It took some 1,250 years to move from the integral of a quadratic to that of a fourth degree polynomial. When we jump too fast to the magical algorithm, when we fail to acknowledge the effort that went into its creation, we risk dragging our students past that conceptual understanding. Source.

By the time Newton and Leibniz were developing Calculus mathematicians already knew how to solve particular problems of derivatives and integration. There was a good understanding on the way to resolve problems, too. Newton's teacher, Isaac Barrow, already had some understandings of the topics Newton would push to the perfection. Source.

For me, pre-calc was plugging and chugging trigonometric, geometric, and algebraic equations. Math books follow a rather narrow and strait path across some imaginary border from the province of pre-calc into calculus. As was mentioned already, that historic transition was anything but discreet. Should Pre-Calc teach the foundations of the fundamental theorem of calculus that were known Pre-Lebniz/Newton? I think that would energize what is otherwise a boring exercise in tedious calculation.

A beautiful example of a Pre-Leibniz integration: Who realized $\int \frac 1x dx =\ln(x)+c$?

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    $\begingroup$ You'll be interested in the following article: maa.org/external_archive/devlin/LockhartsLament.pdf $\endgroup$ Commented Aug 1, 2013 at 5:29
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    $\begingroup$ @HenryT.Horton Perhapse we should put some more graphs on wikipedia =). $\endgroup$
    – User3910
    Commented Aug 1, 2013 at 5:39
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    $\begingroup$ You know, if it weren't for the answers, I'd be tempted to vote to close this question: not only is it hopelessly vague (how are we supposedly teaching pre-calc wrong, and where do we teach it like that? and who's "we", anyway?) and subjective, but it frankly reads as little more than a rant disguised as a question ("the way we teach pre-calc sucks, am I right?"). But despite all that, it's generated some nice answers, so I won't. $\endgroup$ Commented Aug 1, 2013 at 18:56
  • $\begingroup$ @IlmariKaronen The question is asking how to improve the teaching of pre-calculus. Specifically how to inspire students. It is most interesting to compare answers here with answers here $\endgroup$
    – User3910
    Commented Aug 2, 2013 at 0:13
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    $\begingroup$ I'm not saying there isn't a good question buried in your post; there obviously is, given the answers it's received. But the way it's currently written, it's almost a textbook example of the kind of question we should avoid asking here. In your comment above you say you want to ask "how to improve the teaching of pre-calculus" and "how to inspire students", but you don't actually ask either of those questions in your post; the only actual questions there are the title and the penultimate sentence. Everything else there is just soapboxing. $\endgroup$ Commented Aug 2, 2013 at 9:57

4 Answers 4


We are teaching math wrong in many, many ways.

For some reason, there has been a historical backlash against abstraction in secondary and pre-secondary mathematics. "I'll never need to use this," say so many students and parents.

As a result, what we learn is a hackneyed attempt to apply the mathematical problems to "real world examples."

So when we learn trig, we start talking about things like leaning ladders against houses, and we give students an impression that in order to do such a thing, they need to compute the inverse sin of the height over the length of the ladder.

Of course, by the time students are 14, 15, or 16 years old, they've probably seen someone lean a ladder against a house and do no such thing.

Pre-calculus mathematics is very interesting because it is the first opportunity one has to apply relationships and definitions to problems and to learn how to transform complicated problems into simpler ones. This technique is fundamental no only in mathematics, but in the real world.

At the same time, real-time discovery is often haphazard and chaotic. We don't teach biology in the order it was discovered, principally because we assumed a whole lot of extremely wrong nonsense for most of human history.

The historical motivating factors are not the same as the present ones. Newton and Leibniz were trying to solve specific problems, and they needed new math. But we've solved those problems now, so those motivating factors have perhaps lost their edge.

Instead, we should look at their process, not their imperative. We should be teaching students to ask whether there exists a meaningful relationship between a function and its slope, and whether this can have an effect on a real-world problem. We should teach students about how we can infer new properties from a handful of well-defined conditions. We should teach students to explore "what if..."

Instead, we teach students about billiard balls, ladders, and two-column proofs, as if this bears any relevance to the real world, the mathematical world, or any world. In short, we waste their time. So yes, we're teaching it wrong.

A side story: As a partial counter-example... When I substitute taught math courses, I often got classes full of "Level 2" students, which was a way of saying "remedial". They hated their class, the lessons, the work, everything. So I used to put the Navier-Stokes equations on the board, and I'd tell them that whoever could solve them would win not just $1M dollars, but eternal fame.

I asked the students to name the most popular people they knew. They'd respond with "Avril Lavigne" or "Brittney Spears" or some such.

I asked them if they knew who Sophia Loren was. Or Dom Delouise. Or Patsy Cline. They never heard of them.

I then asked them if they ever heard of Newton. Of Einstein. Of Riemann. Of Euler. They, in fact, did know their names. I told them that mathematics is one of the few ways that your mark can be left on the world permanently. That if you did something truly great, that the high school students in 300 years would be hearing your name.

Of course, none of them really thought they could do that. But it resonated with them, because it made them think, "wow yeah, I have heard of Newton." It made them realize that maybe there was something important, and that unlike being a famous world leader, it was something that was almost purely product of self-actualization.

This context gave them a glimpse into a world they didn't know existed. None of them wanted to compute how to lean a ladder against a house. They wanted to know why math was important -- not microcosmically, but in the context of great things. These were young kids with great thoughts, not computing machines.

I told those students that "yes, it does not matter if you can compute these numbers, but it does matter if you know how." That concept alone motivated them more than anything else, because it showed them that they controlled their own destiny.

For a teenage kid frustrated with school, that's a wonderful feeling. Math is one of the few fields where we can actually deliver that consistently.

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    $\begingroup$ I think the reason why "there is a historical backlash against abstraction in secondary and pre-secondary mathematics." is directly related to incompetent education majors teaching mathematics as if they knew something themselves. My proposed solution would be to require a masters degree in the subject being taught to teach high-school. $\endgroup$
    – User3910
    Commented Aug 1, 2013 at 5:35
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    $\begingroup$ Indeed. Or maybe do away with "teachers" entirely, and have education be the provenance of domain professionals. If I could teach two math classes a day in addition to my engineering job, I absolutely would. In fact, doing so is one of my biggest career goals. Sadly, the education system won't let this happen. $\endgroup$
    – Emily
    Commented Aug 1, 2013 at 5:40
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    $\begingroup$ @Arkamis I used to put a picture of a scientist or mathematician in my syllabus or webpage and make it worth bonus points if they were the first to figure it out. Then I would tell them about who Ed Witten or C.N Yang etc... was, about how their work would change the future and the present. It was fun... until that reverse image search thing ruined my game forever :( $\endgroup$ Commented Aug 1, 2013 at 6:29
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    $\begingroup$ @JamesS.Cook But, reverse image search wouldn't be possible without a good grounding in math! ;) $\endgroup$
    – Izkata
    Commented Aug 1, 2013 at 12:01
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    $\begingroup$ @JoeHobbit: Here in Finland, they actually do require that. It's been cited as one of the reasons why we're doing so well e.g. in the PISA studies. $\endgroup$ Commented Aug 2, 2013 at 9:34

I like @Arkamis' answer a lot and I'd like to add to it by expanding the ending part of that answer.

Primary and high school mathematics is often seen as a bunch rules and formulas to solve certain problems, and very few teachers give any explanation for those.

For example, a colleague of mine asked her student something along the lines $2^3 \cdot 2^4 = ?$. He was trying hard to remember the formula from his high school, but couldn't. Then she went on with $2^3 = 2 \cdot 2 \cdot 2$, $2^4 = 2 \cdot 2 \cdot 2 \cdot 2$, so $2^3 \cdot 2^4 = (2 \cdot 2 \cdot 2) \cdot (2 \cdot 2 \cdot 2 \cdot 2)$, so we have $7$ twos, so that's $2^7 = 2^{3+4}$. The student's response was, in complete astonishment: "Wow! So that formula actually makes sense!?!".

When P.E. is "taught", it is often customary to learn basics of few sports, do various gymnastic stuff, run around a bit, train a body,... why is mathematics not seen as a polygon to train brain to think, but instead it's a place where they try to pour lots of cryptic stuff in that same brain?

"You'll need it later in life" is a lie for most of the mathematics, unless the student is to become a mathematician, physicist, engineer,... But, thinking in order to solve problems, even if they're not real-life, would be priceless "in life".

And just like the aim of P.E. is not in those results you make there (none of the measured running times or climbing heights will ever be used anywhere), a bunch of theory and formulas should not be the aim of mathematics. We make kids run on P.E. to train their bodies in a certain way; the process of understanding how and why, to train brains, should be the aim of mathematics, not encyclopedic knowledge that can be found on Wikipedia and bunch of other sites.

Students don't need $a^b \cdot a^c = a^{b+c}$, but they do need to understand why it works like that. They need to see that the above "proof" works for the natural numbers, it is good for them to think how would they expand it to integers, maybe even rationals, and then be explained how it is actually done. Sure, it easier to just spit out the formula, but such approach misses the point entirely.

It is worth noting that this is the problem with most schooling. History, for example, is often taught as a combination of dates and names. Lots of boring facts, without concept. In my opinion, it is much more important to explain how Hitler could've happened, than what is the exact date he attacked Poland. It is the understanding of the phenomena that lead to a completely different approach towards Germany than after the WWI. That explanation is important and, more importantly for the scope of this discussion, far more interesting to students.

Geography... I was taught how much coal does Russia export (?!?), but I learned more about countries from "Where in the World is Carmen Sandiego" than I did in school, just like I learned more history in 2-3 years reading of Cracked (a comedy site!) than I did in 12 years of primary+high school. In the end, all I was left with was contempt for that stuff, just like many students are left with contempt for mathematics.

"Why?" is the most important question in the World, and most schools seem to avoid it. Try asking a student "Why?" for an answer (s)he wrote on the blackboard, and they'll immediately try to erase the answer, thinking that "Why?" equals "This is wrong". And that is the true failure of the schooling system.

A bit of context for this answer: I was born and raised in ex-Yugoslavia and Croatia; I've read about schooling systems around the World, but I work at the university, so I'm not part of school education, but I do see the consequences.


Today's SMBC is very appropriate for this discussion, and it'd be a pity not to include it here:

Until you teach someone calculus, they can't even walk finite distances. But they can get reallllllly close.
-- Source: SMBC, 23.6.2016. (extra panel, off-topic for this discussion, here)

There is also a quote I recently read:

Every maker of video games knows something that the makers of curriculum don't seem to understand. You'll never see a video game being advertised as being easy. Kids who do not like school will tell you it's not because it's too hard. It's because it's boring.
-- Seymour Papert [source]

  • $\begingroup$ My summary of your post; School is about delivering what the teacher wants: It should be about exploring ideas. $\endgroup$
    – User3910
    Commented Aug 1, 2013 at 14:59
  • $\begingroup$ More like developing the ability to have and explore ideas. $\endgroup$ Commented Aug 1, 2013 at 15:01
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    $\begingroup$ @JoeHobbit: That’s a bit unfair: quite a few teachers would love to do more exploration of ideas than the syllabus and testing requirements allow. In some classrooms school is about delivering what the teacher wants, but in many it’s more about delivering what the system requires. $\endgroup$ Commented Aug 2, 2013 at 21:37
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    $\begingroup$ In my experience, a teacher formally has the freedom to teach the way (s)he wants. However, the sheer amount of stuff that needs to be taught makes it impossible to be nearly as thorough as I am suggesting. That's why I haven't even touched teachers in my answer. Sure, some are bad (in many countries, many or even most of them, due to the low social and financial status that teachers have there), but even the good ones are more than often powerless. $\endgroup$ Commented Aug 2, 2013 at 22:45
  • $\begingroup$ @VedranŠego: Spot on. One man can't save a ship. $\endgroup$
    – Mike Jones
    Commented Nov 2, 2015 at 16:41

In precalculus the real question is can we fix the bad study habits before its too late. From what I see ( I don't teach it personally) the main danger is apathy. I don't think historical tidbits and/or motivations are really the problem. Sadly, the problem is soceital (here).

Now, when I begin calculus, I take a few minutes to sketch the big idea. It doesn't take too long. The stories of the main characters are interesting and I try to return to them from time to time. That said, think about the problem of making the typical student walk the steps which Isaac walked (or Euler, Galois whoever)... it's just not realistic to suppose the typical student has an inkling of the unbounded curiousity of those individuals.

People who do math and people who make (or discover if you prefer) are not the same. The demarcation is not as stark as black and white, but I know from teaching a few thousand students that the percentage out there which are genuinely interested is very small.

One of the reasons I love this website is that it puts me in contact with that precious minority of students who genuinely enjoy math forsaking some degree-seeking requirement.

I don't want to rain on your parade too much, but I think it's better to spend your energy looking to identify the interested students and work with them one on one.

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    $\begingroup$ How would you respond to Arkamis' concept of self-actualization as motivation? Also, while some will always be uninspired, A teacher can occasionally inspire. $\endgroup$
    – User3910
    Commented Aug 1, 2013 at 5:55
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    $\begingroup$ I think your answer suggests treating the interest of an student in math as an inherent property of the student. Uninterest students are not that way. They were made that way by their previous teachers. $\endgroup$
    – OR.
    Commented Aug 1, 2013 at 5:55
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    $\begingroup$ @RGB Yes, I think some students are infinitely more inquisitive than others. Is this a product of their upbringing or a consequence of their genetic make-up? I don't think I'm going to settle that here and even I could it wouldn't alter the reality that students have to care before they can be taught. $\endgroup$ Commented Aug 1, 2013 at 6:09
  • $\begingroup$ JoeHobbit I tend to agree with Arakamis' way of inspiring. My teaching is somewhat inline. I very much agree that students don't really care about real world applications and they are worthless if they detract from actual math. Ideally, I want them to understand that math is not dead, no, it is alive, active and something which presents an intellectual challenge which is so much more than a game of jeopardy. I try to emphasize consistency and analogies whenever possible and I avoid tricks, gimmicks and edufads with a vengeance. $\endgroup$ Commented Aug 1, 2013 at 6:16
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    $\begingroup$ Some students don’t care about real-world applications. Some students are unreachable without real-world applications. And most are somewhere in the vast middle ground. $\endgroup$ Commented Aug 1, 2013 at 6:23

There is a good amount of maths in the 'shilling arithmethic' of 1951, (a school text), that one see the mathematicians struggle with. For example, a whole chapter is devoted to the idea of various forms of compound arithmetic (multiplication and division of several units without conversion).

The whole idea that you can take a lengths in furlongs and perches, and multiply these by way of a division directly into acres, roods and perches, whthout conversions, seems totally alien in these days. Yet this whole matter is discussed on a single page in an 1895 arithmetic book. The method for doing this might be described as 'invoice arithmetic', applied twice. So, one first finds the area of 1 furlong by $x$, and then finds $x$ by $y$, eg. Here there is no conversion of units.

   Multiply  5 furlongs 22 perches by 8 furlongs 16 perches.

        1 furlong   'costs'   10.0.00     5   costs 50.0.00
       10 chains    'costs'    2.2.00     2   costs  5.0.00   (4
        1 chain     'costs'      1.00     2   costs  0.2.00  (10
                        total                 -------------

        1 furlong  'costs'    55.2.00     8   costs  444.0.00   
       10 chains   'costs'    13.3.20     1   costs   13.3.20  (4
        1 chain    'costs'     1.1.22     6   costs    8.1.12 (10
                                              total  466.0.12
           466 acres, 12 perches.

This is the normal method with dealing with added fractions, without means of conversions. Multiplication by parts is usually dismissed as a kind of 'peasent multiplication', but you see from this example, it is perfectly clear what is going on, and faster than most means of converting back and fourth.

One often sees on this list, 'continued fractions', often written in the quite ugly long vincula. There is very little comment (if at all), on 'added fractions' or 'continued numerators', which provide a direct way of dealing with quite large fractions without conversions. I regularly convert things like sixtyfourths into eights and added eights (eg $\frac {35}{64} = \frac 48 \frac 38 = \frac {4 \frac 38}8$).

Decimals are simply added fractions, rather than increasingly large exponents. That is, when we upgrade $1.61$ to $1.618$, we are adding $\frac 8{10}$ to the end of such a fraction. One sees, especially in money, amounts like $£ 5.15\frac 12$, exactly with this meaning: the fraction is against the unit of the last place.

One talks of the sumerian numbers, but totally ignores the fact, that well into the fourteenth century, that the English used a hundred of six scores, and that the great multitude of weights and measures used in pairs or tripples, is as much to avoid using hundred-numbers (eg a weight of seventeen two = 17-2 = 17 stone 2 lb = 240 lb, avoids the hundred lb issue).

Even the idea of 'english number' is given as absurd. That is, to write eg $720$ as vi C (ie six C, where the hundred is six score), rather than 'DC', is taken as entirely strange. Yet there are plenty of examples of this sort of writing in Zupko's "Dictionary of English Weights and Measures".

The sumerian system is not a 'regular base' as we know it, but a 'division base'. The most significant digit is the units, and subsequent places are divisions or added fractions of sixtieths. Division bases have leading siginficant zeros, so 00.01 means 1/60, and 00,00.01 means 1/3600. Yet you often see people translate $44.26.40$ as $160,000$, rather than its correct value of $44 \frac 49$.

One of course, seriously objects to this absurdity of introducing radians into every discussion of angle. Not only is it possible to do angle calculations without such absurd units, but geometry had gone a long way before anyone decided to invent it.

One can do rather complex non-euclidean geometry (including hyperbolic geometry) without recourse to things like $\tanh$ etc or radians. The coherent angle for higher geometry is the fraction of all space, something i usually write in fractions against base 120, to simplify calculations. In such units, i found the solid angles occupied by the vertices of all of the four-dimensional regular polytopes (polychora), well before the cubic radian was known.

In practice, the use of radians and radii simply impede the process: nearly everyone in the real world measure circles by their diameter. You go an buy screws or plates or pipes or whatever, and it is the diameter that is quoted. The unit circular inch is the area of an inch of unit diameter. It is interesting that the Sumerians, according to Sir Thomas Heath, rated the circumference of 'real' circles (ie ones that you could walk around or hold), as if their diameter were 60, and $\pi=3$, ie 180 ells, divided to 24 digits. It is only circles you stand in the middle of (like the sky), that has $360$ degrees = $2(\pi=3) 60$. Yet i have been caught out here even suggesting that formulae for volumes by dimaeter be discussed.

  • $\begingroup$ It’s not clear that this actually addresses the question. Moreover, some of it is wrong. E.g., the Sumerian/early Babylonian notation transcribed $44.26.40$ is ambiguous: whether it represents $160,000$, $44\frac49$, $2666\frac23$, etc. can only be determined by context. Radians are not in fact ‘introduced into every discussion of angle’, nor are they ‘absurd units’ from a mathematical point of view. Finally, the medieval and Renaissance notations that you find in Zupko are of interest and importance to historians but are at most an aside in the elementary classroom. $\endgroup$ Commented Aug 2, 2013 at 22:03
  • $\begingroup$ The sumerian 44.26.40 is not ambiguois. What is ambigious is our expectations of what it should mean, aganist what it actually does. The sumerian sexagesimal, in this form, refers to a number in calculation, whcih means that it's a division-number: ie the first column is units, and subsequent places are divisions. Medieval notations continue to haunt us in terms of roman numbers, which do the duty of 'second-series' numbers. Thirdly, radians are absurd, in as much as their main use seems to be enumerating functions, and not angles. They are not a coherent unit across dimensions. $\endgroup$ Commented Aug 3, 2013 at 7:37
  • $\begingroup$ This particular answer is a demonstration that mathematics is being taught wrong. Egyptian fractions, which fell out of use by 500 AD, seems to have more impact here than added numerators, which form the basis of modern decimals. That people are down-voting it seems that they have missed the heart of the question. $\endgroup$ Commented Aug 3, 2013 at 10:11
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    $\begingroup$ You’re simply mistaken about Sumerian numerals. Roman numerals survive marginally but have no mathematical significance; other medieval and Renaissance notations — and yes, I’m familiar with them — do not survive. What you call added numerators is nothing more than positional notation, extended in your example to base eight. Positional notation is important, but it should obviously be taught first for integers. Radian measure, like natural logarithms, is a mathematically obvious and inevitable development. Finally, whatever the flaws in the teaching of mathematics, nothing in your answer ... $\endgroup$ Commented Aug 3, 2013 at 10:38
  • $\begingroup$ ... represents an improvement. Oh, and I’ve not seen any emphasis on Egyptian fractions in the schools. I’ve seem them used as supplementary material in the lower grades, something that might pique some students’ interest, and of course they can be a source of problems for courses in elementary number theory or discrete mathematics (as in this question), but those are both appropriate uses. $\endgroup$ Commented Aug 3, 2013 at 10:41

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