Wednesday, January 28, 2009


I've been teaching exponential and logarithmic functions in my Algebra 2 courses for the past few school weeks, and a stark realization hit me.

We're not teaching near as much about logarithms in Algebra 2 as I learned when I was in school.

I don't have a lot of specific memories of my high school classes, but for some reason I remember my old Algebra 2 topics exceedingly well. I remember what we learned about logarithms, and quite a bit of what I learned isn't even covered by our textbook--a textbook, by the way, that meets California's rigorous Algebra 2 standards.

I learned Algebra 2 in the 80-81 school year; calculators may not have been ubiquitous, but everyone had one :-) I don't recall that my teachers had a "classroom set" to lend to students, though. Even though we had access to calculators, my teachers still insisted we learn logarithms the "old fashioned" way--by looking them up in a table, by interpolating for values not found on the table, and making use of the laws of logarithms for those values above 9.99 (the highest number on the table). It was through all that manual work that facility with logs was developed, that deep understanding that even then made me think logs were "easy".

As an example of how things have changed, our Algebra 2 book does not have a table of logs in it. The book assumes students have an electronic calculator. In this way, though, students don't get a "feel" for the magnitude of logarithms; instead, a log is merely a number displayed after pushing a few buttons. There is much to be gained by seeing all the logs arrayed in a table, and that gain goes missing when there's just one log lighting up a screen.

Because there's no table, there's no interpolating. The algebra of ratios, the geometry of similar triangles--the synergy gained by all this practical application is now gone, replaced by a false sense of "exactness" when the calculator displays logs to 10 decimal places.

The depth of understanding gained by fitting all these little pieces together--it isn't there now, because all the little pieces are missing.

Because they didn't learn with a table, students today had a much more difficult time than we did "back in the day" understanding why if log 5.3=.7243, then log 530=2.7243. Because electronic calculators are assumed, understanding is lost. Sadly, this assumption is embedded in the textbook.

It's not that California's math standards insist on calculator use; in fact, quite the opposite is so. And the standards are explicit in what students are required to know about logarithms (see the standards here, and scroll to page 54/72 to see the log requirements for Algebra 2). The state isn't mandating this overreliance on calculators, but since the textbook meets the requirements listed it gets approved--and as I said, this textbook assumes calculator use in the course. And enough teachers support calculator use that students are too dependent on them long before they ever get to my class.

Even worse, I got a call from my sister tonight. She needed to ask me about graphing calculators, saying my nephew has to have one for Algebra 2. I cannot for the life of me see the utility of a graphing calculator in an Algebra 2 course--not if you think it's important for students, not machines, to understand the graphing.

I've heard about "dumbing down" for a long time, but I think this is actually the first time I've experienced it. And it's not like I can blame the students--they'll learn what we teach them. And what we're teaching them isn't necessarily the best.

And before anyone suggests it--yes, I'm required to teach the material in the textbook.


Ronnie said...

I have to disagree, I've went through calculus, vector analysis, linear algebra, and now I'm on differential equations and the only time I've ever seen a log has been many, many natural logs. I understand basic manipulation of logs should be taught but I think the fact that they are so rarely used in modern upper math courses makes it hard to justify learning how to use a table to find any log.

Logs would be very far down the list of things I wish I had mastered more of, I would like to see more "2.0 Students solve systems of linear equations and inequalities (in two or three variables) by substitution, with graphs, or with matrices" and sections 20.0-24.0 dealing with sequences and series.

Aus_Andrew said...

Hi Darren,

First time commenting after a long time reading. We have used Graphics Calculators (such as the TI-83, 83+, 84, 84+) in Australian schools for much the last decade. The argument for them is that it allows students to attempt "real problems" that have non-integer solutions, in other words, "messy" answers. I don't hold to this viewpoint, but it does allow for students to explore graph translations much more swiftly than manual plotting.

However, the evolution has continued, and we are now starting to use CAS (computer algebra system) calculators. These calculators actually can perform algebraic operations with unspecified variables. This means that students are now beginning to use calculators to resolve simple factorisations instead of learning the algebraic techniques. This seems to me to be the start of a dangerous trend.

Mrs. C said...

My son Patrick tells me that at his school, "advanced" is the normal class, "gifted" is the advanced, and "regular" classes are the slow ones. They just don't want to tell you that because it would make you feel bad if you were in the slow class.

I don't know if he's right or not, what with my growing up in the regular classes and all. :]

Darren said...

One of my students has a calculator that will work with logs of bases other than 10 or e. Other calculators do "symbolic manipulation", meaning you can type in an x or y or any other variable; in theory, a student with such a calculator wouldn't have to know any actual math, just how to enter the information into the calculator.

rightwingprof said...

John Napier is an ancestor of mine.

PeggyU said...

in theory, a student with such a calculator wouldn't have to know any actual math, just how to enter the information into the calculator.

I saw it in practice yesterday. In the "Algebra" menu (F2, I think) of the TI-89 are "solve", "expand", and "factor" functions ... yep, if a student finds them, he will use them!

On logs, I think I have something that might be useful, Darren. This guy has written a book that addresses the usefulness of teaching students the history of logs, but that also avoids burying them in details.

I also just found this, which looks like it might be handy.

Anonymous said...

One of the problems with just using a calculator without understanding the process of the problems, is that you can easily mis-input numbers and get an answer that is several magnitudes off form what it should be. I've seen students do that!

I was fortunate in that I learned to use a slide rule back in the 1960s, where I had to keep track of the mantissa and exponent, therefore I could expect an answer in a certain range. I carried that experience over to the calculator age, so if my answer was out of where I expected it, I would re-input the data.

Of course, I also learned to estimate answers using the "SWAG" system--"Scientific Wild Ass Guess." :-)


Erica said...

Your students are lucky, even in my multi-variable calc class we were required to graph by hand.

On the up side I'm quite comfortable thinking about equations as physical objects, which translates into some ability to mentally build and manipulate real objects easily. Though I don't do much higher calc these days, I use spacial manipulation skills constantly in my job.

Darren, sorry to hijack your comments here for a second, but I'm moving to the Dallas area in a few weeks, and want to get a note over to Ellen K as I can't find her contact info. Ellen - I'd love to get together for lunch sometime, send me an email!

Anonymous said...

It's easy to find the log of a number using any base.

log(number) / log(desired base)

Teach that too!

Babbie said...

As an English teacher introducing the word "interpolate" to students today, I told them of the tables of logs we had to interpolate without calculators (back in the Dark Ages of course)in math classes. Of course, they looked as though I had two heads. But then they still marvel that I figure out their test grades by adding and subtracting without a calculator!

Elaine C. said...

I insist that all my pre-algebra kids do NOT use calculators. The first few months are filled with whinging, but after that they get over it. I tell them that they *might* need calculators for Calculus - but pretty much not until then. They go on to not bother with the machines in future classes also!

The truly amusing part happens when they're doing math around the kids from the other math classes... the other kids scrounge around to find calculators. My kids make fun of them (mild joking around), while doing the problems on scratch paper.. (I'm also VERY insistent about showing all work.)

Funniest thing? I teach the lowest kids. I have ALL the RSP/mild SPED kids scattered through my classes. (I really pushed for full inclusion, instead of having them shoved into a SDC room.) The kids who don't have a 504 or IEP are the ones with the lowest CST scores. (Another thing I actually encouraged.) These are also the same kids who make SIGNIFICANT gains at the end of the school year - My kids from last year made up 1/2 the Algebra I classes this year... and a majority of the Honors classes. The other 5 math teachers pretty much evenly share the remaining Alg I slots.

No calculators, digging into the *why* of things, and plenty of practice with the mechanics are definitely important.

David said...

Erica..."On the up side I'm quite comfortable thinking about equations as physical objects, which translates into some ability to mentally build and manipulate real objects easily."

I think there might be some real pedagogical advantages in using *analog* computers--either electronic or mechanical, but especially mechanical--to give students a better and more intuitive understanding of mathematical processes. A couple of professors at Marshall University (WV) have done some work in this direction, re-creating a 1930s-era mechanical differential analyzer, which solves differential equations using wheel-and-disk integrators.

Bob said...

I teach logs in my middle-school classes ... not because one needs to remember how to do them when they are 30, but because interpolation gives practice in almost every arithmetic operator. Besides, being able to estimate an answer is a skill that is absolutely necessary when one is 30 ...

I get a kick telling my kids that their TI-monster-89 graphing calculator doesn't have a button to calculate the 13th root of a number.

Our books don't have log tables in any shape or form whatsoever. I generate them in Microsoft Excell if you're interested. 4 place ...


Anonymous said...

Your Algebra II kids are lucky to be learning anything about logarithms at all. When I taught calculus at a (horrible) state college, over half of my students had never heard of logarithms. They were told on day one that logs and exponentials were assumed prerequisite knowledge, so they had to catch up fast and either sink or swim. Many of them sank.

Oh, and this reminds me of one of my favorite extra credit problems that I had for a college algebra class: find the number of digits in log(8^(8^8)). The look on their faces when they punch the number into their calculator and then get an overflow error is truly priceless.

PeggyU said...

9 digits?

Darren said...

Bob, surely the TI-89 can take the thirteenth root of a number.

Darren said...

I get 8 digits--to the left of the decimal point. Could we argue that, as a transcendental number, it has an infinite number of digits? :-)

PeggyU said...

Or, you could argue that the numbers 0 - 9 are digits, and since log (8^(8^8)) = 15,099,494.4 and the digits 0,1,4,5, and 9 are used, that there are five digits.

Anonymous said...

Oops, I completely goofed up the statement of that problem (that only shows how long it's been since I've taught "college" algebra). The problem should've been: How many digits are in 8^(8^8)? Of course, you can get the answer using logarithms.