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# What's the best way to think about mathematics?

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Test yourself all the time. The way our brains work, you will learn better if you are continuously being tested.

In a classroom lecture situation, that means you will do your best if you have a short quiz

Compared to lazier approaches, this may seem to make more work for the instructor or the instructor's assistants, but you can get around some of it by, for example, having students exchange test papers and grade each other's answers as the instructor walks everyone through the method and shows the answers.

If your instructor doesn't test frequently or if you're working on your own, you need to test yourself frequently with books and programs that will give you a steady stream of sample problems. In this case, make sure you grade yourself honestly and chart your progress.

posted by pracowity at 12:56 AM on January 27, 2012 [1 favorite]

This isn't true if you're not getting the fundamental technique required for learning this stuff in the first place.The problem many of us have with symbolic representation in math is that it is truly abstract - there is no associative meaning other than more abstraction and that's the stuff that involves drilling and memorising to get it to stick. If you are academically strong in other areas you'll be used to learning by (often subconscious) association rather than by rote. This difference is very poorly articulated in most schools.

At school I was academically bright in a multitude of areas except maths. I didn't understand the sheer amount you have to take as read. I was always struggling with the 'why' something was correct, never realising that the answer was usually 'because it is'. A lot of even very basic arithmetic is predicated on logic that is very hard to grasp until you have developed the capacity for true abstraction. Thing is you don't need to understand that stuff, the algorithms required for arithmetic aren't maths in themselves (though the construction of them was), they are the tools. You just have to be able to commit them to memory and they become tools for the creative abstraction that 'doing maths' actually consists of. Having realised this, the learning process became easier.

An interesting read on this very topic is The Maths Gene by Keith Devlin. Some of his theories are a bit woolly but the central argument focusses on how the ability to think mathematically is related to the ability to learn language and there are a lot of examples that might provide the 'Eureka' moment you're looking for.

posted by freya_lamb at 2:22 AM on January 27, 2012 [1 favorite]

Four-pronged. Don't I look silly. I can't even count, lol.

posted by ollyollyoxenfree at 2:21 AM on January 29, 2012 [1 favorite]

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# What's the best way to think about mathematics?

January 26, 2012 6:32 PM Subscribe

What's the best way to think about mathematics? What's the best advice you have ever received about learning math and approaching math problems?

Ok, so I'm terrible at math. Not just terrible, but atrocious. I'm one of those people who proves that innate intelligence doesn't correlate well with all subject areas. In high school and college, I was a great student in almost all subject areas, from biology to English language and literature. However, when it came time to do math, anything with symbols and numbers, I was just atrocious at it. Not just atrocious but almost learning disabled bad.

I think that my stupidity in math isn't something innate however. At some point, I was never able to grasp just how to approach math theory and math problem solving. I am certain this is the case because there were subjects as a youngster that I was terrible at until I taught myself to look at them differently or learn them in a unique way. Let me give you some examples so you'll see what I mean.

1. I was a horrible speller in third grade. Just terrible. One day, I tried a new approach. I looked at any word I needed to spell and I would sound it out emphasizing every single syllable. So for example, if I needed to memorize "automatically" I would say outloud "AU-TO-MA-TEE-CALL-EEE"" and that one simple trick really attuned my mind to spelling in a way that all previous methods didn't.

2. When learning bio, I realized that a good to think of molecular was to conceptualize it as though the whole system was like a Rube Goldberg machine. I would imagine in my head one process X causing Y which result in Z occurring and so on. It wasn't the best approach but it certainly changed how i thought about molecular biology and immunology. Plus, I did well in those classes, becoming one of the best students in that subject matter.

3. This doesn't have to do with scholastic subject matter but it's still worth relating to show how powerful a simple conceptual change can have in our ability to perform tasks. As a kid, I read this book called The Einstein Factor and in it the author related a story about how a kid he knew drastically improved his batting average. The advice the kid received was to imagine a small speck or dot on the ball that was being pitched to him and try to aim for that dot when hitting the ball. As I was playing little league at the time, I tried this technique and the result was miraculous. I started hitting balls that had previously eluded. What was more incredible is that somehow by imagining this small speck or dot, I also learned to lay off the poorly pitched balls and thus build the patience to hit the best pitches to me.

These examples changed my life in one way or another and it taught me the power of concceputalization as a tool for learning and understanding. Does anybody have anything like this to offer me? Some sort of advice or perspective that will help me finally understand the basics of math and be able to solve the math problems. I know I am overlooking something in approaching math and I am certain that it's right there but I am somehow unable to see it.

So the hive mind: Please tell me of any tricks or advice you know about how to approach math (by math I mean GRE Level math)

Ok, so I'm terrible at math. Not just terrible, but atrocious. I'm one of those people who proves that innate intelligence doesn't correlate well with all subject areas. In high school and college, I was a great student in almost all subject areas, from biology to English language and literature. However, when it came time to do math, anything with symbols and numbers, I was just atrocious at it. Not just atrocious but almost learning disabled bad.

I think that my stupidity in math isn't something innate however. At some point, I was never able to grasp just how to approach math theory and math problem solving. I am certain this is the case because there were subjects as a youngster that I was terrible at until I taught myself to look at them differently or learn them in a unique way. Let me give you some examples so you'll see what I mean.

1. I was a horrible speller in third grade. Just terrible. One day, I tried a new approach. I looked at any word I needed to spell and I would sound it out emphasizing every single syllable. So for example, if I needed to memorize "automatically" I would say outloud "AU-TO-MA-TEE-CALL-EEE"" and that one simple trick really attuned my mind to spelling in a way that all previous methods didn't.

2. When learning bio, I realized that a good to think of molecular was to conceptualize it as though the whole system was like a Rube Goldberg machine. I would imagine in my head one process X causing Y which result in Z occurring and so on. It wasn't the best approach but it certainly changed how i thought about molecular biology and immunology. Plus, I did well in those classes, becoming one of the best students in that subject matter.

3. This doesn't have to do with scholastic subject matter but it's still worth relating to show how powerful a simple conceptual change can have in our ability to perform tasks. As a kid, I read this book called The Einstein Factor and in it the author related a story about how a kid he knew drastically improved his batting average. The advice the kid received was to imagine a small speck or dot on the ball that was being pitched to him and try to aim for that dot when hitting the ball. As I was playing little league at the time, I tried this technique and the result was miraculous. I started hitting balls that had previously eluded. What was more incredible is that somehow by imagining this small speck or dot, I also learned to lay off the poorly pitched balls and thus build the patience to hit the best pitches to me.

These examples changed my life in one way or another and it taught me the power of concceputalization as a tool for learning and understanding. Does anybody have anything like this to offer me? Some sort of advice or perspective that will help me finally understand the basics of math and be able to solve the math problems. I know I am overlooking something in approaching math and I am certain that it's right there but I am somehow unable to see it.

So the hive mind: Please tell me of any tricks or advice you know about how to approach math (by math I mean GRE Level math)

The most important thing that mathematics -- of any kind -- teaches you is

Anyone looking for a shortcut will fail. That's the thing you learn from math. There are no shortcuts. You have to understand every step you take.

Some people can't make that breakthrough, and for them math is somewhere between mysterious and terrifying.

posted by Chocolate Pickle at 6:50 PM on January 26, 2012 [9 favorites]

*rigor*. No guesses, no shortcuts. The path of a thousand miles begins with a single step -- and you have to take every single step, one at a time, and complete each step before taking the next one.Anyone looking for a shortcut will fail. That's the thing you learn from math. There are no shortcuts. You have to understand every step you take.

Some people can't make that breakthrough, and for them math is somewhere between mysterious and terrifying.

posted by Chocolate Pickle at 6:50 PM on January 26, 2012 [9 favorites]

Basically all low level math problems are algorithmic. What I mean by that is that once you identify what type of problem it is, there is a set of steps that will get you to the answer for all questions of that type.

The problem lots of people have in math is that they read the question, and then they try to think really hard about what the answer is. This is the wrong way to approach the problem*. Your first step is to identify the question type, think about the steps you need to do to get to the answer (this is basically just memory), then run through the steps one at a time, writing everything out.

So when you're studying, take a general topic like trigonometry and go through the problem sets to figure out what types of questions they ask about trigonometry. Figure out the steps you need to solve each type of problem, practice, then move on to the next type of problem**.

This should go without saying, but maybe not: In math, you *must* learn by doing. Read the explanations in the text, of course, but if you're not actively solving problems then you're not learning anything.

Finally, here's a problem lots of my students have had. They'll work on a problem and get it wrong, then look up the answer. Once they identify what they did wrong, they'll move on to the next problem. This is bad. If you get a problem wrong, even if you know your mistake, go back and redo every step in the problem. This will make it stick in your brain for the next time that type of problem comes up.

None of these are simple tricks, so sorry for that. But those are the big things that have helped students who struggle in math, in my experience. I don't believe most people are bad at math, they've just never been taught the right way to approach math problems.

*I blame elementary school math teachers for this

**This might be easy to do on your own, or it might not. Find help if you can

posted by no regrets, coyote at 6:52 PM on January 26, 2012 [6 favorites]

The problem lots of people have in math is that they read the question, and then they try to think really hard about what the answer is. This is the wrong way to approach the problem*. Your first step is to identify the question type, think about the steps you need to do to get to the answer (this is basically just memory), then run through the steps one at a time, writing everything out.

So when you're studying, take a general topic like trigonometry and go through the problem sets to figure out what types of questions they ask about trigonometry. Figure out the steps you need to solve each type of problem, practice, then move on to the next type of problem**.

This should go without saying, but maybe not: In math, you *must* learn by doing. Read the explanations in the text, of course, but if you're not actively solving problems then you're not learning anything.

Finally, here's a problem lots of my students have had. They'll work on a problem and get it wrong, then look up the answer. Once they identify what they did wrong, they'll move on to the next problem. This is bad. If you get a problem wrong, even if you know your mistake, go back and redo every step in the problem. This will make it stick in your brain for the next time that type of problem comes up.

None of these are simple tricks, so sorry for that. But those are the big things that have helped students who struggle in math, in my experience. I don't believe most people are bad at math, they've just never been taught the right way to approach math problems.

*I blame elementary school math teachers for this

**This might be easy to do on your own, or it might not. Find help if you can

posted by no regrets, coyote at 6:52 PM on January 26, 2012 [6 favorites]

You might try taking a look at What is it like to have an understanding of very advanced mathematics? It's at least entertaining, and it might provide some opening for insights as well.

posted by alms at 6:56 PM on January 26, 2012 [1 favorite]

posted by alms at 6:56 PM on January 26, 2012 [1 favorite]

Former math major here. I got through high school math through sheer repetition and practice. My brain just learned the patterns of "If you see A, do B" - on preview, what

I developed a deeper understanding of math concepts and reasoning while taking a history of math class as an undergraduate. We worked through Boyer's book in order - I say worked through, because in each chapter we had to prove or derive the major achievement using only the mathematical tools available at the time. Strangely enough, for me this made the disparate pieces of math knowledge and skills fall in place in a way that my previous math education had not.

(Random aside: I was always bad at doing word problems. I had to think of math as being its own thing, and attempts to relate it to the "real world" as in word problems or presenting things to me in terms of "if you have 10 apples and give 3 apples to Joey, how many will you have left?" only confused me terribly.)

posted by needled at 7:03 PM on January 26, 2012 [3 favorites]

**no regrets, coyote**said.I developed a deeper understanding of math concepts and reasoning while taking a history of math class as an undergraduate. We worked through Boyer's book in order - I say worked through, because in each chapter we had to prove or derive the major achievement using only the mathematical tools available at the time. Strangely enough, for me this made the disparate pieces of math knowledge and skills fall in place in a way that my previous math education had not.

(Random aside: I was always bad at doing word problems. I had to think of math as being its own thing, and attempts to relate it to the "real world" as in word problems or presenting things to me in terms of "if you have 10 apples and give 3 apples to Joey, how many will you have left?" only confused me terribly.)

posted by needled at 7:03 PM on January 26, 2012 [3 favorites]

I'm a Basic Skills math teacher with 7.5 years of experience teaching at the elementary and middle school level.

The first thing I recommend is to think about concepts as STRUCTURES rather than as PROCESSES, and learn them by looking at the relationships among their elements and their relationships to other things you know. When learning fraction multiplication, for example, don't just learn 2/3 x 5/8 = ?, try a problem like 4/? x 3/11 = 12/55. Think about other examples of multiplication you know, like whole number multiplication, and see 5 x 3 = 15 as being equivalent to 5/1 x 3/1 = 15/1.

The next thing I recommend is to write a ton of shit down each time you approach a problem. Problem solving is really a "leap and the net will appear" sort of thing, where the understanding comes AFTER you start writing things down. Become an expert in organizing your information, with the goal being to make it easy for you to check your work over in multiple ways. Don't make the mistake of skipping over steps using mental math; write down as much as you can.

I talk to my students a lot about "leakage." "Leakage" is when one concept "leaks" into another concept with similar features because a student hasn't been put into a position where they have to "shift gears" between the different problem types. A lot of the time, when a student doesn't have a good concept of fraction multiplication, it's more like "what we're doing with fractions this week," which makes it easy to confuse with fraction addition, or whatever they did last week. The way around this is to mix up the problems you practice so you practice SELECTING THE SKILL. (Standardized test prep books are good for this.) It also helps to be mindful of the fact that you may be experiencing "leakage" when you get a problem wrong, and try to look for the other concept that's being mixed up.

Make drawings of word problems rather than using word tricks to try to work them out.

Talk about your thinking.

When working with a textbook, try to solve all problems with solutions in a given problem set before you move onto the next one. CHECK YOUR ANSWERS AFTER EVERY PROBLEM, NOT AFTER ALL OF THEM, so you have a chance to catch your mistakes.

Look for positive feedback loops, and try to get into them. The more you enjoy math, the better you'll get at it, and the better you get at it, the more you'll enjoy it, etc. The more you look for multiple solutions to a given problem, the more accurate your answers will be, and the more flexible your thinking will be when you see a new problem, and so on.

posted by alphanerd at 7:04 PM on January 26, 2012 [11 favorites]

The first thing I recommend is to think about concepts as STRUCTURES rather than as PROCESSES, and learn them by looking at the relationships among their elements and their relationships to other things you know. When learning fraction multiplication, for example, don't just learn 2/3 x 5/8 = ?, try a problem like 4/? x 3/11 = 12/55. Think about other examples of multiplication you know, like whole number multiplication, and see 5 x 3 = 15 as being equivalent to 5/1 x 3/1 = 15/1.

The next thing I recommend is to write a ton of shit down each time you approach a problem. Problem solving is really a "leap and the net will appear" sort of thing, where the understanding comes AFTER you start writing things down. Become an expert in organizing your information, with the goal being to make it easy for you to check your work over in multiple ways. Don't make the mistake of skipping over steps using mental math; write down as much as you can.

I talk to my students a lot about "leakage." "Leakage" is when one concept "leaks" into another concept with similar features because a student hasn't been put into a position where they have to "shift gears" between the different problem types. A lot of the time, when a student doesn't have a good concept of fraction multiplication, it's more like "what we're doing with fractions this week," which makes it easy to confuse with fraction addition, or whatever they did last week. The way around this is to mix up the problems you practice so you practice SELECTING THE SKILL. (Standardized test prep books are good for this.) It also helps to be mindful of the fact that you may be experiencing "leakage" when you get a problem wrong, and try to look for the other concept that's being mixed up.

Make drawings of word problems rather than using word tricks to try to work them out.

Talk about your thinking.

When working with a textbook, try to solve all problems with solutions in a given problem set before you move onto the next one. CHECK YOUR ANSWERS AFTER EVERY PROBLEM, NOT AFTER ALL OF THEM, so you have a chance to catch your mistakes.

Look for positive feedback loops, and try to get into them. The more you enjoy math, the better you'll get at it, and the better you get at it, the more you'll enjoy it, etc. The more you look for multiple solutions to a given problem, the more accurate your answers will be, and the more flexible your thinking will be when you see a new problem, and so on.

posted by alphanerd at 7:04 PM on January 26, 2012 [11 favorites]

I was a real math nerd in my youth, and I have to say that the single most valuable tool for me in really comprehending math, not just memorizing it, was the Montessori-inspired (I think) wooden blocks that I got to play with in school. They were kind of like these and these.

What I took away from those blocks and still use EVERY DAY in my head as a grown-ass adult, without even knowing it, is this: Imagine that there's a fixed amount of space that is 10 units long. Now lay a red rod that is 8 units long in it. When you try to add a green rod that is 5 units long to it, laying them end to end, there will be two green units that "fit" in the 10-space, and three green units that hang off the end. That's how and why 8+5=1

It sounds like that's only really useful in arithmetic, but honestly, that concept of an x-units-big-space and how different variables interact with one another within it (as if they were real physical entities with

Try the blocks! (And please let me know if my explanation is more confusing than helpful, I'm happy to try to clarify). Best of luck.

posted by argonauta at 7:10 PM on January 26, 2012 [2 favorites]

What I took away from those blocks and still use EVERY DAY in my head as a grown-ass adult, without even knowing it, is this: Imagine that there's a fixed amount of space that is 10 units long. Now lay a red rod that is 8 units long in it. When you try to add a green rod that is 5 units long to it, laying them end to end, there will be two green units that "fit" in the 10-space, and three green units that hang off the end. That's how and why 8+5=1

**3**.It sounds like that's only really useful in arithmetic, but honestly, that concept of an x-units-big-space and how different variables interact with one another within it (as if they were real physical entities with

*size*, not just symbols), was absolutely critical to how my brain processed algebra, geometry, trigonometry, calculus, linear algebra, number theory, logic problems, etc..Try the blocks! (And please let me know if my explanation is more confusing than helpful, I'm happy to try to clarify). Best of luck.

posted by argonauta at 7:10 PM on January 26, 2012 [2 favorites]

There's no one answer that's right for everyone. For instance, I think "you have to take every single step, one at a time, and complete each step before taking the next one" is wildly different from the way I think about mathematics (I am a mathematician) but it may be right for some people. I very much doubt you'll find a magic technique that has the kind of effect you report regarding 1.2,.or 3. But who knows? I have never in my life had my ability to execute a cognitive task change so dramatically with a shift of technique, and it's happened to you three times, so there may just be something phase-shifty about the way you approach problems.

What I can say is this: it will help if you tell us what kind of problems give you trouble, and what happens in your mind as you try to solve them.

posted by escabeche at 7:18 PM on January 26, 2012

What I can say is this: it will help if you tell us what kind of problems give you trouble, and what happens in your mind as you try to solve them.

posted by escabeche at 7:18 PM on January 26, 2012

Without pre-reading the previous answers..

The most basic key to doing math for me, is to hold the numbers inside my head. I'm not sure if I can explain it properly, but when I'm doing a series of calculations, I keep all the numbers in my head and can explain how I got from one to another. It's like holding a handful of cards, and some get put in piles together (for example a six and a nine go together in the fifteen pile but they're still a six and nine in my head). Others get categorized by type (ie, all the x's go into one group and 'others' go into another.. so 3x +5x +10 equals an 8x group and a 10 group).

I also find it unusually easy (for me) to recall the number from a previous calculation for use in my current one. This suggests to me that I really am holding them in some part of my memory that is longer than short-term, but certainly shorter than long-term memory. For example, if I calculate the contribution margin for a product, do a series of other calculations for that product, and need the contribution margin a few minutes later, I can easily recall it. However, trying to remember a number given to me (say, the fixed cost for the week) requires me to look it up every time.

posted by valoius at 7:25 PM on January 26, 2012

The most basic key to doing math for me, is to hold the numbers inside my head. I'm not sure if I can explain it properly, but when I'm doing a series of calculations, I keep all the numbers in my head and can explain how I got from one to another. It's like holding a handful of cards, and some get put in piles together (for example a six and a nine go together in the fifteen pile but they're still a six and nine in my head). Others get categorized by type (ie, all the x's go into one group and 'others' go into another.. so 3x +5x +10 equals an 8x group and a 10 group).

I also find it unusually easy (for me) to recall the number from a previous calculation for use in my current one. This suggests to me that I really am holding them in some part of my memory that is longer than short-term, but certainly shorter than long-term memory. For example, if I calculate the contribution margin for a product, do a series of other calculations for that product, and need the contribution margin a few minutes later, I can easily recall it. However, trying to remember a number given to me (say, the fixed cost for the week) requires me to look it up every time.

posted by valoius at 7:25 PM on January 26, 2012

I had trouble with math, and I suck at memorizing. Mnemonic tricks never worked for me.

What worked for me was to try to understand why the equations work the way they do. An informal way to do this is to play around with the equation you're trying to learn. Try plugging in different values for the variables and see what happens. What if

If you want to go into more depth, learning about where different types of equations come from and what they're used for will help a lot, and math will make a lot more sense.

posted by nangar at 7:45 PM on January 26, 2012

What worked for me was to try to understand why the equations work the way they do. An informal way to do this is to play around with the equation you're trying to learn. Try plugging in different values for the variables and see what happens. What if

*a*is a small number and*b*is a large one? What about the reverse? What if they're both small? Why does that make a difference? After a while you can get what the operations are doing. And once you've got it, you can remember it.If you want to go into more depth, learning about where different types of equations come from and what they're used for will help a lot, and math will make a lot more sense.

posted by nangar at 7:45 PM on January 26, 2012

I went from being a mediocre-to-poor math student to a very good one. This quote from a NY Magazine article summed up how I got there:

"Learning math is not about learning math. It's about weightlifting. You are pumping the iron of math."

Take a problem and take every single variation of that problem, and do them over and over again until you recognize the problem and understand how to do it. Then check and double check your work for mistakes.

Other techniques, though, to gain an intuitive "feel" for what's going on is to do things like plug in a really large value to an unknown variable in a function and see what the result is, and then plug in a really small or very negative value. Then you can get an intuition for how a function acts.

posted by deanc at 7:55 PM on January 26, 2012 [1 favorite]

"Learning math is not about learning math. It's about weightlifting. You are pumping the iron of math."

Take a problem and take every single variation of that problem, and do them over and over again until you recognize the problem and understand how to do it. Then check and double check your work for mistakes.

Other techniques, though, to gain an intuitive "feel" for what's going on is to do things like plug in a really large value to an unknown variable in a function and see what the result is, and then plug in a really small or very negative value. Then you can get an intuition for how a function acts.

posted by deanc at 7:55 PM on January 26, 2012 [1 favorite]

As a negative data point, I did well enough in math classes in school--took AP calculus in high school and did well--and I took the purely algorithmic approach. Result? I understood almost *nothing* of what I was doing. I got good at recognizing patterns however. To this day, I think the reason I liked geometry, trig, & calculous so much more than algebra was that we had to do proofs, which forced me to understand the underlying concepts; these things became concrete. Like most learning, YMMV.

posted by smirkette at 8:05 PM on January 26, 2012 [2 favorites]

posted by smirkette at 8:05 PM on January 26, 2012 [2 favorites]

You want GED Level math advice? Here you are. Here is the Very Basic mind change that happened to me. I could NOT do math, and I hated it. I'd learned a few tricks, in particular one that goes: if you want to add 9 to a number, increase the first digit and decrease the second digit, each by one. Helpful because I didn't have the multiplication table memorized, and this was years before calculators. But I approached any math the way I approached spiders. Deal with it if I have to, but have as little contact as possible.

Then I read a book, "How Children Fail," by John Holt. He was a math teacher who talked about how children don't learn math, they learn to manipulate the teacher -- I was deeply into alternative schooling, Summerhill type schools at the time, and found the descriptions of learning fascinating.

Then one day, right after I finished the book, I was riding my motorcycle through Golden Garden Park and .... just like your examples, a tiny thing happened that changed my thinking about math. I idly thought to myself: That thing I do with 9's. It's not a magic trick with 9's. I'M ADDING TEN AND SUBTRACTING ONE! .... There are no magic tricks in math! And none of the wretched "rules" with all the damn exceptions that English presents. I could do that "trick" with 8! Add ten and subtract two! Whahoo!! Numbers are just there, with no secrets, no emotional components. Just lovely, lovely numbers that work together, and I can figure them out if I go slowly and pay attention to details.

And once you're presented with x+y sorts of problems, there are formulae to work them out. Slowly, carefully you CAN UNDERSTAND them.

Numbers have been my friends ever since. And then I recently discovered "The Numbers Game" by Michael Blastland and Andrew Dilnot, I am actually finding it fun to think about numbers. If the newspaper claims "1 person dies of choking on potato chips every 10 minutes in this country" I sit down and figure "how many people are dying every hour? How many die every month? How many in my state? Does that really sound like a true number?" It's very empowering.

Sorry to go on and on, but I just want to encourage you to continue to look for a way into numbers. It's so impressive that you're trying. Almost everyone just shrugs and says "I don't like math" and let other people do their figuring -- and their thinking -- for them. Good on ya!

posted by kestralwing at 9:21 PM on January 26, 2012 [4 favorites]

Then I read a book, "How Children Fail," by John Holt. He was a math teacher who talked about how children don't learn math, they learn to manipulate the teacher -- I was deeply into alternative schooling, Summerhill type schools at the time, and found the descriptions of learning fascinating.

Then one day, right after I finished the book, I was riding my motorcycle through Golden Garden Park and .... just like your examples, a tiny thing happened that changed my thinking about math. I idly thought to myself: That thing I do with 9's. It's not a magic trick with 9's. I'M ADDING TEN AND SUBTRACTING ONE! .... There are no magic tricks in math! And none of the wretched "rules" with all the damn exceptions that English presents. I could do that "trick" with 8! Add ten and subtract two! Whahoo!! Numbers are just there, with no secrets, no emotional components. Just lovely, lovely numbers that work together, and I can figure them out if I go slowly and pay attention to details.

And once you're presented with x+y sorts of problems, there are formulae to work them out. Slowly, carefully you CAN UNDERSTAND them.

Numbers have been my friends ever since. And then I recently discovered "The Numbers Game" by Michael Blastland and Andrew Dilnot, I am actually finding it fun to think about numbers. If the newspaper claims "1 person dies of choking on potato chips every 10 minutes in this country" I sit down and figure "how many people are dying every hour? How many die every month? How many in my state? Does that really sound like a true number?" It's very empowering.

Sorry to go on and on, but I just want to encourage you to continue to look for a way into numbers. It's so impressive that you're trying. Almost everyone just shrugs and says "I don't like math" and let other people do their figuring -- and their thinking -- for them. Good on ya!

posted by kestralwing at 9:21 PM on January 26, 2012 [4 favorites]

If you know how to read and write proofs, they can be really valuable in learning why something is the way it is, and helping you to never forget how to do it (in the algorithmic sense). I've only studied analysis, but once you understand the big picture, smaller operations are simple to strategize (if not immediately to solve). Learning to write proofs (in simple number theory and Calculus) hugely improved my confidence with numbers and logic.

I like the pumping iron explanation, because repetition and different types of problems in the same genus are your best friend, especially when it comes to something like integration.

Here's another math-related thing that makes eating out at restaurants so much easier. Say you want to tip 18%, and your bill is $23.40. What's 10% of the bill? $2.34, you can figure that out just by moving the decimal point. What's 20%? $4.68, you know that by doubling 10%. What's roughly in between those two figures? You can totally guess, or you can roughly approximate half of 10% and add it to 10%, which is 15%. Add a little, subtract a little, to get it between 15% and 20%. You can basically figure out any tip to a practical degree of accuracy within a few seconds. Err on the side of tipping a little too much. I see people pull out their phones and tip cards to figure this out, but it's quite simple mental math, and it keeps your arithmetic gears from rusting.

(P.S., Another good conceptualization: when you're turning in a car, instead of looking at the curve of the road, look at a middle distance between car and curve. Your hands will automatically turn the wheel smoothly. I hate driving, but this makes me feel much more confident. Obviously, look at other things around you as they require your attention.)

posted by stoneandstar at 9:35 PM on January 26, 2012

I like the pumping iron explanation, because repetition and different types of problems in the same genus are your best friend, especially when it comes to something like integration.

Here's another math-related thing that makes eating out at restaurants so much easier. Say you want to tip 18%, and your bill is $23.40. What's 10% of the bill? $2.34, you can figure that out just by moving the decimal point. What's 20%? $4.68, you know that by doubling 10%. What's roughly in between those two figures? You can totally guess, or you can roughly approximate half of 10% and add it to 10%, which is 15%. Add a little, subtract a little, to get it between 15% and 20%. You can basically figure out any tip to a practical degree of accuracy within a few seconds. Err on the side of tipping a little too much. I see people pull out their phones and tip cards to figure this out, but it's quite simple mental math, and it keeps your arithmetic gears from rusting.

(P.S., Another good conceptualization: when you're turning in a car, instead of looking at the curve of the road, look at a middle distance between car and curve. Your hands will automatically turn the wheel smoothly. I hate driving, but this makes me feel much more confident. Obviously, look at other things around you as they require your attention.)

posted by stoneandstar at 9:35 PM on January 26, 2012

It would help a lot if I knew what level of math you were talking about. I'm going to assume it's algebra, and related topics of about that level.

For me, I think it's useful to think of math as a game. Like Monopoly. Before you can play for the very first time, you have to open the lid and read that really daunting chunk of small-point text. Lots of tedious rules and regulations to get straight. But once you get them down, you can play the game over and over and over.

For me, math is just like that. There are some basic rules and regulations you have to learn. Techniques and tricks that are valid to use in solving an equation, and other things that seem like they oughta be fair, but aren't. And getting that stuff straight can be tedious and frustrating. But really, it's just the rules of the game. Find a great teacher or tutor who can help you understand the rules. Have them walk through a

And by the way, you have my respect and admiration for wanting to get better in this area. Kudos to you! I hope it works out well, and you find the journey challenging, but rewarding.

posted by browse at 9:52 PM on January 26, 2012

For me, I think it's useful to think of math as a game. Like Monopoly. Before you can play for the very first time, you have to open the lid and read that really daunting chunk of small-point text. Lots of tedious rules and regulations to get straight. But once you get them down, you can play the game over and over and over.

For me, math is just like that. There are some basic rules and regulations you have to learn. Techniques and tricks that are valid to use in solving an equation, and other things that seem like they oughta be fair, but aren't. And getting that stuff straight can be tedious and frustrating. But really, it's just the rules of the game. Find a great teacher or tutor who can help you understand the rules. Have them walk through a

**ton**of examples, making sure you follow the logic. Have them watch patiently while you walk through another ton of examples on your own, making sure you can follow the same rules of the road. I can't emphasize this enough; the right instructor makes all the difference in the world. If you find a bad one, dump them and look for another. Repeat until you find one who speaks the same language you do, and then soak up as much as you can.And by the way, you have my respect and admiration for wanting to get better in this area. Kudos to you! I hope it works out well, and you find the journey challenging, but rewarding.

posted by browse at 9:52 PM on January 26, 2012

Something that totally blew my brain across the room (uh...in a good way) was lattice multiplication.

Why I never learned about it in school, I'll never understand.

posted by Sys Rq at 10:36 PM on January 26, 2012 [3 favorites]

Why I never learned about it in school, I'll never understand.

posted by Sys Rq at 10:36 PM on January 26, 2012 [3 favorites]

One of the things that helped me with learning mathematics is having a detailed concept of the number line in my head. So, when I imagine numbers, they stretch out in line, but the line loops at the 10s and 100s and 1000s. Some people's number lines are more like grids, and others categorise numbers with colours. I have found that the more I was able to visualise the numbers and be comfortable about finding them on the number line in my mind, the easier it was to solve problem with them.

I have never been as good with basic geometry, and I think that's because I don't have a really good visualisation or model in my head of triangles and circles. I literally have to remember each rule individually, even though I know perfectly well that they all fit together in a grand unified theory.

When struggling with mathematics, I find this von Neumannn quote helpful

Makes me feel better, because eventually I just get used to whatever idea I've been struggling with.

posted by plonkee at 11:36 PM on January 26, 2012

I have never been as good with basic geometry, and I think that's because I don't have a really good visualisation or model in my head of triangles and circles. I literally have to remember each rule individually, even though I know perfectly well that they all fit together in a grand unified theory.

When struggling with mathematics, I find this von Neumannn quote helpful

In mathematics you don't understand things. You just get used to them.

Makes me feel better, because eventually I just get used to whatever idea I've been struggling with.

posted by plonkee at 11:36 PM on January 26, 2012

*What's the best advice you have ever received about learning math and approaching math problems?*

Test yourself all the time. The way our brains work, you will learn better if you are continuously being tested.

In a classroom lecture situation, that means you will do your best if you have a short quiz

*every day*towards the end of the lecture and a graded homework assignment

*every night*that is due the next day, and if you start every lecture by getting back the previous day's graded homework and tests so you always know how you stand.

Compared to lazier approaches, this may seem to make more work for the instructor or the instructor's assistants, but you can get around some of it by, for example, having students exchange test papers and grade each other's answers as the instructor walks everyone through the method and shows the answers.

If your instructor doesn't test frequently or if you're working on your own, you need to test yourself frequently with books and programs that will give you a steady stream of sample problems. In this case, make sure you grade yourself honestly and chart your progress.

posted by pracowity at 12:56 AM on January 27, 2012 [1 favorite]

*Test yourself all the time. The way our brains work, you will learn better if you are continuously being tested.*

This isn't true if you're not getting the fundamental technique required for learning this stuff in the first place.The problem many of us have with symbolic representation in math is that it is truly abstract - there is no associative meaning other than more abstraction and that's the stuff that involves drilling and memorising to get it to stick. If you are academically strong in other areas you'll be used to learning by (often subconscious) association rather than by rote. This difference is very poorly articulated in most schools.

At school I was academically bright in a multitude of areas except maths. I didn't understand the sheer amount you have to take as read. I was always struggling with the 'why' something was correct, never realising that the answer was usually 'because it is'. A lot of even very basic arithmetic is predicated on logic that is very hard to grasp until you have developed the capacity for true abstraction. Thing is you don't need to understand that stuff, the algorithms required for arithmetic aren't maths in themselves (though the construction of them was), they are the tools. You just have to be able to commit them to memory and they become tools for the creative abstraction that 'doing maths' actually consists of. Having realised this, the learning process became easier.

An interesting read on this very topic is The Maths Gene by Keith Devlin. Some of his theories are a bit woolly but the central argument focusses on how the ability to think mathematically is related to the ability to learn language and there are a lot of examples that might provide the 'Eureka' moment you're looking for.

posted by freya_lamb at 2:22 AM on January 27, 2012 [1 favorite]

'Rigor above intuition' is what I learned. In understanding other people's proofs and solving problems yourself, you have to follow a rigorous logic that is not the same as the 'if/then' of what to expect from people or social systems. That's why folks here have been saying 'no shortcuts': because the intuition by which you swiftly navigate human interactions often won't get you to the right answer in maths: you have to go step by step until you're confident you've mastered that logical sequence. I had to (kind of) unlearn my humanities education. It can be frustrating at first but it is consistent, so if it seems illogical, cruel or contrary, persevere and you'll master it.

posted by Gomoryhu at 3:58 AM on January 27, 2012 [1 favorite]

posted by Gomoryhu at 3:58 AM on January 27, 2012 [1 favorite]

You're getting a lot of mixed statements here because a) there is no "best" way to think about mathematics and b) the way you think about it, in the the ways in which you can think about it, changes as you get more and more advanced. If you've only learned up through linear algebra and calculus, rules and rigidity may seem like the only course of action. As you learn the subjects involved more deeply, though, one has to bring to bear a strong intuitive sense of what the math means. After all, mathematicians don't come up with theorems by starting with some assumptions and then following a stepping stone of formal techniques, they have an expectation that some statement is true and then use the formal tools of math to show why indeed it must be true. That said, you're not going to get the latter step within the context of GRE math.

To my mind, you're absolutely on the right track with the general idea of conceptualizing math problems in other ways. I have two mental suggestions, and one general tip:

1) Draw everything. Nine times out of ten, the hardest part of a problem is setting it up correctly. All of the details of mathematical operations can be practiced (and even math profs will still screw them up sometimes when put on the spot), but the part of the problem that needs cleverness is often the first statement. In most cases, the easiest way to get this right is to make as good a picture of the situation being posed as you can — not only do you not omit important details, but you can better use the intuition you already have to suggest approaches. It also might help break the problem into smaller component parts.

2) Imagine what the most similar easy problem would be and figure out how you would solve that one. Maybe you have one of those word problems with the two trains starting at the same place and moving in opposite directions, but the speed of one of the trains keeps changing. The problem would clearly be simpler if that one train just kept a steady speed, so think about how you'd solve it in that case. Often, at that point you're only one or two steps away from turning the problem-solving method from the easier case into the one needed for the problem at hand.

General Tip) When you get things wrong, do them again. It's not enough to read the correct solution to a problem you got wrong, you need to write out the solution yourself. As you write it, make sure you have a step by step understanding not only of why what you're doing is correct, but why it's the right next step to take. If you do this a bunch, you will start to see that there are actually only so many general approaches that one uses, almost no matter the problem. Don't be embarrassed, either, ask friends if you don't understand some step. As people said above, practice is crucial, but not all practice is equally useful.

posted by Schismatic at 4:40 AM on January 27, 2012 [1 favorite]

To my mind, you're absolutely on the right track with the general idea of conceptualizing math problems in other ways. I have two mental suggestions, and one general tip:

1) Draw everything. Nine times out of ten, the hardest part of a problem is setting it up correctly. All of the details of mathematical operations can be practiced (and even math profs will still screw them up sometimes when put on the spot), but the part of the problem that needs cleverness is often the first statement. In most cases, the easiest way to get this right is to make as good a picture of the situation being posed as you can — not only do you not omit important details, but you can better use the intuition you already have to suggest approaches. It also might help break the problem into smaller component parts.

2) Imagine what the most similar easy problem would be and figure out how you would solve that one. Maybe you have one of those word problems with the two trains starting at the same place and moving in opposite directions, but the speed of one of the trains keeps changing. The problem would clearly be simpler if that one train just kept a steady speed, so think about how you'd solve it in that case. Often, at that point you're only one or two steps away from turning the problem-solving method from the easier case into the one needed for the problem at hand.

General Tip) When you get things wrong, do them again. It's not enough to read the correct solution to a problem you got wrong, you need to write out the solution yourself. As you write it, make sure you have a step by step understanding not only of why what you're doing is correct, but why it's the right next step to take. If you do this a bunch, you will start to see that there are actually only so many general approaches that one uses, almost no matter the problem. Don't be embarrassed, either, ask friends if you don't understand some step. As people said above, practice is crucial, but not all practice is equally useful.

posted by Schismatic at 4:40 AM on January 27, 2012 [1 favorite]

I have answered this before about math questions but it is true but taking up poker has helped me a lot with math: probability, percentages, logic, arithmetic, etc. and it has also taught me to like math more & see its value in my life.

posted by pointystick at 5:38 AM on January 27, 2012

posted by pointystick at 5:38 AM on January 27, 2012

Continuing the 'this is what worked for me' (ie; like maths, there are many ways to get to the answer)

Dont be dazzled by the misty heights of pure maths. The 'everyday' world and maths intersect in physics. So if you are smart, and are pretty good at understanding most stuff; learn the mathematical patterns behind the world around you. Mind you; this way lies engineering...

I'm saving my 'Calculus and Analytical Geometry' book for a rainy day.

posted by BadMiker at 5:42 AM on January 27, 2012

Dont be dazzled by the misty heights of pure maths. The 'everyday' world and maths intersect in physics. So if you are smart, and are pretty good at understanding most stuff; learn the mathematical patterns behind the world around you. Mind you; this way lies engineering...

I'm saving my 'Calculus and Analytical Geometry' book for a rainy day.

posted by BadMiker at 5:42 AM on January 27, 2012

When people say they are "bad at math", I feel more like they are bad at

If it is indeed calculation that you are bad at, this could be a matter of drilling. However, that is boring and lame, and usually a big reason why people develop an anxiety toward mathematics at a young age. Just because you can't multiply 23*47 as fast as some other student doesn't mean you're worse at math than he is. It means nothing, really.

Someone mentioned above something about "never skipping any steps", but I don't think that's how most mathematicians (including myself) actually think about math. I'm usually guessing at what the right conclusion (or "answer", if you like) should be, then verifying/falsifying it through an argument.

If you want to be better at calculations, you can take the route that pointystick mentioned. Learn about probability and odds. You will get better at calculating stuff on the fly. I also suggest that when you're in the grocery store and you see that there is a bottle of 100 aspirin for 10.99 and a bottle of 150 aspirin for 12.99, do a rough calculation to see how much each pill costs. Check your calculation on your phone. Doing things like this every day (and in a situation where you might care about the outcome, as opposed to the lame drilling for drilling's sake) will make you better.

If you want to get better at developing critical thinking skills, read the books of Raymond Smullyan. They are full of logic "puzzles" and what not, and will get the critical thinking side of your brain working. Play Mindtrap.

Lastly, and I know I link to this often, but read this [PDF link]. It may help you understand why you are where you are in terms of mathematics, and might give you more hints as to how to get better.

posted by King Bee at 6:31 AM on January 27, 2012 [1 favorite]

*calculation*. I find this a lot when I teach my calculus courses. Many of the ideas and theorems (like the Intermediate Value Theorem or the Squeeze Theorem or the Mean Value Theorem) are actually intuitively obvious to the students. So obvious sometimes, that they can't even grok why you would need a*theorem*for something so innate.If it is indeed calculation that you are bad at, this could be a matter of drilling. However, that is boring and lame, and usually a big reason why people develop an anxiety toward mathematics at a young age. Just because you can't multiply 23*47 as fast as some other student doesn't mean you're worse at math than he is. It means nothing, really.

Someone mentioned above something about "never skipping any steps", but I don't think that's how most mathematicians (including myself) actually think about math. I'm usually guessing at what the right conclusion (or "answer", if you like) should be, then verifying/falsifying it through an argument.

If you want to be better at calculations, you can take the route that pointystick mentioned. Learn about probability and odds. You will get better at calculating stuff on the fly. I also suggest that when you're in the grocery store and you see that there is a bottle of 100 aspirin for 10.99 and a bottle of 150 aspirin for 12.99, do a rough calculation to see how much each pill costs. Check your calculation on your phone. Doing things like this every day (and in a situation where you might care about the outcome, as opposed to the lame drilling for drilling's sake) will make you better.

If you want to get better at developing critical thinking skills, read the books of Raymond Smullyan. They are full of logic "puzzles" and what not, and will get the critical thinking side of your brain working. Play Mindtrap.

Lastly, and I know I link to this often, but read this [PDF link]. It may help you understand why you are where you are in terms of mathematics, and might give you more hints as to how to get better.

posted by King Bee at 6:31 AM on January 27, 2012 [1 favorite]

I'm pretty good at math, (I have a math degree from MIT), but even so I have to go reallllyyyyy slowly to understand mathematical concepts. Like, an hour to fully understand a page, slow. I don't move on until I fully understand it.

Here is my fool-proof guide to whether I understand a new concept:

Each incremental bit of new knowledge has to be just close enough to your existing knowledge base that you feel like, had you not just read about this new idea in the book, but rather were trapped on a desert island puzzling it out, you could have come up with it yourself.

And if you ever find yourself thinking "How on earth did anyone figure

For example, algebra is pretty straightforward. x + 5 = 12. I hated how the teacher kept harping on "okay, subtract 5 from both sides of the equal sign." That just seemed kind of arbitrary to me and didn't make intuitive sense.* Instead, I would sit there and think and repeat to myself "Something plus five is twelve. Something plus five is twelve. Some

And here's the thing: you need a solid base at the level just below this to make the connection. If you know that 7 + 5 = 12, 12 - 5 = 7, etc, then kind of juggling that in your mind while repeating the "something plus five is twelve mantra" really helps make the connection.

Once you are feeling some sort of realization dawning, play around with a few more examples and see if you can isolate what exactly is happening. x + 8 = 15 becomes "something plus eight is fifteen, fifteen is eight more than some number" eventually becomes "ah, so in this situation, what i'm doing is subtracting the one number from the other. Got it."

Then you can poke and prod and extend it. x - 8 = 15. "Okay, something minus eight is fifteen. A number, taking away eight, leaves 15. Ah, so that number is bigger than fifteen. That number is eight bigger than fifteen. That number is fifteen plus eight."

Now you've got addition and subtraction in your algebraic equations. You can play around further and work with multiplication and division, and then you can combine them. Then you can think about what if there are two unknowns, and so not quite enough information to determine what they are

So summary:

1) The level of intuition to shoot for is "I could have discovered that myself."

2) Poke and prod a concept, repeating it in your head, changing a few words around at a time.

3) When small intuition dawns, try a new very similar but not quite identical problem.

4) Isolate what all your examples have in common to extract the mathematical idea.

Hope that helps. If you want to provide an example of something which you don't intuitively understand, but seems like it's close to something you do understand well, I can help you figure out how to make the jump to full understanding of the new concept.

*(At first, eventually the concept that if two expressions are equal, then you can do anything to both of them and they're still equal, became pretty cool and intuitive in its own right.)

posted by losvedir at 7:25 AM on January 27, 2012 [7 favorites]

Here is my fool-proof guide to whether I understand a new concept:

**Could I have figured this out by myself?**Each incremental bit of new knowledge has to be just close enough to your existing knowledge base that you feel like, had you not just read about this new idea in the book, but rather were trapped on a desert island puzzling it out, you could have come up with it yourself.

And if you ever find yourself thinking "How on earth did anyone figure

*that*out?" then the gap is too great. There are some intermediate steps you need to know.For example, algebra is pretty straightforward. x + 5 = 12. I hated how the teacher kept harping on "okay, subtract 5 from both sides of the equal sign." That just seemed kind of arbitrary to me and didn't make intuitive sense.* Instead, I would sit there and think and repeat to myself "Something plus five is twelve. Something plus five is twelve. Some

*number*plus five is twelve. Twelve is five*more*than some number. If I started at this number, and counted up by five, I'd end up at twelve." I'd play with it in my mind, reformulating it, sometimes just repeating what I just said, until the idea became clear.And here's the thing: you need a solid base at the level just below this to make the connection. If you know that 7 + 5 = 12, 12 - 5 = 7, etc, then kind of juggling that in your mind while repeating the "something plus five is twelve mantra" really helps make the connection.

Once you are feeling some sort of realization dawning, play around with a few more examples and see if you can isolate what exactly is happening. x + 8 = 15 becomes "something plus eight is fifteen, fifteen is eight more than some number" eventually becomes "ah, so in this situation, what i'm doing is subtracting the one number from the other. Got it."

Then you can poke and prod and extend it. x - 8 = 15. "Okay, something minus eight is fifteen. A number, taking away eight, leaves 15. Ah, so that number is bigger than fifteen. That number is eight bigger than fifteen. That number is fifteen plus eight."

Now you've got addition and subtraction in your algebraic equations. You can play around further and work with multiplication and division, and then you can combine them. Then you can think about what if there are two unknowns, and so not quite enough information to determine what they are

*specifically*but at least maybe find a relationship between the two unknowns - and now you're into lines and graphing. On and on it goes.So summary:

1) The level of intuition to shoot for is "I could have discovered that myself."

2) Poke and prod a concept, repeating it in your head, changing a few words around at a time.

3) When small intuition dawns, try a new very similar but not quite identical problem.

4) Isolate what all your examples have in common to extract the mathematical idea.

Hope that helps. If you want to provide an example of something which you don't intuitively understand, but seems like it's close to something you do understand well, I can help you figure out how to make the jump to full understanding of the new concept.

*(At first, eventually the concept that if two expressions are equal, then you can do anything to both of them and they're still equal, became pretty cool and intuitive in its own right.)

posted by losvedir at 7:25 AM on January 27, 2012 [7 favorites]

This is more a comment on the GRE than on math in general (I'm about to go for a walk and might have an answer when I come back), but the GRE syllabus doesn't actually contain much. You probably did more in high school, since it doesn't have trig. (So there's actually less material on the GRE than on the SAT, if that's any reassurance.) However, the GRE is very good at asking questions that actually require you

posted by hoyland at 8:24 AM on January 27, 2012

*understand*what is happening and it isn't purely algorithmic. As someone in math, I actually found the math section of the general GRE entertaining--it wasn't so difficult I had to expend actual mental energy to do the problems or worry about how I was doing, but I couldn't switch my brain off and plug and chug.posted by hoyland at 8:24 AM on January 27, 2012

My best math teacher in high school insisted on a concept called spiraling. At the beginning of class each day, we would turn our notebooks back ninety days, and redo the problems we were doing then. This helped reinforce the earlier concepts, and kept them fresh and active in our minds. As we went into the current lesson, we could see how the earlier concept had set us up for the day's activity. This changed my entire relationship with math, because it reinforced that you never abandon the earlier concepts, you just use the same concepts in different ways.

She was also just very good at getting concepts plugged into some math-proof young brains.

posted by halfbuckaroo at 9:20 AM on January 27, 2012

She was also just very good at getting concepts plugged into some math-proof young brains.

posted by halfbuckaroo at 9:20 AM on January 27, 2012

This is all great advice. Thanks!

I think my problems in mathematics began in the third grade, oddly enough, when I could not for the life of me understand borrowing in subtraction. To this day, I still have a hard time subtracting numbers that involves a lot of borrowing. Later on, when we got started in fractions, I realized that all hope was gone. In fact, for most of hte people I've talked to, their problem with math almost inevitably began with fractions and how to understand and manipulate them. For one, I couldn't grasp how you could multiply 2/3, for instance, with, say 1/3. After a while, you learn the rules and you move on. That's it.

What I do remember in math class is that I always seemed to ask very good questions, the kind of questions that would stump teachers. I don't mean the teachers would say "Good question" in a patronizing way, but that they actually had to think about it for a while. I never much liked rote memorization in math but the problem is that the kind of questions I would ask had answers that often I couldn't follow.

I remember once in a stats class we were trying to calculate variance and the teacher handed us a formula that we needed to employ in our calculations. One of the operations in the formula, I think, involved us having to square something. Immediately, I grew curious as to why we squared something in this instance. Was it to make sure the number was not a negative?

Anyway, I am still looking for that conceptual insight, that phase-shift as someone referred to it, to help guide me in math. There is something I know I am missing and it's bothering me terribly.

posted by lackadaisical at 11:43 AM on January 27, 2012

I think my problems in mathematics began in the third grade, oddly enough, when I could not for the life of me understand borrowing in subtraction. To this day, I still have a hard time subtracting numbers that involves a lot of borrowing. Later on, when we got started in fractions, I realized that all hope was gone. In fact, for most of hte people I've talked to, their problem with math almost inevitably began with fractions and how to understand and manipulate them. For one, I couldn't grasp how you could multiply 2/3, for instance, with, say 1/3. After a while, you learn the rules and you move on. That's it.

What I do remember in math class is that I always seemed to ask very good questions, the kind of questions that would stump teachers. I don't mean the teachers would say "Good question" in a patronizing way, but that they actually had to think about it for a while. I never much liked rote memorization in math but the problem is that the kind of questions I would ask had answers that often I couldn't follow.

I remember once in a stats class we were trying to calculate variance and the teacher handed us a formula that we needed to employ in our calculations. One of the operations in the formula, I think, involved us having to square something. Immediately, I grew curious as to why we squared something in this instance. Was it to make sure the number was not a negative?

Anyway, I am still looking for that conceptual insight, that phase-shift as someone referred to it, to help guide me in math. There is something I know I am missing and it's bothering me terribly.

posted by lackadaisical at 11:43 AM on January 27, 2012

All hope is not gone! For what it's worth, here's one way to think about multiplying 2/3 by 1/3, using the "give these things physical size" concept I described above (which I think could totally help with the borrowing/subtraction bit too, by the way):

Say you have an apple and cut it into thirds. Put two of those pieces in a baggie. That's your "2/3 Apple." You could imagine having six of those baggies, right? That would be multiplying your "2/3 Apple" by six. If you counted up all those separate 1/3 pieces, you'd see that you have twelve of them. You could then reassemble those twelve "1/3 apple pieces" into four full apples, right? So, multiplying 2/3 x 6 = 4. You get there by saying 2/3 x 6 = 12/3 (multiplying just the numerators of the fraction), and 12/3 (="twelve 1/3 Apple pieces," or 12 divided by 3) = 4.

That's really the same thing you do when you try to multiple a fraction by another fraction, but if you're multiplying by a fraction that's smaller than 1, you're going to have a result that's less than the amount you started with, not more (like when you multiply by 6). Multiplying something by 1/3 is actually the same as dividing it by three. (And multiplying something by 2/3 would be the same as dividing it by three then doubling that amount). So, in the apple example, you'd take your "2/3 Apple" baggie and you want to divide what's in it into three equal amounts, like if you wanted to give equally-sized snacks to three hungry toddlers from that baggie.

So, how can you divide it into three? One of the quickest ways is to take

I hope that helps. I've definitely found that just following the rules of how you manipulate the numbers and symbols is faster than going through this kind of thought exercise every time, but I had to get down the logic of why I was doing what I was doing, and getting the results that I was, by working through how these kinds of calculations would work "in real life." It really makes it easier to test whether you think you're getting the right kinds of answers, too.

Best of luck in your continued quest!

posted by argonauta at 12:27 PM on January 27, 2012 [1 favorite]

Say you have an apple and cut it into thirds. Put two of those pieces in a baggie. That's your "2/3 Apple." You could imagine having six of those baggies, right? That would be multiplying your "2/3 Apple" by six. If you counted up all those separate 1/3 pieces, you'd see that you have twelve of them. You could then reassemble those twelve "1/3 apple pieces" into four full apples, right? So, multiplying 2/3 x 6 = 4. You get there by saying 2/3 x 6 = 12/3 (multiplying just the numerators of the fraction), and 12/3 (="twelve 1/3 Apple pieces," or 12 divided by 3) = 4.

That's really the same thing you do when you try to multiple a fraction by another fraction, but if you're multiplying by a fraction that's smaller than 1, you're going to have a result that's less than the amount you started with, not more (like when you multiply by 6). Multiplying something by 1/3 is actually the same as dividing it by three. (And multiplying something by 2/3 would be the same as dividing it by three then doubling that amount). So, in the apple example, you'd take your "2/3 Apple" baggie and you want to divide what's in it into three equal amounts, like if you wanted to give equally-sized snacks to three hungry toddlers from that baggie.

So, how can you divide it into three? One of the quickest ways is to take

**each**of the "1/3 Apple" pieces in the baggie and cut them into thirds. Each toddler gets two of the new 1/3 pieces of your two original 1/3 pieces of apple. It's just like you'd originally divided your whole apple into nine pieces instead of three, and you now have two of those "1/9 Apple"-sized pieces as your answer. The answer to 2/3 x 1/3= 2/9. In terms of the symbols, you're multiplying the numerators times each other and the denominators times each other (2x1 = 2 and 3x3=9). But this is just a quick mathematical way to represent what the size of one-third of your "2/3 apple."I hope that helps. I've definitely found that just following the rules of how you manipulate the numbers and symbols is faster than going through this kind of thought exercise every time, but I had to get down the logic of why I was doing what I was doing, and getting the results that I was, by working through how these kinds of calculations would work "in real life." It really makes it easier to test whether you think you're getting the right kinds of answers, too.

Best of luck in your continued quest!

posted by argonauta at 12:27 PM on January 27, 2012 [1 favorite]

Interesting approach, argonauta. I like it.

I don't know. I figure I'll post another question next week about some examples of "conceptual shifts" that helped people to understand math or other subjects better. Math is very hard. Or atleast it is for me. Even the simplest problems leave me stymied. Worst part is that people who I know are not that much smarter than me seem to grasp it easily and almost intuitively. Unsurprisingly, these are the same people who can't teach math because it comes so easily to them.

posted by lackadaisical at 1:26 PM on January 27, 2012

I don't know. I figure I'll post another question next week about some examples of "conceptual shifts" that helped people to understand math or other subjects better. Math is very hard. Or atleast it is for me. Even the simplest problems leave me stymied. Worst part is that people who I know are not that much smarter than me seem to grasp it easily and almost intuitively. Unsurprisingly, these are the same people who can't teach math because it comes so easily to them.

posted by lackadaisical at 1:26 PM on January 27, 2012

Most people that struggle in math start to have problems around fractions and decimals, and that lack of understanding carries forward forever. I recently went back to university and had to relearn a bunch of algebra that I haven't looked at for close to a decade. And I mean, I forgot everything.

So I did a huge refresher using the free sources out there, including a heavy reliance on the khan academy to get reacquainted with the basics. And I started low, from scratch you could say. And if I got stuck on a concept, I looked for more sources that would explain it in many different ways until one clicked. There was not one single epiphany, but many different ones as I went through math, depending on how different concepts had to be explained so that I got them.

I will still never be facile with math, and I will never reach the heights of intuitive understanding that people who get into calculus and beyond do, but I have a solid foundation of algebra, stats and general math, which is enough for where I'm at. My point is to look at every challenging concept from several different angles until you 'get' it. And not one source will be the end-all of inspiration.

posted by tatiana131 at 1:30 PM on January 27, 2012

So I did a huge refresher using the free sources out there, including a heavy reliance on the khan academy to get reacquainted with the basics. And I started low, from scratch you could say. And if I got stuck on a concept, I looked for more sources that would explain it in many different ways until one clicked. There was not one single epiphany, but many different ones as I went through math, depending on how different concepts had to be explained so that I got them.

I will still never be facile with math, and I will never reach the heights of intuitive understanding that people who get into calculus and beyond do, but I have a solid foundation of algebra, stats and general math, which is enough for where I'm at. My point is to look at every challenging concept from several different angles until you 'get' it. And not one source will be the end-all of inspiration.

posted by tatiana131 at 1:30 PM on January 27, 2012

I'm in my third year of a math degree. I have a three-pronged approach to thinking about math.

(1) Problem-based approach.

(2) Building intuition.

(3) Proof building.

(4) Computational approach.

(1) is about finding as many problems about a subject as possible. These range from applied problems and word problems; to olympiad-style problems. Find your subject that you want to learn about. Find problem books in that subject. There are many different mathematical series with these problem books. These include the Schaum's series, the Dolciani Mathematical Expositions, and the Springer Problem books in Mathematics series. Finally, practice these problems over and over (and also start looking into books

(2) is finding books and textbooks that give a intuitive feel for the subject. Often these are geometry-based approaches, popular accounts, or historical accounts.

(3) is found in any good textbook. Most textbooks take the definition, theorem, proof approach. Picking up any good Discrete Mathematics textbook will give you a background in how to write proofs (like Epp's textbook). Velleman's

(4) is making yourself familiar with computational approaches in the field you are studying. At my university this is mostly done with MATLAB and R. But learning any other languages would probably help as well (numpy/scipy for python, mathematica).

You need to work (1) to (4) simultaneously. Also, anyone that focuses at one of those approaches at the behest of others is missing out on the big picture. I'm looking especially at people who think math is all about (3). For a counter-point go read Poincare (especially this), and less Bourbaki-influenced approaches. I believe this part of the problem with people that don't get math. You have programmer-types on the Internet that think it is all about (3), and completely discard (1) and (2).

Here is a concrete example of (1) to (4) for Linear Algebra (at least at an undergrad level).

For (1), there are three great books. Schaum's

For (2), Banchoff's

For (3), there are textbooks like Axler's

For (4), there is Golub and Van Loan's

You can do this with pretty much every mathematical subject (at least at an undergraduate level).

posted by ollyollyoxenfree at 8:39 PM on January 28, 2012 [7 favorites]

(1) Problem-based approach.

(2) Building intuition.

(3) Proof building.

(4) Computational approach.

(1) is about finding as many problems about a subject as possible. These range from applied problems and word problems; to olympiad-style problems. Find your subject that you want to learn about. Find problem books in that subject. There are many different mathematical series with these problem books. These include the Schaum's series, the Dolciani Mathematical Expositions, and the Springer Problem books in Mathematics series. Finally, practice these problems over and over (and also start looking into books

*about*problem solving, like Polya's*How to Solve it*).(2) is finding books and textbooks that give a intuitive feel for the subject. Often these are geometry-based approaches, popular accounts, or historical accounts.

(3) is found in any good textbook. Most textbooks take the definition, theorem, proof approach. Picking up any good Discrete Mathematics textbook will give you a background in how to write proofs (like Epp's textbook). Velleman's

*How to Prove it*is good as well. If you are having problems finding a good textbook, the Springer Undergraduate Texts in Mathematics series is excellent. You can find these in your local university library (they are mostly yellow books with UTM on the spine).(4) is making yourself familiar with computational approaches in the field you are studying. At my university this is mostly done with MATLAB and R. But learning any other languages would probably help as well (numpy/scipy for python, mathematica).

You need to work (1) to (4) simultaneously. Also, anyone that focuses at one of those approaches at the behest of others is missing out on the big picture. I'm looking especially at people who think math is all about (3). For a counter-point go read Poincare (especially this), and less Bourbaki-influenced approaches. I believe this part of the problem with people that don't get math. You have programmer-types on the Internet that think it is all about (3), and completely discard (1) and (2).

Here is a concrete example of (1) to (4) for Linear Algebra (at least at an undergrad level).

For (1), there are three great books. Schaum's

*3000 solved problems in linear algebra*, Halmos'*Linear Algebra Problem Book*(Dolciani Mathematical Expositions), and Zhang's*Linear Algebra: Challenging Problems for Students*.For (2), Banchoff's

*Linear Algebra through Geometry*(UTM series) covers a geometry-based approach. There is also Lay's*Linear Algebra and its Applications*, which covers a conceptual approach. Check out his excellent how to study linear algebra as well (PDF warning).For (3), there are textbooks like Axler's

*Linear Algebra Done Right*(UTM series), and Lang's*Linear Algebra*. You could also move onto Golan's*The Linear Algebra A Beginning Graduate Student Ought to Know*.For (4), there is Golub and Van Loan's

*Matrix Computations*, and Bau and Trefethen's*Numerical Linear Algebra*.You can do this with pretty much every mathematical subject (at least at an undergraduate level).

posted by ollyollyoxenfree at 8:39 PM on January 28, 2012 [7 favorites]

*"I have a three-pronged approach to thinking about math."*

Four-pronged. Don't I look silly. I can't even count, lol.

posted by ollyollyoxenfree at 2:21 AM on January 29, 2012 [1 favorite]

Hi

I’m neither a mathematician nor a math teacher; my own schooling in math was only till 12th grade (although this was in India, where we cover all GRE level math in school, and more if we’re preparing for competitive exams for engineering and such). I was taught the same way you were, learn the rules and move on. It wasn’t a problem subject for me, though it was extremely problematic for my younger sister (smart kid, bad at school,

Then a couple years ago, I (now a humanities/social sciences graduate) lived in a place that housed a 10 year old girl as a live-in helper (‘maid’), who knew how to read the alphabet and basically nothing else. She was sharp as a tack, and in her free time (when she wasn’t working

Reading and writing were not hard. You have the alphabet, which is basically symbols denoting sound, which you learn to comprehend separately (this is where she was at), and then you read words (semantic units, let’s call them) made up of those symbols, where you basically plug in that sound. This reveals actual words that we use in our daily life, which we already know the meaning of, and by relating and reading such words we become comfortable with reading comprehension to the point that we can read whole sentences (expressions) which, again, we already use and are hardwired for, and then go on to bigger words in the context of such sentences, drawing out the meanings of the words from context, referencing a dictionary or a friend as needed, and so on.

Not hard, and much easier in Hindi (the language we were using), because it’s a very straightforward script - none of the confounding, convention-based legacy that modern English gets from Middle English and dictionaries - every letter in Hindi has a particular unique sound and is always used as such, and always pronounced.

Math, however, was a whole different ball game. For one, talking about math in Hindi was challenging, since I was taught in English. Also, it doesn’t relate to everyday life - we don’t speak in mathematical expressions, nor think in them - so a fair bit of elementary mathematics is actually the teaching of convention, since the pupil doesn’t have the fundamental framework yet to understand the larger picture, let alone start thinking in these terms. But conveying this stuff to a 10-year old who is both curious about the why of everything and utterly unconvinced (not having had the schooling where you and I learn to sit in our designated place and move on, conceptually and literally, from one thing to the next, from one grade to the next, at designated times) about the value of memorizing stuff that seemed to have no relevance to her real life (all while having to voluntarily miss some of her TV time) required a harder think / study on my part.

For example, you start with counting, and that’s fairly intuitive - you just keep going one by one. Get a bunch of rice grains and start with one, adding one more to it every time and learning the word and symbol for that grouping/set (number): 1, 2 (1+1), 3 (1+1+1) and so on. But then you get to nine, and suddenly you have: 10. Not because we can’t come up with another single symbol to denote 10, but because, well - numbers are infinite and we need some way to manage them. Right here, you break with intuition or logic. You impose a convention - the decimal system. You say, let’s count by 10s. So a ten becomes a "higher unit", which is composed of 10 ones. We denote this by moving 1 to the left, and start using the one-units or digits (0 to 9) again with it to get to 20, then 30, and so on, till hundred, then till 1000. Repeat, ad infinitum (literally). With the decimal convention you run right away into the stumbling block of the zero, which by itself means nothing, but in a 10 or in 100000000 means everything. This is pure convention, and unless it’s separated from the meaning of the number (it’s written like that because we’re counting by 10s, which itself is written like that because we use the place of a digit - its place value - in a number to make larger numbers writeable and manipulable), it can obfuscate everything.

So counting in the decimal system means having to understand, or at the age we teach kids, while they are simultaneously learning words and language itself, having to somehow grasp composition of higher units. Without pointing this out to the learner

If you understand composition of higher units though, subtracting is no longer the stumbling block it becomes when taught in terms of “borrowing”. You don’t

So that when you have to subtract, say, 9 from this number, you simply decompose one 10 from the seven 10s (70) next to the 2, regroup it with the 2, take 9 out of 12, and are left with 39463. You learn to align numbers by place value (in adding, subtracting, multiplying) because that is how the

Similarly, multiplying 4320 by 123 (100+20+3) means multiplying 4320 by 3, multiplying it by 20, and multiplying it by 100 - and adding all the outcomes together. If you understand place value, all you have to do is multiply 4230 by 1, by 2, and by 3, and just move the outcome one more digit to the left each time as you move to the left on the multiplier.

This is very basic, of course. But it is also fundamental. It can be the difference between looking at a number as a plaything, something you can understand the value of (literally) and manipulate as needed, and looking at it as an unweildy, obscure symbol you must plug into some unilluminated procedure of borrowing and carrying and moving. The latter is an oppressive form of teaching, and I knew people in school who

You mentioned fractions. At the elementary level, this needs a conceptual shift from positive integers (or whole numbers, which is what we’ve learned till then) to wholeness itself and to looking at

Actually one can be composed of infinite fractions. You can divide it by 3492390420394, or 1.33333… or

This infinity inside wholeness is well expressed through the number line, which is not a ladder but rather a continuum. Whole numbers, as symbols, are simply multiplications of the abstract concept of wholeness we impose on the number line. In day to day life we treat this wholeness as a given because it is all around us - grains of rice, people, shoes can all be counted (no wonder positive integers are called natural numbers) - but ultimately these are categories that have meaning only in our perception, as is their counting. In math, a number is just an idea - starting with 1 (naturally) and adding, subtracting, multiplying or dividing with that idea itself. So a fraction is always a fraction of a whole, and a whole is anything divided exactly by itself - which is to say, itself. You cannot

Coming to fractions, you don’t have to totally give up on reality. As someone else pointed out above, you can relate it to say, a whole apple, and try to play it out. But that can be limited, as reality is. A better approach is to use the reality-related mathematics of whole numbers - coins, grains of rice, apples etc - to abstract the idea of wholeness and of computation itself, and go from there. Now we know what 2943 into 495 is, and that applies to all numbers on the number line, including fractions, negative numbers (defined by where you put the zero on your number line), and so on. So - let us say I have a fraction: 4/5. This simply means - in relation to wholeness - a whole divided by five, minus one of those parts, or 5/5 - 1/5. To write it as 4/5 (four divided by five) is not convention - it is notation.

(Fractions, in their notation, illuminate division as inverse multiplication. To multiply

So in 4/5, the bottom number (denominator) signifies how many parts it would take to make a whole; the top number (numerator) signifies how many parts there

So when you say you did not understand how you could multiply 2/3 by 1/3, are you asking why we multiply the top by the top and the bottom by the bottom to get the answer? Defining where your difficulty lies will help you seek out an answer that builds on your existing understanding.

Anyway, my pupil learned some elementary math and was quite good at it. Then last year she stole a few-hundred dollar watch from the house (shiny swarovski crystals which she thought were diamonds) and was sent back to live with her own family. I left that city but from time to time still work with young students (aged around 14-17) outside the school context, in small groups, about various issues. We just completed a short get together about the environment, where in the context of ecology the mathematical concept of exponential growth came up (re: exponential increase of forest cover loss, species extinction, human population, urban growth, fossil fuel consumption,

And to understand mathematics as a whole, there is the idea of

Some other recommendations, click through and find out if any of these look helpful or relevant to you:

What Is Mathematics? An Elementary Approach to Ideas and Methods (For actual math help!)

The Housekeeper and the Professor (A novel.)

Journey through Genius: The Great Theorems of Mathematics (Geniuses are just seekers who won't quit.)

I’m embarassed by how much I’ve gone on and on, and I apologize if much of it is too elementary for what you needed, but I hope some of it proves useful. I have not a shred of doubt that you can master elementary mathematics and easily tackle GRE level subjects. Question-asking students like you who chafe under rote memorization and unexplained rules are the bane of our mercilessly talent-harvesting education system and also maybe our most solid argument against it. But as far as succeeding within the system is concerned (GRE et al), I do think you should try to find a good tutor or guide, even if you are very self-motivated. There isn’t just one conceptual shift to be made with math which will solve it for you, there is a conceptual ladder to climb. Text of any kind only goes so far, and as someone above said, mathematical thinking requires rigor, which requires testing the validity of your approaches, which requires help.

I’d be extremely interested to know if there’s anything I wrote that doesn’t make sense to you - feel free to memail me with specific sentences, at that. The only way I know how to keep my extreme pessimism and sense of defeat about the state of the world and our mass compulsory education system at bay is to talk about, against and counter to it, particularly with younger people - but sorry if I over-explained. It’s hard not to, without dialogue. So write back, in case of anything.

posted by mondaygreens at 2:24 PM on January 29, 2012 [5 favorites]

**lackadaisical**! Your question seems to me to be two questions, or at least with two parts - one asking for tips/tricks to approach GRE level math, and the other (the larger part of your question) about how to get some conceptual clarity in elementary math. I’ve skimmed through most answers above, which seem to focus on the former part - problem solving and such - but not much about the second, maybe I missed it but maybe I can add something to it. I’m particularly responding to this part of your follow-up:“To this day, I still have a hard time subtracting numbers that involves a lot of borrowing. Later on, when we got started in fractions, I realized that all hope was gone. In fact, for most of hte people I've talked to, their problem with math almost inevitably began with fractions and how to understand and manipulate them. For one, I couldn't grasp how you could multiply 2/3, for instance, with, say 1/3. After a while, you learn the rules and you move on. That's it.”

I’m neither a mathematician nor a math teacher; my own schooling in math was only till 12th grade (although this was in India, where we cover all GRE level math in school, and more if we’re preparing for competitive exams for engineering and such). I was taught the same way you were, learn the rules and move on. It wasn’t a problem subject for me, though it was extremely problematic for my younger sister (smart kid, bad at school,

*atrocious*at math) and I used to help her try to solve math problems in the ways I could.Then a couple years ago, I (now a humanities/social sciences graduate) lived in a place that housed a 10 year old girl as a live-in helper (‘maid’), who knew how to read the alphabet and basically nothing else. She was sharp as a tack, and in her free time (when she wasn’t working

*and*the house TV wasn’t on) would go through the glossy bits of the newspaper, looking at the photos she liked and trying to put together the words around/under it, by sound. Without anyone to ask or check with, and Hindi newspapers having many words that weren't part of her vocabulary, she hadn’t gotten very far. We got to talking, and decided to put aside some time to just sit down every day and work through basic reading and writing, and also arithmetic, so that she could calculate her income/savings, deal with shopkeepers and basically not get cheated when it came to money. We decided to try, anyway.Reading and writing were not hard. You have the alphabet, which is basically symbols denoting sound, which you learn to comprehend separately (this is where she was at), and then you read words (semantic units, let’s call them) made up of those symbols, where you basically plug in that sound. This reveals actual words that we use in our daily life, which we already know the meaning of, and by relating and reading such words we become comfortable with reading comprehension to the point that we can read whole sentences (expressions) which, again, we already use and are hardwired for, and then go on to bigger words in the context of such sentences, drawing out the meanings of the words from context, referencing a dictionary or a friend as needed, and so on.

Not hard, and much easier in Hindi (the language we were using), because it’s a very straightforward script - none of the confounding, convention-based legacy that modern English gets from Middle English and dictionaries - every letter in Hindi has a particular unique sound and is always used as such, and always pronounced.

Math, however, was a whole different ball game. For one, talking about math in Hindi was challenging, since I was taught in English. Also, it doesn’t relate to everyday life - we don’t speak in mathematical expressions, nor think in them - so a fair bit of elementary mathematics is actually the teaching of convention, since the pupil doesn’t have the fundamental framework yet to understand the larger picture, let alone start thinking in these terms. But conveying this stuff to a 10-year old who is both curious about the why of everything and utterly unconvinced (not having had the schooling where you and I learn to sit in our designated place and move on, conceptually and literally, from one thing to the next, from one grade to the next, at designated times) about the value of memorizing stuff that seemed to have no relevance to her real life (all while having to voluntarily miss some of her TV time) required a harder think / study on my part.

For example, you start with counting, and that’s fairly intuitive - you just keep going one by one. Get a bunch of rice grains and start with one, adding one more to it every time and learning the word and symbol for that grouping/set (number): 1, 2 (1+1), 3 (1+1+1) and so on. But then you get to nine, and suddenly you have: 10. Not because we can’t come up with another single symbol to denote 10, but because, well - numbers are infinite and we need some way to manage them. Right here, you break with intuition or logic. You impose a convention - the decimal system. You say, let’s count by 10s. So a ten becomes a "higher unit", which is composed of 10 ones. We denote this by moving 1 to the left, and start using the one-units or digits (0 to 9) again with it to get to 20, then 30, and so on, till hundred, then till 1000. Repeat, ad infinitum (literally). With the decimal convention you run right away into the stumbling block of the zero, which by itself means nothing, but in a 10 or in 100000000 means everything. This is pure convention, and unless it’s separated from the meaning of the number (it’s written like that because we’re counting by 10s, which itself is written like that because we use the place of a digit - its place value - in a number to make larger numbers writeable and manipulable), it can obfuscate everything.

So counting in the decimal system means having to understand, or at the age we teach kids, while they are simultaneously learning words and language itself, having to somehow grasp composition of higher units. Without pointing this out to the learner

*at some level*, whether with abaci or place values or whatever, all you are getting through to them is - hey, this is how it’s done, this is just how we count, and how we write how we count. Just know it so we can use it to do some calculations, which we will also lay out in simple steps. No need to understand*how*it happens, or*why*we do it.If you understand composition of higher units though, subtracting is no longer the stumbling block it becomes when taught in terms of “borrowing”. You don’t

*borrow*, you decompose higher units as needed. You learn to look at a big number as*one set*expressed as units (digits, with their 10 factor described in terms of their place in the number) of different kinds - 39472 becomes 30000+9000+400+70+2, and so on. These are hidden zeros that give the whole number its meaning and its integrity.So that when you have to subtract, say, 9 from this number, you simply decompose one 10 from the seven 10s (70) next to the 2, regroup it with the 2, take 9 out of 12, and are left with 39463. You learn to align numbers by place value (in adding, subtracting, multiplying) because that is how the

*digits*make sense and align across various numbers, in terms of equivalence. Subtracting 34292 from 239133 requires that we align the digits according to place value, because otherwise the digits themselves (across the two numbers) are not equal in value, so you cannot subtract, say, the 3 of the first number from the 2 of the second number because the first is thirty thousand and the second is two hundred thousand.Similarly, multiplying 4320 by 123 (100+20+3) means multiplying 4320 by 3, multiplying it by 20, and multiplying it by 100 - and adding all the outcomes together. If you understand place value, all you have to do is multiply 4230 by 1, by 2, and by 3, and just move the outcome one more digit to the left each time as you move to the left on the multiplier.

This is very basic, of course. But it is also fundamental. It can be the difference between looking at a number as a plaything, something you can understand the value of (literally) and manipulate as needed, and looking at it as an unweildy, obscure symbol you must plug into some unilluminated procedure of borrowing and carrying and moving. The latter is an oppressive form of teaching, and I knew people in school who

*were*oppressed by math (as was my sister) in a way that was utterly unique to the subject. That oppression comes directly, I think, from unexplained and thus*forced*convention, coupled with the whole “how good you are in school is how worthy you are” tenet that pervades everything, everything. It is especially pronounced with math not just because there are right answers in math like in nothing else, but also because math is*all*about building on basic concepts and abstracting to higher levels in many different ways, so if you fall behind it is difficult to catch up (future concepts assume understanding of previous ones) and so also there are numerous ways to approach a particular problem - there isn’t just one right strategy, nor are they all equally well suited.You mentioned fractions. At the elementary level, this needs a conceptual shift from positive integers (or whole numbers, which is what we’ve learned till then) to wholeness itself and to looking at

*everything*as a fraction. The number “1” is all the numerical idea you need of wholeness, an idea you already have. The number 2 is two wholes (1+1, or 1x2), the number 927 is 927 wholes (1 x 927) and so on. But what is 1, or wholeness itself? We started our counting with grains of rice, so one grain is 1, that’s easy. But the conceptual "1" - it is just the idea of wholeness, a concept of oneness. Am I losing you? How many parts / fractions is this “one” made of? 2? 9? 72329?Actually one can be composed of infinite fractions. You can divide it by 3492390420394, or 1.33333… or

*anything else*, as long and large as you can possibly imagine. To make any fraction whole again - to make it 1 - requires simply to multiply it by as much as it has been divided by. So one is equal to 1/1 (a whole divided by itself stays whole - or rather, to divide by 1 is to basically do nothing),*and*9/9 (9 wholes divided amongst 9 people gives one whole each),*and*72329/72329, and absolutely any number divided by itself.This infinity inside wholeness is well expressed through the number line, which is not a ladder but rather a continuum. Whole numbers, as symbols, are simply multiplications of the abstract concept of wholeness we impose on the number line. In day to day life we treat this wholeness as a given because it is all around us - grains of rice, people, shoes can all be counted (no wonder positive integers are called natural numbers) - but ultimately these are categories that have meaning only in our perception, as is their counting. In math, a number is just an idea - starting with 1 (naturally) and adding, subtracting, multiplying or dividing with that idea itself. So a fraction is always a fraction of a whole, and a whole is anything divided exactly by itself - which is to say, itself. You cannot

*count*one, it is itself. You start counting*with*it.Coming to fractions, you don’t have to totally give up on reality. As someone else pointed out above, you can relate it to say, a whole apple, and try to play it out. But that can be limited, as reality is. A better approach is to use the reality-related mathematics of whole numbers - coins, grains of rice, apples etc - to abstract the idea of wholeness and of computation itself, and go from there. Now we know what 2943 into 495 is, and that applies to all numbers on the number line, including fractions, negative numbers (defined by where you put the zero on your number line), and so on. So - let us say I have a fraction: 4/5. This simply means - in relation to wholeness - a whole divided by five, minus one of those parts, or 5/5 - 1/5. To write it as 4/5 (four divided by five) is not convention - it is notation.

(Fractions, in their notation, illuminate division as inverse multiplication. To multiply

*or*divide something by one is to mathematically do nothing to it. So to multiply something by any number (say, 42) is the same as dividing by the*multiplicative inverse*of that number (1/42). Multiplying by 4/5 is the same as dividing by 5/4. This is also how fractions with common factors in the top and bottom cancel each other out: a whole number is not just its digits but can be written as the sum of its various place values - this is used when we add or subtract -**or**as the product of its factors - this is used when we divide or multiply.)So in 4/5, the bottom number (denominator) signifies how many parts it would take to make a whole; the top number (numerator) signifies how many parts there

*are*. This new fraction is as much a number as any integer - it is just not whole, which itself is something we have conceptualized (wholeness) and defined (in this case it has five parts). It can be multiplied, divided further, added to, subtracted from. Eg, add 3 wholes to this fraction to get the mixed number 3 4/5, or 3*5/5 + 4/5 or 19/5, an improper fraction. And there can be several ways of expressing a fraction - not just a top and bottom but also decimals (add a dot to the top number, and then as many zeros as you need, since they don’t mean anything after the dot, in order to divide by the bottom number to get a decimal), percentages (expressed as parts of 100 instead of 1) and so on. They are the same number, and can be computed like any other number, whole or not, so long as the numbers they are computed with are expressed in the same way.So when you say you did not understand how you could multiply 2/3 by 1/3, are you asking why we multiply the top by the top and the bottom by the bottom to get the answer? Defining where your difficulty lies will help you seek out an answer that builds on your existing understanding.

Anyway, my pupil learned some elementary math and was quite good at it. Then last year she stole a few-hundred dollar watch from the house (shiny swarovski crystals which she thought were diamonds) and was sent back to live with her own family. I left that city but from time to time still work with young students (aged around 14-17) outside the school context, in small groups, about various issues. We just completed a short get together about the environment, where in the context of ecology the mathematical concept of exponential growth came up (re: exponential increase of forest cover loss, species extinction, human population, urban growth, fossil fuel consumption,

*all*consumption), and it was very interesting how unintuitive it was for everyone to imagine exponential growth. We used the wheat and chessboard problem, among other examples. Here is also a video you might find interesting.And to understand mathematics as a whole, there is the idea of

*patterns*. Patterns are inherent, fundamental to how we understand everything, from language to reality to scientific truth to our relationships to our own selves - like, you have a pattern on not succeeding at math, so you might deduce from that that you are bad at math. If thirty out of fifty kids are not succeeding at math (and I don't mean based on standardized scores, but, say, asking students about their ability to think mathematically in a critical way), we might look at the larger pattern, and shift the logic from you to the education system to suggest that it is bad at teaching kids math. (This is not a mathematical pattern, of course. A book that studies this pattern in a small but illuminating social-science way: Knowing and Teaching Elementary Mathematics: Teachers' Understanding of Fundamental Mathematics in China and the United States.) While mathematics is both abstract and a very specific kind of language describing quantity, shape, motion, logic, probability and other hardcore things - it is a fundamentally a study of patterns. You might enjoy Keith Devlin’s Mathematics: The Science of Patterns: The Search for Order in Life, Mind and the Universe. (Or another, less illustrated version of the same book: The Language of Mathematics: Making the Invisible Visible.) It is not designed to help with GRE type math though.Some other recommendations, click through and find out if any of these look helpful or relevant to you:

What Is Mathematics? An Elementary Approach to Ideas and Methods (For actual math help!)

The Housekeeper and the Professor (A novel.)

Journey through Genius: The Great Theorems of Mathematics (Geniuses are just seekers who won't quit.)

I’m embarassed by how much I’ve gone on and on, and I apologize if much of it is too elementary for what you needed, but I hope some of it proves useful. I have not a shred of doubt that you can master elementary mathematics and easily tackle GRE level subjects. Question-asking students like you who chafe under rote memorization and unexplained rules are the bane of our mercilessly talent-harvesting education system and also maybe our most solid argument against it. But as far as succeeding within the system is concerned (GRE et al), I do think you should try to find a good tutor or guide, even if you are very self-motivated. There isn’t just one conceptual shift to be made with math which will solve it for you, there is a conceptual ladder to climb. Text of any kind only goes so far, and as someone above said, mathematical thinking requires rigor, which requires testing the validity of your approaches, which requires help.

I’d be extremely interested to know if there’s anything I wrote that doesn’t make sense to you - feel free to memail me with specific sentences, at that. The only way I know how to keep my extreme pessimism and sense of defeat about the state of the world and our mass compulsory education system at bay is to talk about, against and counter to it, particularly with younger people - but sorry if I over-explained. It’s hard not to, without dialogue. So write back, in case of anything.

posted by mondaygreens at 2:24 PM on January 29, 2012 [5 favorites]

Try "The Elements of Math", a 15 part series in the NY Times. It might give you the insight you're looking for.

Sorry if overlooked that someone already suggested this. I didn't notice it but know that it's been mentioned frequently around here.

BTW, I can't wait to try that "aim for that dot when hitting the [base]ball" tip...

posted by agog at 9:19 PM on January 30, 2012

Sorry if overlooked that someone already suggested this. I didn't notice it but know that it's been mentioned frequently around here.

BTW, I can't wait to try that "aim for that dot when hitting the [base]ball" tip...

posted by agog at 9:19 PM on January 30, 2012

The set of thoughts that helped me out:

If you think you're bad at math, that means - after many years - that you've had far, far less practice than people who didn't think the same way. Practice, work your way back up, and don't get discouraged; learning new things is tough for everyone, especially if their expectations are high.

Set high expectations, set a reasonable pace, and don't discourage yourself.

posted by talldean at 12:05 PM on February 2, 2012

If you think you're bad at math, that means - after many years - that you've had far, far less practice than people who didn't think the same way. Practice, work your way back up, and don't get discouraged; learning new things is tough for everyone, especially if their expectations are high.

Set high expectations, set a reasonable pace, and don't discourage yourself.

posted by talldean at 12:05 PM on February 2, 2012

This thread is closed to new comments.

posted by HuronBob at 6:50 PM on January 26, 2012 [2 favorites]