On what might have been

How many times have you wished that one little thing could change in the past? If that hadn't happened, the effects in the present would be countless and powerfully different. If I'd only chosen to turn right instead of turning left, I would have seen my friend and been able to give him a ride. If she hadn't waited 15 minutes before leaving home, she wouldn't have been crossing the street when that drunk driver came along. We could go on endlessly.

Of course, we don't really know what would have happened. We have gotten pretty good at guessing, but we don't ever know for sure. The scientific method usually involves repeated experiments – and certainly involves allowing others to repeat our experiments. The hope is that if we observe enough things carefully enough, we'll understand cause and effect.

The way that a scientist tries to understand cause and effect is by isolating a single difference. An experiment involves doing two things exactly the same way except for that one crucial difference and observing the results. If changing that one thing consistently gives a changed outcome, we assume that there is (and that we have observed) a causal relationship.

We've become so accustomed to experimentation and so casual as we set up our experiments that we've come to make frequent errors in the way that we see the world. Some of these errors lead us to dangerous conclusions.

Our habit of breaking things down into experiments has trained us to think only of one factor in an outcome at a time. Isolating the different factors is wise, but we must eventually remember the ones we've left to the side. This is especially important because we don't always think of all of the factors when we set up our experiments. The fact of the matter is that many light things fall more slowly than rocks do, although their weight is not the only factor in that measured outcome.

Our legal system has a similar approach to things that can have a similar effect on our thinking: since there are many things that contribute to any one event happening, our law concerns itself principally with a proximate cause. Without this legal doctrine, we'd have no way of assigning any sort of blame to anyone or anything and our laws would be completely pointless. But if we're not careful, this too can train us to think that every outcome has a single cause.

We do well to identify the other causes of the events in life, whether we hope to repeat them or avoid them. We can do so by asking questions like, "what else contributed to this?" and, "what else might I have done in this regard?" When we identify something that we might have done that seems likely to lead to a desired outcome, we can consider if we want to try acting in that way in the future.

It's also a good idea to recognize that, because so many factors contribute to every single event, we rarely have a suitable justification to blame anyone else for anything. Even if the outcome could not have been the same had they acted differently (this is rarely the case), it's usually impossible to prove that another persons actions alone would guarantee that outcome. This is usually impossible to prove because it isn't true.

Similarly, it seems that in most cases where a person tries to influence another's decisions, it's because the other person's actions would contribute to some desired outcome. There's nothing wrong with this, but we'd be a lot less tempted to manipulate each other if we spent more time thinking about what we can do ourselves to contribute to those same outcomes.

On a related note, we usually form opinions too soon about whether or not we want something to happen. I've looked back on several then-happy experiences and wished they'd been different and I've been immensely grateful for many of my unpleasant experiences. In fact, we'd probably all be better off if we made more of our decisions based on our principles instead of what we imagine and hope will be the outcome.

So stop fretting about what would have happened as if you know and figure out how to make the best of the situation you're in instead.

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?