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Senselessness of school math

A Mathematician's Lament is a brilliant essay by Paul Lockhart on the school approach to mathematics and why it sucks the beauty from the art form. (Click the title to read more.)


A mathematician, like a painter or poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made with ideas.


A musician wakes from a terrible nightmare. In his dream he finds himself in a society where music education has been made mandatory. “We are helping our students become more competitive in an increasingly sound-filled world.” Educators, school systems, and the state are put in charge of this vital project. Studies are commissioned, committees are formed, and decisions are made— all without the advice or participation of a single working musician or composer.


Since musicians are known to set down their ideas in the form of sheet music, these curious black dots and lines must constitute the “language of music.” It is imperative that students become fluent in this language if they are to attain any degree of musical competence; indeed, it would be ludicrous to expect a child to sing a song or play an instrument without having a thorough grounding in music notation and theory. Playing and listening to music, let alone composing an original piece, are considered very advanced topics and are generally put off until college, and more often graduate school.


As for the primary and secondary schools, their mission is to train students to use this language— to jiggle symbols around according to a fixed set of rules: “Music class is where we take out our staff paper, our teacher puts some notes on the board, and we copy them or transpose them into a different key. We have to make sure to get the clefs and key signatures right, and our teacher is very picky about making sure we fill in our quarter-notes completely. One time we had a chromatic scale problem and I did it right, but the teacher gave me no credit because I had the stems pointing the wrong way.”


In their wisdom, educators soon realize that even very young children can be given this kind of musical instruction. In fact it is considered quite shameful if one’s third-grader hasn’t completely memorized his circle of fifths. “I’ll have to get my son a music tutor. He simply won’t apply himself to his music homework. He says it’s boring. He just sits there staring out the window, humming tunes to himself and making up silly songs.”




I'm just wondering whether anybody in the group has ever used any of the Mindsprinting programs before, or know of anyone who has? If so, do you have any thoughts/critiques to offer?


Unnecessary is my first reaction. My second reaction is that it's bringing the stress of school home.


My third reaction after taking the test has unfortunately spawned an evilly long rant on how poorly schools approach math and how much damage they do.


Anyone who wants to, try taking the 5th grade test and notice how high your frustration level goes. Notice how bad you feel about your math skills.


I have a degree in electrical engineering. The test told me (as I posed as a 5th grader named Joan) that I was performing on a 4th grade level. Yeah. Right. Down below my rant is their list of the areas I need to work on.


Here's some sample questions. No calculators. (There were a total of 20.)


Divide 6847 by 43.


Remember, no calculators. Why is my question. This is ancient dinosaur thinking that math teachers have passed down to math teachers since the beginning of schools. As a present for my freshman year in college in 1974 I got an early electronic calculator. You had to get them through the mail from Hewlett Packard. It was a "scientific" calculator meaning it could do square roots and squares. They were $150. (How much is that in 2008 dollars? ;-) That was 34 years ago. I haven't done long division for 34 years. Before that people used slide rules.


WHY are kids being made to do these ridiculous calculations by hand? Math teachers will tell you they need to know. As an engineer, I will tell you that's a lie.


What's really useful is for kids to use numbers in ways that are meaningful to them, to get answers that mean something to what's important to the kids. (And I can not understate how useful video games are for using math concepts in real ways!)


While my daughter is not a math head -- she's more of an artist and writer -- she played a lot of video games. I also walked myself through problems when she needed an answer so she could see how I was manipulating the numbers to get to an answer. While her father did show her some cool stuff like Fibonacci numbers, the amount of math she had each week was on the order of minutes compared to hours for schooled kids. When she was 14 she decided it would be fun to take the college math course (Statistics) her father taught. While she did have him on hand to ask questions of while doing homework, she took the tests just like the rest of the students. She was at or near the top of the class. A class full of students 4-5 years older with 12+ years of math. (She went on to take several other math courses for fun, one without him as teacher. Again, doing at or near the top of the class.)


It's useful to be able to get a rough guesstimate to problems like the one given. (The verbal walk through I would have given my daughter is: I would bump the first number up to 7000 and the second down to 40. Then 100 times 40 is 4000 and 200 times 40 is 8000. So the answer is closer to but less than 200. That gives you a ball park answer which is adequate for real life.) If I need an accurate answer, I'm not going to rely on calculating by hand. I'm going to use a calculator.


Calculate the amount of elapsed time in hours, minutes, and seconds from 4:30:15 a.m. to 12:24:31 p.m.


Just looking at that makes me want to pull my hair out. When would anyone need to know down to the second how much time between those two? When would anyone need to know without a calculator?


If someone was in some field where such calculations were common, they probably would eventually figure out tricks to do it quickly. 10 yo kids don't need tortured with such inanities that they're unlikely to ever use.


Hiro would like to visit Japan, the homeland of his grandfather. The trip will cost him 200,000 Japanese yen. How much will Hiro's trip cost him in American dollars? ($1.00 U.S. = 117.51 Yen)


My daughter is big into manga and anime and we've often wanted to know how much some number of yen in a manga is in US dollars.


I'll tell you the trick: 1 yen is about a penny. That gets you close enough to get a feel for what kind of money they're talking about. Just move the decimal point over two places. 200,000 yen is approximately $2000.


But for this problem, first, if the guy is booking a flight from the US to Japan on the computer, they're not going to quote him a price in yen. It's going to be in dollars. Okay, no matter. Say he needs grandpa to wire him the money so needs to tell him how many yen. Maybe he's going through some Japanese booking center. (Grumble. Stupid made up question.)


Anyway, he's got the browser open. He types into the search box in Google:


convert 200,000 yen into US$


and it comes back and tells him:


200,000 Japanese yen = 1 869.2 US$


Quicker, more accurate, with up to the minute currency values than doing it by hand with an out of date conversion.


Find the volume of the frustum.


What the heck is a frustum? Well, one could guess from the picture with the question. (Yahoo blocks pictures so I can't show you but you could type it into Google Image if you wanted to see.) The word itself might actually be interesting for some kids. It has a cool sound. But by this point in the test, I'd bet a lot of kids would feel inadequate that they don't know. Don't worry, kids. Life is an open book test. If frustum *ever* comes up in real life, someone will tell you or you'll be able to look it up.


Select the drawing that shows triangle ABC with median BR using a compass and a straight edge.


I probably got this one wrong.


The problem with the question is that you have no idea why it's being asked and what information you want to extract from what's given.


In real life that wouldn't be true. In real life, you'd understand why the problem is set up that way and why you wanted the answer and what you wanted to do with the answer once you had it.


If it costs $3.50 for each person to swim, how much money did the pool make last weekend?


This is not far off questions that actually come up in real life.


(At least that's what I said originally. On second glance, it's not. The question is stupid.


If the pool charges everyone the same, they're not going to know how many of each they have.


But, it is *vaguely* similar to real life questions.)


The problem is that when kids who aren't grasping math are faced with these word problems over and over, they tend to freeze up over time and their brains shut down at the first hint of something that smells like a word problem. So even when faced with a real life one that they actually care about, they can cringe and feel like they can't do it.


But when things like this are just part of life, when mom walks through *real life* problems in front of the kids while she finds an answer for them (or lets them if they're saying "No, don't tell me!"), the problems aren't scary and kids can see why the problem is worded as it is because the kids understand the real context and they know why they want to know an answer.


(In this particular problem, the pie chart showing how many adults, children and students there were was too small to read.)


Draw an expression for the situation below. Matt has 4 more video games than Sarah. Sarah has x video games. How many video games does Matt have?


To make things really confusing, the answer was "None of the above."


When does anyone ever solve problems like that? They're fun for kids who love puzzles. And you can buy books of them -- *not* workbooks, but *real* puzzle books -- at grocery store checkouts and book stores.


While this problem is a math book version of an algebra problem, again, the huge problem with it is that when would someone ever set up a problem that way and why do they want the answer?


Real life problems have meaning. When my daughter was 10ish maybe -- and without demonstrating any previous need or desire to set up such problems -- she asked, "Do you know how long it would take to drive from Boston to LA?" At the time we were in the Pittsburgh airport. We were waiting for a flight to LA. We had driven from Boston to Pittsburgh a lot to see my father so she knew driving took 11 hours. She knew the flight had taken -- I can't remember now, but say 3 hours. She knew the next leg of the flight was 6 (guessing again.) So she said 33 hours. Absolutely spot on algebraic reasoning.


*That's* real life algebra. She could see how all the pieces fit together. She had a desire for the answer. She found a way to use what she knew to find out what she didn't.


Find the mean of the following numbers: 19, 13, 25, 21, 19, 5

Remember, no calculator. Not tortured enough? Do it again:

Find the median of the following numbers: 14, 19, 17, 14, 24, 14

Find the mode of the following numbers: 10, 3, 23, 20, 23, 13, 13


You *do* of course remember what a mode is, don't you? You've probably used it 100s of times in real life.


No, you haven't.


While some people do use modes (I used them in programming and the professor explained them in a couple of minutes), and probably some branches of statistics. 10 yos don't need modes.


Some kids will get a kick out of them. But there's absolutely no reason to make a 10 yo learn them. There's even more of a reason not to make a 10 yo feel inadequate for not knowing or not caring.


Find the odds against rolling a 5 or a 6 on a die.


I really liked probability in college. It was a nice respite before we hit statistics ;-) While kids might have fun with probability, they don't need to learn it when they're 10. It's easy to grasp, takes minutes to learn when you're 18. Hours and hours when you're 10 if you don't understand why you're being made to do it. (And the danger is that kids will decide they either hate math or that they're dumb at math.)


Well, that was a few of the questions. And here's what the site says I need to work on. Goodness, there were only 20 questions! At most I may have missed a couple of the geometry ones (since I took the practical route and used a calculator). If I were intimidated by math, I'd feel really stupid.


4th Grade


Unit 40: Long Division - Two or More Digits with No Remainders

Unit 41: Long Division - Three or More Digits with Remainders

Unit 61: Measurement - Area

Unit 62: Area

Unit 409: Measurement - 2

Unit 411: Solving Equations

Unit 210: Mean, Median and Mode

Unit 211: Probability


5th Grade


Unit 201: Comparing and Ordering Numbers

Unit 205: Factors and Multiples

Unit 42: Understanding Fractions

Unit 43: Equivalent Fractions with Pictures

Unit 44: Simplifying Fractions

Unit 45: Four Types of Fractions

Unit 46: Adding Fractions - Same Denominator

Unit 47: Subtracting Fractions - Same Denominator

Unit 48: Adding Fractions - Different Denominators

Unit 49: Subtracting Fractions - Different Denominators

Unit 50: Multiplying Fractions

Unit 51: Dividing Fractions

Unit 52: Using BEDMAS with Fractions

Unit 403A: Time, Money, and Temperature

Unit 406: Number Sense - 2

Unit 407: Numeric Patterns

Unit 408: Data Analysis and Probability - 1

Unit 412: Logic and Proofs - 1

Unit 413: Problem Solving - 1

I was doing a math problem the other day and I got it wrong because I couldn't understand what they wanted by the question they asked. Here's the problem if anyone wants to have some fun with it. There are 42 boys and 24 girls in a chess club. How many percent more boys than girls are there?


I think it helps to change it into a problem that you can see the answer to.


If there were 2 boys and 1 girl, how many percent more boys than girls are there?


Then test your theory with some other number combinations that are obvious. 0's, 1's and 2's can give you false clues so it's always a good idea to test it with numbers that aren't even but that are obvious enough to still let you see the pattern developing.


BUT -- and it's an important but -- basically it's a dumb question. It's not the type of information someone would want to know from that kind of situation. Which makes it hard to figure out because the answer doesn't really mean anything. (One of the hazards of learning math from textbooks.) From that type of situation someone might want to know something like:


  • what percentage of the total the boys were

  • how many more boys than girls

  • the ratio of boys to girls


I suppose someone could come up with a reason for wanting to know the percent more boys than girls, but it just isn't information someone would normally ask for so that's why the question doesn't make sense.


Usually when you want to know what percentage more, it is in the context of packaging or price, like that store's price is 10% more than this store's price or you get 20% more (than the regular size) for free.


I think the problem is the same as asking if the 42 ounce package is the same price as the 24 ounce package then what percent more are you getting free?


So if it's 2 lbs for the price of 1 lb, you're getting 1 pound free which is 100% of the regular 1 lb package, so 100% more free. (Not, of course the same as 100% free!) If it's 3 lbs for the price of 2 lbs you're getting 1 pound free which is 50% of the regular 2 pounds, so it's 50% more free.


So it's the amount you're getting free (how much larger the bigger package is than the regular one) divided by the regular size: (42-24)/24 = 18/24 = .75 or 75% more for free.


If there comes a time when it is needed in her life then she will learn it. If she doesn't learn it then maybe consider that it really wasn't "needed." -- Pam G.


I agree. Math learned "just because" is very very hard. And the longer kids are subjected to "just because" efforts the harder it is to learn math because kids then have the weight of all those previous years of tedious math that has convinced them math is hard.


But the only thing those years have proven is that being made to do math by rote is very hard.


Math learned as a side effect of using it is easy. Kids learn to see the big picture and how things fit together and how numbers work.


When kids are made to do pencil and paper math, they get lost in the details. They have to figure out 11/17 of 87 before they have been casually exposed to hundreds of personally meaningful ways fractions are used around them.


I think one of the most helpful things parents can do is to solve everyday problems in their head out loud. It forces you to see things in simpler terms so that you can do it in your head. If one is faced with 103-56 and does it the way you were taught in school, you'll have to juggle and remember a lot of numbers that don't relate to the problem in your head. But if you can see the problem broken down into understandable pieces, then it's much easier and kids get to see how numbers work. (One of the big problems with pencil and paper math is that the numbers feel fixed. You can't alter the problem into something simpler.)


So for 103-56 you might ask how far 56 was from 100. Well, 4 gets you to 60 and 40 more gets you to 100 and 3 more gets you to 103. So 47.



The point is, do we really actually learn anything from formal math classes?


I think we learn something different than what it's assumed we're being taught. We're taught to recognize patterns and to apply the appropriate formula. ("When you see an equation that looks like this, do this to it.") The goal is to "do" math not to understand it. If understanding comes, it's a side effect and generally happens to people who are naturally good at mathematical thinking.


Teachers and textbooks go through the motions of explaining things, but the exercises and tests are designed to test for skill memorization, not for understanding. Not because that what teachers want but because they're trapped by having to show that what's expected to be learned is getting into the kids. Teachers aren't required to show that the information sticks or that the kids understand it. Just that it went in long enough to be spit back on a test.



But I think working with math formulas are much more involved than learning to read music simply because there are so many formulas and all are multi-step and abstract on paper.


Only in school math. Math in real life is done much more intuitively. Educators need to break it down into memorizable steps because kids don't understand the concepts. And since it's really tough to test for understanding, the educators rely on testing if someone can recall something they've memorized.


At the conference I was (not very successfully since I need lots of time to gather my thoughts!) trying to explain how kids get arithmetic. I was explaining the steps my daughter might use to add 2 largish numbers. It would involve changing the numbers around to make them into numbers that are easy to manipulate in your head. But the man I was explaining it to pointed out that it was way more complex than learning to borrow and carry.


Well it is if you were teaching it as a formula. But when it's done because someone understands how numbers work, it's very simple.



But I think working with math formulas are much more involved than learning to read music simply because there are so many formulas and all are multi-step and abstract on paper.


Only in school math. Math in real life is done much more intuitively. Educators need to break it down into memorizable steps because kids don't understand the concepts. And since it's really tough to test for understanding, the educators rely on testing if someone can recall something they've memorized.


At the conference I was (not very successfully since I need lots of time to gather my thoughts!) trying to explain how kids get arithmetic. I was explaining the steps my daughter might use to add 2 largish numbers. It would involve changing the numbers around to make them into numbers that are easy to manipulate in your head. But the man I was explaining it to pointed out that it was way more complex than learning to borrow and carry.


Well it is if you were teaching it as a formula. But when it's done because someone understands how numbers work, it's very simple.


Why, when we are dividing fractions, do we flip the second one over and multiply?


It's convention. It works out that way. ;-) And that's the problem with memorizing procedures to spit back for a test.


Think about it conceptually. If you take a whole (1) and divide it into larger and larger numbers of pieces 1/2, 1/3, 1/27. 1/100, you get smaller and smaller pieces. If you divide a whole into 1 piece, you'll get 1 piece (1/1). If you divide a whole into smaller and smaller numbers of pieces (though the normal way of describing division doesn't work so well -- dividing 1 into half a piece, or taking 1 piece out of 1/2 pieces doesn't make much sense), the pieces should get larger and larger. In other words they have to get bigger than one. So 1 divided by 1/2 is 2 (or 2/1 which is 1/2 flipped over.) 1 divided by 1/6 should be 6, so 2 divided by 1/6 ( 2/(1/6) which is the same as 2x(6/1)) should be twice as much, so 12.


Math needs to describe things that fall outside the realm of physical objects. So, though in some contexts it make sense to divide by numbers smaller than 1, it doesn't make much sense to divide a plate of cookies by 1/2 a child. ;-) But it might help make the concept clearer if you tried! If there are 5 cookies to divide by 1/2 a child, how many cookies does a whole child get? If there are 2 cookies to divide by 1/6 of a child, how many cookies does a whole child get? How many cookies would 3 children get?



But it's just a formula for helping one to remember how to multiply 2 negative-signed numbers, right?


No, it's real stuff!


What would really help me understand is if there is ever a time IRL when there is a need to multiply 2 negative-signed numbers, or a positive times a negative, etc. Is there such an example? Or do these problems exist exclusively on paper?


Usually people explain positive and negative numbers to kids as having and owing money. But positive and negative can represent anything that's opposite. In Pam's example, they represent opposite cardinal directions. They can also represent charges on a particle, whether something is increasing or decreasing, coming after or before or above or below some chosen point.


It's from being taught how to do arithmetic before we've had lots of experience with numbers that has cemented the idea that positives are something and negatives are less than nothing.


Multiplying 2 negatives numbers is just multiplying 2 qualities that come in opposite flavors to their 2 positive counterparts. The positives and negatives sometimes make sense like up and rising being positive and down and falling being negative. Sometimes they're just chosen so everyone is using negative the same way. (The "negative" charge of an electron isn't really negative. It's just opposite the "positive" charge of the proton.)


The numbers can tell you how big something is. The positives and negatives can tell you where it is in relation to some (sometimes arbitrarily chosen) reference (zero) point.


Life came first. Then we invented math as a way to describe life. Teaching math out of the context of what it's describing is like teaching a foreign grammar and vocabulary without ever hearing or using the language.


Math makes sense in context. Out of context it's just memorization. It seems that the positive/negative thing could be useful to just explain relationships between things?




In fact that's what all (or lots) of math is. Math helps us to compare things, to see how two things relate to one another, how changing one thing affects the whole thing.


If, for instance, you're traveling from Tampa to Miami, if you set your reference (zero) point at Tampa, you'll know how far you are in relation to Tampa. That isn't really very useful. But if you set your reference point as Miami, then you know how many more miles you have to go.


With positive and negative numbers you get more information than just how far it is from the zero point. It tells you where -- which side -- in relation to the zero point. So if a train engineer tells the dispatcher they're broken down 10 miles from Miami that isn't enough information for the dispatcher to send out a repair crew. But the plus or minus (or east or west, or before or after (which the dispatcher will need to relate to which way that particular train is heading) -- those are all just different ways of telling where) tells the dispatcher which side of Miami they're at.


Positives and negatives are even more useful when you get x-y coordinates -- which is shut down time for some people ;-) -- so you can locate things on the surface of the earth (or the moon or Mars) rather than on a train track.


Say you wanted to tell a robot how to get to a meteor that hit Mars. (Admittedly not an everyday household use, but nonetheless a real life use. ;-) Telling the robot the meteor hit 5 miles from the robot's landing site on Mars means it could be anywhere on a circle of points that are 5 miles from the landing point. But saying it hit 4 miles east and 3 miles north precisely identifies exactly one of those points. Or 4 miles east and 3 miles south pinpoints another unique point.


(Choosing the landing site as the zero or reference point is just for convenience sake. It's just whatever will make talking about how one thing relates to another easiest. What reference point to choose would be obvious to anyone actually immersed in a real life problem because you'd know what information you wanted and what you wanted to do with it. Which is a major weakness of made up problems because you aren't gathering and manipulating and relating the information for any real personally meaningful reason.)


If we say that north and east are positive and south and west are negative. (Again, choosing that is just another convention because it's less confusing. There's nothing negative about south or west. It's just convention to put North up and East right, South down and West left. And it's just convention to put positives to the top and right and negatives to the bottom and down. Being conventional just makes communicating less confusing.)


So we can say the meteor is at coordinates 4,3 for the north east one and -4,3 for the south east one. (And 4,-3 would put it in the north west quadrant and -4,-3 would put it in the south west quadrant.)


All that does is tell the robot how the meteor's location relates to the zero point set at the landing zone.


Of course nonmathies would say why not just stick with north, south, east and west. ;-) For that example with the robot sitting at the landing site the north-south reference is probably easier to understand than x-y coordinates and therefore better. But if you start needing to know something about -- relate -- the distance between two meteors rather than between one meteor and the zero point, or relate the distance between where the robot is and where it wants to go, or relate how far the robot who has moved from the landing zone is from each meteor to figure out the closest meteor, then the positive and negative numbers make the calculation incredibly easy.



Joyfully Rejoycing
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