Why do many math books have so much detail and so little enlightenment? (2010)
This is something I think about often. Maths is one of those subjects which is taught repetitively rather than philosophically, and yet there couldn&#x27;t be a worse subject to teach in such a manner. It forms a barrier-to-entry which mathematicians probably care for, glancing at some of the quotes already posted here, but doesn&#x27;t help people who would make excellent mathematicians if only they had the encouragement.<p>I always found mathematics self-explanatory. From all the repetitions the understanding came naturally, and I think there is definitely a subset of the population that has what you might call &#x27;aptitude&#x27; for maths. It makes sense, because philosophically it is self-explanatory by definition. LHS = RHS - the trick is to prove it or fill in the blanks.<p>But I dare say the majority of people aren&#x27;t &#x27;apt&#x27; for maths in the way in which it is taught, but at the same time it&#x27;s totally unfair to rule them out as maths-stupid. Many people who have no mathematical knowledge whatsoever still prove rather deft at deducing when their partners are lying or when an argument in a debate is self-contradictory. People who failed their mathematical education go on thinking they suck at math, but when they play video games and figure out optimal strategies and formulae for success in the game, it&#x27;s not all that different. Sometimes the mathematical understanding comes later to people as a direct result of being forced to reason about mathematical problems in an applied setting and discover that they could have been good at maths all along if they had only understood the &#x27;why&#x27; of the exercises and formulae they were doing.<p>I think there are two big stumbling blocks that stop people from taking a bigger interest and investing more concentration and care into maths: firstly people mix up arithmetic with maths, and assume from the fact that they take a long time to divide a couple of numbers (or get the wrong answer when they do so) that they are doomed to be crap at maths.<p>But that&#x27;s utter BS. I&#x27;m hopeless at mental arithmetic, but maths isn&#x27;t about adding numeric values, its about deduction and reasoning, and I&#x27;m certainly not alone in being a terrible arithmetician and a good mathematician.<p>The other issue that people struggle with is that the notation itself isn&#x27;t clearly explained or appears daunting, especially if they didn&#x27;t go beyond compulsory education. It can prove quite distracting because a person might have questions as to why limit notation looks the way it does or what the hell the integral sign means, when really it&#x27;s no different to choosing commas and brackets to denote semantic delineation in written language.<p>Take limit notation or summation. The placement of the various numbers is arbitrary on a fundamental level, it&#x27;s just the standard everybody agreed to use. If someone good enough at math sat down and tried to build up math from first principles, with no knowledge of modern mathematical standards and notation, he&#x2F;she could come up with the same formulae and concepts but an entirely different way of representing them.<p>That&#x27;s exactly what happened really. you had Leibniz and Newton just organizing information in a way that made sense to them, and it stuck. An explanation of how arbitrary the notation is on a fundamental level would probably make a board full of symbols and algebraic letters considerably less daunting for those who could-but-daren&#x27;t understand it.<p>I think there&#x27;s a big problem in education as well. I&#x27;m from Great Britain and growing up I think the biggest mistake in our education system is the dire failure to explain mathematics in a way that is friendly for people who aren&#x27;t tip-top at abstract thinking or imagining.<p>Take the humble function, for example. teacher says &quot;a function takes an input and produces an output. so f of x equals y minus 5x...&quot; - and half the class stares blankly, daydreams for 15 minutes, chatters through the &quot;work your way through the textbook&quot; phase and does one of two things before the teacher goes through the solutions:<p><pre><code>    A. copy answers from the back of the textbook
    B. copies an adjacent boffin.
</code></pre>
I think the much maligned set theory actually provides a very good way to teach about functions, and yet it doesn&#x27;t appear until you hit college, and then suddenly a function is referred to as a mapping, without any preparation or explanation of whether there&#x27;s a difference between &#x27;function&#x27; and &#x27;mapping&#x27;.<p>I was endlessly helping classmates in school and I often think about how the education system could have been better and made math more accessible. The conspiracy theorist might say that society cannot afford for everybody to grasp mathematics and get a &#x2F;good&#x2F; job, but hopefully that&#x27;s not possible.<p>Geometry in education is a big fail as well. when I was growing up, trigonometry wasn&#x27;t explained at all, it was just repetition of applying trig tables. Nothing visually relating right-angled triangles and circles like Thales was even touched upon until college.<p>Secondary education in UK tries to cover to many things. Not that there&#x27;s any way of chopping math down to a single textbook or subject area, but there&#x27;s often little in the way of structure and it&#x27;s about cramming as many formulae down the student&#x27;s throats and then examining them on their ability to apply what they have repeated. but you can&#x27;t apply what you repeat, you apply what you learn and grok.<p>I&#x27;m not sure why there is this enormous failure in education. It could be because the people who write the books and lesson plans already grok mathematics and forget to give people aids to intuition or intrinsically understanding what they&#x27;re doing. Or maybe it&#x27;s because a mathematical education excludes the philosophy of math and set theory which is a mistake. Set theory is perfect to start off with, it gives insight into functions, probability, everything really. Can always discredit the &#x2F;project&#x2F; of founding mathematics on set theory later.<p>Of course, maybe I went to a bad school, and it was a few years ago now so perhaps things have changed - but I don&#x27;t think so. most of my younger sister&#x27;s friends have a worse understanding than the disinterested kids when I was growing up, but that could be down to the whole fast-food, instasnaptwit culture and a hundred other things that seem to be distracting people from anything academic.