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.

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?

Exactly!

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.