by C.J. Uberroth
In teaching high school mathematics for the last seven years, I’ve begun to realize that there are some issues in our field that seem ubiquitous. The first is the death of estimation. Students faced with a simple arithmetic problem will sooner jump to their phone for the answer than think about what answer they should obtain. This presents a serious problem. Since students are not thinking ahead about what the approximate answer should be when multiplying 2.5 by 8, they are just as likely to answer 200 as they are 20. All it requires is missing the decimal button, something that would immediately be corrected, if the student had already estimated the answer.
Secondly, I find myself longing for students who actually understand the core concepts of arithmetic and algebra. By using the calculator, the concepts that relate the different parts of mathematics fall by the wayside. How can students see the linear connection between second differences of a quadratic function or the subtle relationship between the places of numbers and how they interact under multiplication? The logic and beauty of patterns that one discovers breeds a real hunger for further knowledge, and without this hunger, the quality of our mathematicians will decay.
Finally, I have a hard time taking seriously some of the issues on which my colleagues expend a significant amount of their energy. Some of them, like simplifying fractions by their common factors, I understand: reporting a value of ten fourths as opposed to five halves make a person look uneducated. Others seem needlessly obtuse in a technologically advanced world. In the history of mathematics, it was necessary to rationalize denominators of fractions when the denominator contained an irrational number, most often an un-simplified radical. This was important when division was done by hand. If students need to estimate the value of one divided by radical two, they would be faced with a near impossibility due to the never-ending nature of the square root of two. By rationalizing the denominator, you can turn the fraction to radical two divided by two, making the estimation a feasible process.
Today, we have the technology to evaluate these expressions without the hassle of rationalization. This is by my no means to say I don’t see value in learning the process required to achieve this rationalization, but I couldn’t believe that my colleagues would still lose their minds over un-rationalized fractions. The same goes for solving an equation. If a student’s method of solving an algebraic equation is different from the teacher’s, points will be marked off, regardless of the solution.
Recently, my mind wandered to my Grandfather’s slide rule, a fascinating piece of technology. Years ago, I found it in his garage and was immediately enthralled. I set out looking for any information I could find on how to solve problems with this beautiful, plastic rectangle. My labors paid off, and the complexity of device astounded me. Arithmetic on a slide rule is fairly simple: Align the initial number to the secondary number, slide the cursor to the secondary number’s other occurrence, and then read the solution. The important task comes when you realize that the slide rule returns the same answer for 120 * 6 as it does for 1200 * 60: 7.2. It is up to the user to know what this value really represents. The death of estimation that I referred to earlier would make this tool largely useless to students, if it were reinstated in the classroom today, without substantial, remedial education.
The slide rule also does a remarkable job of visually representing the concepts in arithmetic. Each scale is constructed in intervals, so one can visualize how each of the functions on the slide rule change. This visualization is exactly what I’ve noticed students now lack. Understanding that the gap between log(1) and log(2) is an exponential difference; understanding that distance between percentage growth and decay changes as the principal value changes; these concepts are not challenging, but finding students who can make these connections is rare, something I think is detrimental to the quality of math education today.
With all this said, the reality of the slide rule’s passing does not surprise me. After all, we as a culture have come to value expedience above all else. What did surprise me was the age and longevity of the slide rule. It’s origins can be traced back to the early 1600’s. All through the 19th century, various slide rules were created and produced to serve as calculation tools for everything from simple arithmetic to radical estimation to . The power of the slide rule far exceeded its juvenile competitors in the mechanical calculator and early computers. The skill and speed of a trained slide rule user was unmatched. Ultimately, I feel like we’ve lost a great deal of our skill with numbers as this tool disappeared and we have become ignorant of its history.
Even if the sort of remediation required was provided, I don’t think that that replacing calculators with slide rules in math classes would constitute some sort of magical fix for our current educational woes, but I find myself wondering if the quality of our math education might significantly improve if we did. Would students have a larger appreciation for the history of mathematics? Would they be more capable of recognizing and analyzing unknown patterns? If slide rules were the only calculating devices we permitted in class, would the students’ capacity for mental arithmetic be improved? I am inclined to answer “Yes” to all of these questions.
Perhaps it is time to consider a more rigorous fundamental approach to the development of numeracy in our young people. Perhaps it is time to return to the age of the slide rule.
Several slide rules