How Quickly Do We Forget?

I visited an Algebra II class at Northern Virginia Community College last night. On the third floor of a Reston office building, beneath fluorescent lights, more than two dozen students toiled away on radical expressions they have seen once before.

"We are here because we forgot Algebra II," said Liz Weiblen, a 1990 JEB Stuart grad who is majoring in Japanese. She wanted to enroll in a math for liberal arts class, but couldn't go any further until she got the Algebra II credit.

It may not be surprising to lose algebra over time. I don't know when and where I left mine over 14 years. More troubling are the recent high school graduates who have not taken a sustained break from equations and are back at it.

Laura Mays, who graduated from a college-prep school in Georgia in 2006 with calculus on her transcript, was disappointed when the NVCC placement test sent her backwards. Hina Mansoor, a 2008 Herndon High graduate, took pre-calculus last year but is back in Algebra II. And a whole cluster of recent graduates from Westfield High, many of whom took the course within the past few years, are doing it again.

Teacher Jane Serbousek said her students do not always need to re-learn the material, but to recall it.

It seems if you learned it well the first time, it should not vanish so soon. But how quickly do we forget? Is the problem a failure to teach or a failure to apply? How can we teach for the long-term?

By Michael Alison Chandler  |  October 30, 2008; 9:37 AM ET  | Category:  Class Time , Other Math Classrooms
Previous: Algebra in the News | Next: I Have a Tutor!

Comments



good question. here's my take.
things that are *memorized*
are likely to be forgotten
very quickly; things *understood*
can often be safely ignored for years.

i made it a point as a student
to try to *avoid* memory work ...
i'll even admit i took this to
kind of an absurd extreme.
rather than just learn once-for-all
that sin(\pi/6) = 1/2, for example,
i'd have to mentally construct
a certain triangle; rather than
"plug in" certain numbers on
the quadratic formula, i'd use
the "complete the square" process;
so on. once i started actually
*lecturing* on the material, of course
i found it very useful to have
commonly-used facts and formulas
available at a moment's notice;
i also found that students with
the same ability tend to write
better quiz and exam papers ...

with that said, let me emphasize
that, like many another math teacher,
i spend a great deal of time and trouble
trying to convince people that
"knowing the formulas" is only
a *very small part* of what we're about
when we're "doing mathematics".
if it's done *right*, you can *forget*
everything but the "big picture" ideas;
the details will work themselves out.

once miss de baggio showed me
(and my whole 6th grade class)
*how* to see that the area of a triangle
is half the base-times-height
i became (as it were) *incapable*
of forgetting it; a student who
merely memorizes A = hb/2 is likely
to forget something vital (the equation
itself or what the heck it's for at all,
for example). every teacher learns this.

http://vlorbik.wordpress.com

Posted by: vlorbik | October 30, 2008 11:21 AM | Report abuse

I think it's a failure to apply. Personally, I use math every day in my classes. Calculus, algebra, complex numbers, trig, exponentials - they all factor into my daily vocabulary. But things like differential equations (which I avoid like the plague) and some of probability theory are hard for me because I never use them, so I always have to look things up. A word of advice - never sell your math textbooks. You WILL need them one day.

PS - vlorbik, I don't this blog recognizes standard LaTeX escape sequences like \pi. :) Though I did discover the other day that MATLAB does.

Posted by: UVaEE09 | October 30, 2008 12:31 PM | Report abuse

I think it's a failure to apply. Personally, I use math every day in my classes. Calculus, algebra, complex numbers, trig, exponentials - they all factor into my daily vocabulary. But things like differential equations (which I avoid like the plague) and some of probability theory are hard for me because I never use them, so I always have to look things up. A word of advice - never sell your math textbooks. You WILL need them one day.

PS - vlorbik, I don't this blog recognizes standard LaTeX escape sequences like \pi. :) Though I did discover the other day that MATLAB does.

Posted by: UVaEE09 | October 30, 2008 12:32 PM | Report abuse

I tend to agree with vlorbik more than UVaEE09 - it's a lot harder to forget something you understand than it is to forget something you memorize. Or at least, if you do forget something you understand, you can recreate it from simpler things you DO remember. Failing that, if you understand that area is essentially the product of two length measurements, you can at least get close. You might not remember that the area of a triangle is b*h/2, but you'd be pretty sure that it's not b^3*h^1.7.

Familiarity and repeated use does help, though. And I agree, never EVER sell your math books. I used one just the other day. It's holding up my kitchen cabinet. :-)

By the way, I just posted a solution to mathlete's cow problem in the marathon blog 2 posts ago. Still hoping to hear from anybody who would like to take a stab at mine.

Posted by: tomsing | October 30, 2008 1:16 PM | Report abuse

Also, regarding how to teach for the long term: if you accept that the way to learn is by really understanding, rather than being able to recite, then if you want to find out whether a student is learning, you've got to test for that understanding, not for the ability to recite!

The problem is, some people are going to get it quickly, and some people aren't. Some people have a natural gift for math just like some people have a natural gift for basketball - they're built for it. Conversely, some people have to work very hard to even be passable at math, just like basketball. And then there are people in the middle, who will do very well if they work hard. But we're not all going to be in the NBA, and we're not all going to be math majors.

So you can't promote someone on to the next level of math just because they passed a memorization test. There's no point to it if there's no understanding. That means that you spend the necessary time to get a student to truely understand basic arithmetic before moving on to higher level things, and you hold them back if necessary. In lower grades, that's not easy to do, because repeating math doesn't mean you also need to repeat reading or social studies. So maybe look at a model more similar to middle and high schools.

Posted by: tomsing | October 30, 2008 1:56 PM | Report abuse

The question is why are colleges requiring students other than those in scientific, health, or technical fields to pass Algebra II. I haven't used any algebra beyond solving very basic equations (the kind covered in my 7th grade pre-alegbra class) since high school.

Posted by: CrimsonWife | October 30, 2008 2:26 PM | Report abuse

There are two aspects to memorization that I think are relevant to the discussion. They are elaborate encoding and repetition.

Repetition at regular intervals is best after: 1 minute, 1 hour, 1 day, 1 month, 3 months and then once a year after the material is learned should be enough. This method is based on the classic observations of the rate at which people forget by Ebbinghaus.

http://en.wikipedia.org/wiki/Forgetting_curve

The aspect of understanding that helps you remember things is that it is a form of elaborate encoding where you memorize more than you strictly need to know and you memorize it in a way that is highly redundant which allows you to reconstruct the original information more accurately.

The elaborate encoding system need not be related to the material the way understanding the material would be. There a many memory systems related to translating numbers into letters, lists into stories and so on. While understanding is a good way, it might not always be the fastest way.

Posted by: mathlete | October 30, 2008 2:33 PM | Report abuse

I have some sympathy with Crimsonwife. I want to know more about what the end goal is. Memorization can be sufficient in some cases and understanding in others. It's a continuum. Even professional mathematicians don't necessarily understand the details all the mathematics that they might use. It all depends on how deeply the idea needs to be understood to be of use. We all have finite amounts of time and attention as well as different goals.

Posted by: mathlete | October 30, 2008 2:37 PM | Report abuse

Two thoughts:
(1) Math in school teaches how to actually solve a problem--how to approach it methodically, figure it out and produce an answer. Writing a creative essay in English makes similar demands--but not many other subects do. A lot of social studies, history, literature etc--even some science--focus on memorizing. In math, you must actually produce something. That skill is valuable by itself.
A neighbor's kid once asked me to check him out for a test on the Pythagoras theorem. I did, and he knew it well. I then told him, "here is something you can actually do with it" and showed him how to calculate the distance to the horizon (http://www.phy6.org/stargaze/Shorizon.htm). His reply: Wow! This is the first time someone showed me how to USE this formula for something!

(2) Math may not be intrinsically hard. What it is, however, is UNFORGIVING. Usually, you must understand early lessons to continue past them. In history, if you were out with a cold while the class studied the French and Indian wars (say), you can still catch up with the US revolution. In math, if something is not transparent (say, the concept of unknown numbers), it is likely you will stay confused about equations etc
.
That is the reason why math also has a reputation for being boring. The teacher knows she or he cannot afford any attrition in math class, so the subject is drilled again and again, while the majority which got it the first time is bored to tears. The solution is to provide optional material, and make it interesting to whet interest (how do you tell a number is divisible by 9? by 3? by 11?; how do you write numbers in binary notation, and what are the rules for addition, multiplication etc? In a pythagoras triangle, say (3, 4, 5), show at least one number is divisible by 5. also 3 and 4?)

Should algebra be taught to all? Yes, at least elementary algebra, to get the taste. Look up
http://www.phy6/org/stargaze/Smath.htm

Posted by: phy6guy | October 30, 2008 3:12 PM | Report abuse

of course phy6guy means
http://www.phy6.org/stargaze/Smath.htm
(dot not slash before "org").
some cool stuff here!

Posted by: vlorbik | October 31, 2008 12:14 PM | Report abuse

I still hang on to math books. Going to used book sales, I notice math books are scarce. There isn't a great selection, usually, in local book stores. I browse Amazon.com and Powells.com looking for bargains and interesting reads.
Best advice I ever read, in review of a math book on Amazon, offered by the reviewer who tutors math for SAT preparation: Do some math every day.
I do! It's usually algebra because that's what I enjoy and all algebra books are NOT alike. Math every day is like studying a little bit of French every day. You do get better. And like French, it is memorization but you also gain in fluency quickly.

Posted by: KathyWi | November 1, 2008 11:48 AM | Report abuse

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