I’ve spent the better part of Friday and today grading GE 103 papers. In the process, I’m becoming more and more convinced that the single biggest roadblock to understanding mathematics, at least at the college level, is the ability to read and write. I don’t mean basic literacy; I mean the ability to consume and produce written information and use information effectively to solve problems. If a person can read and write effecitvely, s/he can make up whatever deficiencies may exist in their math background. If not, then not even the best prerequisite preparation is going to translate into success. Here are some cases in point:

- A question on the latest GE 103 lab homework, which covers voting methods, gives a balloting situation and asks, “Use this example to show that the Hare system does not satisfy monotonicity.” Many of the answers consisted of: “The Hare system doesn’t satisfy monotonicity because [
*insert definition of monotonicity here*].” Even if you don’t know the terms, it ought to be clear that just stating a definition proves nothing. You could ask me whether or not the sky is the color of an eggplant, and I could say, “Yes, because an eggplant is purple.” But it doesn’t work as an explanation, and of course is in fact a false statement. But for some reason many students don’t get that.
- The latest GE 103 test covered probability. On the test is a clearly labelled instruction that says, “You must
**show all work and explain all your reasoning** in order to receive credit for an answer. **Answers that have no supporting work or explanations will receive 10% of the possible point value if they are correct and no credit at all if they are incorrect**.” The bold face appears in the original on the test; this was also verbally announced at the beginning of the test. I have also repeatedly told the class that I am not grading their answer; I am grading the thought process which leads to the answer. Still, when asked to determine the probability of an event for which the answer is 1/2, well over the half the class just puts “1/2” with no justification. Is this is a failure of the ability to read (they glossed over the instructions) or the ability to write (they don’t realize that just writing a fraction doesn’t constitute a justification), or a linear combination of those? Some students lost tons and tons of credit on this test due to simply not justifying an answer after they were told to do so, even though justifying an answer is really easy and short and they were warned of the consequences of not doing so. It’s baffling. Plus, there are going to be a lot of unhappy people when they get their tests back and I am going to have to explain all over again why a disembodied answer isn’t enough.
- I’m also seeing students make the same costly mistakes on tests that they made on homework, when I corrected the homework and told them what the mistake was. For example, one student lost points on a finance problem because s/he turned the exponent in the compound interest formula into a multple (A = P(1+i)n instead of A = P(1+i)
^{n}.) This is a nonzero but inexpensive error when caught on the homework level, and I made a clear comment on the homework about what the error was. 2 points lost. But now the same student has done the same thing on the test; 6 points lost. This is a reading problem; the student has the information to make the correction but then doesn’t covert it into actual correction. (Or maybe it’s a motivation problem; the student considers the effort spent on making and understanding the correction to be less valuable than the effort spent on something else.)

I single out GE 103 here, but of course this problem is pandemic. You get the same thing all the time in calculus, linear algebra, modern algebra, etc. (How many times have I asked, “Is the ring of integers mod 6 a field?” and gotten the answer “Yes, because every nonzero element is invertible”, ignoring the actual behavior of the ring?)

I’m strongly considering formatting future test and homework items in a two-tiered approach. Each problem would have two blanks, one labelled “Answer: ” and the other labelled “Justification: “. In a 10-point problem, the answer would be worth 1 point and the justification 9. That might succeed in getting across to students that the justification is what I am really grading here, and you can’t expect an answer to be self-evident.

And I will note that I can make this stuff as plain as day to students, but unless the individual student takes up the responsibility to act on the parameters that are set for them in their education, it won’t matter.