According to Aristotle (*Posterior Analytics*, 1.2), knowing a thing in the truest sense involves knowing the causes that make the thing what it is. I think this is quite evident in high school mathematics. There is a great difference between students who memorize a formula and know roughly when to apply it, and students who know how the formula is derived and so know the exact circumstances under which it can be used. Indeed, my first-year university calculus professor said that if a student wanted to know if they had a good grasp on the material, they should try his multiple-choice questions rather than his short answer questions, because the former tested most strictly whether students knew the assumptions behind each theorem and when exactly they could be used.

Although high school and first-year university math may seem trivial, this actually has had life-or-death consequences. For example, being able to make a right turn at a red light has been widely accepted in North America, at least in part due to studies which concluded that there was no “statistically significant difference” in accidents at intersections before and after allowing right-turn-on-red. Therefore, lawmakers concluded that right-turn-on-red was not a significant source of danger. However, this is a misunderstanding of how statistical tests work. If a test says there is no statistically significant difference between two numbers (e.g., crashes before and crashes after allowing right-turn-on-red), that does not necessarily tell you there is no difference between the two in reality, nor even that there is no important difference (it is a bit silly to expect a statistical test to tell you how many accidents can be tolerated). Such a result sometimes occurs when there is in fact a difference in reality, but the sample size of the test was not large enough to detect it. And this is what happened: larger studies did confirm a difference. But the lawmakers understood none of this, because they did not have knowledge of statistical testing in the truest sense. This is not really the lawmakers’ fault, but surely there should have been someone to help them properly understand statistical results. Furthermore, equally grave problems have arisen when DNA tests are similarly taken to be magic without any understanding of the processes involved in producing the result.

So maybe it is not so trivial after all to emphasize to students the importance of knowing the assumptions and limitations of theories, tests, formulas, etc., which I believe comes most easily through knowing a thing by the causes which make it what it is. Such knowledge will certainly help give students the “deep learning” that educationalists often talk about, and perhaps will help them avoid fatal errors later in life.