What do we want math education to be?
Every child is a mathematician. Students can (and should) be educated as mathematical thinkers, not calculators. Mathematics education with this intention must focus on the development of two key elements: strong number sense and mathematical reasoning. We want children to see quantity, to experience the myriad of ways that a given collection can grow or can be decomposed, and to experiment and take risks in the learning arena. Fundamentally, students need to understand that math makes sense - math is figure-out-able.
Mathematics is a fundamental part of human thought and logic, and integral to attempts to understand the world and ourselves. From the earliest stages, children show a natural interest in and enjoyment of mathematics. In play and throughout their daily activities, children explore mathematical ideas and processes. They sort and classify, compare quantities, and notice shapes and patterns. Mathematics helps children make sense of the world around them, and children are naturally inclined to use mathematics in this way (“Look how many shells I found !!” “That won’t fit in there—the box is too small”). Our goal is to empower children by capitalizing on such moments and, through the years, work to cultivate and extend that inherent mathematical understanding and interest. Through rich (and often challenging!) mathematical tasks and opportunities, students develop the reasoning skills to determine and apply effective strategies as they work to better understand and attempt complex problems.
The Ongoing Debate: Calculation vs. Conceptual Understanding
Let’s get some vocabulary on the table. Number sense, math facts, calculation, and computation. What is number sense? It is the basic understanding of quantity and its relationship to our number system that allows children to work flexiblly and fluently with numbers - to understand what numbers mean, how our number system works, and how numbers relate to one another. Children with strong number sense can appropropriately select and apply a range of mathematical strategies, spot and utilize patterns in the number system, manipulate numbers to make calculations cleaner, and can assess the reasonableness of a solution. Number sense develops over time through ample opportunity to see, explore, and play with quantity. Strong number sense is the essential foundation that all of our children need.
Certainly, as kids grow, calculation skills (deliberate mathematical processes) are an important component of mathematics pursuits. However, in every stage of math learning, we need to encourage reasoning and the development of number sense, not rote learning. When I can see that 7 + 5 equals 12 because I took three from the five to build to ten and had two left over, I am exercising number sense. I am not only able to solve that one problem, but am able to apply that same understanding to 17 + 5 and 127 + 5. And, as I develop that deep understanding, I become quite fluent and even automatic with my math facts. The ultimate goals of computational flexibility and fluency come from strong number sense - and this comes from time to explore, investigate, and reason through mathematical problems. Replacing exploration with memorization or procedures is a mistake that too many make. It is a short term solution that robs students of long-term mathematical confidence and facility. Like in so many areas of development, it is not about who does it first or how quickly they memorize it. This type of comparative, check it off and move on view of development needs to shift.
If children develop strong number sense and understand the value of differing approaches, then, as they progress through the years and confront new problems, they have the ability to effectively and flexibly select and apply appropriate approaches. Think about this expression: 103-98. A child who has learned to approach math through memorized procedures may stack the numbers vertically and go about the laborious process of shifting or even “borrowing” from one column to the next. Compare this to a child who understands quantity and takes a breath to consider the relationship of these quantities before jumping into practiced procedures. What is the difference between 103 and 98? Well, 98 plus 2 is 100 and 100 + 3 is 103. The difference is clearly (and efficiently) 5. This child is thinking rather than mechanically (and often inefficiently) calculating.
Mathematics is a subject that allows for precise thinking, but when that precise thinking is combined with creativity, openness, visualization, and flexibility, mathematics comes alive. The goal is for children to develop strong number sense and, through reason based approaches, strong calculation skills, so that they can tackle the complex problems that they will face - not just in “school math” but in the real world math happening all around them now and in their futures.
Here are a few tips to remember as you engage with your child:
Offer questions rather than answers.
Focus on the how and why.
Avoid a focus on right/wrong answers.
Remember that mistakes are part of the path towards a solution.
Let your child take the lead - let them show you, share, and express their understanding and excitement for all things math.
Avoid labelling your child as a math person or not a math person
Students should come to know:
Everyone can do well in math.
Problems can be solved with many different insights and methods.
Our brains think about math visually.
Mistakes are valuable: they encourage brain growth and learning. They are steps along the often windy path towards a solution.
Mathematics will help them in their lives, not because they will always see the same types of problems in the real world, but because they are learning to think quantitatively and abstractly and developing an inquiry relationship with math.