What It Means to “Really” Answer a Math Problem: A deep dive into the math skills we should expect from our students

By Dr. Mona Tauber, Math Specialist at The Langley School

Math Classroom


This discussion, that can be heard in upper elementary classrooms, reveals much about children’s mental math skills, as well as what children understand about numbers. This school year, in the short time I’ve been at The Langley School as a newly hired math specialist, I am not surprised to hear teachers and parents discuss a similar theme: math facts. This topic may be new for some households and yet has been a common point of contention in mathematics for years. I hear both individual frustration around the “drill and kill method” or what is often referred to as “rote memorization” and simultaneously a joint desire for all children to learn their math facts.  

We know children need to learn their math facts and the goal is to eventually recall them quickly. Yet, the discussion with students above shows that understanding numbers is about more than just “knowing your math facts.” It is part of what we call computational fluency and we know that a deeper understanding of whole numbers, along with fractions and proportional reasoning, has been shown to prepare children for algebra. So what does it mean to be “fluent in math?”

Three Parts to Demonstrating Computational Fluency
Computational fluency commonly refers to building and using efficient and accurate computation methods. Children develop computational fluency when they demonstrate:

  • flexibility in the methods they choose (e.g., using different mental math paths as shown above, using different algorithms like the traditional or area model, or using drawings or tables)
  • understanding and can explain these methods
  • an ability to calculate answers accurately and efficiently

Children should use computational methods based on mathematical ideas that they understand well, including the structure of the base-ten number system, properties of multiplication and division, and number relationships.

How to Develop Computational Fluency in Students
Developing computational fluency, including quick and accurate recall with the basic facts, is a developmental process. Children move through stages, starting with counting, then to more efficient reasoning strategies and eventually to quick recall. Therefore, our instructional support must mirror these same phases, including opportunities to practice and build their basic facts. That is why at The Langley School we use strategies like counting, skip counting, breaking apart numbers into friendlier numbers students know, estimation, and mental math starting in junior kindergarten and continuing in the upper elementary grades when students are studying multiplication and division. 


The student discussion at the start of this blog and this video from Jo Boaler (or this related one for parents and teachers) provide you some insight into why Math in Focus (the curriculum currently used at The Langley School) and other curriculum today support children’s flexible thinking as they develop their basic facts. For example, children may not yet recall that 7 x 9 is 63, but if they see that 7 x 10 is 70 and that 9 is 1 less group of 7 or they know 5 x 9 is 45 and 2 more groups of 9 is 18, they can figure out that the product is 63. Guessing does not build the same math self-confidence and empowerment that we want for all children. Think back to your days of studying only through flashcards!  

A sample of fourth-grade students at The Langley School sharing how they mentally solved two different multi-digit multiplication problems.

Changing Our Approach to Meet Today’s Expected Math Skills
Practice is important and supports automaticity, but what we call “rote memorization” or “drill and kill” as the sole method is one of the least effective ways to reach fluency. Memorizing facts and procedures before developing an understanding of the concepts inhibits flexibility. Trying to imagine the traditional multiplication method like Student #1 in the scenario above only works up to a certain point.

If you would like to learn more about the strategies The Langley School uses to support computational fluency, please contact your child’s classroom teacher or Langley’s math specialists. We can work together to help children develop a deeper sense of numbers. From my perspective, there is great power in changing our collective view and instructional practices to broaden children’s views of mathematics and help them approach and make sense of the mathematics.

Change is inevitable and when welcomed, it can be the catalyst we all need to grow in new ways together alongside the children. 

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