Technology influences both the content and the methods of math education, but the impact is slow and gradual, not sudden and dramatic. This is in part because it takes time for technology to reach the classroom, but it is especially because school and societal culture develops unevenly. In this post, I think about some specific examples — past, present, and future.
I am 72. Here are some topics I learned in high school which have vanished from the curriculum because of the impact of technology.
How to use a slide rule. Using a slide rule effectively was connected with some math content (understanding significant digits, logarithms, and no doubt other topics). The advent of the scientific calculator relegated the slide rule to the closet or the trash can.
An algorithm to calculate square roots with as many digits as you need. This was a long-division-like process carried out on paper. Understanding the algorithm was not a primary goal of instruction — the goal was to get the answers efficiently. Obviously, the calculator was even more efficient, so the algorithm is no longer taught.
Log and trig tables. One side benefit of using those was developing some skills in interpolating between values. But in the end, this was another tool that was displaced by the calculator.
In all three cases, we replaced one tool with another, and this was not particularly controversial. Yet this process did not continue to move forward as technology advanced. I will discuss three specific instances, in chronological order, based on the introduction and availability of the relevant technology.
We are told that schooling is intended to prepare you for the job market. And also that it is important to know how to use the traditional algorithms for calculations. Yet, there is not a single line of work that requires facility or even basic competence in paper-pencil arithmetic. This is true in retail, health care, manufacturing, finance, engineering, etc. Everywhere in society (outside of schools) arithmetic calculation is done electronically, and students know it.
In defense of those algorithms, we are told they help students develop number sense. This is preposterous: the whole point of these algorithms is to facilitate fast and accurate computation. It has nothing to do with understanding numbers. Quite the opposite: thinking about numbers is a distraction from carrying out the step-by-step instructions, which need to be 100% automatic. This is why it is crucial, in that view, to automate basic addition and multiplication facts, irrespective of the math that is embedded in there.
And yes, there is much interesting math embedded in the addition and multiplication tables, in long division, and so on. It is useful and interesting to learn about that math, and it is best accomplished through number talks, base 10 blocks, cuisenaire rods, one-on-one conversations with students, and so on. Analyzing algorithms, including the traditional ones, also offers opportunities for interesting mathematics. But aiming for speed and accuracy in paper arithmetic just doesn’t make sense any more. It takes a lot of time spent in mindless practice, and it is so boring and frustrating that it turns many students off to mathematics.
We can get to many important ideas by moving mental arithmetic to center stage. Sharing, discussing, and practicing various mental arithmetic and estimation strategies is a powerful way to develop number sense, and at the same time it is vastly more useful in daily life than paper-pencil algorithms — in and out of the classroom. Meanwhile, routine calculator use in word problems will not harm students’ understanding, and can in fact be combined with the use of mental arithmetic and estimation.
Many math educators consider the solving of linear equations to be the heart of algebra. The topic is often introduced early, atomized into many special cases (one-step, two-step, etc.), and the subject of endless mind-numbing practice. For many students, this carries little meaning, other than “I’m supposed to get x by itself”. This is counterproductive. Algebra is about many things: structure, including especially the distributive law; modeling real world phenomena; functions; exponents; and so on. Equation solving is much better taught in concert with all these other dimensions of the subject. But note that of all the conceivable equations, only linear and quadratic equations can be solved with paper and pencil. If exponential, trigonometric, or logarithmic expressions are involved, Algebra 1 techniques are useless.
On the other hand, any equation at all can be solved graphically. This was already true once the graphing calculator entered the scene, and is only more so in the age of Desmos and GeoGebra. What would be lost, and what would be gained if we allowed the default equation-solving tool to be electronic graphing?
Probably the biggest loss would be that students would lose much possibly useful practice in symbol manipulation. I can think of two ways to make up for that. First, I suggest equation talks: solving linear equations mentally, and discussing what strategies make that possible. That would help center meaning, it would help the teacher get a sense of student understanding, and it would reinforce the idea that there are many ways to proceed. Second, I suggest more work on literal equations. Those are quite useful in manipulating formulas in science classes, but they are notoriously difficult for students. Part of that is because so little time is spent on them.
The gains of allowing graphical / electronic equation solving as the default would be substantial. It would increase access, as more students would readily be able to use those techniques. It would make room for more applied and modeling problems, which would help increase motivation. It would require a better understanding of graphing and functions, which are crucial in subsequent courses. And it would not preclude the use of well-chosen paper-pencil problems in reasonable amounts, as well as lessons that focus on making the connection between graphical and symbolic methods. (See such a lesson in Add Till It’s Plaid.)
Speed and accuracy in paper-pencil computation is no longer a valid goal.
GeoGebra has a Computer Algebra System (CAS) built into it. Here is an example of CAS in action:
To me, this means we should spend much less time on factoring (as a skill) and put some of that time into understanding factoring, as a concept. Part of that is accomplished with the so-called area model, which is best introduced with the help of manipulatives, and developed with the help of “the box”. (See for example my Introduction to the Lab Gear videos.) I’m especially distressed when I hear of teachers spending a ridiculous amount of time teaching arcane factoring techniques and shortcuts. Some trial-and-error factoring of trinomials is a good thing, as it helps reinforce understanding of the distributive property. Beyond that, time is better spent on other things.
(And yes, CAS also makes it possible to solve equations at the touch of a button. I don’t know whether that dooms my proposal to prioritize graphical approaches.)
The fact that too much time in Algebra 2 was spent on such manipulations has led some in our profession to call for the elimination of the course. I fear that this will end up increasing educational inequalities, as the children of the well-off will still have access to Algebra 2, which is prerequisite to so many topics in math, science, engineering and statistics. In my view, we should continue the work we have been doing to bring math to all students. Going all in on technology, as suggested in this post, is a crucial part of that mission. It can help with understanding, motivation, and equity.
The concepts underlying the topics discussed in this post are still fundamental. Long division may die, but division is not going anywhere. Equations can be solved by machines, but they remain essential in STEM fields. Understanding the distributive law is needed if students are to use formulas and symbolic representations in any quantitative discipline. However these concepts should not be confused with the way they were addressed at different stages in the history of technology. The task facing us is to figure out how to teach the concepts, while simultaneously deemphasizing paper-pencil manipulation.