & Alibali, M.W. (2005), Why Won't You Change Your Mind? Knowledge of Operational Patterns Hinders Learning Performance on Equations. Child Development, 76(4), 883-899.
About the author
Julia Almaguer is a math specialist at The Summit for grades 1-6 and coordinator of the Lower School Math Department. She has bachelor’s degrees in economics and French from Duke University and a master’s degree in education from Ohio State University. A teacher for 22 years, she was a key innovator of The Summit’s signature Conceptual Math program.