Slide Rules, Affordances, Constraints, and Trig Identities

When I’ve taught trigonometry identities in the past, I sometimes have given students the prompt:

One day you spill a coffee on your calculator and afterwards the cosine button stops working. How would you calculate cos(79˚)?

This constraint leads students to use the complementary angle identity. In the language of design, I removed an affordance from the tool. In an interview with Nat Banting I just heard, he talked about constraints in the classroom, and the pedagogical usefulness of obstructing students. That sounds strange, but constraints are generally acknowledged to foster creativity. That observation resonated with me, and reminded me of this problem.

This week, I was messing around with slide rules, to show students how great they have it now, and how annoying it used to be to find trig function values. While they filled in a table of values with a scientific calculator, I used the giant demonstration slide rule gathering dust on a shelf in a neighboring classroom. I was surprised and impressed with myself that I could calculate values of sine with a precision of about ±0.002.

One thing that intrigued me was the slide rule actually realized the constraint I had used in the past. Many slide rules have three scales for trigonometry: S for sine, T for tangent, and ST for small values of both sine and tangent (since they are nearly identical within the precision of the instrument). Therefore, finding cos(79˚) requires looking at the 11˚ mark of the S scale.

Conveniently, the complementary scale is often marked as well in red. Here’s an image from an online emulator (

The hairline here shows that cos(79˚)=0.191 [Note that the user must understand 1.9 on the C scale to be 0.19 when doing trig calcluations]

This excursion into an obsolete calculating device reminded me how the history of calculation has profoundly influenced what is contained in the standards and curriculum of school mathematics. Even trigonometry itself once had much greater practical purpose. It was the toolbox needed to do calculations for navigation and astronomy, two areas of science and engineering which drove innovation across instrumentation and mathematics. In the era of GPS, far fewer people need to understand spherical geometry calculations. Someone needs to program navigational systems, but practitioners need different knowledge now, because the tools have different affordances.

The technology we teach with has a profound impact on what is considered important. If you always have access to a Computer Algebra System (CAS), then knowing how to find exact polynomial roots by hand is mostly unnecessary. You can argue for the benefits of learning factoring or certain formulae, but you can’t say that the average person, or even an engineer strictly speaking needs this knowledge. The question of when and how to provide students with a CAS is a fascinating one to me. The research I’ve examined indicates that having consistent access to CAS does not actually reduce procedural fluency. What it does do is significantly reduce computational burdens to doing advanced exploration of algebra. But perhaps selectively removing CAS access could force students to be creative in different ways.

As we continually gain greater access (free, on almost any platform) to ever more powerful technological tools for doing calculations, procedures which once seemed essential become vestigial. In a world where division by two is far easier than taking the reciprocal of the radical, rationalizing denominators has a use. Now it serves less of a purpose when a calculator can just as easily find the decimal value when required. Being able to algebraically manipulate a radical expression is definitely useful, but making the “simplified” form necessarily have a rational denominator is mostly pedantry at this point in time.

What other standards or pieces of content in the curriculum are holdovers from the necessity of calculation using tools with fewer affordances?

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