Other useless school trivia: the quadratic formula

I have two young boys and I have decided to pay attention to what they are learning in school. Beside basic writing and reading skills, mathematics feels like the next most important academic topic. I realize that relatively little has changed in 30 years regarding how mathematics is taught.

As a young kid, I liked mathematics but it became less enjoyable in high school. After a time, I learned to love algebra as it made it possible to solve difficult word problems more easily. To this day, I still do algebra to way I was initially taught, and I often get the right answers.

Later in my schooling, I came across the quadratic formula. That was quite a disappointment. I will not even reproduce it here because it is an ugly hack. Algebra all made sense up to this point, and then we had to memorize a complicated formula. I was not impressed and there were many students to complain about it too. I attended a Catholic school at the time, and I remember that the teacher told us to take it on faith.

Relying on faith to do mathematics did not square well with me. Maybe you can guess that I never did memorize the quadratic formula. Indeed, it is almost trivial and much more satisfying to derive it from first principes. Our teacher was kind enough to rush through the derivation on the blackboard once: it was too fast for any of us to understand, but sufficient for me to figure it out on my own later. How could he derive it at first and then ask us to take it on faith? I guess he thought that having us learn to figure it out on our own would take too long.

I would argue that there is little sense in having people who cannot derive the quadratic formula, memorize it. If they cannot derive it, it is very unlikely that they will grow up to become adults who use algebra in a non-trivial manner. And if they are never going to do algebra, why would they need the quadratic formula?

I think that most people at ease with algebra can derive the quadratic formula quickly. The first thing to note is that there is no reason to find the roots of a x2 + b x + c. You should divide throughout by a.

So, really, all you need to do is to figure out the roots of x2 + b x + c. It is good enough. It is already a more elegant problem.

Then you just need one more trick… it is called “completing the square” and it says that x2 + b x = (x + b/2) 2b2/4. Of course, there is no need to memorize the completion of the square formula…

From this, solving for the quadratic formula is easy… you start from…

x2 + b x = –c

and you complete the square…

(x + b/2) 2 = –c + b2/4

And that is all!

Admittedly, it can take me longer to solve for the roots of a quadratic polynomial using this derivation than someone who has memorized the formula. I would guess that it takes me an extra 5 seconds. But how often do you have to solve for the roots of such a polynomial without the assistance of a computer?

I will also reiterate my argument against memorizing multiplication tables… Reasoning out that 6 times 8 is 48 is more important than the fact itself. Similarly, figuring out the binomial formula from first principles might help you approach a wider range of problems with confidence…

To be fair to contemporary education, the American Common Core seems to recommend doing exactly as I did as a kid regarding the quadratic formula.

Of course, the quadratic formula was only the first of many disappointments to come. Next we learned trigonometry, and then we had to suffer through analytic geometry…

The pattern would repeat itself endlessly: “You have to learn that that the square of the secant minus the square of the tangent is 1 [replace by your favorite piece of trivia].” Then I would find a way to “route” around the problem. For example, in trigonometry, I eventually figured out that most things could be derived from Euler’s formula. That formula itself was intuitive enough if you could remember that the cosine and sine give you the coordinates of a point on the unit circle given an angle. In fact, almost all of trigonometry can be derived from this simple observation.

We still insist on presenting mathematics as a collection of definitions, facts and routines to be memorized. Thankfully, the emphasis is somewhat lesser than in my days… but if you scratch beyond the surface, too little has changed.

Evidently, the purpose of such mathematics in schools is primarily to rank students. It is not very different from the Imperial examination of the Song dynasty. We are teaching to the test because only the test results matter… the test only serves to separate the “good students” from the bad… under a pretence of “merit”.

And then, wait for it, we find out that many of our graduates lack skills that would make them employable! It does not matter because employers only look for degrees and diplomas… except when they do not or cannot.

My boys love to play Minecraft. They build all sorts of crazy devices. By my estimation, this is probably a better preparation for the “real world” they will face in 10 years than the math that they are learning in school. Sadly, I am quite serious.

Further reading: Several people have pointed out that many of my comments are related to a Mathematician’s Lament by Lockhart.

11 thoughts on “Other useless school trivia: the quadratic formula”

  1. What terrible word: trigonometry, geometry, algebra ).

    When I was in school, I liked math, then algebra and geometry. But then, I became interested in the computer much, and then, due to a large Number of missed lessons behind the school curriculum.

    When did you start to analyze the curriculum you like what you see?

  2. Working out from first principles may be better, but it’s a lot slower. Memorization is a tool that frees up some mental CPU cycles, so you can work on even harder problems.

  3. Daniel, I agree that memorizing the quadratic formula is useless and
    inflicting it on students instead of showing them how to derive it is
    a crime. But, as you noted in passing, current mathematical education
    thinking tries to move away from such memorization towards
    understanding. If you research current mathematical education
    thinking, I suspect that you will find much to agree with.

    The question of why is education not moving fast enough in the
    direction you (and many others) suggest, is mostly the result of
    inertia, both from the parents (hence the politicians) and the
    teachers, not the researchers in education. For anecdotal ‘proof’ of
    the former claim, search FB, for instance, for parents who will
    publicly and strongly berate any teacher trying to get their beloved
    progeny to understand (not memorize but understand) our algorithms for
    addition, or multiplication, by, among other techniques, comparing
    them to other algorithms. Typical reaction: “Why does my kid need to
    know that. I never learned it.” (As if what they learned was the
    pinnacle of education!)

    And as ‘proof’ of the latter claim, consider, for a moment, who
    becomes teachers in the US. Not the brightests nor the bests. In the
    US (in stark contrast to Europe), teachers are not respected
    (understatement of the day) and therefore, most who go into the
    profession do so after other doors have closed. Who am I to say this:
    a mathematics professor who has been teaching future teachers for
    decades. If you ever go to a parent teacher meeting, ask your son’s
    math teacher to derive the quadratic formula (or any other tidbit
    forced onto your kids) When he fails to do so, you will understand why
    he will not teach how to derive it. (At least this is what would
    happen, most of the time, if you did this experiment in the US.
    Although a Canadian, I have left the country years ago, so I cannot
    say for certain that the reasons are identical in Canada, but I
    suspect they play a role.)

  4. The problem with what you’re saying in this series of posts is that you’re much more adept than most at algebra and far more adept than most at algorithmic approaches to problem solving. It’s part of why you’ve got your job and your interests and you learn new programming languages the way you do.

    Teachers in high school, and lecturers in university teach as a statistics game in most situations. The largest number of students learn this item on the syllabus most effectively this way, so we will teach it this way. About half of the students that don’t like this technique will have another perfectly good technique (like you do) that they remember just as well. The other half we’ll spend some extra time with if we can to try and help get up to speed.

    And while I agree the specific skill of solving a quadratic equation without a computer may be something a small subset of people use regularly, school has the job of preparing people for those careers as well. It’s also a nice concrete example of a skill to manipulate polynomial equations for the first time, a skill which is going to get built on and used a lot in maths, physics and engineering.

    There is a legitimate debate about the entire structure of education and the curriculum in every country. I wouldn’t start from where you are though. For example, why do we rely on exams as an assessment of student ability when their ability in a job will never be anything like their ability to perform in an exam? That seems like a much bigger target than you’ve gone for and just as legitimate as picking on little bits of the maths curriculum that you’ve learnt and remembered in a different way.

  5. @Eloise Nice point about exams, which I totally agree to. However, I see two separate (though related) approaches (yours and Daniels, where Daniels is mainly the one from the previous two articles) we need to take in parallel, not in any order.

    @Daniel, while I fully agree with your points on the previous two memoization posts, I only partially agree with this one for several reasons

    (1)
    I faced education in german high school (Gymnasium) up until 2011.

    In the earlier grades used to get somewhat bad (many 4s on a scale from 1-6 where 1 ist best) marks from other teachers because I didn’t want to (add|multiply|divide|cancel) giant fractions with 5-10 numbers of 6-8 digits each without a calculator. Others got 1s (best mark) because they memoized (as you call it).

    I didn’t care becuase math was fun to me since I was 3 while school was no fun. Whatever I did in school was not math for me but rather some strange thing that had to be done at the time.

    In grade 8 or 9, when we were taught solving quadratic polynomials, I had a teacher who didn’t care about parents complaining that their children got bad grades because he asked for understanding and not for memoization in the exams. He introduced the concepts in math by deriving them from whatever we learned before.

    Those who had 1s before suddenly got 5s or even 6s (which are generally quite rare these days) while I scored almost exclusively 1s and had some fun even in school.

    However, of the 28 people in my class, almost no-one besides me listened at all, because it was too complicated for them. They did not learn something from this at all.

    You might argue that they are not interested in the inner workings of algebra etc. While this is certainly true, it has a reason. They don’t see a real-life application neither for solving quadratic polynomials nor for understanding the inner workings of something they most certainly won’t ever use.

    This is exactly the same argument you made in comment 11 here
    http://lemire.me/blog/archives/2015/03/09/the-mathematics-we-teach-our-kids/#comments
    about solving polynomial division problems in [Insert your favourite math software here] rather than manually.

    At the time I actually had a Casio calculator that could solve an equation for a variable.Everyy time I had a real issue involving quadratic polynomials I used that instead of any formula or algorithm (computers are usually not allowed at school).

    I conclude that while we might share an interest in inner workings of algebra (yours might extend more than mine, however), most people do not and that’s for a reason. That goes for both “completing the square” and the formula mentioned in your article. The effect of this is that they just waste time by not listening to mathematics around quadratic equations (you can probably generalize that).

    That being said, I don’t believe that general education should be generally oriented towards “us” (i.e. those who are interested in those inner workings) but try to teach useful skills in a fun way to the maximum number of people possible.

    (2)

    In german high school, we usually learn both “completing the square” and the formula. In the exams we usually can choose which algorithm to use.

    I have seen a lot of exams, but no-one who learned the formula used “completing the square”. This goes for both the “normal” people and the ones (like “us”) who are actually interested in algebra.

    Why? Because even IF you can do “completing the square” (which is not always the case) it involves significantly more (useless) effort than the formula.

    Given the current fact that you have to solve the equation using pen&paper without some miracle solve-equation tool it is quite hard to justify using the more complicated of two algorithms which has no advantage over the easier one.

    While “completing the square” might be nice as a teaching tool (compare to http://lemire.me/blog/archives/2013/02/11/the-big-o-notation-is-a-teaching-tool/) I consider it utterly useless for actual tasks of solving equations — just as [the skill of] solving equations using pen&paper is utterly useless [except maybe as a teaching tool] if you have Wolfram Alpha (or similar) available.

    I conclude that people will naturally choose — for any given task — the one associated to the least effort from the set of algorithms deemed fit for the purpose of solving the task. Completing the square it not one of them if you know the formula, therefore it serves no purpose once you have either learned the formula or an equation solver available.

    While I used to argue that people should now what a polynomial is and how to solve it, I am not so sure of that any more. Today, I believe that this is irrelevant not only because (as you say) maths teaching currently is about teaching definitions (which are irrelevant), but mainly because people only listen if either a) the concept seems absolutely relevant to them or b) they are actually interested in the subject itself.

    While polynomials are a somewhat extreme example — many of my classmates will actually remember what a polynomials is even if most have never used the concept after school — I am a strong advocate of teaching concepts only after you presented a real-live usecase that can not be solved by the pupils without knowing the concept.

    Implementing this approach would IMO result in polynomials not being teached in school. Once you are free to start teaching useful things in school, you won’t have any issue quickly introducing concepts like polnyomials for mathematicians and physicists in university — because those are the people that might actually have a use for this concept.

    I believe that teaching stuff without presenting real-life use-cases is at least one of the roots of the educational evil. Without using a concept people will forget it if they comprehend it at all. Without having a usecase in the first place, it is overwhelmingly likely that they simply don’t care about it. Exams do not seem to be a valid usecase for me.

  6. All this (and your previous post) begs the question: shouldn’t we really rethink what should be taught now that your smartphone will answer most of these questions if you (literally) ask it.

    Isn’t it time to focus on why we want to solve the quadratic equation instead of how. As someone once said: I don’t need to know what is under the hood.

  7. @Jan

    Isn’t it time to focus on why we want to solve the quadratic equation instead of how.

    …. which is why I think we should focus on solving interesting word problems…

    This being said, 99.99% of the adult population never needs to solve a quadratic equation…

  8. @Eloise

    you’re much more adept than most at algebra and far more adept than most at algorithmic approaches to problem solving. It’s part of why you’ve got your job and your interests and you learn new programming languages the way you do.

    If I think that what was taught to me was absurd, too complicated, boring… and I was an A student… can you imagine how the bottom of the class felt?

    It’s also a nice concrete example of a skill to manipulate polynomial equations for the first time, a skill which is going to get built on and used a lot in maths, physics and engineering.

    I do not object to algebra at all as it is a generally useful language. It is hard to use spreadsheet software (like Excel) fully without a working knowledge of algebra. It is also quite reasonable to learn how to solve simple equations without a computer…

    Some students, the interested ones, should learn how to solve quadratic equations, and as I have demonstrated, it is almost trivial to do so if you have a working knowledge of algebra…

    For example, why do we rely on exams as an assessment of student ability when their ability in a job will never be anything like their ability to perform in an exam? That seems like a much bigger target than you’ve gone for and just as legitimate as picking on little bits of the maths curriculum that you’ve learnt and remembered in a different way.

    It seems to me that if the curriculum is ridiculous, as it often is, the relevance of the grades is obviously in question… no matter how you get to the grades. But who says that grades matter? It is not like most people will even attempt to get admitted to Harvard.

    I am a lot more worried about the 30%, sometimes 50%, of all boys who drop out of high school because it is too boring.

  9. @Uli

    I also got bad grades some of the time in primary school. I failed kindergarten. Then I did much better in high school, routinely finishing first.

    Interestingly, many academics believe that grades are a direct consequence of your IQ which is, itself, determined by your genes. You would think that you could not go from last to first in your class within a few years…?

    It seems obvious to me that it is hard to be smart when working on problems that do not interest you.

    However, of the 28 people in my class, almost no-one besides me listened at all, because it was too complicated for them. They did not learn something from this at all.

    I am not sure why we teach most kids how to solve a quadratic equation… seems pointless. But if you are going to do it, I would argue that deriving the solution from first principles as I have done is more instructive than memorizing the formula.

    I have seen a lot of exams, but no-one who learned the formula used “completing the square”. This goes for both the “normal” people and the ones (like “us”) who are actually interested in algebra.

    I suppose that if you are going to be tested on it, you might as well memorize the formula for the duration of the tests, and then you can quickly forget all about it.

    Still feels pointless though.

    While “completing the square” might be nice as a teaching tool (compare to http://lemire.me/blog/archives/2013/02/11/the-big-o-notation-is-a-teaching-tool/) I consider it utterly useless for actual tasks of solving equations — just as [the skill of] solving equations using pen&paper is utterly useless [except maybe as a teaching tool] if you have Wolfram Alpha (or similar) available.

    Of course, it is useless in practice… no argument there.


    I believe that teaching stuff without presenting real-life use-cases is at least one of the roots of the educational evil. Without using a concept people will forget it if they comprehend it at all. Without having a usecase in the first place, it is overwhelmingly likely that they simply don’t care about it. Exams do not seem to be a valid usecase for me.

    I agree…

  10. I love math, and I attribute that to my inability to pay attention in class. I was daydreaming about math rather than letting them beat the love out of me.

    Terrible grades, but I still love it.

  11. Generally I agree with you. I never use any trig except the visualization of the sine and cosine wave and its relation to coordinates of a point traversing a unit circle on the origin. 5 minutes of instruction, 10 minutes tops, using the right aids, like a 4-stroke engine crank shaft. On the other hand, trig proofs were a good introduction to proofs and logic in general.

    I differ with you on memorizing the times tables. I use them pretty often and having it in my head helps.

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