The new way of multiplying that's exactly the same as the old way

There's an article that's been kicking around the intertubes for the last couple of days. You'll find it here (as far as I can tell, this is the original version).

The gist is that there's an easy way to teach multiplication – that the Japanese have figured something out that we haven't in America. I presume that people are sharing because they wish they'd been taught that way in school.

To multiply 12 x 23, we draw these lines:

Now we count the intersections of the lines in each corner:

Here, we can add the two numbers in the middle column and put the numbers together to get 276, which is the correct result for 12 x 23.

But if we restructure the lines slightly:

And remove the lines:

We have exactly the same multiplication with the exact same calculations as we would have done had we simply done as taught in school.

Now to my objections.

The "proof" given that this gives the same result is only a demonstration that the result worked for a particular example. To prove that a method is correct, it's necessary to show that it will always work. If our students could actually demonstrate why these two methods always produce the same results, we'd be going somewhere.

I fail to see how this is any simpler. It actually takes longer to do. I suppose some students might object to the typical American method because it seems so arbitrary and they don't see why it would work, but the same applies just as easily to this visual method.

Try using bigger digits (not my own thought; I found this one online).

Most importantly, though, all of this misses the bigger point. It's not just about being able to perform computation. We have calculators for that. It's about understanding what a computation is, what it means, and how it works. This could be a useful teaching tool but could also be a crutch that prevents another generation of students from understanding one of the most fundamental operations in mathematics.

So why is this any different or better? Maybe I've missed something?

All is not as it seems

Philip Guo recently published a memoir of his experience as a PhD student in computer science. His is a story of self-discovery and triumph. Getting a PhD is hard, especially for someone who hasn't already discovered a burning passion for a subfield, which Guo had not done when he started his PhD.

Having completed two years of PhD school in computer science myself and not having identified a subfield for which I have a consuming passion, I strongly sympathized with Guo as I read his memoir. Accordingly, I read closely and imagined myself in his situation. I came away hopeful that my experience would be similarly serendipitous. I also came away more acutely aware of the problems with the academic peer review system. Of particular note is the fact that Guo submitted papers that were rejected because his writing didn't meet the arbitrary expectations of current experts in the field, not because his research was unoriginal or uninformative. I believe in good writing and recognize that good writing necessarily reflects familiarity with an audience. I'm not taking issue with the need for good writing, but I believe that some of the requirements for publication are detrimental to the academic community as a whole.

Allow me to explain.

Early in his graduate career, Guo submitted a paper to a conference and was rejected. His research was, as the memoir tells, of a similar caliber to other research that was published. Guo's failure was in convincing the entrenched researchers in that community that his work was original and useful. That is, it wasn't the quality of work that mattered for publication. What mattered was the apparent quality of his work. Guo's paper went unpublished because the system evaluates whether or not research seems to be good, not whether or not the research is good.

To be fair, this is a scientific community and it strives to be objective. There isn't a way to determine if research is good objectively, so the community makes do with the best solution that it has found to date.

The problem with peer review is compounded by the fact that the quality of a scientist is estimated by his or her publication record (see, for example, h-index). This leads to problems such as this, where a scientist manages to falsify peer review in order to seem like an effective scientist.

Unfortunately, problems of this kind are found everywhere. We get a job not by being the best candidate for a job but by seeming to be the best candidate. Dating works along the same lines. Sports revolve around what the officials perceive, so they have precisely the same issue. Political discourse clearly emphasizes seeming over being.

Being something allows us to act. Other people's behavior towards us, however, depends on what we seem to be. It isn't possible (or desirable) to avoid seeming to be something. Instead, we should all try to seem to be what we are and to avoid seeming to be what we are not. If you find yourself trying to seem in a certain way, check to make sure that it's actually true. Are you really confident or do you seem that way? Are you actually good at what you do or do you just have a killer resume? Keep in mind that self-deception is not just possible but commonplace.

This is not to say that we should not aspire for greatness beyond what we have yet attained. And reaching for greatness requires emulation. We should try to improve by emulating the best that we see in other people. The purpose of this emulation must be for us to acquire positive characteristics, not to seem better than we are. And when others try to improve, we should encourage them instead of calling them hypocrites.

Since we interact with others, the question of our own characteristics is insufficient; we must also question our perceptions of others. Do you and I assume that the things we perceive are reality? When someone seems to be self-absorbed or quiet or happy, do we assume that this is generally the case? Do we disregard others' ideas when they aren't presented as we are used to hearing ideas or when they come from unusual or unproven sources? Do we assume that people remain the same or do we believe that people can change for the better?

The most important thing that we can do is personal: each of us can work at becoming better. We can concern ourselves less with how we seem and more with how we are. And we can give others the benefit of the doubt. We must also encourage societal change to value actuality over mere appearance. We must consciously choose to value genuineness over the appearance of virtues. We can expect people to be good but not extraordinary, freeing them from the pressure to seem to be good enough for our unrealistic expectations. Relatedly, we must put less pressure on people to be something that they are not; introverts and extroverts and everyone in between are good people. The same goes for scientists and liberal arts majors. In other words, we should encourage people to grow but not to be untrue to themselves. We also need to encourage virtues, such as honesty, industry, and kindness – but never assume that we see them clearly in others. Hardest of all, we must strive as a society and especially as individuals to acquire these virtues.

You may say I'm a dreamer

I think the valedictorian at my high school had a GPA of over 4.5. I remember that I had friends who would avoid taking a class because it wasn't honors or AP and would bring down their GPA, even if they got an A. My approach was to take the hardest (and therefore most beneficial) classes that taught things I wanted to learn. Needless to say, I did not have a 4.5.

But when I went to college, my GPA was too low to even apply for the scholarship I wanted. I worked hard to get good grades. Towards the end of my undergraduate education, I started to realize that I'd made a huge mistake: I was trying to get good grades instead of trying to learn. I began focusing on internalizing the material instead of just succeeding. I learned more, enjoyed it more, and my grades were at least as good as before.

I'm glad I learned that lesson as early as I did but wish I'd learned it earlier. Since then, I've learned a lot about how I learn and about how others learn. I've also learned about how our educational systems could improve. This post describes the problems as I see them and my suggestions for how to improve them.

Read the whole story...

When I grow up

I started writing a blog post about different fields of study and jobs that I've tried or thought about and how none of them has worked out. I realized that much of my introductory material is essentially the same as this post, which I wrote 16 months ago. There are, of course, some differences in what I would emphasize or how I'd phrase things, but it's mostly the same.

The bottom line is that I'm still not really into my research. I suppose it might be a little different if I were in another job. If I were on an assembly line, I could do good work even though I didn't think much of it. But as a researcher, I can't do high-quality work without being fully invested in it. So far, I haven't been fully invested and it's crippled my ability to perform.

Read the whole story...

This has gone far enough

Author’s note: I am biased. For religious and philosophical reasons, I reject the idea that any person is incapable of anything. Some people have predispositions towards or against certain ways of thinking or performing and some people have disabilities. But I still believe, at least in general, that anyone who wants to learn something can learn it.

There’s a paper that has been quoted and linked quite a bit for the last little while. It describes an attempt to understand the bimodal distribution computer science teachers have found, no matter how they teach, for many years. All of the evidence seems to indicate that some students can program and some can’t.

The authors designed a test, showing a very small, very simple excerpt from a Java program, like the one that follows:

int a = 10;
int b = 20;
a = b;

The test essentially consisted of recording students’ responses to questions about the effects of the program. Of course, the authors were not so simple as to try to test students on their programming abilities to determine if they could become capable programmers. Instead, their conclusions stated that students who demonstrate consistency from question to question were more likely to succeed in an introductory programming class after the fact. By consistency, the authors mean forming a mental model for what each statement does and using that same model for each statement in each program, in order.

The paper shows tabular results from several different tests. The tables validate the authors' conclusions - and several other conclusions. Importantly, they show that students who have prior programming experience are likely to do very well on the test - much more likely, in fact, than those who have no such experience.

Since the test includes no instruction about programming, students are left to wonder how the statements interact with each other. Most students who have no experience in programming are likely to use algebraic rules to interpret the program; after all, algebra is the only system most of them would know that uses notation that looks like this. An algebraic evaluation, however, leads to a contradiction for our sample program (as 10 is not equal to 20). There is also no reason to assume that students who have no prior experience would assume (correctly, in this case) that the statements take place in sequence and not all at once. It is likely, but not certain, that students who do assume sequence will assume (again, correctly) that the sequence occurs in top-down order.

In short, the test seems to favor those who already know the answers. Those who don't are unlikely to succeed in programming classes.

It seems likely that a host of factors play a part in these findings. Students who do well in math are generally less intimidated by mathematical notation and so can think more clearly on the test and in the class. They also are more likely to have prior experience. Those who have prior experience are more likely to have habitual thought patterns that match other programmers; as long as programmers typically come from some demographic, their work will reflect their polity. This means that language design will favor those who use it because it has to make sense to them. This also means that professors in computer science, who are themselves successful programmers (we assume), are likely to think along the same lines. They are likely, then, to teach in a way that reflects their habitual thought patterns. This favors like-minded students and makes life more difficult for those who approach things differently.

To be sure, programming requires people to learn to think in new ways. It's hardly as if people are born with the right thought patterns in mind. But some students have a head start because of predisposition and prior experience. Those who don't, it seems, tend to do poorly in their classes.

There's quite a bit of irony here. Programmers are, by necessity, logicians (some better than others). We ought to be able to spot logical fallacies easily. But no one seems to have noticed that this study essentially begs the question: if students know programming, they'll do well in the class that ostensibly teaches them programming. If they don't know it, they won't do well.

It seems obvious to me that the problem is in the way we teach; after all, the students who don't know programming but sign up for an introductory class are really the ones the class should target. For example, do we make enough time for the students who don't know about programming to learn? Do we assume that all students will make the same assumptions we make (after decades of training) about sequence? Do we make other similar assumptions? Do these assumptions detract from our teaching? Does the large number of students in an introductory lecture make it impossible for the students who don't know how to program but who want to learn to ask questions and get answers?

In short, does our educational system fail to help those who need its education most?

I think it does.

Chew on this

If we chew food thoroughly, we digest it better. We can swallow it just fine in larger chunks, but it can occasionally cause indigestion and frequently causes our body to avail itself of only some of the proffered nutritional benefit.

Something similar happens when we learn - intellectually or spiritually. Some of us digest our food but do so quickly. We can't be bothered to think about it more than is necessary. We learn what we must in a class to pass the test. Spiritually, we may learn the lessons that are taught to us explicitly but fail to learn the lessons that are available to us but that aren't called to our attention.

Read the whole story...

Different ways of looking at programming

People are frequently mystified by the fact that I do things with computers. From their reactions, you'd think that because I program, I'm some sort of intellectual demigod. When I talk with them a little more, it becomes clear to me that many people don't know what programming is like, so they imagine something much more difficult and specialized than it is (it turns out that your computer is not made out of magic, despite this amusing article).

As is the case with any profession, it takes a lot of work to do it right. I'm not minimizing the training that other programmers and I have received. But we're not superhuman - even if most of us claim to be.

Read the whole story...

The attention span I never had

I was bored a lot as a kid. And I do mean a lot. It seemed like adults always just wanted me to be quiet and not bother them. It chafed, but I wasn't in much of a position to argue. As the fictional Valentine put it in Ender's Game, "They have a word for people our age. They call us children and they treat us like mice" (p. 127).

My mind works really fast. It always has. I'm sure this contributed to the interminable nature of each car ride, church meeting, class, or social gathering. I'm reminded of something Data tells Captain Picard: "0.68 seconds sir. For an android, that is nearly an eternity." (Star Trek: First Contact -  http://www.imdb.com/title/tt0117731/quotes#qt0455868). Maybe all the other kids were just as bored as I was. I'm not sure. But having a quick mind definitely didn't help.

It may be surprising to some of my readers that school bored me. Read the whole story...