Before formal mathematics, there were numbers, algebra for solving non-linear problems, and rules for working with triangles, rectangles, and circles. There were also cookbooks of formulas derived from numerical-logical reasoning, enabling its practitioners to work out problems of every day life. These abilities from over four millennia ago form the basis of a state of mind and skill set we call numeracy.
Numeracy is essential to a functional populace, and increasingly so as numbers, data, and statistics govern more and more of our lives. Every child needs to be shown the way of numbers, how numbers work for us, how numerical reasoning provides answers to life’s quantitative questions, how one can have fun while learning math, solving problems, and taking the measure of our world.
Many adults seem to understand the need and benefit, yet millions of young lives are being stunted by our failure to do this simple job competently.
Math Begins With Numeracy – We All Gotta Have Some
Our everyday decision-making and problem solving is aided by good number sense and the ability to think logically. Once that numeracy pill is ingested, a lifetime awaits of being right about stuff.
Number sense is a tool to assist in navigating through our world, and does double duty as a self-defense. We are continually bombarded by devious manipulators, trying to delude us with false logic, fake choices, and corrupt statistics. By getting us to think wrongly, they enhance their own hidden agendas. And even in the absence of malfeasance, media too often mistakenly present misinformation, due to absence of math competence on their staffs.
Everyone is born with a primal number sense. Primary grades are tasked to stretch and reinforce this sense for everyone. Our numeracy/math teachers can make this activity fun and social, instilling a sense of discovery that can lead to strong interest and eventual passion. Interested students will seek further math advancement, now selling themselves on math.
We have a mathematician in our family, so I can verify that once hooked on math, students run with it, pushing into mathematics to the limits of their talents. Our primary schools need to set this hook, creating willing participants in mathematics study.
Jordan Ellenberg, in his book How Not To Be Wrong[commercial site], explains: “Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is the science of not being wrong about things.”
Ellenberg illustrates the successes of mathematical reasoning with a series of examples from real life. I don’t want to give away the plot, but I will observe that the math guys and gals always win. Their brains are trained to make correct inferences from observations and data. They develop invisible armor that protects them from being wrong. They can develop instinctive ‘sanity checks’ to quickly tell if a calculated result is approximately correct. Their X-ray specs see through the obfuscation of unsound or dishonest arguments swirling around staged data, as is typical in advertising, economics, politics, or bad science.
Math Salespersons Needed
Judging by the small count of math enthusiasts versus the large count of math-phobes in general circulation, mathematics is apparently not enjoyable for most people. Among all subjects taught, perhaps it’s the math teachers that need to be the best sales people, to overcome negative bias and put excitement and fun into the equation.
Mathematics should be able to sell itself, once these negative preconceptions have been debunked and enlightened teaching practice implemented. Then we can open up the world of mathematics to students and help prepare them to want to buy what’s inside. It should be an easy sale, a product with utility, beauty, uniqueness, excitement, fun, power; and it doesn’t cost much (until you have to buy your math books).
Teachers of numeracy themselves must have numeracy competence. Someone who teaches only from a cookbook lesson plan will not be able to sell the subject or present it intelligibly. After demonstrating numeracy competence, teachers must be trained to teach it. Our math instruction problem cannot be solved without competent teachers, teachers who have been trained in the same path to numeracy that they will be offering/selling to their own students. This task is not being done well enough now.
Competent math teachers can be taught to tap into the latent promise of mathematics, bootstrapping a continuing transformation of phobes into fans. What a step up that will be from our current propensity to form battle lines, desperately throw money at our schools, then play the blame game over the failure of our math education.
Properly trained, our teachers become among our most valued resources. Imagine adding these special subject skills to all the other required skills of an effective teacher: classroom manager, social worker, advocate for the student (the customer), manager of expectations of the parents (the customer’s agents). We should pay them appropriately, and support them well by promoting administrators from the ranks of effective teachers, placing teacher advocates in charge.
How might today’s primary/secondary math curricula be made more effective? Perhaps an idea from teaching trends in higher mathematics might be effective. Enlightened college math professors now teach to the real world and the problems it offers, downplaying the prior math for math’s sake formalisms that filled blackboards for decades and ruined the appetites of many students for mathematics. This pedagogical pivot is due to wider recognition that fundamental mathematical developments have been motivated by our desire to more deeply understand reality and to make the physical world work for us.
When accompanied by revised textbooks, teaching mathematics in the primary and secondary levels will surely benefit from such a paradigm shift, embedding mathematics instruction within motivating experience, while dispatching from the curriculum most requirements for rote memory and forced abstraction (recalling the dubious practice 50 years ago of introducing elementary arithmetic through set theory).
To overcome math resistance due to fear of failure, lack of social acceptance, and questions of future relevance, young students from middle school on will benefit from meeting together socially and talking about activities related to logical reasoning: how to solve problems, how to code algorithms, how to interpret/critique statistical arguments, how to estimate quickly, and various other topics designed to foster mathematical creativity.
The reinforcement, both psychological and pedagogical, derived from associating with fellow travelers through mathland, will foster can-do attitudes and success.
Developing Math Chops
There are no limits to the demands that mathematics study can place on a student. One’s limits must be chosen based on what one wants from mathematics. Prospective mathematicians are well-advised to go all in. Others may chart an easier course, but complete mastery of numeracy is the minimal objective. Even those limiting their math contact after the secondary grades can enjoy being right a lot more often, seeing the world more clearly, and perhaps even enjoying mathematical recreational pass-times.
You don’t have to be a world class talent to have success with mathematics. Those who choose math as their college major subject, as in my own case, will receive in return a desire for continued involvement, together with appreciation, satisfaction, and insight into the world that others are missing.
Mathematics learning is a process. Since it is easy to feel out of our element sometimes on this journey, one needs to focus on the process to get some positive strokes. If there’s something you don’t get, and available assistance couldn’t help, don’t be discouraged from forging ahead with hopefulness. Keep seeking assistance until someone can explain it in an understandable way. And then attempt to teach it to others, using your experience as a guide.
Students without higher mathematical aspirations can use college math to prepare to work in technical fields requiring mathematical skills: science, engineering, teaching, data and statistical analysis, and programming. These occupations are as close to a certain future job market as can be predicted. Hopefully some of these talented young folk will find that salesman’s position is worth another look, and will make their objective the teaching of mathematics.
A few college students with special talents for abstraction, visualization, or computation will have developed strong facility with abstract spatial and logical thinking, good instincts for what tool to use and when to use it, and broad skills at problem solving. These then can move on, with good prospect for success in doing advanced mathematics.
Only a very small number will participate in and contribute to mathematics at the highest levels. Their rewards are great satisfaction, pleasure, and recognition in their peer community. They earn these rewards through their well developed talents, derived from years of hard work. After realizing the effort required to reach top rank status, most students will set lower objectives. The same dynamic operates in sports or music; participants develop their talents to appropriate levels according to their objectives and their tolerance for work.
For the seriously motivated students exhibiting early interest, development of mathematical instincts ideally begins in primary school and is typically boosted by programs outside the regular school day. Just as athletes and performing musicians train outside of classroom hours, so should serious mathematics students. In elementary and secondary grades, well-supported schools will offer extracurricular mathematics study focused on competitions.
Typically, a master teacher will guide the program and mathematically-inclined parent volunteers will provide support. Teams are formed and they work together to learn to solve a variety of problems. They then demonstrate their proficiency by competing among themselves, then with other schools. It is through such continual problem analysis that math instincts are formed. Without such instincts, math will become an increasingly difficult slog in its advancing levels.
There are no shortcuts on the path to mathematical enlightenment. Continual engagement in the subject is maintained by working on applied and theoretical problems. The best preparation for a life of serious mathematics is early intense math gong fu, problem solving repetitions preparing the mind for the eventual state of wu wei, the free flow of reasoning that is guided by instinct and intuition to maximize the leverage of conscious effort.
Just as the athlete or musician performs without expending mental energy on the mechanical details of performance, so does the mathematician construct hypotheses and logical trains of thought, informed initially by well-developed instinct.
Reaching Out Across Social Barriers
Most of what was discussed above is already happening in enclaves of highly educated people across our country. But in schools outside such enclaves, it is a struggle to meet minimum standards for mathematical achievement. But we know such schools will have a number of students that would be successful with mathematics had they been born under different circumstance.
Improving math education in such districts will continue to be a huge challenge, due to social distrust and apathy of the parents, and hence their children, (and sometimes their teachers) who do not understand, do not feel motivation for, and cannot develop affinity for mathematical thinking.
Rather than condemn students of mathematical promise to wasted talents, by leaving them undiscovered or underdeveloped in such failed programs, private capital is needed to put forward programs of discovery and development. Some such programs are already up and running, with heartening and promising results. We need to cookie-cutter these successes and apply them everywhere they are needed. Who better to benefit from the American Dream than our most deserving talents. Help them blossom, wherever they are.