Algebra is optional

In 1999, California legislators decided that all student should pass basic algebra or an “equivalent course” to earn a high school diploma, starting in 2004. But it’s not happening. Districts across the state are getting waivers, claiming they forgot to tell students they needed algebra to graduate. Jennifer Nelson writes in a San Francisco Chronicle column:

Everyone who felt the requirement was onerous came forward to request waivers for regular students, special-education pupils, adult learners and kids in continuation schools. Nearly 5 percent of the state’s high school seniors have not successfully completed Algebra I, yet they expect to march down the aisle and receive their diploma this spring.

. . . The curriculum director for the San Jose Unified School District, for example, told the San Jose Mercury News, “The law says every student must take and pass an algebra course. That isn’t going to happen.” Why not? It’s the law!

San Jose Unified has won kudos for requiring all students to pass college-prep classes, which include algebra, geometry and advanced algebra/trig. They must be waiving the requirement for a lot of students if they can’t even get kids past basic algebra.

A teacher at a continuation school in the Huntington Beach School District told the Los Angeles Times, “I hate this requirement. The downside is for the bottom percentage of students. They’re just not there conceptually, and the [graduation requirement] is like pounding them over the head with a hammer. They were frustrated with algebra before, and now we’re just ratcheting up the pressure some more.”

Let me understand: Kids who don’t like school and aren’t particularly good students may feel pressure to learn, so we should just not ask them to meet any higher standards?

By the way, it’s impossible to evaluate the effectiveness of 19 math curricula funded by the National Science Foundation, says a report by the National Academies’ Mathematical Sciences Education Board. Evaluations of the math programs “fall short of the scientific standards necessary to gauge overall effectiveness.” So, NSF is funding math “reforms” without demanding valid studies of what works. By the time useful research is done, a lot of students will be finding algebra unpassable.

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  1. A _very_ long time ago (mid-1940s) my mother entered high school and was encouraged to take algebra. She lasted all of a day — it was too intimidating. The abstraction of representing unknowns with letters and manipulating them was just too bizarre.

    Oddly enough, she was an ace with the toughest problems I remember from introductory algebra, the terrifying “word problems”. She could do them in her head. She just had a wonderful, intuitive grasp of the process that I could never fully understand; for me, manipulating the symbols came much more naturally.

    So I have reservations about the “everyone will take algebra” mindset.

  2. Karen,

    Sounds to me as if your mother could handle algebra as long as the problems were worded a certain way.

    Letting students graduate from high school sounds like an equation for disaster. Do commenters here think that algebra is absolutely essential? I found math absolutely torturous in high school and did calculus only so that I wouldn’t have to do math ever again. 15 years later, I’ve forgotten all of my math except algebra which I use all the time, and I wish I remembered more geometry. So I consider algebra to be essential.

    But I don’t like to fall into the trap of assuming that my level of knowledge (or ignorance*) should be the standard that all students should achieve, so I’m interested to hear other perspectives, particularly less pro-math ones.

    (*This argument takes many forms: e.g., “I read all these capital-G Great Books so all children should too.” By this logic, all people should study Sanskrit since I did.)

  3. I remember none of my algebra skills. So what? It taught me to think in a logical manner and I sort of remember that. I worry that by not requiring kids to be competent in basic math, you are putting them at a disadvantage later, in terms of employment. I wonder how many of these kids are otherwise going to end up at the bottom of the employment ladder because of lack of preparation for the real world. I suspect the lack of algebra is a symptom of an even greater lack of education.

  4. Another thing people forget about the stuff they forget (=cough=) is that it tends to be much, much easier to relearn what you’ve forgotten than to learn it in the first place. I forget how to code a QR decomposition and what it’s used for, but when I need it for a project – wayhey! There’s a book on my shelf! Or, to reference knowledge I’ve not yet used on the job, I don’t remember all my kana anymore, but when I come across some katakana signs, I can usually figure enough out to determine what the sign says (well, and usually the sign has the English version on it, too, as I’m in the U.S.)

    I think algebra is a good course to require, but I think either algebra or high school geometry would do it. I don’t see why algebra should come before geometry in the curriculum — indeed, it may be a good way to introduce people to algebra by giving them drawings to look at. You’ve got two angles that together make a right angle, and one is 25 degrees – what’s the other one? I think people can figure that one out without writing down an x. Some could try figuring it out by actually drawing the diagram. But then you can show how this is equivalent to solving 25 + x = 90.

    I think both algebra and geometry are equally useful. A couple years ago, I taught my guitar teacher (who’s also a luthier) about how the Pythagorean theorem could help him figure out some designs. And I showed him some simple compass-and-straightedge constructions. He liked it because he had been reading up on the golden ratio, and its aesthetics. Then I showed him how he could construct the archimedean spiral. Really cool stuff, and it’s not that difficult to show.

  5. Most people fail to understand that higher level math (which algebra is the first step) teaches critical thinking, analysis, and problem solving.
    (all of which are relevant in the world of work).

    Having a parent suggest the kid not take algebra because they might get a low grade in it is a crime, IMO. The real benefit is just perhaps, the student learns to think on their own, and then the real learning can start to happen.

    Just my two cents worth…

  6. Our wonderful School district in DC (that’s sarcasm you can smell) has only just announced that you’ll have to successfully pass Algebra 1 in order to get a diploma starting in 2007. Frankly, how people who claim to be educators can set a standard so low boggles the mind.

  7. Cousin Dave says:

    Amritas, I do think algebra is essential. My opinion is that if you don’t understand algebra, you really don’t understand math itself. Without the ability to work out how to determine an unknown quantity from known quantities, the math user is restricted to only being able to solve “canned” problems that have already been laid out by someone else, or are identical to problems solved before. Math without algebra is a bit like what whole-language reading instruction does to reading skills.

    I’m stunned that this is going on in San Jose, right smack in the middle of Silicon Valley. Jeez, why don’t they dispense with all that hard work with classes and schools and teachers and just automatically issue every child a diploma on their 18th birthday? It’d be a lot cheaper, and the effect would be about the same.

  8. Being a high school math teacher and part time math instructor at the local college, I naturally fall on the side of more math.

    Have we so soon forgotten the space race? We are not in a space race today but if we don’t equip our students with as much math as possible then we doom them to the lower end of the job spectrum. Employers harp all of the time of the lack of math skills and critical thinking skills that our kids lack. I say push for more.

  9. In general, I support kids having a basic algegra course – as pointed out, it helps develop logical thinking skills. However, given the diversity of the human race – “by definition” this will not work for ALL kids, some will, some won’t.

    HOWEVER – how many of you feel that the following problem is appropriate for a “regular” 7th grade class (non-GATE) ? This is a real problem from a real public school here in the Silicon Valley.

    In a shop that builds and repairs bikes, there are 118 vehicles (bicycles, adult tricycles, and tandems) with 135 seats and 269 wheels. How many bicycles, how many tricycles, and how many tandems are there ?

  10. 17 Tricycles
    49 Bicycles
    52 Tandems
    118 total

  11. Mike McKeown says:

    The paragraph about not being able to judge the effectiveness of NSF-funded curricula from the published literature is interesting.

    I seems to support the the criticisms of the “Open Letter to Richard Riley”

    regarding “Exemplary and Promising” Math Programs.

  12. Algebra is incredibly elementary and should be a part of every high school graduates curriculum. How are kids passing the rest of their requirements without algebra? Didn’t they have a science requirement? I can’t imagine doing chemistry or even biology(exponetial cell growth) without a basic understanding of algebra. We’re letting the system churn out irresponsible adults who aren’t even mathmatically equiped to handle credit. Mathmatical ineptitude is just as serious a problem as illiteracy.

    Furthermore, to remain competitive with the world we need to expand the reach of calculus in high school. To make it possible for more students to be on track to be able to take calculus in a high school environment, geometry should be displaced. Pure Euclidian geometry is not particularly useful, and there are better introductions to mathmatical proofs. There’s a reason why colleges reserve geometry for the math majors. I also don’t get the point of the so-called pre-calculus. It’s all trigernometry and logarithms which become much easier to do and understand when you have the calculus foundation they are based on. A better track would be Algebra, Foundations of Calculus, and Applied Calculus wich would spend develop the reasoning for trigernometry, logarithms. Finally I would offer analytical geometry or financial calculus for the brightest students that completed the track early. Analytical calculus would focus on using integral calculs to calculate the surface area, volume, etc of real objects (i.e. this would be good prep for engineers and computer scientists). Financial calculus would focus on investments and beat students over the head with net present value. (for the future M.B.A.s) I may sound a bit crazy, but, hey, Star Trek TNG had kids doing their calculus homerwork for elementary school.

  13. Mike, that MathLand program is scary. How can you not teach kids the standard shift and add multiply algorithm?

  14. Harvey, I would much more concerned if your answer were to be found in that math class.

    (Hint) tandems has two seats and two wheels while trikes have one seat/three wheels and bikes have one seat/two wheels

    since this problem allows straightforward elimination of variables, it is not too complex (if that was your fear) here is how my 4th grader figured it out

    “only tandems have more than one seat … so 135 seats – 118 vehicles = number of tandems = 17

    lets forget now about the seats and take away the 17 tandems. I have 101 vehicles and 235 seats. All of them have two wheels and some have three. If I give 2 wheels to each of the vehicles, that would use up 202 wheels and I would have some left over. 235 – 202 = 33 wheels left over. they must all be trikes.

    so the number of bikes is 101 – 33 = 68 bikes”

    33 tricycles
    69 bicycles
    17 tandems

    The problem does speak to old chestnut about assumptions e.g. childrens tandems have 1 wheel and 1 seat (the ride=behind bike) but that’s not so likely here because then the # of seats = # of vehicles. Likewise, any academic prankster (I was guilty of that) might slyly and politely ask what if one of the vehicles were in the repair shop for missing wheels?

  15. The algebra is really easy. I don’t see where Harvey got the answers he reported, though.

    Here are mine:

    x = # of bikes (2 wheels/1 seat)
    y = # of trikes (3 wheels/1 seat)
    z = # of tandems (2 wheels/2 seats)

    Equations are:

    (1) x+y+z=118
    (2) x+y+2z=135
    (3) 2x+3y+2z=269

    Subtracting eq. 1 from eq. 2 gives

    Substituting z=17 back into eq. 2 (just pick an equation; simplest is best) and solving for x gives

    Substituting x=101-y and z=17 back into eq. 3 gives

    Since x=101-y, substituting gives

    x=68 bikes
    y=33 trikes
    z=17 tandems
    for a total of 118 vehicles.

    A bit redundant, I know, but I’m used to showing every step of my work for kids, since combining two steps into one can confuse those not comfortable with the technique. In fact, it took me at least 3 times as long to type this message as it did to solve the problem.

  16. It seems to me that discussions such as this are a tangle of conflicting good ideas, with some not-so-good ones mixed in:

    1. Good idea: A HS diploma should be evidence that the student learned certain basic skills, including algebra.

    2. Supposedly Good idea: everyone should graduate from HS, since without those skills your job choices will be terribly limited, and you definitely don’t belong in college.

    3. Fact: Some kids just aren’t going to learn algebra. They may be capable of learning the non-mathematical skills behind other good jobs.

    This creates a certain conflict, eh? But it gets worse:

    4. Terrible idea: Because we can’t hire enough competent algebra teachers, we’ll eliminate the requirement to pass algebra and try to discourage kids who might not make it from taking the class in the first place.

    5. Terrible idea: Lower standards until all our kids can graduate high school, then get them all into college where they might get a real education.

  17. The only problem with #5 is that by lowering standards in high school, the graduates won’t have the background to handle college and succeed.

    The education trust just published a study asking who starts and finishes college within 6 years (and the numbers are very interesting to look at).

    Also, another study shows that the chances of college completion go down based on the number of remedial courses required for that student (once a recent high school graduate needs 3 or more remedial courses, the chance of them finishing drops to less than 10 percent).

    I happen to be one of those persons who now believe that a quality education is needed to succeed in college, the trades, or in life in general.

  18. Cousin Dave says:

    George, you’ve got some interesting ideas there. I don’t know that I agree that Euclidian geometry is “worthless”, but I get your point: there is way too much time spent in the average high-school geometry class on things that don’t help build the foundation for what will come later. I’d like to see someone try an abbreviated geometry class (one semester) that introduces the trig concepts, and particularly the basic trig functions of sine/cosine/tangent, integrated with the geometry. I think that would put more substance on the geometry, and the converse is that the geometry helps teach the trig. I think the trig makes more sense to students if it is initially taught using Pythagoras and triangles, vs. starting with the series expansion the way a calculus class would.

    The pre-calc class I took in high school spent some time on limits, but it was also a grab bag of introductions to stuff that I’d see later in college, everything from linear algebra to numerical methods to differential equations. Now that you have made me think about it, I realize that it was a pretty shotgun approach, and I don’t think I recalled much of it by the time I took the college courses several years later, so it didn’t help much. Limits and L’hopital’s Rule can be done pretty quickly at the beginning of a calc class. I like the idea of spending the time instead going off into a few different tracks of more application-oriented math so that students get a feel for how all this math stuff applies to the real world.

    My pet math crusade is to see more students get at least an introduction to abstract algebra. It teaches one a lot about logic and about how to think about math in general (it tends to be a “head expanding” topic that can get students starting to think more like college students do, kind of like what a modern literature course can do). And, one aspect of it is Boolean algebra which is the mathematical basis for all modern computers. However, I will admit that I don’t know where I would find time to cram it into a good high school curriculum.

  19. I think requiring algebra for high school graduation is unfair to students who are intelligent but just don’t understand that type of math. Students should be given more options. A much more relevant type of math would be a finace class. Instead of abstract math, students would learn about the stock market, retirement planning, buying a home, investing, and money management. All students need to have this knowledge. You would be surprised how many students graduate college and don’t even know how to balance a checkbook. Some people cannot understand algebra no matter how hard they try or how many hours they put in studying.
    I am an example of someone who is good at math but just doesn’t get algebra. I have a 3.0 GPA in college and will be graduating next year with a double major in economics and business administration.
    I could never understand algebra. I went to tutors all through high school and got C’s and D’s in high school algebra. Then I went to college and took algebra 3 times with 3 different professors. The highest grade I got in college algebra was a D. This was with 15 to 20 hours of studying per week and tutoring 3 to 5 times per week. I would study more for algebra in a week than my 4 other classes combined.
    I then transfered to a four year university and was able to take staistics instead (which is supposed to be a harder class by the way) and got an A. I have also taken an advanced economic and business statistics course and an MBA level accounting class and did quite well in it. My point is that there are many competent students out there that just aren’t going to understand algebra and can be successful in college and later in life. Schools put way too much emphasis on abstract math.