We’ve seen a lot of construction so far, and in all the slowing down and merging we’ve been doing I had an epiphany: maybe there’s something about this construction that can help my students learn to write proofs.
In the interests of full disclosure, I should open by saying that I hate the way we teach proof in high school mathematics. I suffered through a ninth grade geometry class in which I struggled to “prove” something that seemed to be glaringly obvious, and the only apparent reason for doing so was that this was supposed to somehow help me appreciate the beauty of mathematics. It didn’t. Fifteen years later when I taught geometry we were still doing it the same way. To me, there’s something wrong with that. (For a great explanation of what’s wrong with the way we typically teach proofs in geometry, I highly recommend Ben Orlin’s post, “Two-Column Proofs that Two-Column Proofs are Terrible.”)
In my previous life, I worked as an engineering consultant. One of the things I did was work with construction specifications. These are the written requirements that accompany a set of plans. The plans and the specifications together describe what the final product should look like. Here’s an example of construction specifications from the New Hampshire Deparment of Transportation:
3.12.1 Method Requirements.
184.108.40.206 Immediately after the hot asphalt mix has been spread, struck off, and surface irregularities adjusted, it shall be thoroughly and uniformly compacted by rolling. The initial rolling shall be done with a static or vibratory steel-drum roller. Intermediate rolling shall be done by a pneumatic-tired roller. Final rolling shall be done with a static steel-drum roller or a roller of the steel-drum three-axle type, locked. The completed course shall be free from ridges, ruts, humps, depressions, objectionable marks, visible segregation, or irregularities and in conformance with the line, grade, and cross-section shown in the Plans or as established by the Engineer […]
220.127.116.11 Pneumatic-tire rollers shall be self-propelled and shall be equipped with smooth tires of equal size and diameter. The wheels shall be so spaced that one pass of a two-axle roller accomplishes one complete coverage. The wheels shall not wobble and shall be equipped with pads that keep the tires wet. The rollers shall provide an operating weight of not less than 2,000 lb (900 kg) per wheel. Tires shall be maintained at a uniform pressure between 55 and 90 psi (380 and 620 kPa) with a 5 psi (35 kPa) tolerance between all tires. A suitable tire pressure gauge shall be readily available.
If you managed to read the entire section, then you likely noticed a few things. First, there is a lot of specific detail here, but this isn’t a step-by-step guide. Second, there’s a lot of construction-specific vocabulary here that one must know in order to make sense of this. Third, some words are used that don’t have a specific defnition but are widely understoood in the industry, like the word “suitable” that is used to describe the tire pressure gauge.
So what does this have to do with proof? First of all, specifications aren’t proofs, but they hold many of the same characteristics: specific detail without being step-by-step guides; use of specific vocabulary; and some language that is not well defined but is widely understood. Second, it has been said that the reason we prove things is not to sow that they are true but to help us understand them better. The same is true of specifications; they don’t tell the contractor how to do every step of the work, but they help him/her better understand what work must be done. Third, construction specifications are difficult to write in just the right way, so that there’s the right amount of information and everything is clear without being too detailed. The same is true of proofs.
As I think about the new integrated math currciulum we’re developing, I know that I’d like to push formal proof later in the curriculum. I’d like the students to be a little older and have a little more background so that we can work on proving both algebraic and geometric relationships. I’ve been thinking about what we might have the students do in the earlier years to prepare them for proof writing, and I think I now have the answer: let them write specifications.
We might start with writing instructions for making a peanut butter and jelly sandwich, and once we’ve learned how to do that we could move onto more complicated things with growing levels of mathematical focus. By the time the students are ready to write proofs, they will have plenty of practice with writing clear, technical information. If others have ways they’ve built up to proof writing I’d love to hear about it.