Adapted from Wendy Ward Hoffer’s All Minds on Mathematics.
All our students can be math people; agentic lifelong learners who relish the grapple of mathematical meaning making. We have the opportunity to guide each student in their efforts toward becoming capable mathematicians who possess the following:
- productive dispositions
- number sense
- conceptual understanding
- problem-solving capabilities
These are not distinct abilities but instead they flow across one another and can each be honed with conscientious effort.
1. Productive Dispositions
A student with a productive disposition understands that math is relevant, meaningful, and accessible, and perseveres in their efforts to “Make sense of problems and persevere in solving them” (Common Core State Standards Initiative n.d., 6). By the time they reach us, too many students have decided they’re not strong in math. Maybe their parents or siblings threw their game, or a previous teacher passed along a fear or bias. We may never know what rattled their confidence, but it is never too late to build their identities as problem solvers who believe in their own competence. As Stanford math education expert Jo Boaler describes in Mathematical Mindsets, “No one is born knowing math, and no one is born lacking the ability to learn math” (2022, 6).
We can model a math-positive stance, frequently sharing our own high regard for math and reporting on all the ways math helps us navigate our lives. I often invite teachers to notice and tell students stories of our own mathematical exploits: how we calculate a tip, measure mileage on a trip, act on an investment tip, decide how much to chip in for a meal—and more—are all matters resolved with math. Further, we can also each report on times when we grappled and persevered as mathematicians and otherwise, illuminating this virtuous quality of character.
Catching students in the act of persevering reminds other learners how persistence looks. Delight in learners embodying productive beliefs. When we praise and honor growth, we imbue in students an appreciation that their determination is of value and inspire more of it. Learners are listening.
2. Number Sense
Another aspect of becoming a capable mathematician is the development of number sense: the ability to understand and work with numerals, grasp them and number systems as abstract representations of concrete quantities, and know how to compare and combine numbers in various ways. This is but one aspect of math, not its entirety. When we have number sense, we are able to think flexibly about numbers and how to compose and decompose them with a variety of operations.
Teaching number sense involves exposing students to many opportunities to be creative and versatile. Number talks in which learners explore various ways to represent a quantity, number sense routines that invite students to think and talk about math, whether estimating or inferring, and other quick, daily activities can be useful tools for students to rehearse their acuity with numbers. These invitations can work well at the opening or closing of a math workshop, or in those few minutes we find at school between one scheduled event and another.
These skills of understanding bare numbers and their relationships are critical to the development of numeracy and yet are not an end in themselves; number sense prepares learners for more complex, applied challenges as problem solvers. Students with number sense can step back from a solution and notice whether it seems accurate and makes sense, given the context and the mathematical operations at play.
3. Conceptual Understanding
An instructional focus on conceptual understanding allows students to remember a mathematical idea and to apply it flexibly, as needed, with confidence and precision. Our quest to scaffold our learners’ understanding is an ongoing, up-growing spiral with stages of progress along the route. When a student truly understands, according to Understanding by Design authors Grant Wiggins and Jay McTighe (1998), they can explain, interpret, apply, provide perspective, demonstrate empathy, and reflect on their self-knowledge.
When we comprehend mathematical concepts, we can generalize from a specific case to a set of principles or patterns, and vice versa, in ways that increase efficiency and foster discussion. We leverage our understanding of prior concepts—say, multiplication with whole numbers—to build our understanding of more advanced concepts such as multiplication of polynomials. Understanding builds the foundation for future learning and lifelong mathematical success.
Understanding is an active process, requiring higher order thinking across time. To internalize an understanding, brain research tells us, we must work with a concept ourselves. This process of connecting learners’ minds with ideas and vice versa can be achieved in a variety of ways, including through modeling, open-ended tasks, and conversations.
4. Problem Solving
When we, as math learners, have can-do attitudes, when we understand the nature of the numbers at hand and the operations or concepts we are being asked to apply to them, we can effectively solve problems. University of Delaware mathematics education professor James Hiebert described the importance of problem solving for our learners in his book Making Sense:
| Today’s students need flexible approaches to problem solving methods that can be adapted to new situations, and they need know-how to develop new methods for new kinds of problems. Nowhere are such approaches more critical than in the mathematics classroom. (Hiebert et al. 1997, 1) |
While solving problems in math is certainly a worthy calling and one with important life applications ranging from paying bills to calculating taxes, life is full of seemingly un-mathematical problems that also call for solutions: We lose our keys, get a flat tire, clog the toilet. While solutions to some of these adult foibles don’t necessarily require math, they do call upon us to use some of the same skills—productive dispositions and conceptual understanding to solve. We can remind learners that the qualities and insights they are developing as mathematical problem solvers will serve them well throughout their lives.
While teaching math, we are building habits of mind. As true problem solvers, capable mathematicians are able not only to check the right box on a multiple-choice test, but also to persevere and prevail through mathematical and real-life challenges.
