<story><title>You never did math in high school</title><url>http://j2kun.svbtle.com/you-never-did-math-in-high-school/?_nospa=true</url></story><parent_chain><item><author>omegaham</author><text>The thing is, I can&amp;#x27;t think of another way to teach the curriculum.&lt;p&gt;Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You&amp;#x27;ve counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you&amp;#x27;ve done that, you abstract all of arithmetic with algebra.&lt;p&gt;And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have &amp;quot;learned&amp;quot; everything up to calculus, but they don&amp;#x27;t actually understand what&amp;#x27;s going on. They just know the formulae and how to plug-and-chug.&lt;p&gt;How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood &lt;i&gt;why&lt;/i&gt; the formula was used just as much as &lt;i&gt;how&lt;/i&gt; it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they&amp;#x27;re fucked.&lt;p&gt;In any case, though, I think that the current curriculum is as good as it&amp;#x27;s going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren&amp;#x27;t going to learn calculus unless you understand algebra, and you aren&amp;#x27;t going to learn algebra unless you understand arithmetic.</text></item><item><author>Jtsummers</author><text>&amp;gt; Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.&lt;p&gt;For the author&amp;#x27;s analogy, music is &lt;i&gt;not&lt;/i&gt; being taught like what you describe. In his analogy it&amp;#x27;s being taught as several years of learning to read and transcribe music, without listening to or performing it.&lt;p&gt;Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching &amp;quot;real math&amp;quot; once the students reach 13 or 14 years old. And throughout this, it&amp;#x27;s rarely hinted at &lt;i&gt;how&lt;/i&gt; this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don&amp;#x27;t realize its real potential because the solution to the &amp;quot;problem&amp;quot; is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) &amp;#x2F; 2&amp;#x27; and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won&amp;#x27;t and will get to college thinking they&amp;#x27;re &amp;quot;good at math&amp;quot; and then fail horribly because they don&amp;#x27;t have the skill set for college mathematics, they don&amp;#x27;t realize what college mathematics entailed (so many jokes about my &amp;quot;modern algebra&amp;quot; textbooks, &amp;quot;We took that in 9th grade!&amp;quot;).&lt;p&gt;EDIT: Grammar.</text></item><item><author>matthewmacleod</author><text>Hmm. I&amp;#x27;m not sure I agree.&lt;p&gt;It seems that &amp;quot;solving algebra problems and doing two-column geometry proofs&amp;quot; is a necessary step on the road to &amp;quot;generating your own questions about whatever interests you and trying to answer them&amp;quot;. That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.&lt;p&gt;Contrary to the point made, we &lt;i&gt;do&lt;/i&gt; teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.&lt;p&gt;Mathematics has always seemed the same to me. I don&amp;#x27;t really use much of it day-to-day, but occasionally I&amp;#x27;ll come across a geometry problem or something when I&amp;#x27;m building software; maybe I end up doodling triangles, and using basic trig and algebraic manipulation to understand more or solve my problem.&lt;p&gt;Much of our teaching processes focus on skills, rather than a more abstract notion of &amp;quot;education.&amp;quot; There&amp;#x27;s been much said about why this is a bad thing; I&amp;#x27;m rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don&amp;#x27;t teach math, but we do teach the skills that are required to &amp;quot;experience&amp;quot; math at a later date.&lt;p&gt;So maybe I&amp;#x27;ve convinced myself of the validity of the title, if not the individual arguments.</text></item></parent_chain><comment><author>ef4</author><text>&amp;gt; The thing is, I can&amp;#x27;t think of another way to teach the curriculum.&lt;p&gt;But there&amp;#x27;s already a rich literature of real-world results for better ways to teach mathematics. See for example Seymour Papert&amp;#x27;s &amp;quot;Mindstorms&amp;quot; as a starting point (and much has been done since it was written ~30 years ago).&lt;p&gt;All children already learn quite a lot of fairly deep mathematical intuitions. We just take them for granted because everybody learns them.&lt;p&gt;For example: conservation of volume, the concept of &amp;quot;integer&amp;quot;, order independence of cardinality, projecting orientation onto other reference frames, the equivalence between ordinal and cardinal numbers.&lt;p&gt;Everybody learns these things because they&amp;#x27;re embedded in our environments, and we can learn them playfully as children. When we create environments that embed even richer concepts, children learn those concepts just as easily. This is the explicit design goal of LOGO, and the whole family of descendants it has inspired.&lt;p&gt;Teaching in this way requires a degree of freedom and play that normal schools generally don&amp;#x27;t tolerate, which is why these proven, powerful tools still haven&amp;#x27;t taken over the world.</text></comment>
<story><title>You never did math in high school</title><url>http://j2kun.svbtle.com/you-never-did-math-in-high-school/?_nospa=true</url></story><parent_chain><item><author>omegaham</author><text>The thing is, I can&amp;#x27;t think of another way to teach the curriculum.&lt;p&gt;Math is abstraction on top of abstraction. You start off with counting. Once you get counting down, you abstract it with addition (You&amp;#x27;ve counted 5 things, and you want to add it to a group of 7 things) and subtraction (Count 5 things and take them away from a group of 7 things). Then you abstract addition with multiplication, and then abstract that with division. Once you&amp;#x27;ve done that, you abstract all of arithmetic with algebra.&lt;p&gt;And, well, it goes from there. You need some abstraction of algebra to do trig, calculus, and geometry. And to be able to abstract it, you need to understand it. This is where the disconnect happens - you get kids who have &amp;quot;learned&amp;quot; everything up to calculus, but they don&amp;#x27;t actually understand what&amp;#x27;s going on. They just know the formulae and how to plug-and-chug.&lt;p&gt;How do you get these kids to understand? My dad would relentlessly quiz me on the concepts, and he was ruthless in making sure that I understood &lt;i&gt;why&lt;/i&gt; the formula was used just as much as &lt;i&gt;how&lt;/i&gt; it was used. Many kids just learn the latter, and when it comes to any sort of independent thought, they&amp;#x27;re fucked.&lt;p&gt;In any case, though, I think that the current curriculum is as good as it&amp;#x27;s going to get. You can teach these concepts in a horribly boring manner, or you can teach them in an engaging, interesting manner. Either way, you aren&amp;#x27;t going to learn calculus unless you understand algebra, and you aren&amp;#x27;t going to learn algebra unless you understand arithmetic.</text></item><item><author>Jtsummers</author><text>&amp;gt; Contrary to the point made, we do teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.&lt;p&gt;For the author&amp;#x27;s analogy, music is &lt;i&gt;not&lt;/i&gt; being taught like what you describe. In his analogy it&amp;#x27;s being taught as several years of learning to read and transcribe music, without listening to or performing it.&lt;p&gt;Taking this analogy back to the reality of math education, the first 6 or 7 years of the standard US math curriculum is dedicated to arithmetic. Hell, it takes 4 or 5 years (3rd or 4th grade) to get to long division. The notion of variables is covered some time in middle school (6th or 7th grade) with pre-algebra (a watered down version of algebra with simple algebraic statements) being commonly taught in 7th or 8th grade, and algebra proper only showing up for 8th or 9th graders. That means we only start approaching &amp;quot;real math&amp;quot; once the students reach 13 or 14 years old. And throughout this, it&amp;#x27;s rarely hinted at &lt;i&gt;how&lt;/i&gt; this subject can be applied. Most of the real world examples are contrived, or simple enough that the students that get it don&amp;#x27;t realize its real potential because the solution to the &amp;quot;problem&amp;quot; is practically handed to them. Showing how the sum of the angles in polygons can be determined by the number of sides and [developing a formula] via induction is a college topic in the US. Showing the sum of the first n positive integers is `n * (n + 1) &amp;#x2F; 2&amp;#x27; and how to arrive at that is shown in a freshman or sophomore discrete math course. Bored, smart students (like I was) will recreate the tools like induction and develop these things themselves, but most won&amp;#x27;t and will get to college thinking they&amp;#x27;re &amp;quot;good at math&amp;quot; and then fail horribly because they don&amp;#x27;t have the skill set for college mathematics, they don&amp;#x27;t realize what college mathematics entailed (so many jokes about my &amp;quot;modern algebra&amp;quot; textbooks, &amp;quot;We took that in 9th grade!&amp;quot;).&lt;p&gt;EDIT: Grammar.</text></item><item><author>matthewmacleod</author><text>Hmm. I&amp;#x27;m not sure I agree.&lt;p&gt;It seems that &amp;quot;solving algebra problems and doing two-column geometry proofs&amp;quot; is a necessary step on the road to &amp;quot;generating your own questions about whatever interests you and trying to answer them&amp;quot;. That is, an understanding of the concepts and established mechanisms for dealing with abstract reasoning and patterns is required in order to have any hope of moving further in mathematics.&lt;p&gt;Contrary to the point made, we &lt;i&gt;do&lt;/i&gt; teach students music in school by explaining and using the established tools we use to create music. We teach notation, rhythm, keys, harmonies… we then exploit that to compose, perform or understand music.&lt;p&gt;Mathematics has always seemed the same to me. I don&amp;#x27;t really use much of it day-to-day, but occasionally I&amp;#x27;ll come across a geometry problem or something when I&amp;#x27;m building software; maybe I end up doodling triangles, and using basic trig and algebraic manipulation to understand more or solve my problem.&lt;p&gt;Much of our teaching processes focus on skills, rather than a more abstract notion of &amp;quot;education.&amp;quot; There&amp;#x27;s been much said about why this is a bad thing; I&amp;#x27;m rather ambivalent on it myself, seeing from casual observation how much benefit skill-focused education can offer to those who would otherwise simply learn nothing. Of course, this works better where self-motivated students are not stymied by too-strict adherence to curricula. IOW, perhaps we don&amp;#x27;t teach math, but we do teach the skills that are required to &amp;quot;experience&amp;quot; math at a later date.&lt;p&gt;So maybe I&amp;#x27;ve convinced myself of the validity of the title, if not the individual arguments.</text></item></parent_chain><comment><author>nitrogen</author><text>A few weeks ago there was an article here on HN that suggested kindergarten students might do better learning an intuitive form of calculus and algebra before arithmetic.  Yes, math is built out of layered abstractions, but we can rotate the entire conceptual space to use a different foundation and still get a complete picture in the end.</text></comment>