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According to a recent experiment conducted by user Steven Gubkin, nearly one half of his students in a senior level Real Analysis course do not have any idea how to prove the quadratic formula. Is this a problem in our education of students majoring in mathematics? Or are we alright with students obtaining bachlor's degree in mathematics without knowing the proof of the quadratic formula?

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    $\begingroup$ This question seems primarily opinion-based. $\endgroup$ – Gerald Edgar Jan 30 at 17:19
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    $\begingroup$ I wonder how much of this is explained by the setting. Were these students actually sitting in their real analysis course, fresh from solving problems involving Lebesgue integrals and Banach spaces? I felt my sphere of understanding pretty compartmentalized at that point in my education, so a question like this one (or "prove the Pythagorean theorem", or "construct a regular hexagon with compass and straight edge", or any other math classic) might have seen me flounder a bit. I suspect these students will remember this experience, some taking the opportunity to memorize this proof. $\endgroup$ – Nick C Jan 30 at 17:25
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    $\begingroup$ Also, at what point in the basic curriculum is/should this proof taught? I have seen it taught only in a pre-college level course, or taught in the first term of college algebra and never again. $\endgroup$ – Nick C Jan 30 at 17:28
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    $\begingroup$ For the record, I don't think I was ever explicitly taught about completing the square, other than as a clever trick the teacher did just before telling us to memorise the quadratic formula. I only really learnt it years later, when I decided to see if I could derive the formula for myself and work out what the trick had been. $\endgroup$ – timtfj Jan 31 at 14:44
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    $\begingroup$ @GeraldEdgar I don't think that sharing the results of a quiz question is shaming. I think that, especially on a website devoted to math education, we should be able to share both student success and failures. I do think that the fact that so few of the seniors at my institution can derive the quadratic formula is concerning, and is a symptome of the fact that we do very little "backwards" integration of knowledge. We keep learning more and more new, but little integration of old. $\endgroup$ – Steven Gubkin Feb 6 at 18:38
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We cannot reduce the assessment of mathematics education to knowledge of how to prove a specific theorem. To be sure, the inability of so many students to provide a proof for such a fairly simple mathematical proposition (in the context of asking the problem of senior math majors) is troubling, and reflects the fact that we don't give students enough time to truly understand many of the key underlying ideas of what they are learning.

At the same time, mathematics as a study serves a wide population of students. If a student intended to go to graduate school in mathematics and could not prove the quadratic formula, I'd be very skeptical of their chances at success in their chosen endeavor. But I would not necessarily feel the same away about a student planning to work in actuarial sciences or engineering. Indeed, one could design a bridge brilliantly without knowledge of a proof of the quadratic formula and conversely (as I can personally instantiate), one can know how to prove the quadratic formula and have not the slightest clue how to properly design a bridge.

The fact that so few senior math majors can prove the quadratic formula brings up a lot of issues, but I don't think this single fact itself is a problem. Or certainly not the root problem. It really depends what the goals of mathematics education are and many students in the same mathematics class may have very different goals for what the course is supposed to help them achieve. If a student does not know how to prove the quadratic formula, depending on context, that could be a sign of a gross failure in their education or an indication that their studies were focused in a different direction more relevant to their future aspirations.

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Prove it, or derive it?

I had to derive it as my final task in 9th-grade Algebra I (1982). That is, I had to show

If $ax^2+bx+c =0$, then $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$.

I memorized it, forgot it, then re-learned it years later as a high school math teacher just to say I could do it, which many of my colleagues couldn’t.

Algebra and Calculus are getting less and less paper-and-pencil manipulative, which I view as a good trend generally.

That probably means that techniques like completing the square aren’t taught as much, and they certainly aren’t reinforced very much, which lets them spill out of a 21-year-old brain pretty fast. I’m not sure I could derive it without that technique.

While working at a major statistical software company, I polled our (hundreds of) statisticians, asking when was the last time they used certain things. The quadratic formula was one of those things, and it was universally unused after algebra class, with an occasional Calc I response. Quadratic firms? All day. Quadratic equation? Nope.

Overall, I wouldn’t be too concerned about it.

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The way things like this are usually presented is problematic, as is the pressure to move too fast. Many students memorize too much, and figure things out for themselves too little.

I don't have a phd, so I don't teach at that level (I'm a community college math prof), but my calculus course (1974, U of Mich) was an honors course, which made it close to real analysis. What I remember best about the course is the challenge problems Professor Piranian offered us, which required creative mathematical thinking. If this particular question were framed as a challenge problem, maybe more students would get it, knowing to give themselves plenty of time to explore.

The usual way involves division (by a) at the beginning. It is much easier when done the way James Tanton describes in his videos (though there may be an extra step for some problems, which he doesn't address). His method of completing the square does not feel like a trick to me. There are 3 videos: Completing the Square, Part 1, Completing the Square, Part 2, and Deriving the Quadratic Formula.

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I didn't realize until coming across the earlier comment thread at Are students majoring in pure mathematics expected to know classical results in mathematics very well by their graduation? that the students only had 10 minutes to do the quiz. This is a little bogus. How many students walked in late and barely had time to look at the question before their time was up? Even the full 10 minutes is not really a lot of time to set one's thoughts in order on something that is a total surprise on some random topic. I'm not sure that 10 minutes would be long enough to tie my shoelaces if it was a morning class and I hadn't yet had my second cup of coffee.

I was exposed to completing the square in 7th grade (roughly age 12 in the US system). This was the only time I ever saw the technique. I remember that it seemed kind of complicated and kind of like a trick. I didn't like learning tricks unless they seemed like they were generally useful. This one didn't seem generally useful, so I didn't bother studying it.

For many students, remembering that this was how the quadratic formula was derived a long time ago would probably be a disadvantage, because they may have just seen Gubkin's test as a test to see whether they remembered this random procedure from a long time ago. If a student didn't remember that this was how the quadratic formula had been derived for them during their distant childhood, their brains would probably be freed up to attack the problem in the obvious way, which would be simply to take the known quadratic formula, plug in, and verify that it gives solutions.

One thing I've noticed about student psychology is that the idea of finding a solution and then checking that it's a solution is not one that most of them will ever do naturally unless the problem specifically says to. E.g., I could give them a system of 3 nonlinear equations in the unknowns $(x,y,z)$ and ask them to solve the equations. They proceed to solve the equations by successively eliminating variables. At the end, none of them will spontaneously plug back in to see if it's right. This may be partly because they're oriented toward carrying out algorithms and trying to do the required algorithm correctly, partly because they don't really care about whether their answers are right, and partly because they have never received any instruction in the general topic of debugging a calculation. (At many schools, you can get an A in many STEM courses without ever solving a single moderately complicated problem completely correctly.)

It would be interesting to know whether this was a school with selective admissions, and whether the students who couldn't do this task were students who were doing well. If they're C students at a school with nonselective admissions, then this might be about the expected result. It's the kind of thing that I would expect from students who have received instruction in which they're told that the point is just to memorize problem-solving algorithms -- and many math teachers at nonselective schools teach math entirely with this philosophy. They produce people with math degrees who are competent enough to run a 7th grade math class reasonably well. This is not actually a terrible outcome when you consider that in many low-socioeconomic status school districts in the US, math is being taught by people who have no degree in math. The principal literally pulls the gym teacher off the field and puts them in a math classroom.

But of course if this is Harvard, and these students have A averages and are applying to grad schools, then these results are pretty scandalous.

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I would not be able to prove the quadratic formula if you told me "prove the quadratic formula", as I'm not entirely sure what it is (there are lots of formulas involving squares, after all).

After googling it, I know that I never once saw a proof of it, simply because my math teacher back when I was around 14-15 said that it is easier to divide by $a$ first and only memorize the $pq-$formula. If you give me the formula, I think I'll be able to prove it, but I can well understand if many students have forgotten the proof after such a long time.

So the first question I would ask Steven Gubkin is if he only said "prove the quadratic formula" or if he actually gave the formula to be proven. Next, it would be good to know how much time passed from them seeing the proof to him asking for it, how much time they were given to come up with a proof (e.g. was it a homework or did he ask in class for a complete proof on the spot). It would also be good to know how much proofs the students had to do up to that point, if they were used to writing formal proofs yet or not, etc.

Of course we can discuss about students not learning the right things for ages, but such a short message without background is not really representative.

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  • $\begingroup$ The questions in paragraph 3 would probably be more appropriate as comments to the original question. $\endgroup$ – Daniel R. Collins Feb 8 at 14:40

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