# Algebra Is Like Sunscreen

Every year around this time, references to that damn sunscreen speech pop up again, as people start thinking of graduations. It’s in the air (Union’s graduation is this Sunday, and I don’t think I’ve ever been happier to see the end of an academic year).

And, of course, I have actually been asked to give a graduation speech. Which leads naturally to thinking about what one piece of advice I would give to a high school student who came up to me and said “I plan to study physics in college. What one thing should I study?”

(Hey, it could happen…)

My one-word piece of advice for students planning to study physics (or any other science, really, but mostly physics): Algebra.

If you have the slightest interest in physics, learn to do algebra. Learn it backwards, forwards, upside-down, and sideways. Get comfortable with x and y, and all the other variables.

Make up equations and solve them. Practice solving systems of two equations for two unknown quantities. Generalize that to N equations with N unknowns. Find unknown quantities in terms of other abstract symbols.

Solve quadratic equations, factor polynomials, learn to manipulate trig functions and logarithms. If you’re feeling really ambitious, play around with complex algebra– let i be the square root of -1, and go to town.

Yeah, there’s plenty of calculus in physics. But calculus, we can teach you in college. It does you no good to know how to take a derivative if you don’t know how to do algebra once you’ve got the answer.

The top-of-the-list, number-one-with-a-bullet way that students go wrong in introductory physics is by failing to do algebra correctly. If there’s one thing you need to know going into college, algebra is that thing.

Sunscreen is overrated. Stay in the shade, and learn to do algebra.

1. #1 dean
June 9, 2009

“The top-of-the-list, number-one-with-a-bullet way that students go wrong in introductory physics is by failing to do algebra correctly. If there’s one thing you need to know going into college, algebra is that thing”

Poor algebra skills will haunt students in pretty much every technical area: statistics (which I teach), finite math, etc. Excellent post.

2. #2 Brandon Goodell
June 9, 2009

Absolutely. Just not the pure-math kind of algebra – entirely different animal.

I’ve always told people I’ve tutored – if you are wrong on a physics problem, it’s not because of the calculus, but the algebra. Foil foil foil

3. #3 ScentOfViolets
June 9, 2009

Yep. And the number one thing calculus students get wrong is . . . algebra. I’ve had students attempting to, say, integrate a six-term polynomial over a four-term polynomial(usually with a Hail Mary integration by parts after a u substitution fails) without once checking to see if there is a common factor. I get the impression that they think that calculus is ‘important’, while algebra is just that stuff they had to learn three times over in high school.

4. #4 ScentOfViolets
June 9, 2009

Yep. And the number one thing calculus students get wrong is . . . algebra. I’ve had students attempting to, say, integrate a six-term polynomial over a four-term polynomial(usually with a Hail Mary integration by parts after a u substitution fails) without once checking to see if there is a common factor. I get the impression that they think that calculus is ‘important’, while algebra is just that stuff they had to learn three times over in high school.

5. #5 ScienceWoman
June 9, 2009

Echoing dean. Algebra is essential for all of the sciences. It slays me when my students get problems wrong (or, worse, refuse to attempt them) because they can’t do the algebra.

6. #6 FuSchmu
June 9, 2009

It’s the same way in ecology, except I think that we fight an even more uphill battle. While the physics students have likely resigned themselves to the fact that they’ll need to know have a handle on algebra and calculus, the ecology students often seem to have picked the subject because they believed it to be a mathless field. They’re always very distraught when I begin teaching population dynamics and they discover that they’ll actually need the math they’ve tried desperately to forget.

Anyway, I totally agree that algebra is an important tool for students in all of the sciences. The hours devoted to mastering algebra will be well spent.

7. #7 Uncle Al
June 9, 2009

American education is a confabulating disaster and algebra is the rotten heart of it. By insisting there exists a singular objective solution to a problem, 40 years of enlightened social policy is swept away like flushing a toilet. By insisting there exists an empirical solution to a problem, the whole of religion is gainsaid. By insisting there exists an achievable solution to a problem, professional management is rendered superlfluous.

Algebra must be vigorously banned and quashed. Algebra denies every voice has equal merit. Algebra denies salvation can be achieved through penance. Algebra denies truth is negotiable. Algebra is hate language. (1 + 1) can never be allowed to equal 2.

4 – 10 = 9 – 15
4 – 10 + 25/4 = 9 – 15 + 25/4
Write sides as complete squares,
(2 – 5/2)^2 = (3 – 5/2)^2
Take the square root of both sides
2 – 5/2 = 3 – 5/2,
2=3

Truth and morality are defined by convenience of the moment. That is why unmarried teenage single mother Bristol Palin is the primary spokeshominid for “no sex through abstinence.” The Church of Rome is built upon that simple concept – and never wavers.

8. #8 Abby
June 9, 2009

Yup. I see this all the time with psychology students, trying to get through stats class. (Not just undergrads, either, sadly, although the grad students have usually resigned themselves to gritting their teeth and struggling through. It helps that by then we are doing very little work by hand.) Like FuSchmu’s ecology students, psych undergrads seem to labor under the belief that there would be no math here – or worse, that there are fields one can enter that do not require algebra.

9. #9 joemac53
June 9, 2009

Mr. High School Math/Physics teacher is going to quote you around town. They always badger me about these things, and I tell them they have to speak the language. Algebra is spelling and grammar.

10. #10 Alistair Wall
June 9, 2009

(XML|sunscreen|algebra) is like violence: if it doesn’t work, you aren’t using enough of it.

11. #11 Bill
June 9, 2009

Wow. Uncle Al (comment #7 above) may have attended a lot of English classes, but obviously missed Algebra class quite a few times. Perhaps he was demonstrating the point of this article for us?

Did Uncle Al actually work through that problem on paper first? Because it makes absolutely no sense the way he wrote it. Square and then square root? That did absolutely nothing according to the work he showed.

12. #12 dean
June 9, 2009

All Uncle Al has done (besides reinforcing the fact that he never has anything concrete to add) is neglect to place absolute values around the left side of his “work” immediately after taking the square root (he needs to have 2 – 5/2 inside absolute value).

Whether he came up with this on his own or copied it from somewhere I couldn’t say.

13. #13 Eric Lund
June 9, 2009

Bill: you fell for the troll. “Uncle Al” resides under yonder bridge, waiting to grab the unwary. The logic error in his post is exactly the sort of thing Chad referred to in today’s dog dialog as a “2=1” proof: assuming that because two numbers have the same square they are equal.

14. #14 Uncle al
June 9, 2009

(2 – 5/2) is -1/2, hence the invalidity of the “proof”. Learn some algebra. A civilization whose technology is engineering and whose social policy is bullbleep cannot sustain. Eventually the bleep from the latter is put in charge of the former and it all comes crashing down. The future arrived in late 2008 and now there will be no getting shut of it.

Uncle Al scored 750(verbal) + 750(math) = 1500/1600 on the Graduate Record Exam when it identified the functionally competent as opposed to the politically convenient. When Galileo brought Sidereus Nuncius to Tommaso Baglioni (Thomam Baglionum) in late February 1610 it was a death warrant for Church of Rome Aristotelian dogma. The dead refused to die but were no more alive for it.

15. #15 E. Lawlor
June 9, 2009

Help Wordniks! How do you spell canceled (cancelled)?????

16. #16 E. Lawlor
June 9, 2009

Help Wordniks! How do you spell canceled (cancelled)?????

17. #17 mpatter
June 9, 2009

It’s just so true. I was lazy about algebra in school, at the age of 17 or so when I should have been getting fluent with it. Six years and a degree later, I’m doing physics research, but I’m still paying in blood for being slow at algebra.

18. #18 dr. dave
June 9, 2009

I give the students the example of my 4 year old kids learning to read. They see a word like “bigger” and they have to sound it out phonetically… B…II…G ERRRR… but you and I don’t have to do that, we look at it and READ it all at once.

I tell my students that most of them are in the same place with algebra that my 4 year olds were with reading. A lot of them aren’t quite literate yet. So they see 4x^2 = 16, and they have to work through a bunch of steps to find the answer. When you can look at that and say “x=2”, then you are algebraically literate.

Then I tell them that trying to get students who are still doing algebra “phonetically” to learn physics or calculus is like trying to get a 4 year old to read Shakespeare. They’ll spend so much effort sounding out the words that they won’t have any hope of understanding anything they are reading.

19. #19 Dennis
June 10, 2009

Also, if you plan on studying science, you won’t see much sun anyways.

20. #20 meichenl
June 11, 2009

Aren’t we shooting pretty low if we tell future physics students to focus on algebra in high school? You should pretty much have algebra down in middle school, at least in terms of solving quadratic equations and “finding x”. Then you can reinforce in by using it constantly to do analytic geometry, trig, and calculus in high school. Ideally by the end of high school, you’re starting to understand how algebra lets you understand geometry better, and how geometry lets you understand calculus better, and in general how each new bit of math you learn can fit into the picture of things you already know.

If someone had entered my undergrad program feeling confident because they were good at algebra, they wouldn’t have made it through even the first two years’ core math curriculum. Rather than algebra, I’d like to see high school students do well in geometry, or possibly even receive an introduction to some basic ideas from abstract and linear algebra in high school, so they get used to handling math (as opposed to cranking computations).

At least, those things would have been more beneficial for me in high school than the courses I did take senior year (which were ostensibly linear algebra and differential equations, but were basically cookbooks of algorithms to mindlessly crunch through when certain types of problems were presented).

Without some more abstract mathematical thinking in high school, it was very difficult for me to understand why anyone would want to prove the “obvious” intermediate value theorem, or why you’d play some funny game with deriving properties of vector spaces from a set of axioms. I had to go back later in college and re-learn all the of freshman/sophomore math I didn’t pick up on the first time through, since I wasn’t prepared for that level of discourse.

I also think it would have been helpful to have had more high school preparation in understanding where approximations are useful. Almost every day I make approximations like 1/1+e = 1-e, sin(x) = x, (1+e)^2 = 1+2e, etc. You can be staring at a completely intractable problem for a long time if you don’t know how to look at things to first order, and know when it is and is not valid to make such approximations. I think physics would have been easier if, as an incoming freshman, I’d have been able to do something like

(101)^1/2 = (100 + 1)^1/2 = 10(1 + .01)^1/2 = 10(1 + .005) = 10.05, and then been able to approximate the error as less than -1/8(.01)^2

I hope this response doesn’t sound elitist, but it seems like you’re going to have a very rough time if your goal is to become a physicist, but you only get as far as algebra in high school. It might be something a lot of students don’t do, but most students struggling through introductory physics aren’t trying to major in it, at least in my (admittedly limited) experience.

June 11, 2009

Aren’t we shooting pretty low if we tell future physics students to focus on algebra in high school? You should pretty much have algebra down in middle school, at least in terms of solving quadratic equations and “finding x”. Then you can reinforce in by using it constantly to do analytic geometry, trig, and calculus in high school. Ideally by the end of high school, you’re starting to understand how algebra lets you understand geometry better, and how geometry lets you understand calculus better, and in general how each new bit of math you learn can fit into the picture of things you already know.

It may not be the ideal situation, but having students enter our intro classes being really comfortable with algebra would be an improvement over the current situation. As I said in the post, the number-one most common failure mode I see on homeworks and exams is a failure to do algebra correctly– factoring 1/(a+b) into (1/a) + (1/b), and that sort of thing. That happens even in the honors section, where we try to get all the prospective majors together.

22. #22 ScentOfViolets
June 11, 2009

Agree with Chad, especially the part about the so-called honors section. Part of the problem may be that the subtopics of algebra are covered too quickly in high school; and covering algebra too quickly twice in a row, even if the material is exactly the same, is not substitute for a single two-year course.

The other problem may be(at the high end) a certain lack of theory. While a lot of this stuff is only going be mastered by brute repetition – I like to compare it to playing scales or running the same pick option over and over – the brighter students seem bored without the theory. Why is zero times anything zero? Why is division by zero not allowed? Really?

23. #23 MRW
July 23, 2010

I year late, but I don’t think I saw this until Chad referred to it today. I just wanted to chime in and say this goes for chemistry, too. It’s frustrating that the problems students in my 101 class have are so often related to algebra rather than chemistry.

Relatedly, I read a study once on how taking AP courses in high school affected grades in science and math classes in college. If I remember correctly, taking AP bio didn’t affect college chemistry grades, taking AP chem didn’t affect college physics grades, and so on. Or, at least, it was a small effect. The only large effect was that taking AP Calc correlated with improved college science grades. I suspect it was less about the calculus itself than the algebra underlying it.