My brother, who teaches economics at Texas A&M once asked me, “Why is it that some of my students don’t recognize massive mistakes in their computations? Numbers will be going up when they should be going down. Sometimes answers are off by orders of magnitude, and the whole idea slips unnoticed.” It all comes down to what we think of as number sense – making sense of numbers.
As educators, it is essential that we recognize number sense deficiencies, perhaps most essentially at the younger grades, but throughout all grades.
Although number sense is often thought of as being the domain of the primary classroom, this thinking that numbers must make sense is something that needs to infuse math education all the way through from elementary through postgrad economics. Number sense must be nourished at every level of mathematics.
What is Number Sense?
Number sense is generally thought to start as an innate intuitive sense for quantity and magnitude. It is universal to all societies everywhere. We have a system for numbers that has evolved with us and became, of course, more sophisticated, but it is this primal sense that is at the core of number sense. Lakoff and Nunez in “Where Mathematics Comes From” present the idea that numerals and numbers are essentially metaphors the mind uses to make sense of and communicate ideas related to groups of objects, size and magnitude, length, and motion along a path. These metaphors, they propose, form the basis of how we relate abstract symbols to reality. This idea that a number must reference a tangible or imaginable reality is key to the idea of number sense.
A recent “TED” talk from Conrad Wolfram entitled, “Teaching Kids Real Math with Computers” breaks down math into four phases:
- Asking the right question.
- Real world -> math formulation
- Math formulation -> real world, verification.
In his presentation, Wolfram suggests that we ought to focus our mathematics instruction on steps one, two, and four of this process and lessen our focus on step three, computation. What I find interesting about this is that I would suggest that steps one, two, and four, asking the right questions, making mathematical formulations of real world situations, and making sense of and verifying for the original context, all share number sense as the essential ingredient. This is exactly where my brother’s graduate economics students fell apart. They were unable to apply their computations, whether done by hand or via computer, back to the real world context they were attempting to mathematically model.
Of course the idea of making sense of mathematics for the real world is nothing new. It is for exactly this reason that there is no Nobel Prize for mathematics. Brilliant mathematicians like Richard Feynman and John Nash received their prizes in physics and economics, not mathematics. Because Nobel felt that mathematics need to be applicable to make sense.
Number Sense across all Levels
In the younger grades, where it all starts, we recognize number sense when students realize that quantities are represented in groups of objects or as distance or length increasing or decreasing. As the child’s insights are attached to symbols representing both numerosities and operations, and as events and situations like four birds leaving a group of five are interpreted into abstract symbols like 5-4=1, number sense begins. Making sense of the symbols of mathematics is where number sense takes shape, or conversely mathematics become increasingly senseless, as it does for far too many. It’s because of this crucial stage in a student’s mathematical development that number sense is often considered as the domain of primary classroom.
However, number sense can fall apart at any stage in the learning process. I have had far too many conversations with people for whom mathematics started to make no sense once they hit fractions. Thus we end up with ridiculous adages like, “Ours is not to wonder why, just invert and multiply.” For others, like my son, it falls apart at algebra when his text began to completely remove all sense of number from the procedures, and contexts, which should have been increasingly meaningful, were stripped away from the mathematics. Suddenly his number sense, which had been remarkably strong up until this point, was no longer recognized, fostered, and harnessed for him to make sense of the procedures he was learning. The sole purpose of mathematics became passing an exam.
Start Making Sense
As educators, it is essential that we recognize number sense deficiencies, perhaps most essentially at the younger grades, but also throughout all grades. Because when mathematics stops making sense, it is more difficult to learn, but more importantly it becomes useless. How often do we hear our children and students ask, “When am I ever going to use this?” The lessons we choose should support, foster, and help students develop number sense, especially as the mathematics become increasingly abstract.
Lessons that support number sense most often begin with contexts that support “meaning making.” I prefer to talk of contexts rather than manipulatives for materials. Because manipulatives, although often very well designed for teaching mathematical concepts, do not contain the contact steps within themselves, the meaning comes from the student making meaning of the materials and often contexts which are relevant to students need to be present for that student to truly make sense. Number sense starts and continues in the stories that students model with numbers. And thus we must use stories, contexts, situations, images, and materials, all to support number sense development.
This was the issue that my brother saw among his undergrad and graduate students in economics. The numbers had become completely disconnected from the stories, and yet his students failed to notice. They didn’t see the sense that the numbers should be making. Although we must ensure that students’ initial interactions with numbers lay a foundation for number sense that ensures early success, lessons throughout all levels of mathematics should continue to build number sense to ensure that we graduate students ready to use mathematics in their lives and in their chosen careers.