Consider the calculator. We may take for granted those little devices that help us complete tedious computations (and maybe a little trigonometry), but when I started college in 1971, I paid $150 for my first Texas Instruments calculator, almost as much as my car at the time, and way more reliable.
In the intervening decades, I’ve grumbled aloud more than once at how using this device (and its digital successors) has made it nearly impossible for me to do commonplace arithmetic “in my head,” whether it is gauging a tip, comparing grocery item costs, or making metric conversions. But using a calculator still requires that I am able to apply what mathematicians call standards of reasonableness to the results: Does that calculation look odd? Should I trust that I entered the values properly and used the right operator?
Checking for reasonableness in math involves applying a process for making an independent estimate of a problem and then evaluating the estimate to help check the actual solution for accuracy. It comes in handy when you don’t have a calculator available, or if you want to check the calculator’s work.
Reasonableness standards are not innate, so how do we teach and learn reasonableness? This is a deceptively straightforward question, but the answers might also help us to teach and learn how to recognize cultural blind spots.
When I look at the literature on teaching and learning reasonableness in math, I appreciate several things. First, we have to start early in teaching number sense and math reasoning skills. Next, there is a difference between getting the correct answer to a math problem and tackling the problem in a reasonable manner. And third, teaching strategies have developed to get students in the habit of asking and answering questions of themselves related to the reasonableness of a solution or problem-solving strategy.
Math learning starts early and accumulates wisdom with age-appropriate lessons about number lines, set theory, Venn diagramming, and everyday pattern recognition activities.
It used to be that development of math reasoning skills reflecting number sense relied heavily on fact fluency, often achieved by memorization and drilling for speed and accuracy. Math facts remain important, but an emphasis on memorization has given way to knowledge gained by using facts regularly and developing an understanding of numerical relations.
More attention is now given to showing your process and to the questions teachers ask about the work being shown. Teachers’ questions have a great deal of influence on student actions and perceptions, and question paths can help students get in the habit of explicitly asking themselves if their answer makes sense and then checking their process. Research has also shown that learning environments can unintentionally mute some voices, especially when large group discussions lead to some voices getting lost or never being heard. Small groups, peer-to-peer questions, and a host of other approaches to structuring productive learning environments are among the promising practices that have emerged.
Replace “math” with “culture” in the previous paragraphs and consider whether there might be useful lessons for anthropology. The more I read about strategies of math instruction, the more I am convinced that a parallel can be constructed in age-appropriate fashion about recognizing cultural blind spots. Why am I preoccupied with the familiar? Am I exoticizing the unfamiliar? Did I over-generalize from limited observations? Did I cherry-pick from my observations to prove a point I was already convinced was right? Did I conflate a social category with a community? Have I essentialized a whole category or community in a reductive fashion? Does the way I have framed this problem say more about me than about my collaborators? “I don’t know,” or “I don’t get it” are not sufficient in assessing how one’s judgments may potentially be imposing harms, wittingly or not. One can learn to be more specific about where understanding falls short. And the earlier we begin to cultivate awareness about the reasonableness of our judgment processes, the more likely we can strengthen our powers of reflection and empathy.
AAA has recently introduced a deeply discounted membership category for K–12 educators, and we are hoping to pilot test a summer workshop, if not this summer then next, to bring together educators from primary and secondary grade levels with anthropologists. The aim is to draw on the local knowledge of how to introduce new teaching and learning materials related to key anthropological concepts, with interested anthropologists on hand to suggest resources that can be adapted for age-appropriate use. I know it is a calculated risk, but since math generally is taught in the primary grade levels by the same teachers who cover language, geography, history, and life sciences, I am enthusiastic that teaching strategies can be applied in ways that help expose students to key concepts and recognize cultural blind spots.
Cite as: Liebow, Ed. 2021. “What We Can Learn from Math Teachers.” Anthropology News website, April 14, 2021.