Basic Logic Errors

I constantly wish that students were taught rudimentary logic at an early age (links: one, two, three). Just musing about that today, here are three common stumbling blocks I see in different classes due to not being able to read logical statements properly:
  1. "If" Statement. In a basic algebra class, we have the rule "If the base is negative, then even powers are positive, but odd powers are negative". Immediately after that, I'll always have some students incorrectly evaluate something like: −5² = 25 (or worse, 2³ = −8) . Note that the base of the exponent is not negative, but some students overlook the check required for the "if" qualifier.

  2. "Or" Statement. In an elementary statistics class, we have the rule "To estimate a population mean, we must have either a normal population or a large sample size." Then when I ask the class "Do we need a normal population?", the entire class will always incorrectly respond with "Yes!" the first time. Of course that's not true; they're overlooking that only one case of the "or" needs to be satisfied -- most commonly by a large sample size. It takes several sessions of quizzing them on that before they are sensitive to the question being asked.

  3. "And" Statement. In practically any class, we might have the policy, "To pass this class you need at least a 60% weighted average, and a 60% score on the final exam." This constantly causes confusion and aggravation. Testy "So, the final exam doesn't count?", or "So, only the final exam counts?" are questions that I routinely have to address. Obviously, students are unclear on the fact that each of two requirements must be satisfied for an "and" statement like that.


  1. I have trouble with students being unable to read logical statements in LOGIC classes. Sigh. But a few comments on these examples:

    #1) To my chagrin, I can't figure out what's wrong, here, and I know I'm reading the logic right. Apparently, my definition of "base" is incorrect. Could it be that some students are having the same problem?

    #2) That one, I hope, arises from the equivocal nature of "need" in the vernacular, where it is often conflated with "desire" or "want." Of course, I have noticed that students struggle a lot more with the "necessary/sufficient" distinction than I would have thought.

    #3) That one is just sad.

    1. Actually, in revising my basic arithmetic lectures, I just wrote a quick primer on these issues (in a 1-hour block that usually floats around the schedule as an "optional" lesson). And in my algebra classes I introduced a diagnostic where the first question is: "If Alice is happy, then Bob is sad. Which of the following cannot happen? (4 permutations)", which turned out to be one of the hardest things on that diagnostic. If that turns out to be correlated with final results, then I may start injected that same 1-hour logic primer into all my remedial classes.

      Anyway, in item #1 above, I'm assuming you can see 2^3 = -8 as being nutso (and we do get it). The thing with -5^2 = 25 -- which I claim is the single most common, simple error in algebra -- is that for a negative base you need to write it in parentheses (a statement and example of which is always physically on the board as I'm quizzing students about #1).

      A juxtaposed negative sign is the same as multiplying by -1, so in truth: -5^2 = (-1)5^2 = (-1)25 = -25 (by order of operations: exponent happens first, then the negative multiplication). If someone wants to square -5, then it must be written: (-5)^2 = (-5)(-5) = 25. Even students with calculators are likely to overlook this, type it into the calculator without parentheses, and so when asked "what is -5 squared?" come back with the incorrect (but correct for what they typed) -25.

  2. Oh, wow! Yeah, I got that 2^3=-8 was crazy, but I legit could not see the problem with -5^2=25. I had no idea that for a negative base, you needed to enclose the whole thing in parentheses, else it would be assumed that you'd multiply the final exponentiation of the positive base by -1. I assume I knew that at some point, since I started as a math major and made it through a year of that before switching... but then again, my grades weren't great! (grin) Thanks!

    1. My pleasure! I get to clarify that a few hundred times a year, so I've got that pretty well practiced. :-)