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.

Two Eternal Tensions

It occurred to me while reading Kevin Kelly's What Technology Wants that there are two constant (related) tensions that play a big role in civilization (by this I do not mean that these are the only forces, I simply am focusing on two of them). One is the tension between the desire for the benefits of government and the desire to be as free as possible. The other is the tension between the desire for the benefits of technology and the desire to have as much control over our lives as possible, to feel that it is we that use technology and not the other way around.

The relationships between these notions are complicated. Government is a type of technology, and ideally enhances freedom in that it enables a civilization in which we have a much wider range of choices, giving us a net gain of freedom. One of Kelly's main points is that this the typical effect of technology. So far, so good.

But reality is more complicated, because government and technology in general both tend to overreach. Government, unchecked, will seek to empower itself at the cost of civil liberties. Technology, unchecked, will seek to empower itself at the cost of our well-being and our environment. However, both are worth the cost, so instead of asking how to do away with government or technology, we must ask how to steer it.

To check government (inevitably world government) will always be a struggle. Desired methods will often go beyond what is legal (civil disobedience) but, at least for the more democratic governments, will rarely approach anything resembling revolt. To check technology, we must simultaneously embrace it and be vigilant against its harms. Kelly makes an argument that the Precautionary Principle, whereby a technology or policy must be proven to do no harm before it can be used, is a terrible notion to apply. Global warming denialists like to latch on to criticism of the Precautionary Principle, but let us not be dissuaded from examining it critically by them. One can't help but assent to the observation that the Precautionary Principle is self-contradictory, since in order to apply it, we would first have to prove that it can do no harm. And this is really Kelly's point, the harm it does is precisely in preventing technological progress. He recommends a program of careful adoption and vigilant oversight.

But to me the parallel is more clear in my day-to-day life, at least on the technological side. I am very into technology but I try to be careful that the technology I use helps me rather than harms me. For example, I've found Twitter to be mostly distracting, something that is too tempting to check every ten minutes, and mostly full of content I don't actually get that much out of. I've tried it a couple times, but I doubt I'll go back for quite awhile. I'm on the fence about RSS readers for the same reason. Of course I'm not always successful, I've gotten sucked into computer games or, worse, television shows, many of which provide mindless entertainment.

So I guess my point is that we benefit greatly from government, even as we (well, some of us) often find it necessary to check its overreach, in the same way that we benefit greatly from technology, even as we struggle to live our lives independent of its grasp.



I have felt the urge to publish various thoughts on matters sparked by What Technology Wants and have chosen not to restrain myself to the book review format, where I put them all in one post. This will probably be how I share my thoughts while reading from now on.

Anti-science sentiment on the Left

It's no secret that many conservatives, or at least their congressional representatives, are at odds with facts on which scientists agree, from evolution to global warming to whether raped women can get pregnant. But the left should hardly claim the scientific high ground without addressing certain anti-science movements in liberal ideology.

One common belief is that we liberals are so in touch with our bodies that we know when something hurts/helps us even when medical science disagrees. Examples of this include the gluten-free movement, advocates in the existence of chronic Lyme disease, and the belief in homeopathy. Intelligent individuals allow advocacy of these unscientific diseases and remedies to become such an integral part of their identity that changing their opinions in the face of contradicting evidence (or lack of supporting evidence) would amount to losing part of their sense of self. I do not have proof that backers of these ideas skew to the left politically, and if presented with evidence that this isn't true, I would have to change my mind. But it's been true in my experience.

Another common belief is that people were better off in the past. Strains of this belief run through a lot of environmentalist thinking. While liberals are all for moral progress, they often are hesitant to support technological progress. Kevin Kelly recently wrote an excellent book on technological progress (review will come when I finish it), which addresses concerns that such progress brings more harm than good. I will defer to him in countering these arguments, but I want to point out that this hesitancy is purely conservative, even if held by liberals. For being "liberal" is not the same as being "progressive".

It seems to me that progressives accept change as inevitable and try to push it in as good a direction as possible, while conservatives try to slow it down or even reverse it (note that by "conservatives" I do not necessarily mean political conservatives, though those groups often do overlap). Note that both groups see a potentially dangerous future which they try to avoid. Historically it seems clear that conservatives cannot succeed, but that does not ensure success for progressives.

Right now the generic liberal position seems to be progressive morally but conservative technologically. This would pair nicely with the (politically) conservative position being conservative morally (which it is) and progressive technologically, but this last position certainly isn't uniform. While capitalism is an element of technological progress, so is the science through which we learn about global warming and evolution, so the (politically) conservative position seems split on technological progress.

This does not seem to bode well. We can only hope that we will do our best to steer the technological stampede, instead of planting ourselves in its way, only to be trampled.

Glasshouse, by Charles Stross

(Originally written for Math Graduate Board Newsletter)

It's the 27th century, but Robin has chosen to wake up from his memory excision in an orthohuman body. You could say he's old-fashioned that way. After all, when death just means reversion to your last backup, it isn't hard to tell the assembler to put you in just about any kind of body you can imagine. Sticking with an orthohuman body isn't just a choice, it's a statement.

But when Robin is recruited to participate in a sociological study recreating the pre-Acceleration dark ages (20th/21st centuries), "orthohuman" gets a little too realistic. There are no assemblers to rearrange her atoms in case of injury or illness, the NPCs there to aid in her transportation drive taxis instead of pointing her to the nearest T-gate, and experimenters have gone to the extreme of making the participants fertile (Robin has been female orthohuman before, but she is not happy with the weakness of her new body, and fertility is just a kick in the stomach). And then there's the society in which she's imprisoned; Were morals really so irrational, customs so arbitrary, life so uncomfortable? But she's got more to worry about when she begins to remember what, through memory excision, she tried so hard to forget...

In Glasshouse, Charles Stross foresees a future in which a mind can be backed up, printed into the physical world in any body and in as many copies as desired. The driving question behind Stross' book, then, is the question of identity. If "you" are not your body, what are "you"? This classic philosophical question gains both complications and insights in Stross' future, in which one's consciousness can be not only split or merged, but also tampered with. If someone reaches in and swaps a few memories, or tweaks a few personality dials, what's left?

Stross manages, as well, to pack in allegories of and commentary on no less than religion, gender roles, death, war, government surveillance, and more, all in what is certainly a page-turner. The best science fiction not only provides a specific vision of the future, it offers sharp critique of the present, and is, if not quite self-fulfilling prophecy, at least a cog in the engine of progress. Glasshouse fits the bill as well as anything.