Computer-Graded Math Homework: Why Not?

In recent decades there have been more and more efforts to introduce computers into math education, with varying levels of success. The University of Iowa now offers several different computer-based courses, but they have not, for the most part, replaced the traditional lecture and discussion courses that likely look the same as they did fifty years ago. One area that I think is underdeveloped is computer grading.

The first widespread use of computer grading of which I'm aware was the introduction of the 'e-rater' in grading GRE essays. Interest in computer grading has grown recently with interest in Massive Open Online Courses (MOOCs). Grading an essay is not an easy task for a computer, and current algorithms tend to rely on quantitative notions like word count and average word length to score submissions. Many academics consider this somewhat absurd, and I tend to agree with them. Verbosity and use of advanced vocabulary are only helpful when succinct, clear prose won't do.

But I expect that grading math, say, college algebra and possibly calculus, would be considerably easier for computers. There are currently software packages that allow professors to post quizzes in which students are given a box to enter the answer, or possibly multiple-choice options. But I think that computers could actually check and give feedback on work, this being the type of feedback that students actually find useful. Furthermore tools currently in place in computer-grading software for essays could be helpful, such as a process through which students could continually revise their homework and watch how their score changes. This would encourage revision of work, and would encourage students to keep changing their work until they get it right, something that is important in math, but rarely incentivized in a realistic way. These same methods could be used to grade exams, and would probably be an improvement over the current multiple-choice tests given in large universities.

These ideas aren't new, they've been implemented for essay grading. But it seems to me they lend themselves more to grading math.

Book Review: The Emperor's New Mind, by Roger Penrose

The Emperor's New Mind is ostensibly a response to proponents of strong AI and a vehicle for eminent mathematical physicist Roger Penrose to lay out his perspective on the physics underlying consciousness. While that part of the tome is interesting, what is most valuable is the first three quarters of the book, in which Penrose explains key issues in theoretical computer science, the philosophy of mathematics, logic, and a great deal of physics. Curious minds looking for clear introductions to subjects such as the Church-Turing thesis and computability, Mathematical Platonism, Godel's incompleteness theorems, classical or modern physics, and basic neurobiology would be hard pressed to find better reading than the relevant chapters contained in this book.

Penrose does return, in the last chapter or so, to the question of whether machines can think, or dually, whether our minds operate algorithmically. He answers this question negatively, using fairly plausible arguments incorporating computability theory and Godel's incompleteness theorem. He then proceeds to lay out very speculative ideas concerning the nature of consciousness and how we might eventually come to understand the phenomenon physically. Penrose believes that the a more complete explanation of consciousness will only come with a theory he calls "Correct Quantum Gravity". True to his scientific training, Penrose very clearly states when he is speculating, but the conclusion of the book is nevertheless weakened by such whimsical trains of thought. One almost gets the impression that Penrose is simply smart enough to draw whatever conclusions he desires. But that keen intellect is exactly what saves the last chapter, as even the less organized thoughts make for thought-provoking fare, and illustrate how relatively the relatively disparate fields covered in earlier chapters can be brought to bear in novel ways. While the reader will likely remain skeptical of his conclusions, the journey through Penrose's ideas is the true genius of this book.