I feel obliged to run my kids through exercises like these two, but I don’t like them. Speculate:  Why do they rub me the wrong way?  Am I right not to like these?

1.
$log_b2 = x$,
$log_b9 = y$.
Express $log_b24$ in terms of x and y.

2. Write as a single log:   1 + log v – 3log w

What do you think? I usually like what I teach. But I really didn’t like teaching these sorts of questions.

(This is the third of my mini-posts on teaching logarithms in Algebra II. Here are the first and the second)

1. March 5, 2010 am31 8:19 am 8:19 am

I feel your pain. I don’t like em either. Maybe it’s not the same reason as yours, but I just think they’re totally pointless. Easy enough I guess if you memorize log laws, but hard if you’re trying to come to it from a place of understanding, and there’s a legit “who cares” factor here.

2. March 5, 2010 am31 11:26 am 11:26 am

The first one is very textbook-y; I don’t think it would be seen outside of its natural enviornment of the math class.

The second one I’ve had to actually do for a real-life job. (Calculating the running time of a computer program.) So it bothers me less.

3. March 5, 2010 am31 11:32 am 11:32 am

Hmm… I couldn’t do them until I pulled out a pen and paper. Then , for me, they were fun little puzzles. If they’re that hard for the teacher, then they’re probably too hard for a student to do with understanding. I assume the goal was to give students practice with the properties of logs, but these exercises won’t work because they pull the student toward mindless use, which will put those properties in short-term memory but won’t build the connections necessary for long-term memory. (‘Necessary’ may be too strong a word. I don’t know enough brain science to be sure here…)

The textbook author or curriculum developer was either lazy here, or not clear enough about the pedagogical issues. One thing that would get in the way is the trouble most people (me included) have with realizing how hard something is, once they’ve learned it.

This makes me think about the issue Jo Boaler explains in What’s Math Got to Do With It? (she may be quoting someone else, I can’t look it up right now) of compression, where complex ideas get simpler as they become clearer to you, and get compacted. Ball also mentions this in an article titled “Interweaving content and pedagogy in teaching and learning to teach: Knowing and using mathematics”. (I was able to download a pdf of it.) Here’s Ball:

“…Mathematicsis a discipline in which compression is central. Indeed, its polished, compressed form can obscure one’s ability to discern how learners are thinking at the roots of that knowledge. … Because teachers must be able to work with content for students in its growing, not finished, state, they must be able to do something perverse: work backward from mature and compressed understanding of the content to unpack its constituent elements.”

So, if our goal is to get students practicing with these properties, what would be good exercises (or projects)? I have a project I call the Murder Mystery, but it really only practices the property that allows you to pull out the exponent.

March 29, 2010 pm31 10:07 pm 10:07 pm

Sue, can you share your murder mystery?

• March 29, 2010 pm31 10:22 pm 10:22 pm

Sure. I’ll blog about it, hopefully by Friday. If you’d like something earlier, contact me by email, suevanhattum on hotmail. (I may be offline for a few days. My son’s school has spring break this week, so he and I are going to go away overnight tomorrow.)

• April 3, 2010 pm30 12:47 pm 12:47 pm

Done. Here it is.

• April 5, 2010 pm30 9:37 pm 9:37 pm

Cool – thanks for sharing!

4. March 5, 2010 am31 11:36 am 11:36 am

Jason, can you tell us more? That running time of a computer program sounds good to me.

• March 5, 2010 pm31 12:16 pm 12:16 pm

It’s hard to explain everything in a single blog comment, but:

When implementing and comparing algorithms it can be essential to know how many steps a particular algorithm will take; it can mean the difference between two hours or two days or two years to run a particular program. So a lot of what computer scientists do is characterize the running times of programs and try to optimize them.

Example: suppose you have a program that plays a game of “guess my number” (computer guesses a number, human tells computer if it’s high or low, keep going until the right answer). Let’s say you’re playing with numbers from 1-1000. One way to implement the algorithm would be run through every possible number starting at 1, giving an maximum running time of 1000 steps, “linear” running time. However, it’s possible to improve by starting at 500. Then if the response is “too low” the program next guesses 750, or if the respnose is “too high” the next guess is 250. In general the program picks the halfway point between the number just guessed and the remaining possible numbers. This takes a maximum of log_2(1000) steps (each time the total set of numbers we are searching for gets divided by 2), “logarithmic time”.

Of course, the “logarithmic time” (maximum of 10 steps) is better than the “linear time” (maximum of 1000 steps).

Now, in a real algorithm, this subdividing of search spaces tends to happen quite a lot, so you might have multiple logarithms being added together; going back to the example, let’s say “guess my number” is played three times, once between 1-1000, once between 1-950, and once between 1-4125. Then the maximum running time is
log_2(1000) + log_2(950) + log_2(4125)

of course algorithms are implemented generally, so we won’t know what exact values people are using, so really it should be
log_2(x) + log_2(y) + log_2(z)

or log_2(xyz)

The “guess my number” example is pretty popular as a starting one in CS classes, because the same principle can be used to make a sorting algorithm. Sorting a list of names into alphabetical order can be considered an extended version of the game.

5. March 5, 2010 pm31 1:10 pm 1:10 pm

Thanks!

6. March 5, 2010 pm31 4:16 pm 4:16 pm

The first one is a little bit artificial. Second one I have no problem with; you need to do this sort of stuff to, for example, simplify the solutions to some separable differential equations, most obviously exponential growth/decay.

7. March 5, 2010 pm31 5:35 pm 5:35 pm

What is the bigger truth these exercises lead to? I looked at them straight and then with my head slightly tilted, and I did not come up with any. The point of working with examples should be to understand some more systematic math pattern…

• March 5, 2010 pm31 10:47 pm 10:47 pm

The point I am trying to make: in order to be able to love exercises mathematically, you need to situate them where students can reasonably notice patterns, make conjectures, and otherwise “gain a level of understanding.” I don’t immediately see what such bigger understanding will be with these exercises, but I trust it’s there somewhere, and can be found.

8. March 5, 2010 pm31 9:57 pm 9:57 pm

I am concerned that the kids will face these on our state test. It bothers me, but I want to make sure they’ve run through them at least once.

Interesting that the views diverged slightly on #2.

By the way, I don’t always think artificial is bad. Sometimes artificial feels like a little game. This week I meant to, but time prevented it, I meant to ask them to express $0.3\overline{a}$ as a fraction in lowest terms.

It’s just as artificial, but it feels fun.

9. March 5, 2010 pm31 9:58 pm 9:58 pm

(But I agree with the consensus that has been established in this space – they are icky questions.)

• March 6, 2010 pm31 3:31 pm 3:31 pm

Hmm. I really don’t. I guess it’s all a question of taste, though :).

10. March 6, 2010 pm31 12:09 pm 12:09 pm

Warning: Shameless self-promotion. Nevertheless, try my video
The laws of logarithms. In additional to the surprise at the beginning, there is a serious discussion of function inversion from the point of view of its linguistic complexity.

• March 6, 2010 pm31 7:53 pm 7:53 pm

I think the word “self-promotion” plus the link were enough to get the spam filter to grab your comments. Sorry. All better now.

11. March 6, 2010 pm31 3:59 pm 3:59 pm

You might find this video on logarithms to be amusing. It is also quite serious. I often tell students that the definition of a logarithm is “tortuous” since any inverse idea is circuitousy defined.

• March 23, 2010 am31 2:25 am 2:25 am

I don’t think that’s necessarily true. I think it can be explained using some simple abstraction:
First, consider multiplication as a binary operator:
a*b = b*a = c
Then, division is the inverse of multiplication:
c / a = b; c / b = a

Exponentiation is another binary operator, but which is not commutative:
a^b != b^a
if a^b = c, then the bth root of c or c^(1/b) = a
The other inverse involves logarithms:
log (base a) of c = b.

In one case, you solve for the base, in another, you solve for the exponent. (This also explains why we call the base of a logarithm, the base.)

• March 23, 2010 am31 10:35 am 10:35 am

You have explained nicely. But that doesn’t mean students can really understand it – a nice explanation is often not enough. What I know is that when I started teaching logs, I didn’t really understand them. (Teaching is what has really taught me math.) I think proving the properties of logs is one of the least straightforward things I do with my below calc level classes. I do agree with both tortuous and circuitous.

Perhaps playing with Joshua’s rulers would help me teach it more effectively. (I graduated high school in 74, just as calculators were coming out. For a long time, I thought it was a problem that I didn’t know how to use a slide rule. I finally got over that. This discussion is making me think it might be good to learn.)

• March 27, 2010 pm31 3:24 pm 3:24 pm

That’s fair – I explained it this way to a fellow graduate student who never got logarithms when they were first taught to her, and she immediately understood the connection. For first-time learners of logs, additional context would be necessary. I can see how learning the properties of logs can seem arbitrary, especially if they are presented without showing the corresponding property of exponentiation (for which we can hope the student remembers!).

March 7, 2010 am31 1:36 am 1:36 am

From my perspective, the first one is a puzzle and good for stretching the brain; the second is more (directly? generally?) useful. I am teaching first-year calculus, just finished logarithmic differentiation. It becomes useful to learn how to split up and combine logs. (Yes, this problem asks it backward; that’s fine by me. I don’t think it’s a big step to go from log(a) + log(b) = log(ab) to log(ab) = log(a) + log(b).)

Seems to me that it’s good to learn to manipulate logs using their properties because it’s these properties that make them so useful – yes exponential growth and decay, but they’re also transform-like. Have to differentiate a complicated function? Taking the logarithm lets us break the function up into smaller functions that are easier to differentiate. It reminds me of using Laplace transforms to solve differential equations – map your problem into easy-land, solve it, and map it back.

I’ve never taught precalculus, so I can’t comment on the relative ease of the two problems in comparison to each other. The second seems simpler to me, but to fresh eyes it may well be easier to slap some cold hard integers into the problem.

March 7, 2010 am31 5:30 am 5:30 am

The first one is a bit puzzle-y but it also seems a bit like what you’d need to do if you were building your own log table way back when.

I’d rather have them use the same basic skills to build a slide rule. I tell my students to assume 2^10 = 10^3 and then see how accurately they can place all the rest of the marks. The fact that you can copy the marks onto another sheet, and then slide to multiply, gives a nice check (and helps them catch all the times they assumed 5 is halfway between 1 and 10 or whatnot). It’s also some fun number sense: for instance when you’re trying to place 3, it helps to notice that 81 is very close to 80. (In fact, with 2^10 = 10^3 you get log 2 = .3 instead of .301, close enough, and then you get log 80 = 1.9 and thus since 81 > 80, log 3 is a little more than .475, and it’s actually .477. Probably their slide rule will be accurate to within the thickness of their pencil lead.)

My point being that the skills of #1 aren’t totally useless, and you can prove it by asking the kids to make their own slide rule.

March 7, 2010 am31 5:34 am 5:34 am

As for the second one, I’d put it in context by looking at things like y = a * b^x and y = a * x^p and making them linear by taking the log of both sides. At least you can get something fairly similar to that kind of exercise.

My day with that kind of thing usually involves handing out log-log graph paper and plotting the orbital radius and period of the planets. The line really leaps right out. Then we use the logs to make the “same” line on their calculators by plotting y = log T vs x = log R.

One of the difficulties I always have in this activity is deciding what units to use for T and R. There’s a nice benefit to using years and AU, and I think that might be the right choice for a first example, but it also might mislead by passing through (1,1) and thus in log-land passing through (0,0).

15. March 7, 2010 am31 10:45 am 10:45 am

Josh, These are fabulous! What sorts of groups have you used them with?

March 7, 2010 am31 11:05 am 11:05 am

Thanks! The majority of my teaching experience for these kind of standard topics (as opposed to math circle topics) is with algebra2/trig honors students, mostly 10th graders, at Castilleja school in Palo Alto. I stole some of these ideas, especially the emphasis on linearization as a way of motivating the use of log properties like #2, from Richard Sisley. I’m not sure where I stole the slide rule idea but it ended up being my tradition to do on “Grandparent’s Day” at Castilleja – we would always get an engineer or two who had some good first-hand experience (and often they were fascinated to learn more about how you make one and why it works, and sometimes they already knew that too).

March 10, 2010 am31 11:24 am 11:24 am

The first one makes me uncomfortable because it doesn’t have “an answer,” it has infinitely many answers (one of which is relatively easy to stumble upon). One perfectly reasonable way to solve the problem would be to note that $\log_b 24 = \log_b 2^{\log_2 24} = (\log_2 24) \cdot (\log_b 2) = x \cdot \log_2 24$, and a student following this (completely correct) procedure would not find her (correct) answer among the choices on the Regents exam. The “lesson” of problem 1 is something extremely artificial about writing 24 as a power of 2 times a power of 9; the cleverness lies in this unnecessary step, and the fact that this works is part of the big lie of math textbooks that every question has a nice answer.

(I’m certain that I’ve pinpointed the mathematical aspect of the question that I don’t like; I’m not sure whether my explanation for why I don’t like this aspect is perfect.

March 10, 2010 pm31 12:11 pm 12:11 pm

I see what you mean about the artificiality, but I think it’s clear enough from the problem that the solution is to be in terms of x and y, and not in terms of any other logarithms except those two. Maybe it’s not clear enough, and they should say “in terms of x and y and no other logarithms”. Still that makes the solution unique, I think.

19. March 10, 2010 pm31 3:23 pm 3:23 pm

I understand what you’re saying, but it doesn’t make me feel any better about the problem: the “intended answer” is $3x + y/2$. Where did 3 and 1/2 come from? Well, $24 = 8 \cdot 3$ and $3 = \log_2 8$ and $\frac{1}{2} = \log_9 3$. In other words, the answer is necessarily in terms of some other logarithms, it just happens that there is one such expression in which those logarithms turn out to be rational numbers (specifically 3 and 1/2). (Incidentally, yes, the expression is unique, because $\log_2 3$ is irrational.) There’s no good mathematical sense in which this one expression is privileged over the others, and such an expression only exist because someone consciously chose some contrived values to make it “nice.”

By the way, what do you think of $\log_7 \left(7^{1 + \log v - 3 \log w}\right)$ as an answer to the second question?

I don’t think that either question is ambiguous in a way that should confuse students, i.e., I agree that it’s more-or-less clear what the desired steps are in both questions. Of course, it should be, since these are exercises of the “learn how to use log rules by doing ten questions of simple applications of log rules” variety. But I agree with Jonathan that these questions rub me the wrong way, and I think that the problem I have with them has to do with implicit assumptions about what makes a “good” answer to these questions. If the first question asked for *two different* expressions for that number in terms of x and y, I would be instantly mollified. (It would also be a very different problem!)

March 10, 2010 pm31 3:49 pm 3:49 pm

How about if the first problem were posed like this:
“A slide rule has marks at one end corresponding to the number 1, at a distance of x cm from that end for the number 2, and at a distance of y cm from that end for the number 9. How many cm from that end should 24 be placed?”

I think, practically speaking, you could measure 3x + y/2 in a way that you couldn’t measure any other crazy log relationship. Heck, you could construct that point with straightedge and compass, and in general you can’t construct irrational logs that way.

I’m not saying I love this kind of problem, just that I wouldn’t describe it as ill-posed the way you were a couple comments back.

For the second one, “as a single log” implies to me that it must be written as log(an expression that doesn’t contain any logs).

I think of #2 as the inverse process of something useful: you have some kind of power-law or exponential relationship and you want to linearize it (say, so you can compute a least-squares line or something). Then you want to recognize the slope and intercept of your line in terms of the original expression. So you start with y = a * x^p and you get log y = log a + p log x, so you see a slope of p and y-intercept of log a and want to reinterpret those back into the original power law. If you do it going “forward” like this, then that’s fine, but if you have log y = m log x + b, you need to recognize that as y = 10^b * x^m, which is the same sort of thinking as in this problem.

But I agree that divorced from the kinds of contexts I suggest here, these problems can very quickly become meaningless, arbitrary drill, where the “right” answer is just “the one that makes the teacher happy” instead of something that shows you understand what’s “really” going on. Perhaps the bigger problem is that there’s really nothing going on at all when the problems are posed in this way.

March 13, 2010 pm31 4:11 pm 4:11 pm

Yes, I think we agree that these problems can be given enough context to make them seem worthwhile (and I think you’ve suggested some really excellent contexts in which to place them) but that they really need that context in order to be meaningful.

(I still think that as a purely mathematical puzzle about logarithm rules, the first one is ill-posed. But you have convinced me that asking for a rational linear combination is not a meaningless endeavor, at least in context.)

March 15, 2010 pm31 5:18 pm 5:18 pm

Nice discussion!

They’re not good questions because (at least as they were presented, without context), they are (transparently!) only being asked to determine whether one has internalized the log rules. As such, they’re boring. And worse: I think questions like those are what kill students’ interest in mathematics. If math is learning these rules and then showing that you can use them, then of course it’s boring. I wouldn’t like it either.

March 26, 2010 am31 1:03 am 1:03 am

I serendipitously wandered into this discussion (I am not a teacher), and the exercises caused me to recall a couple of things that I have not thought about for a couple of decades – an interesting, if not particularly applicable, experience. My opinion of their worth is close to that of Natasha, so I was curious to find out what is so wrong with them.

The objections seem to fall into two camps: on the one hand, artificiality; on the other, a lack of unique solutions. These are somewhat at odds, because there is something artificial about a world in which all problems do have unique solutions. Is there not something of a teaching opportunity in problems that have one straightforward solution, and others that are reached from a different perspective? The discussion on how to remove all ambiguity over which answer is ‘correct’ strikes me as pushing the questions further in the direction of formulaic artificiality.

I realize that the discussion of this was driven by a concern over what might appear in the Regents exam. If that body is in the habit of posing ambiguous questions and only accepting the approved answer, I understand why teachers have objections to it.

On the other side, DavidC says their artificiality makes these questions boring. Repetition would be boring, and not very enlightening if the goal were merely to teach the students to recognize question-answer patterns that will show up in the exam, but my experience is that a few exercises like this can help a student get a feel for how the rules work and a clearer understanding of why they do. I often find that I can follow each step of a logical argument, but not really understand the conclusion until I have played around with it a bit.

More generally, how much relevance can you hope to give to logarithms for the majority of students, who will never use them after they graduate? Some will use them in even more abstract math before they graduate, but that’s a rather self-referential and self-justifying argument. If practicality is the measure of worth, one might as well question why, and to whom, any math is being taught (to be clear: I am not arguing against math education; I am questioning whether practicality is a necessary attribute of anything that is to be taught or used in teaching.)

24. March 27, 2010 pm31 3:13 pm 3:13 pm

The history of logarithms and exponents explains a lot of why students don’t get it about logarithms. It’s a long story, explained in podcasts, applets, and lecture notes at
Go to Mathematical Intentions. There are podcasts on square roots, logarithm tables, and slide rules. The history of fractional exponents is in the Wallis section. (But it makes more sense if you listen from the beginning.)

An oversimplified short version:

Logarithm tables were developed starting in around 1600, for calculations, especially of values of trigonometric functions for navigation, and other practical purposes. The printing press was fairly new at that time. Suddenly you could accurately record and reproduce a vast table, and other people wouldn’t have to repeat your work. The attraction was that you would never have to multiply and divide again (just add and subtract) if you bought these tables.

The tables (starting with Briggs) were generated as an arithmetic (additive) sequence paired with a geometric (multiplicative) sequence. The values in the multiplicative column are interpolated by taking square roots. Think of the additive in the left column, and the multiplicative in the right column. Moving to the right is taking the logarithm, to the left is taking the antilogarithm. Nobody gave a name/formula (such as log(x)) to this process. Certainly nobody said that the logarithm is the inverse function of an exponential function: these hadn’t been invented yet.

Descartes was the first to use (positive integer) exponents for shorthand for repeated multiplication. It wasn’t until Newton, following up on groundbreaking work by John Wallis, that anybody thought what x^1/2 or x^2/3 should mean. This was about the time that Newton was developing the calculus.

Oddly, we expect middle schoolers to accept and understand the meaning of negative and fractional exponents, with no justification beyond “It’s the rule” or “It follows the pattern.” It wasn’t obvious to the top mathematicians of the 17th century.

25. April 10, 2010 pm30 7:04 pm 7:04 pm

Wow!! I’m really relieved to know I’m not the only person asking these questions. Great discussion!!

Aside from the potential superficiality of just internalizing log rules, I recently posted on my blog an easy way to remember how logarithmic notation works. Many of my students have found this a useful way to remember what goes where. And it’s fun to call the subscript area the “basement.”

http://bit.ly/cUzvjF