When you think of a high school math class, what images come to mind? Abstract symbols? Pages of problems? Are these images coupled with anxiety and dread, or excitement and wonder? As educators, we are constantly evaluating the methods we use to reach students, especially in math.
There are two main approaches to the concept of teaching and learning mathematics, which I will call traditional and progressive. The traditional approach is best summed up by a conversation I once had with my brother. The school that his children attend was working to define the phrase “A mathematically powerful student is, ...” and he derisively responded “someone who gets the problem right.”
On the surface, this is an appealing thought, which cuts through the jargon and simplifies the idea. Moreover, it is easy to assess as standardized tests are quite good at determining computational proficiency.
However, when looking a little deeper, there appear to be some problems with this approach. Take, for example, the following problem: Simplify: 31n(x) + 21n(y) = 1n(z)
How many adults have seen anything like this since they left formal math education? As a math teacher, I can list countless examples of these types of equations at every level of math – and your kids can, too. What they may not be able to list are the reasons how this seemingly obscure equation applies to the everyday life of a typical adult.
Contrast this with a more modern approach. A few years ago, I approached my principal for guidance, what exactly did she want me to teach our seniors? Her answer was brilliant in its directness and simplicity – “What do you want them to know, that they will need in their future?” To answer this question, I had to question the purpose of the subject – why do I value mathematical knowledge? My personal response stems from the fact that mathematics is the purest form of logic and critical thinking that exists. It is a formal structure that is designed to get correct and unquestioned answers. It is a clear structure for approaching complicated questions. Mathematical logic and reasoning can be applied across all disciplines, and can help us understand complex structures with more precision.
Too often in schools we deprive students of the chance to think creatively and solve true problems. We give them dry algorithms and expect them to repeat them until fluent, test fluency and wait for them to forget. Instead, schools should explicitly teach problem solving and critical thinking. Schools should give students challenging problems that require them to build solutions, ask them to challenge the solutions of their peers and defend their work. We need to have them solve problems of consequence in their lives, where knowing the solution will change their views or behaviors.
This is not to say that knowing algorithms is unimportant for students. Understanding exponential laws when making financial decisions, geometry when planning a house project, ratios for cooking recipes are among many clear practical applications of math skills. It is simply time to recognize that this is not the sole purpose of a mathematical education. Today’s math students must learn to connect the concepts they learned with the real world application, integrating the math and science curriculum so students can truly learn about how their mathematical formulas can be applied in real world situations: half-lives in radioactive isotopes, logarithmic population growth of invasive species, mass and acceleration in motor vehicle accidents.
In addition, the progressive approach is to pose abstract questions that are intuitively interesting. These serve the purpose of training students to think critically, while also building familiarity with mathematical tools. For example, imagine eight people sitting around a circular table. Each person flips a coin. Those who get a tails, stand up. What is the probability two people next to each other will be standing? This seemingly simple question leads students down a variety of intriguing paths, involving combinatorics, binomial expansion theory, sequences and series, and Fibonacci numbers, all in the context of solving a puzzle. While the final answer is not really important, the process that leads to it is, and both train students in critical thinking, as well as exposes them to new mathematical structures.
So when you reflect on the quality of education of our children, consider what is important. Is it standardized test scores, the ability to apply a formula, understanding a complex mathematical model, use of logic to solve a problem?
There are many ways to define success and failures, but as an educator, I believe the most important element is getting our students engaged and thinking critically, while developing an appreciation for the beauty of math.
Kyle Edmondson teaches mathematics at Animas High School. Reach him at kyle.edmondson@animashighschool.com.
