Can We Make Proofs Less Tedious?

My goal for the school year was to write one blog post each week. I made it two weeks before failing to reach this goal. I originally started this post in late September, and then school happened. I’ll try to be better, starting with the completion of this post.

One of my colleagues came into the office a few weeks ago and told me a story about his statistics class. They were working on samples and populations, and his class was struggling to understand the difference between the population and the population size. (For the record: the population is a group of interest, such as students at school, and the population size is the number of objects in the group, which in this example would be 502.) Frustrated, he related this struggle to his wife, who immediately identified the conflict. She noted that if a student is asked in history class what the population of France is, the student is going to respond, “66 million,” not “all the French people.”

It occurred to me that this is similar to something I happened to be thinking about this morning. I was thinking about geometry and proofs, and thinking about some of the hoops we ask students to jump through in completing two-column proofs (for the record: I do not like two-column proofs). A common occurrence in proofs is to see two steps like this:

  1. Angle A is congruent to Angle B | Given
  2. The measure of Angle A is equal to the measure of Angle B | Definition of congruent angles

To a mathematician, the difference between angle congruence and angle measure equivalance is an important and substantive distinction. To a high school student just learning geometry and proofs, this seems like a distinction without a difference – a lot like population versus population size. To them, if the angles are congruent then of course they have the same measure, and vice versa, so why does it matter which term they use?

Students are routinely put off by proofs in geoemtry, and one reason for this is the “necessity” of these steps and others like them. I’m not sure there’s any value in emphasizing such a distinction , or requiring it in proofs. What would be the harm in cutting out some of the rigorous detail in these proofs and focusing instead on the big ideas? More importantly, what would be the benefit? In calculus, we prove things without resorting to epsilons and deltas, so why can’t we mke a similar decision about geometric proofs? At some point, of course, students do need to learn that more detail can be given, that more subtle distinctions exist (and are important). But to a student new to the idea of formal proofs, this level of detail only serves to get in the way.

As we work toward our integrated math curriculum, we’ve been working with two important questions: what should we prove, and when should we prove it? To these, I’ll now add a third: how much “proof” is sufficient? Getting this right seems to me to be one of the keys to enticing students to dive deeper into math instead of scaring them away from it.