Nearly every subject has a shadow, or imitation. It would, I suppose, be quite possible to teach a deaf ... child to play the piano. ... [The child] would have learnt an imitation of music, and would fear the piano exactly as most students fear what is supposed to be mathematics.I think that idea, of a shadow subject, will stick with me, and become more powerful for me over time.
What is true of music is also true of other subjects. One can learn imitation history - kings and dates, but not the slightest idea of the motives behind it all; imitation literature - stacks of notes on Shakespeare's phrases, and a complete destruction of the power to enjoy Shakespeare.
Logarithms
I've told my students logarithms were invented in a time when calculators didn't exist, and scientists were looking at lots of data about the planets, trying to discover patterns. Napier invented a way to do multiplication by adding and division by subtracting, a second application of which allows powers and roots to also become questions of addition and subtraction. I don't think this is enough of an introduction to this strange concept.
How did Napier dream this up? Sawyer gives us a glimmering of the sort of inspiration Napier might have had, with this marvelously concrete model for logarithms:
We are all familiar with machines which [we] use to multiply [our] own strength - pulleys, levers, gears, etc. Suppose you are fire-watching on the roof of a house, and have to lower an injured comrade by means of a rope. It would be natural to pass the rope round some object, such as a post, so that the friction of the rope on the post would assist you in checking the speed of your friend's descent. In breaking-in horses the same idea is used: a rope passes round a post, one end being held by a person, the other fastened to the horse. To get away, the horse would have to pull many times harder than the person.
The effect of such an arrangement depends on the roughness of the rope. Let us suppose that we have a rope and a post which multiply one's strength by ten, when the rope makes one complete turn. What will be the effect if we have a series of such posts? A pull of 1 pound at A is sufficient to hold 10 pounds at B, and this will hold 100 pounds at C, or 1000 pounds at D.
Thus, 108 will represent the effect of 8 posts. ... The number of turns required to get any number is called the logarithm of that number. ... So far we have spoken of whole turns. But the same idea would apply to incomplete turns. ... Accordingly, 101/2 will mean the magnifying effect of half a turn. ... The logarithm of 2 will be that fraction of a turn which is necessary to magnify your pull 2 times. (page 70)
I had to put the book down here, to ask myself why half a turn wouldn't magnify the pull 5 times - half of ten. As I thought about that, I wanted to know if there would be an easy way, either a thought experiment or a very simple physical experiment (i.e., no special equipment), to prove that this relationship must be multiplicative. That is, how do we know the friction of the rope doesn't just add to our pulling force, so that a certain amount is added at each turn? (Can anyone help me with this?)
If we've decided that the relationship must be multiplicative, then we know that two half turns must multiply to have the effect of one whole turn, and that would mean we need the number that multiplied by itself gives ten. To get to this thought, I had to imagine two posts near one another, with the rope halfway around one, and then halfway around the next.
Why haven't I seen this before?!
I haven't read any more of the book yet, because I keep needing to think more about this cool idea. I look forward to more pedagogical delights as I keep reading this book, and maybe others he wrote. (One list is at the bottom of this page.)
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*W. W. Sawyer wrote this book in 1943, long before feminists began to analyze the effect of using the male for the generic. Although Sawyer uses 'man' and 'he' in a generic sense in other sections (which I've taken the liberty of changing in the second quote I've used), perhaps he was trying to avoid that in this story by calling the deaf child of his music example 'it'. I had real trouble with that, and didn't know how to fix it without messing with his meaning, so I left the meat of the example out. You can go here to see it.
Posts
Hmm, neat!
ReplyDeleteBut also: Why should two half turns (on different ropes) have the same effect as a whole turn?
I'm picturing one rope, similar to the situation pictured. Two half turns would give the same amount of surface for friction as one whole turn, right?
ReplyDeleteWell, I just don't understand these things. It makes sense for surface area to be involved, but I also figure force is involved, and maybe angles make their way into that... makes sense that it would all work out the way you said, but I don't understand these things.
ReplyDeleteI am actually highly skeptical that the physical relationship being described here really is multiplicative. It works as a thought experiment but I feel like it's fudging the physical reality.
ReplyDeleteYou could get an honestly multiplicative model with pulleys, but it wouldn't have the virtue of supporting the fractional exponents.
I haven't experimented with it myself but it actually seems to me that your historical-motivation intro to logs has the potential to be enormously powerful. In order to have this idea deliver fully, I think you'd need to figure out a sequence of questions you could ask that would lead your students to generate the idea that multiplication can be reduced to addition, or at least to become incipiently aware of it without needing to be told directly. For example, giving them an extensive set of multiplication problems involving only powers of 10? And helping them realize that they can solve them quickly just by amalgamating the powers? Like, "what's 100 x 1,000? what's 1,000 x 100,000?" etc. And then highlighting for them the fact that in order to solve these problems really they are solving addition problems. And then maybe another sequence of problems involving powers of 2? So that 32 x 8 becomes a simple matter of 5 + 3? Or something. I'm just brainstorming here. But the idea would be to get them to recognize, in the context of whole-number multiplication they can already do other ways, that if you can see numbers as powers of a certain base, then you can multiply them by adding.
I agree that this may be difficult to model theoretically. The only thing left is experiment. I will try this out next week and see what I get.
ReplyDeleteRhett has done the experiment, and posted at his blog. (Thanks, Rhett!!)
ReplyDeleteIt looks like it's probably multiplicative. Ben, I have no good instincts for physics. I want this to be either additive or multiplicative, and the data come closer to multiplicative. Does it feel to you like it might be more complex? Can you say why?
Rhett, you said, "The normal model for friction says that the frictional force is proportional only to the force the two surfaces are pushing against each other. Not sure if that works here." Can you say more about how friction works?
I was convinced it was multiplicative by imagining what happens with a rope going 0 turns around. Obviously that multiplies the force by 1, not 0.
ReplyDeleteBut then I thought, wait a minute, that's assuming it's multiplicative already!
Maybe it's clear enough that the change in tension in the rope as it wraps around a bit of pole is proportional to the tension?
There's a good treatment of this with free body diagrams and all that physics stuff over at http://www.leancrew.com/all-this/2010/04/aye-aye-capstan/
ReplyDeleteJoshua writes: Maybe it's clear enough that the change in tension in the rope as it wraps around a bit of pole is proportional to the tension?
Yeah, this is the key to the argument that the relationship must be multiplicative. The friction from each bit of pole is what allows the tension to change along the length of the rope (otherwise the rope would relax so as to equalize the tension). But the friction is proportional to the normal force between the rope and the pole, which is proportional to the tension. And a function which is proportional to its own derivative is exponential.
Thanks, Orawnzva. I'm going to print out the post at that site, and study it.
ReplyDeleteIncidentally, my LJ handle is related to my real name by a wekll-known substitution cipher.
ReplyDeleteI'm a CS graduate student interested in teaching, having been so far blessed and challenged by my work as a teaching assistant and frustrated and underwhelmed by my work as a research assistant. I added your blog (and other math teaching blogs) to my reading list recently, so you'll probably be seeing more comments from me on scattered posts, current and past, until I catch up.
Hi, I'm the author of the post Orawnzva linked to. As you figured, the behavior is multiplicative, and Orawnzva's explanation is quite good.
ReplyDeleteIt's actually not that difficult to model theoretically; you just have to have some experience with the notion of slicing objects up into differential chunks and applying Newton's laws—and in this case, Coulomb's law—to a typical chunk. Once you've done that, it's just a differential equations problem.
Although my original article was about a single pole, it's easy to extend the analysis to several poles, which I just did.
Hope to have one copy of that Math Delight book to be able to see how I could use it to explore more my knowledge in math. Thanks for the link.
ReplyDeleteThe last comment just brought me back here. I've just now printed out Dr. Drang's post to study it. I would really like to understand this, and I don't yet.
ReplyDelete