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.
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