NOTE - This post may include some mathematically technical material. I will attempt to give warning by bolding parts that need some extra background.
(continued from Part 3A)
Facebook (like other social networking sites) is a great technology. It enhances personal relationships, allowing you to do so much more than you could do otherwise. The ability to share photos, write on people's wall, send people messages, comment on stuff, etc. allows you to do a lot of interactions with people who are physically distant - things you could never do without the technology of the internet.
But imagine the following scenario. Two friends, Alice and Bob, are sitting in a coffee shop, having a conversation. At one point, the conversation turns to Bob's recent vacation to Bali. Alice is curious about what it's like. Bob says, "hey, I've got some pictures on my iphone, do you want to see them?" When Alice says yes, Bob says "hang on, I'll upload them into a private album on Facebook only visible to you and me. Once they're finished uploading, you'll be able to see them with your iphone and tell me what you think!" And the conversation goes silent as he sets to work selecting the photos to upload. A couple minutes later, he finishes, but the photos won't upload for another 15 minutes, so he puts the phone down, and they keep talking. When they finally finish, the conversation is far from Bali, and so there is a jerk in the flow of dialogue as Bob informs Alice that she can see the photos now.
The jerk in the flow of dialogue happens to be named Bob.
Why do I tell this story? Because I believe it might inform our thoughts about the way math is learned - in particular, how to deal with the technology of counting.
As mentioned in Part A, once you have counting down, you can learn just about everything else in mathematics from there. It truly is a powerful technology. But what I'm wondering is... is it the best way? Is it the most efficient way?
Take division - which might just be the root of all the conceptual problems people face in elementary school math (problems with memorization and following rules are a different story). How do we get to division? well, we start with ratios, install counting so we know about whole numbers, learn that adding is extended counting, learn that multiplying is repeated adding, learn that division is the opposite of multiplying, and that from division, we can get ratios.
What the function.
I don't know about you, but that seems a lot like Bob relying on Facebook to show Alice photos when he could have just held up his phone to show her.
My big question is this - are we relying too much on the technology of counting? Are we trying so hard to base everything on the discrete, when the continuous is just as, if not more, natural to us? Just because a technology works doesn't make it the most efficient way to get to every point. Maybe we should bring the ratio into school much earlier - not as a consequence of division (which is a consequence of multiplication (which is a consequence of addition (which is a consequence of counting))), but as something much more fundamental.
I remember reading a blog post once about multiplication, and how it's always taught as "repeated addition," when it's really something more basic than that. In fact, if ratios are the basis of our mathematical reasoning, then multiplication is actually more fundamental than addition is! It's actually an amazing truth that multiplication can be achieved through repeated addition - but this isn't a definition. Defining multiplication as repeated addition is like defining trees as "a source of building material" or defining typewriter as "the longest word you can type using the top row of keys on a typewriter" - both of these are in fact true, but neither count as definitions. Just look at it geometrically - adding is "sliding" the number line, while multiplication is "stretching" the number line. Is stretching equal to repeated sliding? No! In fact, looked at it this way, it's pretty incredible that repeated adding would give you multiplication.
And if multiplication could be made more basic, so could division - as it should, since division is where we get ratios, the most basic mathematical facts we know. Perhaps we should do it the other way around - from ratios, introduce multiplication and division? And introduce the "repeated addition" approach only after the concepts are solid and it's time to learn how to actually compute things? No idea what this would look like in the classroom - but I'm sure some basic ratio games and activities could be introduced very early on, not just to prepare them for "when they'll learn multiplication in second grade," but to actually teach basic multiplication and division before addition has even settled in. Teach it from the geometrical side, stretching, shrinking, perhaps even give them basic number sense activities like the one I showed in Part 2A - show the connection to discrete "five groups of three" questions later on. This will appeal to students' basic number sense and develop that, giving them a better intuition about ratio, and hence about multiplication and division, which depend on ratio; then, once it's time to actually calculate things, like adding two fractions, that intuition about how it should work will be there much more strongly.
Counting is powerful - but are we pushing it too far? Perhaps it's time to bring the ratio, the continuous, back into early early education. Don't just teach students the names of geometrical objects - teach them how to interact with them. How they grow, how they fit inside each other. How every polygon can be split into triangles. Imagine the wonder on a first grader's face when they realize that incredible fact - a wonder totally lost on middle schoolers who have learned long ago that math is all about calculating and rule-following and are taught that same fact just as something else to be memorized. And this is much more possible now, with the ipad or other touch-screen technologies, than it would have been just five years ago. The discrete is incredibly important - but we can't let it be superior to the continuous.
(It's actually an interesting irony that our K~12 education system works almost in reverse of the way our minds do. Right away, from the outset, get students to be thinking about the discrete, and as they go through education, they'll start putting the continuous back in, little by little, but always basing it on the discrete - through decimals and fractions, through the real numbers, through geometry, through functions, until the climax: calculus, the study of continuity, of infinitesimal ratio)
On a very similar note, I've heard some people wonder if we should somehow introduce the concept of complex numbers from an early age, since they are in some way "more natural" than the real numbers, and the only reason people have trouble with complex numbers is being over-familiarized with the reals. I can't agree with that as easily, since complex numbers, despite their mathematical consistency, don't appear to have the same strong connection with our natural number sense as ratios do. But I'd love to hear evidence that such a natural connection to complex numbers does actually exist.
So those are some of my thoughts on the monopoly of counting. What do you think? To those who have been thinking a bit about the subject - Maybe I'm going too far? Maybe not far enough? Maybe my theory is good, but completely impractical? Maybe there just isn't any other way to develop advanced concepts without the firm, clear properties of whole numbers? Maybe my ideas would have been good a couple decades ago, but now that we're into the digital age, we have to focus completely on the discrete?
To those who don't think about learning math all that much, and somehow just survived through these posts - what do you remember about your early learning experiences? What concepts did you struggle with the most? How did you manage to work through some of your difficulties (if you ever did)? Do you think there's some other topic that's a bigger problem?
To those who study number sense in depth - have I got any of my research wrong? Am I missing any important results? Have I made some bad conclusions? Are there other people who have written about the same topic, confirming, denying, or just discussing the ideas?
Thanks for reading. Wow, this is long. O.o
Disclaimer - throughout this series of posts I've used certain terms in ways that aren't mathematically accurate. The main one being continuous, which I roughly use to mean "measurable; not countable," in contrast to discrete. For example, I state "volume is continuous" as an obvious fact, though this would actually be quite a controversial claim using the definition of continuous as being "infinitely subdivisible." I'm sure there are other words used like this as well.
Tuesday, August 30, 2011
1, 2, 3... not the best place to start? (Part 3A - the crucial step)
(continued from Part 2A)
Let's return to the myth I brought up in Part A.
Let's return to the myth I brought up in Part A.
Myth: counting is the foundation of everything we learn about mathematics.
FACT: Ratios are the foundation of everything we learn about mathematics.
Remember the steps to learning math I presented? I'll repeat the first two here.
STEP 1: You learned a list of words, and practice pointing at different things while saying these words.
STEP 2: You begin to apply that list to actual things you see - you realize that the word "five" can be applied both to "five apples" and "five people." This is actually a pretty major step - there are many young children who can count blocks "one, two, three," but don't know how many blocks there are. This is where children learn one of the most important properties of math, abstraction - that a single concept (like "five") can apply to many completely different areas.
I would argue that Step 2 is actually a whole lot bigger than just learning abstraction. In fact, you may have learned abstraction long before counting had any meaning for you - how else could the word "dog" refer to so many different-looking animals, or "blue" refer to things that look completely unrelated? (I went to a fascinating class at Mathcamp about this, actually - the way people see patterns and create mental categories. It's amazing how even individual words can be studied mathematically). Surely it's not a big step to realize "three"can describe both a group of pencils and a group of cookies.
No, the big step is embedding these discrete words into a ratio-based number sense, where ratios are inherently continuous things. The same number sense that could tell you which picture has more dots could tell you which of two lines is longer, at a glance, without measuring them. Or which of two shapes is bigger, without lining them up beside each other. These are all geometric, continuous problems. You could have a line that's exactly 2.54 times the length of another. You can't have 2.54 dots*.
So we have this continuous number sense based on ratio, and this discrete list of counting words. How can we possibly put the two together?
The crucial step, I think (and this is where I'm departing from what I've learned or picked up, though I wouldn't be surprised if other people have reached the same conclusion before me), both in understanding abstraction and in applying a discrete technology to a continuous intuition, comes in realizing that all the words you learned define ratios.
"Wait, what?! You said that ratios are continuous, but these number words are discrete! You've been saying all along that somehow counting is completely different from the continuous process that goes on in the brain! How can these words define ratios?"
Bear with me.
Fairly early on in math education, we're introduced to units. A difficult concept for many people to master is that units must be treated almost exactly as numbers are - for example, 6 km ÷ 2 hours = 3 (km÷hour), or 3 kilometers per hour. (2 boxes) + (3 boxes) = (2+3) boxes (by the distributive property) = 5 boxes. (2 pencils) x (3 pencils) = 6 pencilpencils (?!?). And units cancel out if there's one in the numerator, one in the denominator, just like numbers: If you have 20 cars with 4 seats per car, you get (20 cars) x (4 seats÷car) = 80 seats x cars ÷ car = 80 seats xcars ÷ car = 80 seats.
So let's say you have a group of apples, and you want to see how many apples you have. What you're really asking is - how many single apples go into this group of apples?
This is a ratio. You're comparing the group to the apple, so you have (answer) = (group)÷(apple). Then we say, hey, this group is actually made up of apples... so we'll put in a random descriptive word showing that the group is somehow related to apples, and get (answer) = (lima apple)÷(apple). But now we can cancel the apples, and get answer=lima. In other words, the word lima is what you were looking for - the ratio of group to apple. Lima IS the ratio. The word "five" works just as well as the random word ("lima" is just "five" in Indonesian). "Five" IS the ratio of a certain group to the individuals in the group. The ratio of "B B B B B" to "B" is also five - so you call the first group "five Bs." You're multiplying the word (a ratio) by the unit, to get something that many times bigger than the unit. And this ratio - that's something your instinctive number sense understands.
Sorry, the last couple paragraphs made it sound more complicated than it really is... essentially, the crucial step children go through is to realize that when you say "five apples," you're first of all imagining a certain ratio from your number sense (the one you connect with the word "five"), and you're saying that "the ratio of this collection to a single apple is the same as this ratio I'm imagining."
Numbers as ratios also deals with the problem of abstraction. Because the ratio of "B B B" to "B" is the same as the ratio of "dot dot dot" to "dot" or "dog dog dog" to "dog" or "mom dad brother" to "person" or "yard" to "foot." If you link "three" with a set of blocks, you'll never be able to use it on people without an incredible disregard for their humanity; but if you link "three" with the ratio of the set of blocks to a single block, you can use it anywhere.
You can't compare apples with oranges and here's why. Let's say I have 15 apples and you have 5 oranges. The ratio is "3 apples/orange." That tells you something, but it's not a direct measurement of anything because of the extra units. But if I have 15 apples and you have 5 apples, the ratio is "3." Numbers appear when the units cancel - when you're working with the same type of object. And then you can make direct comparisons - I have exactly three times what you do, and I can understand "three" just using my instinctive number sense - I know the ratio it refers to.
So the crucial step is to take the words, discrete words, and connect each one with a (continuous) ratio that your basic number sense can understand. Once you do this, you've basically installed the "counting technology" into your brain in a way that allows it to interact with your number sense (via the bridge of ratios), not just your language center. And using the fuel of number sense, counting technology is truly powerful, and it drives the rest of most people's math education. Powerful stuff, counting.
(continued in Part 4A)
*"Of course you can have 2.54 dots! Just have three dots, and cut one out so it's 0.54 times the area of the original!" Ah, but see what you did there - now you're not counting dots, you're measuring area. You've switched over from a discrete process to a continuous one. When we say something in everyday life like "I ate three and a half apples," we're not really counting up to 3.5. We're actually counting two different things - apples (three of them) and half apples (one of them), where half apples aren't necessarily exactly half of full apples - it's just another phrase we use to describe something we see in our life. Once you try to make it exact and measure to exactly the halfway point, you once again switched from counting to measurement - discrete to continuous. Apples are discrete - their volume is continuous.
FACT: Ratios are the foundation of everything we learn about mathematics.
Remember the steps to learning math I presented? I'll repeat the first two here.
STEP 1: You learned a list of words, and practice pointing at different things while saying these words.
STEP 2: You begin to apply that list to actual things you see - you realize that the word "five" can be applied both to "five apples" and "five people." This is actually a pretty major step - there are many young children who can count blocks "one, two, three," but don't know how many blocks there are. This is where children learn one of the most important properties of math, abstraction - that a single concept (like "five") can apply to many completely different areas.
I would argue that Step 2 is actually a whole lot bigger than just learning abstraction. In fact, you may have learned abstraction long before counting had any meaning for you - how else could the word "dog" refer to so many different-looking animals, or "blue" refer to things that look completely unrelated? (I went to a fascinating class at Mathcamp about this, actually - the way people see patterns and create mental categories. It's amazing how even individual words can be studied mathematically). Surely it's not a big step to realize "three"can describe both a group of pencils and a group of cookies.
No, the big step is embedding these discrete words into a ratio-based number sense, where ratios are inherently continuous things. The same number sense that could tell you which picture has more dots could tell you which of two lines is longer, at a glance, without measuring them. Or which of two shapes is bigger, without lining them up beside each other. These are all geometric, continuous problems. You could have a line that's exactly 2.54 times the length of another. You can't have 2.54 dots*.
So we have this continuous number sense based on ratio, and this discrete list of counting words. How can we possibly put the two together?
The crucial step, I think (and this is where I'm departing from what I've learned or picked up, though I wouldn't be surprised if other people have reached the same conclusion before me), both in understanding abstraction and in applying a discrete technology to a continuous intuition, comes in realizing that all the words you learned define ratios.
"Wait, what?! You said that ratios are continuous, but these number words are discrete! You've been saying all along that somehow counting is completely different from the continuous process that goes on in the brain! How can these words define ratios?"
Bear with me.
Fairly early on in math education, we're introduced to units. A difficult concept for many people to master is that units must be treated almost exactly as numbers are - for example, 6 km ÷ 2 hours = 3 (km÷hour), or 3 kilometers per hour. (2 boxes) + (3 boxes) = (2+3) boxes (by the distributive property) = 5 boxes. (2 pencils) x (3 pencils) = 6 pencilpencils (?!?). And units cancel out if there's one in the numerator, one in the denominator, just like numbers: If you have 20 cars with 4 seats per car, you get (20 cars) x (4 seats÷car) = 80 seats x cars ÷ car = 80 seats x
So let's say you have a group of apples, and you want to see how many apples you have. What you're really asking is - how many single apples go into this group of apples?
This is a ratio. You're comparing the group to the apple, so you have (answer) = (group)÷(apple). Then we say, hey, this group is actually made up of apples... so we'll put in a random descriptive word showing that the group is somehow related to apples, and get (answer) = (lima apple)÷(apple). But now we can cancel the apples, and get answer=lima. In other words, the word lima is what you were looking for - the ratio of group to apple. Lima IS the ratio. The word "five" works just as well as the random word ("lima" is just "five" in Indonesian). "Five" IS the ratio of a certain group to the individuals in the group. The ratio of "B B B B B" to "B" is also five - so you call the first group "five Bs." You're multiplying the word (a ratio) by the unit, to get something that many times bigger than the unit. And this ratio - that's something your instinctive number sense understands.
Sorry, the last couple paragraphs made it sound more complicated than it really is... essentially, the crucial step children go through is to realize that when you say "five apples," you're first of all imagining a certain ratio from your number sense (the one you connect with the word "five"), and you're saying that "the ratio of this collection to a single apple is the same as this ratio I'm imagining."
Numbers as ratios also deals with the problem of abstraction. Because the ratio of "B B B" to "B" is the same as the ratio of "dot dot dot" to "dot" or "dog dog dog" to "dog" or "mom dad brother" to "person" or "yard" to "foot." If you link "three" with a set of blocks, you'll never be able to use it on people without an incredible disregard for their humanity; but if you link "three" with the ratio of the set of blocks to a single block, you can use it anywhere.
You can't compare apples with oranges and here's why. Let's say I have 15 apples and you have 5 oranges. The ratio is "3 apples/orange." That tells you something, but it's not a direct measurement of anything because of the extra units. But if I have 15 apples and you have 5 apples, the ratio is "3." Numbers appear when the units cancel - when you're working with the same type of object. And then you can make direct comparisons - I have exactly three times what you do, and I can understand "three" just using my instinctive number sense - I know the ratio it refers to.
So the crucial step is to take the words, discrete words, and connect each one with a (continuous) ratio that your basic number sense can understand. Once you do this, you've basically installed the "counting technology" into your brain in a way that allows it to interact with your number sense (via the bridge of ratios), not just your language center. And using the fuel of number sense, counting technology is truly powerful, and it drives the rest of most people's math education. Powerful stuff, counting.
(continued in Part 4A)
*"Of course you can have 2.54 dots! Just have three dots, and cut one out so it's 0.54 times the area of the original!" Ah, but see what you did there - now you're not counting dots, you're measuring area. You've switched over from a discrete process to a continuous one. When we say something in everyday life like "I ate three and a half apples," we're not really counting up to 3.5. We're actually counting two different things - apples (three of them) and half apples (one of them), where half apples aren't necessarily exactly half of full apples - it's just another phrase we use to describe something we see in our life. Once you try to make it exact and measure to exactly the halfway point, you once again switched from counting to measurement - discrete to continuous. Apples are discrete - their volume is continuous.
1, 2, 3... not the best place to start? (Part 2A - number sense)
(continued from Part A)
What do babies know about math?
From this point on, I'm going to be working off memory a lot... I watched a presentation about this once, very fascinating, but I don't know if I can find the data.
So basically, some researchers wanted to know if babies could distinguish between different numbers. So they took infants, less than 1 year old, into a laboratory, and asked them if they knew what the difference between 17 and 18 was.
The babies looked around and started making gurgling noises so the researchers decided they had to be a bit more clever. So instead, they showed these children two pictures with different numbers of objects, and marked which picture held the babies' attention. I'm sure it's a bit more complicated than this, but essentially they tried to find out what babies knew about numbers.
They knew the difference between 0 and 1. Between a blank screen and a screen with a dot on it, the dot is obviously more exciting.
They knew the difference between 1 and 2. Between a lonely dot and dots that weren't lonely, the babies wanted to join the more social group.
The did NOT know the difference between 2 and 3. To them, it looked basically the same - a few friendly dots. Similarly, between 3 and 4, 4 and 5, etc... a difference of 1 was meaningless after the 1~2 jump.
However, they DID know the difference between 2 and 4... between 4 and 8... between 12 and 24... But not between 12 and 18, or 40 and 50, etc.
The result: the RATIO is what the infants recognize. As long as one picture had at least twice as many dots as the other, it captivated the baby's attention.
Further study was done. This time, children of different ages were studied. They found that after a couple months, babies (or toddlers now, perhaps) could distinguish between 2 and 3. And a couple months later, between 3 and 4. Then all of a sudden there's a jump, and in an incredibly short length of time, the children recognize 4, 5, 6, 7, 8...
To people who study progress over time, a sudden jump almost always reflects a new technology. You invent the printing press, and the number of books explodes. Develop the cotton gin, the textile industry skyrockets. Here too, there is a new technology - the technology of counting.
I call counting a technology because it is NOT something humans inherently have. Maybe you've heard the news stories of tribes of people (the Pirahã are an example of this) who don't have words representing different numbers, just concepts of "smaller amount" and "larger amount." They didn't discover the technology of counting, so they couldn't distinguish between, say, 5 dots and 7 dots.
What humans do inherently have is something often called "number sense." The definition on Wikipedia says that number sense is "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Note that the process of counting is not referred to, nor are specific names of numbers - here "number" just refers to an understanding of amount. Once counting is learned, one's number sense drastically improves - reflecting the dramatic growth in a toddler's ability to recognize amounts. But you have number sense even without counting - in its most natural state, number sense is based not on counting, but on RATIOS.
As you grow, whether you learn counting and names of numbers or not, your number sense sharpens - while babies need a 2:1 ratio to be able to recognize something, most adults can recognize smaller ratios, 1.5:1 or lower, and curiously, many people who are very good at higher level math tend to also be very good at this very fundamental, basic number sense, recognizing ratios as small as 1.1:1 or lower (i.e. being able to tell the difference between 11 dots and 10 dots without counting them, just at a glance)
I found this site where you can test your basic number sense for yourself - it works because you don't have time to count the dots, so you have to rely on your more basic sense of ratio. Unfortunately it doesn't tell you your ratio, but if you keep track of the numbers of blue and yellow dots in each question and which ones you got right or wrong, you can figure out what ratio you need to consistently get it right. Unfortunately, after a bit of time of getting used to it, I got the right answer every time, so I'll need a better tester...
Actually, I'm inspired now to make a swf file which does the same thing, but actually calculates the ratio of dots you can distinguish. I'll post it to the blog when I'm done.
(Edit: I'm done.)
Ratio keeps on coming up in the way we think... for example, intuitively, we think $1 billion and $1.01 billion are extremely close. But that difference is more money than most of us will ever own in our entire lives. We don't think about the difference, which is huge - we think about the ratio, which is tiny.
So how, then, does counting, which focuses on sums and differences instead of ratios, fit in?
(continued in Part 3A)
What do babies know about math?
From this point on, I'm going to be working off memory a lot... I watched a presentation about this once, very fascinating, but I don't know if I can find the data.
So basically, some researchers wanted to know if babies could distinguish between different numbers. So they took infants, less than 1 year old, into a laboratory, and asked them if they knew what the difference between 17 and 18 was.
The babies looked around and started making gurgling noises so the researchers decided they had to be a bit more clever. So instead, they showed these children two pictures with different numbers of objects, and marked which picture held the babies' attention. I'm sure it's a bit more complicated than this, but essentially they tried to find out what babies knew about numbers.
They knew the difference between 0 and 1. Between a blank screen and a screen with a dot on it, the dot is obviously more exciting.
They knew the difference between 1 and 2. Between a lonely dot and dots that weren't lonely, the babies wanted to join the more social group.
The did NOT know the difference between 2 and 3. To them, it looked basically the same - a few friendly dots. Similarly, between 3 and 4, 4 and 5, etc... a difference of 1 was meaningless after the 1~2 jump.
However, they DID know the difference between 2 and 4... between 4 and 8... between 12 and 24... But not between 12 and 18, or 40 and 50, etc.
The result: the RATIO is what the infants recognize. As long as one picture had at least twice as many dots as the other, it captivated the baby's attention.
Further study was done. This time, children of different ages were studied. They found that after a couple months, babies (or toddlers now, perhaps) could distinguish between 2 and 3. And a couple months later, between 3 and 4. Then all of a sudden there's a jump, and in an incredibly short length of time, the children recognize 4, 5, 6, 7, 8...
To people who study progress over time, a sudden jump almost always reflects a new technology. You invent the printing press, and the number of books explodes. Develop the cotton gin, the textile industry skyrockets. Here too, there is a new technology - the technology of counting.
I call counting a technology because it is NOT something humans inherently have. Maybe you've heard the news stories of tribes of people (the Pirahã are an example of this) who don't have words representing different numbers, just concepts of "smaller amount" and "larger amount." They didn't discover the technology of counting, so they couldn't distinguish between, say, 5 dots and 7 dots.
What humans do inherently have is something often called "number sense." The definition on Wikipedia says that number sense is "an intuitive understanding of numbers, their magnitude, relationships, and how they are affected by operations." Note that the process of counting is not referred to, nor are specific names of numbers - here "number" just refers to an understanding of amount. Once counting is learned, one's number sense drastically improves - reflecting the dramatic growth in a toddler's ability to recognize amounts. But you have number sense even without counting - in its most natural state, number sense is based not on counting, but on RATIOS.
As you grow, whether you learn counting and names of numbers or not, your number sense sharpens - while babies need a 2:1 ratio to be able to recognize something, most adults can recognize smaller ratios, 1.5:1 or lower, and curiously, many people who are very good at higher level math tend to also be very good at this very fundamental, basic number sense, recognizing ratios as small as 1.1:1 or lower (i.e. being able to tell the difference between 11 dots and 10 dots without counting them, just at a glance)
I found this site where you can test your basic number sense for yourself - it works because you don't have time to count the dots, so you have to rely on your more basic sense of ratio. Unfortunately it doesn't tell you your ratio, but if you keep track of the numbers of blue and yellow dots in each question and which ones you got right or wrong, you can figure out what ratio you need to consistently get it right. Unfortunately, after a bit of time of getting used to it, I got the right answer every time, so I'll need a better tester...
Actually, I'm inspired now to make a swf file which does the same thing, but actually calculates the ratio of dots you can distinguish. I'll post it to the blog when I'm done.
(Edit: I'm done.)
Ratio keeps on coming up in the way we think... for example, intuitively, we think $1 billion and $1.01 billion are extremely close. But that difference is more money than most of us will ever own in our entire lives. We don't think about the difference, which is huge - we think about the ratio, which is tiny.
So how, then, does counting, which focuses on sums and differences instead of ratios, fit in?
(continued in Part 3A)
1, 2, 3... not the best place to start? (Part A - background and history)
NOTE - This post may include some mathematically technical material. I will attempt to give warning by bolding parts that need some extra background.
How did you learn math? Almost certainly, you would say it started with counting.
STEP 1: You learned a list of words - "one two three four five six seven eight nine ten" perhaps, or maybe "一 二 三 四 五 六 七 八 九 十" or something similar, and practice pointing at different things while saying these words.
STEP 2: Soon after, you begin to apply that list to actual things you see - you realize that the word "five" can be applied both to "five apples" and "five people." This is actually a pretty major step - there are many young children who can count blocks "one, two, three," but don't know how many blocks there are. This is where children learn one of the most important properties of math, abstraction - that a single concept (like "five") can apply to many completely different areas. From there, the natural direction to go is
STEP 3: Comparing numbers (6 is bigger than 4 because you have to count past 4 to get to 6)
STEP 4: Addition (you have 1 2 3 blocks here, and 1 2 3 4 5 blocks here - how many altogether?),
STEP 5: Subtraction (you have 1 2 3 4 5 6 blocks, and 1 2 3 4 are in a special group - how many are not in the special group? Or, if you've already learned addition - what number added to 4 makes 6?) and then
STEP 6: Multiplication is taught as doing addition of the same number several times.
But then along comes division. And EVERYONE struggles with division, at least the first time they see it. "It's just the opposite of multiplication," right? So if multiplication is repeated addition, does that mean division is repeated subtraction? (The answer is yes, kind of, but it's not very intuitive). Division is also the first time that you kind of have to leave the whole numbers - adding and multiplying whole numbers give whole numbers, and if you always make sure take the smaller from the larger (which is simple enough to check), same with subtracting. But you would have NO idea if dividing 46 by 4 would give you a whole number or not. So this is where decimals and fractions start becoming necessary. But fractions are confuuuuuusing... Why do we use two numbers to represent one number? Why can't you just add the numerator and denominator? Why do we use such big words? What happens when you divide by zero? Why do we need to learn this? Can we go to recess early?
A bit more background about myself - I love math, and I'm going to study pure mathematics at university... but I've been very interested in developmental math. How do people learn math, and what gets in their way? This interest first came up when I went to Mathcamp, a 5-week summer math program in the U.S., in the summers of 2009 and 2010. Some of the visiting speakers there talked about how people learn math (or just about the psychology of math in general) - talks by Dan Zaharopol, Josh Tenenbaum, and Catherine Havasi were particularly inspiring, and some of the ideas in this post can be attributed to them. And over my 12th grade year, I researched math anxiety and its causes, giving me an even bigger interest in the topic. Which leads me to this post - I'm not trying to push any particular viewpoint, I'm just speculating - what if a completely different approach to math, from a very early age, would address a lot of problems that currently face students learning math?
Myth: counting is the foundation of everything we learn about mathematics.
FACT: It's not.
It's true that a counting process can be used to form a rigorous basis for all mathematics - once the natural numbers have been constructed, it's simple enough to construct the integers, rationals, reals, and complexes (complices?) via equivalence classes of ordered pairs and Dedekind cuts or Cauchy sequences... but I'd like to propose that perhaps this isn't the (only) natural way to learn mathematics.
A bit of history... the Greeks were very good at math. Except for calculus and some ideas about trigonometry and limits, pretty much all elementary, middle, and high school math was known to the Greeks. But they had two completely separate sections of math - geometry and number. They did find ways to connect the two sometimes (such as through the Pythagorean theorem, connecting a right triangle to an equation about numbers), but they started with each separately, and kept the two separate. Why? Because the two are completely different. Their study of number was a discrete process - where discrete basically means "can be counted." Their study of geometry was a study of continuous things - things that "can be measured." And for most of history, the two were kept completely separate. Again, people found many ways to connect them, but you would always learn them separately, then show that they could be joined together.
It wasn't until about a century ago that this changed. Basically, people went crazy proving and discovering all sorts of wacky things, it was getting harder and harder to understand what was going on, and finally someone decided "this is too messy. We need to have a common ground to base all mathematics on, so we can all agree on what's actually happening." And so, almost arbitrarily, some mathematicians chose to make number the foundation for geometry, and proved that all basic facts about geometry could be explained just using facts about numbers.
Was this a wise decision? I don't know. But it certainly had major impacts. For example, one of the most famous mathematical works of all time, Euclid's Elements, which basically sets foundations for geometry, stopped being popular around that time. In the past, Elements was an important part of Western culture, both in mathematics and out; major non-mathematicians such as Thomas Hobbes, Bertrand Russel, and Abraham Lincoln all worked through and loved the Elements. According to John Stillwell, "Perhaps the low cultural status of mathematics today, not to mention the mathematical ignorance of politicians and philosophers, reflects the lack of an Elements suitable for the modern world." And it's quite likely that its drop in popularity came from people thinking "yeah, geometry's great, but let's start with learning about numbers first."
And from the little I know about developmental psychology, it seems like the choice to start with number impacted a lot more than just the popularity of a particular book.
(continued in Part 2A)
How did you learn math? Almost certainly, you would say it started with counting.
STEP 1: You learned a list of words - "one two three four five six seven eight nine ten" perhaps, or maybe "一 二 三 四 五 六 七 八 九 十" or something similar, and practice pointing at different things while saying these words.
STEP 2: Soon after, you begin to apply that list to actual things you see - you realize that the word "five" can be applied both to "five apples" and "five people." This is actually a pretty major step - there are many young children who can count blocks "one, two, three," but don't know how many blocks there are. This is where children learn one of the most important properties of math, abstraction - that a single concept (like "five") can apply to many completely different areas. From there, the natural direction to go is
STEP 3: Comparing numbers (6 is bigger than 4 because you have to count past 4 to get to 6)
STEP 4: Addition (you have 1 2 3 blocks here, and 1 2 3 4 5 blocks here - how many altogether?),
STEP 5: Subtraction (you have 1 2 3 4 5 6 blocks, and 1 2 3 4 are in a special group - how many are not in the special group? Or, if you've already learned addition - what number added to 4 makes 6?) and then
STEP 6: Multiplication is taught as doing addition of the same number several times.
But then along comes division. And EVERYONE struggles with division, at least the first time they see it. "It's just the opposite of multiplication," right? So if multiplication is repeated addition, does that mean division is repeated subtraction? (The answer is yes, kind of, but it's not very intuitive). Division is also the first time that you kind of have to leave the whole numbers - adding and multiplying whole numbers give whole numbers, and if you always make sure take the smaller from the larger (which is simple enough to check), same with subtracting. But you would have NO idea if dividing 46 by 4 would give you a whole number or not. So this is where decimals and fractions start becoming necessary. But fractions are confuuuuuusing... Why do we use two numbers to represent one number? Why can't you just add the numerator and denominator? Why do we use such big words? What happens when you divide by zero? Why do we need to learn this? Can we go to recess early?
A bit more background about myself - I love math, and I'm going to study pure mathematics at university... but I've been very interested in developmental math. How do people learn math, and what gets in their way? This interest first came up when I went to Mathcamp, a 5-week summer math program in the U.S., in the summers of 2009 and 2010. Some of the visiting speakers there talked about how people learn math (or just about the psychology of math in general) - talks by Dan Zaharopol, Josh Tenenbaum, and Catherine Havasi were particularly inspiring, and some of the ideas in this post can be attributed to them. And over my 12th grade year, I researched math anxiety and its causes, giving me an even bigger interest in the topic. Which leads me to this post - I'm not trying to push any particular viewpoint, I'm just speculating - what if a completely different approach to math, from a very early age, would address a lot of problems that currently face students learning math?
Myth: counting is the foundation of everything we learn about mathematics.
FACT: It's not.
It's true that a counting process can be used to form a rigorous basis for all mathematics - once the natural numbers have been constructed, it's simple enough to construct the integers, rationals, reals, and complexes (complices?) via equivalence classes of ordered pairs and Dedekind cuts or Cauchy sequences... but I'd like to propose that perhaps this isn't the (only) natural way to learn mathematics.
A bit of history... the Greeks were very good at math. Except for calculus and some ideas about trigonometry and limits, pretty much all elementary, middle, and high school math was known to the Greeks. But they had two completely separate sections of math - geometry and number. They did find ways to connect the two sometimes (such as through the Pythagorean theorem, connecting a right triangle to an equation about numbers), but they started with each separately, and kept the two separate. Why? Because the two are completely different. Their study of number was a discrete process - where discrete basically means "can be counted." Their study of geometry was a study of continuous things - things that "can be measured." And for most of history, the two were kept completely separate. Again, people found many ways to connect them, but you would always learn them separately, then show that they could be joined together.
It wasn't until about a century ago that this changed. Basically, people went crazy proving and discovering all sorts of wacky things, it was getting harder and harder to understand what was going on, and finally someone decided "this is too messy. We need to have a common ground to base all mathematics on, so we can all agree on what's actually happening." And so, almost arbitrarily, some mathematicians chose to make number the foundation for geometry, and proved that all basic facts about geometry could be explained just using facts about numbers.
Was this a wise decision? I don't know. But it certainly had major impacts. For example, one of the most famous mathematical works of all time, Euclid's Elements, which basically sets foundations for geometry, stopped being popular around that time. In the past, Elements was an important part of Western culture, both in mathematics and out; major non-mathematicians such as Thomas Hobbes, Bertrand Russel, and Abraham Lincoln all worked through and loved the Elements. According to John Stillwell, "Perhaps the low cultural status of mathematics today, not to mention the mathematical ignorance of politicians and philosophers, reflects the lack of an Elements suitable for the modern world." And it's quite likely that its drop in popularity came from people thinking "yeah, geometry's great, but let's start with learning about numbers first."
And from the little I know about developmental psychology, it seems like the choice to start with number impacted a lot more than just the popularity of a particular book.
(continued in Part 2A)
Monday, August 29, 2011
A couple of links
This song makes me happy. http://freeplaymusic.com/search/download_file.php?id=4126&dur=0&type=mp3
Adorable moment at 2:34~3:15. http://www.ted.com/talks/cynthia_breazeal_the_rise_of_personal_robots.html
That's it for now.
(quick life update - currently living in Calgary, but packing my stuff to get ready to fly to Toronto for university in just under a week)
Adorable moment at 2:34~3:15. http://www.ted.com/talks/cynthia_breazeal_the_rise_of_personal_robots.html
That's it for now.
(quick life update - currently living in Calgary, but packing my stuff to get ready to fly to Toronto for university in just under a week)
Sunday, August 28, 2011
A slightly more sane introduction
Watch the video in the post before this one if you haven't already.
Now you're probably at least slightly confused. I know I am. So hopefully this post clarifies some things.
So the first question many of you may have is the title of the blog - why "Bruce Lee and a Blue Chicken?"
Another question you might have is what exactly the purpose of the blog is. In the video, I said that I wasn't planning on giving "day-to-day descriptions of my thoughts, opinions, and activities." This is false. In fact, the main reason I have this blog is so that people who want to can keep track of how I'm doing. But I'm probably not going to have too many posts of that sort, and they won't be very long - this isn't a diary, I'm not going to share my entire life with the whole internet. If you want to know more specifics about something I post about, I'd be glad to talk to you about it. So I'll have a couple posts every now and then that will "paint my life in broad strokes" (source: cliché generator) and the rest of the time... I'll just have fun doing whatever I feel like. And you can watch me do it! :D
Now some of you are probably still wondering about the title. But some of you may be wondering instead about the tools I use to make my videos. The last one was shot entirely on an iphone 4 camera, and edited using Corel Videostudio Pro. Just whatever I happened to have. I'm not too interested at this point in getting into the high quality equipment, as some of my friends are (and making incredible videos as a result) - maybe later?
Anyways, there's an interesting story behind the title of this blog. But a bit of background first, for those of you who may not know me that well - my name is Jonathan Love, and at the time of the writing of this post, I am just about to enter university at the University of Toronto. I spent most of my school years at an international school in Tokyo called Christian Academy in Japan (although a few years here and there were spent in Canada). I'm interested in a wide range of things, but my "primary" interests include, math, music, and Jesus. Any and all of these could come out in any given post, but it's just as likely that I'll talk about something completely unrelated (politics, basketball, sewing, friendship, etc). That's basically enough about me for now.
All right, back to the title. Basically, it comes from the
Now you're probably at least slightly confused. I know I am. So hopefully this post clarifies some things.
So the first question many of you may have is the title of the blog - why "Bruce Lee and a Blue Chicken?"
Another question you might have is what exactly the purpose of the blog is. In the video, I said that I wasn't planning on giving "day-to-day descriptions of my thoughts, opinions, and activities." This is false. In fact, the main reason I have this blog is so that people who want to can keep track of how I'm doing. But I'm probably not going to have too many posts of that sort, and they won't be very long - this isn't a diary, I'm not going to share my entire life with the whole internet. If you want to know more specifics about something I post about, I'd be glad to talk to you about it. So I'll have a couple posts every now and then that will "paint my life in broad strokes" (source: cliché generator) and the rest of the time... I'll just have fun doing whatever I feel like. And you can watch me do it! :D
Now some of you are probably still wondering about the title. But some of you may be wondering instead about the tools I use to make my videos. The last one was shot entirely on an iphone 4 camera, and edited using Corel Videostudio Pro. Just whatever I happened to have. I'm not too interested at this point in getting into the high quality equipment, as some of my friends are (and making incredible videos as a result) - maybe later?
Anyways, there's an interesting story behind the title of this blog. But a bit of background first, for those of you who may not know me that well - my name is Jonathan Love, and at the time of the writing of this post, I am just about to enter university at the University of Toronto. I spent most of my school years at an international school in Tokyo called Christian Academy in Japan (although a few years here and there were spent in Canada). I'm interested in a wide range of things, but my "primary" interests include, math, music, and Jesus. Any and all of these could come out in any given post, but it's just as likely that I'll talk about something completely unrelated (politics, basketball, sewing, friendship, etc). That's basically enough about me for now.
All right, back to the title. Basically, it comes from the
Introducing... (feat. Carl Sundigger)
A: "Hey check out my blog!"
B: "Blog?"
A: "You know, like, web log!"
B: "I can take logarithms without relying on the internet, thank you very much."
Subscribe to:
Comments (Atom)