August 04, 2012

Is Algebra Necessary? Of Course It Is!

Last week an op-ed piece--"Is Algebra Necessary" by a Poltical Sciences Professor Andre Hacker--was published in the New York Times (link here).  The thesis of this note was that schools should stop teaching algebra as a matter of course, that algebra is really not necessary for most careers anyway, and that poor performance in algebra is having a terrible impact on high school and college graduation rates.

This article went somewhat viral:  securing over 450 online comments at the NY Times site, and was noted in newspapers, online media, and radio/TV stations around the nation.  I must have gotten a few dozen emails about it and talked to quite a few of the local media about it.

I have two major points to share.  First, the idea of dropping algebra is one of the most foolish ideas I have heard in a long time.  Second, the fact that the media took this wacky proposal seriously says something about where we are as a nation...and I will let you judge whether that is good or bad.

Algebra is fundamental for virtually all technical subjects and careers:  engineering, science, medicine, and technology.  It is required for finance and business.  The list is a long one.

Now even if we made the huge assumption that the only use of algebra is for technical and math-oriented careers, there is an obvious reason why we need to teach this subject to virtually all our children:   we don't know in junior high or even high school which kids are going into such careers!   If we deny a child access to this subject we are determining that they NEVER will be able to go into a technical or financial subject.  Parents--would you accept this?   Would a student like their future circumscribed in such a way?  Of course not.   We live in an increasingly technological society and many of the careers of the future require math competence.  It is inconceivable that we would deny our children the ability to participate in this new world.   Our international competitors  (e.g., Chinese, Japanese, Germans, to name only a few) are pushing their kids to achieve in math--they laugh at the intellectual disarmament that hacker suggests.  Today, foreign nationals are taking over the technical and mathematical subjects...even in our own universities.  Folks, this is bad, real bad.  Because our nation will not be able to swipe the cream of the worldwide talent pool forever.

But algebra is more than a foundation for key careers of the 21st century.  It is one of the prime achievements of our civilization and one that has greatly contributed to our ability to describe mathematically the world around us.  It is no less important that Shakespeare or Homer.  (any often do we use classic literature in our daily lives...would anyone suggest dropping these subjects?)

Learning math and algebra is a central part of the education of every student.  Not only do they learn how mathematics can describe the world around them and can do so many useful things, but they learn to organize ideas in a logical, ordered way--extraordinary training for anyone, of any career direction.

Hacker makes a big point that a lot of students are not gaining mastery of algebra and for many it is a frustrating experience.   But that is more of a measure of the failures of the U.S. math education enterprise, not the inadvisability of teaching algebra.  The U.S. math education establishment decided, with no empirical evidence, to switch math education to a "discovery" approach roughly 15-20 years ago.  It has been a complete failure, with many schools now switching back to the direct instruction approach used by the nations best in teaching math.  Seattle is a prime example of the failure mode, with poor math curricula at all levels.  We need teachers who have deep knowledge of the math topics they teach, which is often not true in elementary and sometimes middle school.  In short, we need to fix the basics.   But the basics seems to be ignored by many districts and education boosters (like the Seattle Times, Wa League of Education Voters, Gates Foundation) who push quick-fix panaceas (charter schools, Teach for America, restricting teacher flexibility, etc.) that don't deal with the real problems.  Other major culprits are the education schools, who push trendy, theoretical approaches rather than making sure new teachers know their subjects and effective approaches to teaching.  

If students aren't learning algebra well, you can drop the subject--as Hacker suggests-- or learn how to teach it better...which would you prefer?


  1. I wonder what the ancient Arabs and Greeks would think of this? They realized that the language of mathematics describes the natural world. Without a basic grasp of mathematics, a person would be truly ignorant.

  2. My daughter teaches English to pre-school children in Korea. One of the students has demonstrated, to her, on the board, the fundamentals of algebra. He is seven years old (by US reckoning).

    Further, in the math segment of the class, my daughter uses a US 2nd grade textbook, which the students complain is too simple.

    It is not the case that algebra is hard. It is the case that we're simply not trying to teach our children.

  3. Sorry, Cliff, but you're totally off base on this one. I'm now retired but had a very successful business career, managing large amounts of information and data. I never once - not once - needed to use algebra in that career. And I was an A student in algebra - what a waste!

  4. I am glad you commented on this subject. I read the article and was incredulous. I would prefer better teaching, of course. We bailed on the Seattle Public Schools a long time ago and I am sad about it but I'm not sacrificing the quality of my child's education. I would argue that algebra is vastly more important than Shakespeare, but that is another story (Shakespeare is but one amazing writer). I appreciate the thesis about not limiting the career path of a 14 year old. When I was 14 I barely passed algebra but later went on to take it in college and became a physician. My husband took math and physics while majoring in theater and was able to switch his career to engineering after finishing college. it is critical to keep your options open. A 14 year cannot predict her ultimate carrier choices! Algebra can be taught to our kids -- we just need to make it happen.

  5. I read that too, and my first thought was: "How would one be able to run a spreadsheet like Excel if they didn't know at least some algebra.

    My second thought was that Hacker should be sacked. How can anyone that clueless be a professor?

  6. Great article. It is deeply upsetting that dropping Algebra could be taken seriously by anyone. Doing/learning math is good for your brain, and for kids who are developing those brains it is extremely important that they are learning math. It feels like Hacker himself lacks some critical thinking, and should head back to school himself.

  7. Holy smokes batman, algebra should be considered part of a classical education. This is absurd. I'm at a loss of words to discribe my outrage at this absurd notion. Part of the problem with students inability to handle algebra is because of the same absurd ideas about any technical subject or any subject that isn't immediately easy to everyone.

  8. I'm struggling to think of a skilled pursuit that does not use skills taught in an algebra class.

    What is algebra, but a system for describing real things in terms of abstract placeholders that can be manipulated so as to produce new knowledge?

    Here are some other "opinions", analogous to, "Algebra is useless, because how often do I have to factor polynomials in my work?"

    1) Weightlifting is useless, because how often do I have to pick up a barbell in my work?

    2) Reading the classics is useless, because how often do Dostoevsky's protagonists come up win my work?

    3) Biology is useless, because how often do I have to dissect a frog in my work?

    Algebra's detractors are conflating the learning process with the skill being learned.

  9. I got an A in high school algebra. If you asked me today to solve an algebra problem, I wouldn't be able to do it because I haven't used it since that class almost 50 years ago. I've had 3 careers: in nursing, public relations and landscape design. Algebra never figured into the practice of any of them.

  10. I use numerous flavors of math daily as a physical scientist, but 99% of the rest of the population never has to evaluate an integral or perform a derivative - why torment the large fraction of people who would rather be learning to write well, learn how government works, or learn some foreign or computer languages?

    I've been amazed for years that math remains a required subject for so long, with 3 or 4 years in high school required to graduate and gain admission to many colleges. The level of detail and repetition in my daughter's homework problems on the trajectory of a batted baseball were mind-numbing (even if she had any interest in baseball), for just one example, when she would rather have a creative writing class or a third language.

    Of course students aiming for careers in math, science, finance, or several other directions need more, but requiring lots of math from ALL students wastes everyone's time and crowds out more useful learning.

  11. Completely agree that kids don't know what they're going to be in junior high or high school... I thought I was going to be a journalist. Took algebra in high school only to meet the minimum requirement for college entrance. Ended up dropping out of college, living in the "real world" for a while, then returning after deciding to become an engineer. Still needed 7 quarters of math classes to meet engineering requirements, but had I not taken high school algebra, it would have been even harder!

  12. I can cite where this "algebra isn't necessary" thinking negatively impacted my education... I entered algebra in 7th grade as one of the more advanced students. Got a C on one of my first tests, which prompted a phone call from the teacher to my mother. Teacher said I "did not belong" in algebra. No effort was made by teacher or parent to encourage or support me to succeed in math. I was told it really wasn't that important, which set the tone for the rest of junior high and high school. Ended up taking algebra in 11th grade and limped through. When I did return to school in my mid 20s, I had great community college teachers with a passion for math. My average grade in the calculus series was 3.9! We need teachers and parents to support kids in math, not discourage them by saying it's just difficult and unnecessary.

  13. It's a good thing Nurse Marie moved on to careers in PR and landscaping. Guess she never had to check anyone's dosage or convert measurement units or read a graph in her nursing job. Doesn't landscaping require area calculations and coverage/quantity estimates--even unit conversions? Algebra doesn't begin and end with finding "x".

    Businessman Barry must have had really good subordinates who could keep track of all that data for him. I suppose he had complete faith in his finance folks who had to calculate cash flow problems. Someone else must have interpreted the graphs and charts for him.

    Many people don't recognize when they are using math skills or how much they have to take on faith the work of people with math skills.

    John, the physical scientist, thinks his daughter was hampered by too much math. I think he should look at the actual requirements for HS graduation in this state. The math requirements are pretty minimal and do not include calculus as his comment implies. I'm sure his daughter could have fitted creative writing and foreign languages into her schedule--depending on how motivated she was. Oh and good luck learning a computer language without a good background in algebra.
    Paul Middents

  14. I object on principle to this discussion. Students do not have infinite time to master every worthwhile subject. Of course there is benefit to learning algebra, but is this the need with greatest marginal utility?

    As I look at the American political scene, I think a greater deficiency is the poor understanding of economics (nonsense attitudes about taxes, also the pitfalls of not saving) and politics (interpretation of events in DC). From looking at the products of students, writing is another subject requiring more effort than algebra. From within my own field, learning about the Earth and the environment should take precedence (running out of resources, what is and isn't threatening due to pollution and climate change). One can go deep without fancy math, as Cliff's blog demonstrates.

    Furthermore, math is changing. The years I spent learning analytical solutions are now largely wasted - advances in ability to compute numerical solutions, differentiate them, and graph them make most obsolete. Long division, hand calculation of square roots, the necessity of encoding Runge Cutta solutions, knowledge of assembly language for PDP-11s - all as obsolete as the typewriter.

    Curricula need updating, and I agree with Cliff about the absurd "Discovery" fad, but not solely from specialists calling for more of their specialty.

  15. The idea of teaching our children less, rather than more, is the result of poor thinking. As I tried to explain to my English students, their brains will never explode because of too much knowledge. It's never happened.

    I loved algebra. I went into the humanities and now spent my time as a fiber artist and poet. I still use algebra to figure things out. Math is liberating to the spirit.

  16. Cliff, I had exactly the same reaction when I read about that unconscionable piece. I would add that the suggested substitute (something like "social statistics") would add disaster to a train wreck. As a probabilist/statistician, I see a lot of trashy statistics "arguments" pop up everywhere and stats without a quantitative foundation is definitely worse than no stats at all - at least you won't talk nonsense "authoritatively ".

  17. Obviously algebra is necessary. But that is not the point. Having a full-year algebra class is what is unnecessary. The "barrier" between algebra and other types of math (and other life skills) is an obsolete artifact of having too many mathematicians designing math curricula for high schools. There is no reason why a 9th grade math class couldn't teach the essential elements of algebra, geometry, statistics, and even calculus in a one-year class, if they stick to what is most important and fundamental to our everyday lives. In fact, even kids destined for engineering fields would benefit from a good practical math class to start their careers. The idea of a 9th grade class on practical math could work great if the silos could be torn down between math and personal finance, social studies, and science, so those classes reinforce what was done in practical math.

  18. I am amazed at those people who never claim to use algebra since school. I can't imagine being an adult and not living within a budget, using the compound interest in both savings and home loans, calculating how many cans of paint needed for a room, the number of compost bags for a section of my garden, which sale item at the store is better, how to reduce or increase a recipe and on and on.

    Life is a story problem, and often the solutions involve algebra. Even if you are not aware of it.

  19. It's not hard to teach it better.

    I taught electronics one year at a small and rather sleazy tech school. The students were post-high-school, but most had very poor HS records. I found that they could master the basics of algebra quickly when the algebra was slipped in as the necessary method to solve the electronics problem they had just confronted in the lab.

    If math is brought in naturally as the best way to get through a frustrating REAL-LIFE problem, with definite occupational relevance, almost anybody can learn.

    If math is taught in the usual way as pure abstract theory, almost nobody will learn.

  20. I agree. Any subject that is too hard for students or might interfere with their creativity should be dropped. OMG no use teaching english anymor as texting has replaced normal speling. No use noing goegrafy bcuz who cares bout any other dum cuntry anyway.

  21. I am amazed at the people who have said they never use algebra. I have a nontechnical job, but I still use algebra pretty regularly. Budget projections, for instance.

  22. I agree -- most people DO use algebra every day, even though they'd may not recognize it as such. In fact, I'll wager that most people use calculus every day without realizing it -- every time you determine that you can accelerate to speed on the street before the oncoming car reaches you, you're solving calculus problems in your head!

  23. Paul, it isn't necessary to understand algebra to read a graph. And if you knew anything about medicine you would realize that it is not a nurse's job to calculate dosage. But even if it were, good old fashioned arithmetic would do the the job just fine.

    I agree with pdt, students should be exposed to higher math and a general course is a good place to start. If a student resonates with that, then they can go on to the next step. If not, there should be other educational opportunities offered. This one size fits all approach to education is one of the most stultifying aspects of our current system. The system should be there to meet students where they are and bring them forward. Not push them into whatever mold is convenient for those who want to teach or to cater to the conceits of those who fancy themselves "educated."

    Don't tell me that mathematics is THE language that describes the natural world. There are many ways to connect with, describe and understand the natural world. If math does it does for you, fine, but realize that you are likely in the distinct minority. Plenty of people find math the complete antithesis of the natural world. As a nurse, a mother, a grandmother, and a horticulturalist I feel very much in tune with the natural world. And algebra has nothing to do with it.

  24. People use algebra without knowng it. But teaching it requires recognition that there are two, wholely opposed traditions: formal math and practical math. Each side sneers at the other but the distinction is of vital importance.

    I use algebra, geometry and trigometry almost every day in practical situations without having any recollection of how these various equations were derived. And I didn't care at the time I took math classes.

    In other words practical math can be taught as if the formulations were useful, like a cooking recipe. But if you learn cooking by following recipies soon you begin to understand what each step means, how they're related and what the results might be as you change the process.

    Do I need to know how Pythagoras derived his theorem to use it? No. But this idea is anathema
    to classically trained mathematicians.

    The idea is to get the majority of students into the tent. Find out who the real mathematicians are who will go on to careers in engineering, etc. and help the rest learn some useful applications. Even a few of the latter will want to know more. This is a winnowing out process, not another "one size fits all" solution.

  25. Being in the IT field personally, I use some form of algebra almost every day. But even bigger than that, and a point I think you miss here Cliff, is that Algebra is typically the first real introduction students have to logical problem-solving.

    I could probably get by even in my career without knowing quite as much math as I do, but there is 0 change I could have gotten to where I am without having very solid problem-solving skills.

    Math is something that really helped introduce that to me and taught me to develop those skills. And being a great problem solver is good in any area of life. It's sad that we would even consider dropping such a basic math from our list of requirements. This country will not continue to remain a super power for much longer if we don't start doing things the right way, and if we don't find some way to separate politics from education.

    There is NOTHING more important than education. It's what differentiates our children from the children of parents hundreds of years ago. None of us would be doing what we're doing right this very moment were it not for increases in education across the board historically.

  26. << But that is more of a measure of the failures of the U.S. math education enterprise, not the inadvisability of teaching algebra. >>

    Um. Yes. But on the positive side, the Discovery math approach has opened up the door wide for tutors and great teachers to teach kids Algebra one-on-one to students. Unfortunately, ALL children should have access to great learning, not just the ones that can afford to pay extra.

  27. Not jumping in the algebra debat, thought some weather discussion would be a fun distraction.

    Saw an untypical, possibly mammatus, cloud formation today over Bainbridge Island / Rich Passage.


  28. I think the reason so many people say they don't use algebra is school trains us to think of algebra as a series of abstract memorization skills -- remembering how to factor polynomials, and such.

    In my real life I rarely have to do these things. When I do, I tend to use a computer tool to assist.

    However, I use other algebra skills all the time...for example, every time I convert from centimeters to inches, I'm doing a ratio problem. Unfortunately these weren't really covered effectively in my algebra courses, because they were too abstract; it wasn't until engineering class that I was taught how to set up ratio problems and cancel out units to make sure they matched.

    I think we spend too much time teaching the history and derivation of algebra with abstract problems, instead of showing how algebra can be applied to the real world.

  29. I wonder to what degree this is a function of compensation. Someone proficient in math and tech subjects can typically be more financially successful outside the school system. In my (foreign) experience, three quarters of those who studied math, physics and computer science education ended up as programmers.

    Those sorts of incentives leave two kinds of teachers in the system: those who could not make it in the well-paying "tech trades", and those who love teaching for teaching's sake. You want your child taught by the latter type, but both types occur.

    Now, even the proportion is difficult to estimate as no teacher will publicly admit not being in the latter group.

    But consider the situation in humanities - teaching is The good job there. Perhaps this is a big part of why instruction quality may be genuinely higher in those subjects.

  30. The most important thing you can learn in algebra is how to look at a problem that seems unsolvable, take a deep breath, deconstruct the big problem into small solvable ones, put it all back together, and marvel at what you were able to do with patience, logic and diligence.

    And that is a set of skills everyone would benefit from having.

  31. Marie McKinsey: Paul, it isn't necessary to understand algebra to read a graph.

    On this page Cliff included a graphic that says "y = mx + b". That is the equation of a line, used in graphs. Knowing how to use it was required by the elementary statistic course my daughter took at a community college... So Prof. Hacker was a wee bit off base saying statistics did not need algebra.

    When you calculate dosages, it is not just simple arithmetic because you are not just adding and subtracting numbers. You are using ratios, and an unknown. That is actually algebra.

    So, really, even if you do not realize it: you are using algebra.

    And anyone who has tried to calculate if it is better to get a 15 year mortgage versus a 30 year mortgage has used algebra. The fact that there is a very cool relationship between "e" (2.71828...) and compound interest is just a bonus.

    It is a real shame that too many people get turned off by math, and get so flummoxed by ideas like "imaginary" and "transcendental" numbers that they quit (silly names for number types that have very mundane explanations). They miss out on so much of the cool stuff. Like how often Pythagoras' theorem shows up, Euler's formula (discovered with a power series!), then there are Fourier series, and on and on... but not to infinity.

    A good layman's intro to some of this is Ian Stewart's 17 Equations That Changed the World. Another good author of general math history is William Dunham. I recently read his A Mathematical Universe. And a really fun one is Mathematical Cranks by Underwood Dudley. All of these I have checked out from our local library.

    I used to be a structural vibration engineer, but had to quit to take care of kids (especially one that has several health issues). So I actually read math type books for fun. The book in my avatar is one by Paul Nahin, his books require you have a pencil and paper on hand, his books are not "easy reads."

  32. As far as I'm concerned, Algebra is a basic math skill and some Trigonometry is not far behind. I say this simply as a carpenter. I use basic algebra and trig' whenever I layout patterns for framing a roof. The more complex the roof, the more I need to predict rise, run, angle, and intersection. Math is the tool I use to draw out what I'm trying to build, even if the realities of saws and lumber leave things a little bit off.
    Sure, many carpenters use traditional rule of thumb methods instead. Some are really good at that, but I think most have no idea what's behind the rule of thumb, and are quickly lost when unusual situations come up.
    It's bad enough that a lot of tradespeople have poor math skills. It's ludicrous, and sadly decadent, that our broader society would even consider such a fundamental math and mental skill as Algebra unnecessary!

  33. America is clearly losing the global battle for leadership in science and technology--India, China, Russia, and Israel are kicking our butts. So our response should be to quit teaching algebra?

    Kids need algebra to help their brains develop skills in dealing with abstraction and symbols. Forget about direct application to careers. As said, people don't lift weights so they can make a living picking up barbells.

  34. Slightly off topic, but my professor in an Algebra II class told us this story: he was in his last semester of obtaining a degree in history and he had to complete a
    math course. He got so interested he switched majors-to math! Not an easy decision, but he did it.

  35. Algebra, as taught, is near useless. If if was taught as formulas and calculators were taken out of the picture as well as the hypothetical numbers aka letters people would be able to relate to it.They would understand it and it may even be used in their future. I have worked in construction and have helped create 5 start up companies. Except college and high school algebra classes I have never used algebra. Mathematics is a necessity, formulas are needed in life. Algebra is too hypothetical to be useful to 98% of people.


Extraordinary Halos over the Region Tuesday Night

Before and during sunset last night, there were some amazing halos visible around the region. Here are two views from Ellensburg, the first ...