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.