## Thursday, May 03, 2007

### The Big "L"

There are some students who, no matter what, can’t seem to comprehend what a logarithm (when treated like an operation) is doing. I see students that:

1) Cancel the log.

2) Multiply by log.

3) Ask where the 2 went when log2(8) is simplified to 3.

These mistakes indicate that “log” is being perceived as some sort of quantity to be manipulated, not as an operation. This may be due to the fact that “log” is the first time students are exposed to an operation that is represented as a word instead of as a symbol or other numerical notation. Texts apparently assume that this is a natural transition, not even worth mentioning, but it’s pretty clear that it is not as obvious as one might think.

To help students see what is going on, I’ve tried expressing other operations in a similar manner and drawing parallels. For example, take a look at roots and powers:

Logarithm does not have a symbol; our initial idea was to therefore rewrite exponentiation in terms of the “word operation" exp. We then explained that logarithms are the inverse of exponentiation, and that they undo each other, just like addition and subtraction, multiplication and division, and powers and roots.

This seems to have worked moderately well in terms of getting students to be able to evaluate and solve the log problems that they encounter on the STAR tests. However, I don’t think it’s really helped them to understand what a logarithm is, and their ability to apply the concept flexibly is quite limited.

I’m wondering now if going the other direction would have been better. Instead of rewriting exponentiation as a “word operation", we could have invented a symbolic representation for logarithms – say, a big L. (Not to be confused, of course, with the L formed by thumb and pointer finger, raised to the forehead!).

Inverse operations could then be modeled like this:

When I ask my students what “the third root of 8” means, they are pretty good about saying something like “what number to the third power gives you 8”.

When I ask them what “the log base 2 of 8” means, they rarely can say “2 to what power gives you 8”. I wonder if using a symbolic representation of logs will allow this meaning to be clearer. After all, when you think of a log in this way, it’s not really that much more confusing than a root.

I’d be interested in hearing any thoughts on this. Would a symbol for log be helpful? Confusing?

Anonymous said...

I can't use the subscript here, but I find this helps students translate logs.

I tell them to remember that logaB = C is a math problem that needs to be translated.

I write on the board:

logaB = c (a is subscript, or base)

Below it I write Base Result Exponent and point each one to a, b, and c. I tell them it's always in that order (and if base is missing, it's 10).

Base to the Exponent = Result.

Then I give them 20-30 problems of increasing difficulty.

When I was learning logs, it took me a long time to be able to look at a log equation in log form and easily solve it, so don't shoot for that. Just make sure they know it's a math problem, and how to translate it. Then they can work it from there.

Dan Greene said...

The method you describe is similar to what I do in my honors class. I always use "b" as the subscript, because I want them to remeber that it is the base when converting to exponential form. I also use "x" as the argument, to prepare them to think of log as a function of x.

I also tell them that log is an operation that isolates an exponent.

In converting from exponential form, then, they should use the base of the power as the base of the log, and the exponent should be isolated on the other side of the equal sign.

In converting from log form, they should always read it as "the base to the exponent equals the argument".

This seems to work pretty well for the honors class. We didn't try this in the regular classes this year because it seems not to have worked as well in the past. So I'm wondering if this standard method is the best way to do it, and it just takes a lot of time and practice, or if introducing some sort of symbolic notation will help those who aren't getting it this way.

bogusia said...

I like your idea with the symbolic representation, but I don't know whether it will do the trick. Logs are just hard to understand, and I think the way anonymous explains it is the way I would do it too (and I always use the b as base as well). Even now, I sometimes have to go back to the original assumption, the original equation to figure out the value of a log. But on the other hand, I never had someone think of the representation as you did. Try it with your next batch of students, and see how it takes. I would like to see how it works.

Another thing I do is use the calculator to show them that log is an operation and not an argument. This is kind of a cheap way of doing this, but for the not so strong student, it is somewhat visual and it works.

By the way, for my students, sin, cos and tan are the same as log for your students (they cancel them out, they write no argument, etc.). Drives me nuts!

Dan Greene said...

I agree that logs are just hard to make sense of sometimes. MRC and I were talking about this a while ago, and we discussed the idea that, in any pair of inverse operations, one is always the "forward" one, and the "backward" one is defined in terms of the forward one - often in a type of "guess and check" process.

To do subtraction, you think addition. To do division, you run through multiplications. To take a root, you think of multiplication or powers. To take a log, you think of exponentiation. Maybe this reverse process is always going to be more cognitively complex, and maybe logs are the king of those!

I was actually just thinking about the trig functions.. we have switched the order of our courses so that algebra 2 comes before geometry. Maybe gaining comfort in one form of "word operation" will translate to better success when it is seen again.

But now that I think about it further, the trig functions are the first time students are exposed to an operation with only one operand. Even in logs, you have the base and the argument, so you can see how they might be related. With sin, cos, and tan, students need to connect the "word" operating on a number to that number representing an angle in a right triangle to the ratio of the appropriate sides in that right triangle. That's actually quite a cognitive load when you think about it... No wonder students hate logs and trig most of all!

Cal said...

I'm anon--forgot to put my id in.

"So I'm wondering if this standard method is the best way to do it"

Yeah, I'm pretty sure it is.

In all honesty, though, I don't think anyone but honors students should be dealing with logs anyway. (By "honors", I mean students who are ready for precalc or calc in high school.)

If you're describing it as "functions" and "bases", that might be part of the problem they have in understanding it.

I make it very clear when teaching any student who doesn't instantly grasp logs: logaB = c is an equation that he can translate and solve. I don't use words like "function" or "isolate". It's just a math problem. See log, do this. I go through all the rules the same way.

Not only wouldn't symbolic notation help, I think it would add an additional level of complexity to something they aren't getting in the first place.

I'm not much on teaching kids the "why" unless they are up to handling it--and any student that is cancelling out logs is not up to handling it. I want them to handle it as a calculation and nothing more. Many of them will get it later. Anyone who doesn't get it later will in any event remember the operation rules if they learn them properly. Teaching them anything about the "why" will only cloud the issue, I think.

It's funny; I don't see trig functions as having any relationship to the issue with logs. I can relate trig functions to a trig circle and (x,y) values.

Bogusia said...

Hey Cal, I disagree with you completely!

First of all, logs are useful, and not only to "honors" students. Of course not everybody will use it, but if approached the right way, 90% of the students will get the concept. Then when they are faced with any logs in anything in the future (and there can be many situations where non math or non engineering professions have to at least know about logs), they won't shy away.

Second of all, the "why" is exactly what we should be teaching the kids. If they don't learn the "why", then we are not teaching math.

You said: "Many of them will get it later. Anyone who doesn't get it later will in any event remember the operation rules if they learn them properly. Teaching them anything about the "why" will only cloud the issue"." When later? In someone elses class? Or are they to figure out how it's done on their own, understand the "why" just from knowing the rules? Maybe I don't understand what you're getting at.

DAN: I agree with you, in terms of having a forward and backwards operation. That makes a lot of sense. I haven't thought of it in this way, but I bet that's exactly why the "backwards" operations are so much harder than going forwards.

Cal said...

"Then when they are faced with any logs in anything in the future (and there can be many situations where non math or non engineering professions have to at least know about logs),"

On what planet would that be, exactly?

"If they don't learn the "why", then we are not teaching math."

Millions of people get through life just fine without knowing the "why" of logs. But math still has value--it teaches problem solving based on absolute rules.

Not all kids will get the "why" at first, and some won't get the "why" at all. Until then, I want them to know how to solve problems using rules.

I don't confuse my needs with theirs.

H. said...

"I want them to handle it as a calculation and nothing more. Many of them will get it later. Anyone who doesn't get it later will in any event remember the operation rules if they learn them properly."

As far as I can see this approach is problematic both pedagogically and in terms of justifying math requirements. First, the pedagogical issue: Remembering algorithms without understanding them is terribly difficult! A course like Algebra II would become a collection of impossibly many factoids and rules with special cases. Second, if Algebra I and II are reduced to computations and lists of steps - what on earth is the purpose of putting all students through that? Much of high school math is a little hard to motivate anyway - the applications in the textbook margins are often pretty contrived - and practicing mathematical reasoning skills is one of the strongest arguments in favor of plenty of math for everyone. Throw out that aspect of the courses, and what is left?

Dan Greene said...

We were reviewing logs in my algebra 2 regular class this past week. One student presented a problem at the board - her written work was perfect, and she got the right answer, but it was clear from her verbal explanation that she was just "following the steps" and didn't really understand what she was doing: she kept saying "you multiply log base 3 times x".

Now, some may argue that as long as she can perform the steps and get the right answer, everything is fine. But, I disagree with this because, with such fragile knowledge, it is doubtful that she will be able to apply logs correctly to a different or more complex problem. There will be no way to extend her knowledge in the future, as her original knowledge is totally off base. Additionally, it shows her "understanding" to be no more than temporarily memorizing a series of symbols to write - the chances of her remembering how to even do that by the time the final rolls around is next to none.

So after her presentation of the problem, I went to the front of the room and told everyone that her work was perfect, but that there was something wrong with what she was saying that we needed to identify and correct. No one had caught the mistake, so I asked her to repeat the explanation and I pointed out the problem. Of course she thought it was no big deal. I'm always trying to correct students' verbal errors that change mathematical meaning (i.e. saying 2^x as "two-x" or the 4th root of x as "4 square root of x") and their response is usually something like "same thing!"

After I reminded them that we are not "multiplying by log" they actually began to ask why, which is a step in the right direction. We had a discussion about what kinds of things you can multiply together. I asked them questions like "What is 2 times plus? What is 4 times square root? What is division times division?".

Of course, I heard responses like 2-plus, 4-square root, and division-squared! At a certain point, I couldn't tell if they were just messing with me, as they saw my visceral reaction to "division-squared". But I suppose that must be perfectly logical for some of them. I think that, after discussing this for a few more minutes, most of them got the idea that the two things you are multiplying must be *numbers*, not operations. Then I reminded them that "log base 3" is an operation, just like "the third root" or "division". I hope that this discussion has helped move this class forward a bit, but it really shows me that next year, I have to approach things differently.

Darren said...

I repeat ad nauseum that a logarithm is an exponent. Then I have students read log problems the following way:

log-base-2 of 8 = 3

"The exponent of 2 that gives me 8 is three."

It makes sense, because a logarithm is an exponent.

Dan Greene said...

Darren - do you have any thoughts on representing this symbolically?

My students are usually ok when they are just trying to evaluate a single, simple log expression. It's when they start doing more complicated things like solving equations and using the log properties that they get confused. I'm wondering if a symbolic representation will help ease the cognitive load as the problems get tougher.

Cal said...

"First, the pedagogical issue: Remembering algorithms without understanding them is terribly difficult! A course like Algebra II would become a collection of impossibly many factoids and rules with special cases."

Do you think most people understand the requirements for commas and semicolons? How many times have you (or people you know) undergone a grammar and punctuation test and just put your best guess at what sounded right?

Algorithms are rules. For many people, just understanding *how* to apply the rules will get them further than they've ever been in math before. They can figure out *why* later, if they have enough success to justify it.

"Second, if Algebra I and II are reduced to computations and lists of steps - what on earth is the purpose of putting all students through that?"

There's two ways to frame high school subject selection.

First, that school subjects are directly related to real life. Then, teachers spend hours and hours creating tortuous explanations to kids who know full well that it's all bs. Most *professionals*, much less secretaries or dental assistants, don't think about logs, trig, or geometry for a split second after the last time they're forced into it in high school. So this way is a lie, but it's the favored method of most teachers. I really don't know why.

The second way to think of high school math is that it provides an excellent construct for *any* sort of problem solving. In the real world, we often have to deal with constraints and requirements that we don't understand. We learn the rules of the situation and later, if we're lucky, we find out why these idiotic rules exist. Maybe we even agree that they make sense.

In that framework, a teacher is not required to create absurd rationales that everyone knows are nonsense. Math is the means, not the end. It is an outstanding means because it has clear, unambiguous "right" answers that demonstrate whether or not the student is able to master the rules.

It's not enough to get the right answer, which is why Dan's example is inapt. You have to get the right answer for the right reasons. That's "how". But it's not "why".

Dan Greene said...

Cal,

I agree with your point about not trying to convince students that math is something that is used "in everyday life". I do like to show real examples of applications, though, so that they can start making the connection between math and the real world. With parabolas, for example, I discuss headlights and satellite dishes - not because they are all going to become engineers, but because it helps them see how math might be important to society. Many of my students perk up when I show them these types of things, because it gives them a window into why these seemingly meaningless things we force them to do might actually be relevant.

On a daily basis, however, my answer to "why are we learning this" is threefold. One: we are learning it because you need it to pass algebra 1, algebra 2, and geometry if you want to go to college. Two: we are building a foundation of skills that will be necessary for you if, in the future, you decide to study any sort of technical field, and I want you to have as many options open to you as possible. Three: learning math improves your ability to think logically and critically, to spot patterns, and to solve problems.

"It's not enough to get the right answer, which is why Dan's example is inapt. You have to get the right answer for the right reasons. That's "how". But it's not "why"."

I have to admit, I don't understand what you are saying here, or what example you are talking about. My point is not that students should understand the "why" of everything on a fundamental level - since math is a formal system built upon axioms, there really is no "why" after all - just a collection of "how"s. When I say I want my students to know why something works in math, what I really mean is that they can understand how this new piece of knowledge connects with and/or is derived from other bits of knowledge they already have. Instead of teaching every single concept as a new, isolated rule, I want them to see how the pieces fit together.

For example, when learning the rules of exponents, if you memorize that a^b * a^c = a^(b+c), but don't understand that this is simply a result of expanding out the powers, you are much more likely to forget how to apply the rule. My students often forget if they are supposed to add or multiply the exponents; if I don't help them draw the connection between what they already understand about exponents, and this new piece of information, then I don't think I'm doing a very good job as a teacher. When a student asks me for help on a problem like this, I don't repeat the formula. I ask them to think about what it means, or to expand it out, or I make up a simpler related problem for them to work through, and they always conclude that it must be addition - and can explain why that makes sense. This is more powerful and longer lasting than what they would get if I just told them the formula yet again.

According to you, all that your students need is a clear explanation of what the rules are and sufficient practice of those rules. That may work for your students, but I can guarantee you that it does not work for the majority of mine, no matter what you believe.

Cal said...

"My students often forget if they are supposed to add or multiply the exponents; if I don't help them draw the connection between what they already understand about exponents, and this new piece of information, then I don't think I'm doing a very good job as a teacher."

Presumably you taught them the first time what the connection was, and they still forgot.

So when the student has forgotten the first time, you do nothing more than repeat the method that didn't work the first time.

Give them the power laws as a group of five or so that they need to memorize, and they will be less likely to forget. And if they forget, they have a framework to use in order to remember it.

"That may work for your students, but I can guarantee you that it does not work for the majority of mine, no matter what you believe. "

How can you guarantee that? You haven't tried otherwise, and by your own admission, your students aren't remembering the rules now. You're operating from belief just as you suppose I am.

Dan Greene said...

"So when the student has forgotten the first time, you do nothing more than repeat the method that didn't work the first time."

No. The first time, I teach students the rules, and also how they are connected to what they already know. There will always be students who later on get stuck on problems, regardless of how you teach. When this happens, I am able to help them access their knowledge and get unstuck by simply saying something like "expand it out". For most students, I don't have to show the whole thing again, and I don't have to write the rules for them.

There are indeed some who do need to see it again, but that doesn't mean it's the wrong method. I don't share your belief that students must be able to learn and retain a new concept the first time that they see it, and if they don't, that they will be unable to remember or make sense out of it. I may need to show a student 2 or 3 times how something works, and that's fine. Eventually they remember it on their own. This does not work when I just reteach them the rule.

The reason I know it does not work my students is that I have tried it. In the first couple of years, I did a lot of writing of rules, practicing, and watching students fail assessments again and again. As a response to this repeated failure, I changed my approach and thought more deeply about how to present various topics. Each time I did this, students' success rates with those topics dramatically went up. So I'm confident that I am moving in the right direction with my students. If you don't agree, that's fine.

Dan Greene said...

Anyway.. this post has gotten off topic. I am really interested (if anyone has read down this far) in the original subject - the possible implications (good and bad) of using symbolic notation for logarithms.

Darren said...

I don't see how you'd do this symbolically. I stick to "math" as much as possible and resist "tricks", because I haven't seen much evidence that students (or anyone else, for that matter) can easily translate from the trick to the math.

If you want to try bogusia's idea with a calculator, I show:

7*5=35
Then we discuss what the exponent of 10 should be that would give 7, 5, and 35 (less than one, less than one, and between 1 and 2, respectively), and we punch the LOG keys on the calculator and get those values. Using the laws of exponents--again, because logarithms are just exponents--that means that 7 (10 to the .845) times 5 (10 to the .699) must, because of the laws of exponents, be 10 to the 1.544. The exponents on the left add, as do the logarithms.

Dan Greene said...

It isn't a trick to express an operation symbolically. Even what we now think of as the most fundamental symbols (+, -, =) were only invented in the 14th and 15th centuries. New symbols were frequently the key to moving mathematics forward. See this site for a brief history of various symbols:
http://members.aol.com/jeff570/mathsym.html

So why is using the word "log" more "mathy" than using a symbol? We use symbolic notation for roots, exponents, fractions, and just about everything else. The idea of the Big L is not to hide what a logarithm is, but to use notation that allows students to more easily make sense out of it. Everything else they recognize as "math" involves symbols, so maybe this could make the idea of logarithms seem less foreign and daunting.

zac said...

Hi Dan. I'm with you on the idea of a different notation for logarithms.

In fact, your post inspired me to write something that has been brewing in my mind for a while: Towards more meaningful math notation.

Not quite the same problem you are trying to solve, but worth considering, I hope!

Miles said...

Thanks for the great post. I actually came up with a notation like this recently because I am tutoring an algebra 2 student who is awfully confused by logs. I remember being quite confusing myself for a long time.

I finally realized, though, that just as division is repeated subtraction (a/b is "how many times can you subtract b from a before getting zero") which makes it the inverse of multiplication, logarithms stand for repeated division (log_a(b) is "how many times can you divide b by a before getting one?") which makes it the inverse of exponentiation.

Furthermore, logs act a lot like fractions. 1/log_a(b) = log_b(a) (check it!). Also, log_a(b)*log_b(c) = log_a(c) (imagine "canceling" the b's).

So, I propose using the "Large L" notation above for natural logs, but moving the base to the "denominator" for logs with other bases. That is, log base 2 of 3 would be:

|3
---
|2

(that's my attempt to draw a "sideways T" with a 3 on top and a 2 on bottom - if you then take the convention of leaving out the base when it's e you get back the "large L" notation)

|x
--- =
|e

|x
----

lrw757 said...

I'm Glad Cal wasn't my math teacher.

Sue VanHattum said...

I know that weird notation slowed me down sometimes. In calculus, seeing tons of Greek letters I didn't know the names for made understanding those proofs way harder. (I'd read them as squiggle this and squiggle that, and get all muddled up.)

I think either your big L or Kate's power would be a great way to start. I'm trying to figure out which to use. Any thoughts?

DevonMcC said...

[I also posted this comment on http://www.intmath.com/blog/towards-more-meaningful-math-notation]

I use a programming language called J which is an executable mathematical notation. The creators of the language have chosen the symbols of the language very carefully and a particular set of them is relevant to this discussion.

The symbol for "power" is "^" and the symbol for "log" is "^." so the related ideas are clearly linked. As are all J symbols, these both have both a monadic (single right argument) and a dyadic (left and right argument) use.

The language is interactive, so if I enter
^1 2 3
(monadic power assumes base "e"), I get
2.71828 7.38906 20.0855
as my result.

Similarly, entering
2 ^. 2 4 6 8
returns the base 2 log of these numbers:
1 2 2.58496 3

(You may have noticed from these examples that J handles arrays naturally and simply.)

The language is free, open-source, and runs on Windows, Linux, and Mac. You can find out more about J at jsoftware.com. Explanations of the use of the power and log symbols is on their site under "help/dictionary/d200.htm" and "d201.htm".

Anonymous said...

Hey everyone, I am a pre service teacher. my little bit is that I'm not sure introducing new unique notation is a good idea but i can definately see how it can help the original point. the idea of treating log as an operation is very important in gaining an understanding of what is going on. rather than create a new notation I would still use logs but simply write every operation in a different colour while introducing the topic. that way the notation of log() can stay the same but students will still immediately recognize the log as an operation. * / + - ^ log..... all in a different colour