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How's that for an abstract title? Sorry! If you're a philosopher, follow along. If you hate abstract ideas, please avoid burning my house down in frustration.
I want to start by saying that people can say things, and people can do things. What they say does not always match what they do. Do you agree?
Here is an example. If a group of math students has been taught to say that an equals sign (=) means "the value on the left side is the same as the value on the right side," they will likely say just that. Ask them what an equals sign means and see for yourself.
But there are two concepts they might actually have for an equals sign. These concepts are illustrated by what they do with it, not what they say about it.
One concept is that an equals sign is like the 'equals' button on a calculator. If you put in some numbers and plus signs, and then hit 'equals,' you 'transform' those numbers into their sum. For someone who thinks of the symbol this way, it is a transformation function; it changes the thing on one side into a different thing on the other side.
Another concept is that an equals sign is like the center of a balance. If you add or subtract from one side, you can no longer have 'equals' in the middle, because it is off balance. You could have greater than or less than, sure. But not equals. If you subtract from the right-hand side, you must subtract the same amount from the left-hand side. If you double the left-hand side, you must double the right-hand side. Doing so ensures that the balance is always maintained.
Can you imagine which concept is more useful if you are trying to learn algebra? Or any other mathematics beyond arithmetic? Clearly the second one is better; it makes algebra intuitive and automatic. It makes mistakes unlikely.
Someone who is learning math might tell you that an equals sign means 'the same value on both sides.' But then they might demonstrate a misconception. For example, they "move" a +5 from one side of the equation to the other. Since they took away +5 from one side and added +5 to the other, the balance is now off by 10; it's illogical. But people make mistakes just like this. And they make them because their true grasp of the concept lags behind what they say about it.
I think the discrepancies between words and actions, spoken definitions and demonstrated concepts, only increase as we start using fancy words and intricate concepts.
In business, people might talk about revenue and profit and cashflow and making money. But then they make choices, and design a system, that makes money for everyone but themselves. Do they really demonstrate the concepts they talk about?
The possibility of saying something that is true, and following it up with actions that are false, can escalate to the point of a tragic flaw. The problem is that we often believe we are acting on our own words, on our promises, on what we say are our beliefs. But in fact, our actions may be decoupled from our words. Even from the words we think privately to ourselves.
Because of this potential flaw, I think one of the greatest gifts we can receive from another person is to have them point out one of these misconceptions, a discrepancy between our words and our actions. What could be more powerful, more potentially life-changing, than that?
But do I really believe that? If you point out a misconception to me, according to my own words, I should be grateful. If I get defensive instead, I'm demonstrating that I don't truly believe the concept behind those words. How complex we are!
I want to start by saying that people can say things, and people can do things. What they say does not always match what they do. Do you agree?
Here is an example. If a group of math students has been taught to say that an equals sign (=) means "the value on the left side is the same as the value on the right side," they will likely say just that. Ask them what an equals sign means and see for yourself.
But there are two concepts they might actually have for an equals sign. These concepts are illustrated by what they do with it, not what they say about it.
One concept is that an equals sign is like the 'equals' button on a calculator. If you put in some numbers and plus signs, and then hit 'equals,' you 'transform' those numbers into their sum. For someone who thinks of the symbol this way, it is a transformation function; it changes the thing on one side into a different thing on the other side.
Another concept is that an equals sign is like the center of a balance. If you add or subtract from one side, you can no longer have 'equals' in the middle, because it is off balance. You could have greater than or less than, sure. But not equals. If you subtract from the right-hand side, you must subtract the same amount from the left-hand side. If you double the left-hand side, you must double the right-hand side. Doing so ensures that the balance is always maintained.
Can you imagine which concept is more useful if you are trying to learn algebra? Or any other mathematics beyond arithmetic? Clearly the second one is better; it makes algebra intuitive and automatic. It makes mistakes unlikely.
Someone who is learning math might tell you that an equals sign means 'the same value on both sides.' But then they might demonstrate a misconception. For example, they "move" a +5 from one side of the equation to the other. Since they took away +5 from one side and added +5 to the other, the balance is now off by 10; it's illogical. But people make mistakes just like this. And they make them because their true grasp of the concept lags behind what they say about it.
I think the discrepancies between words and actions, spoken definitions and demonstrated concepts, only increase as we start using fancy words and intricate concepts.
In business, people might talk about revenue and profit and cashflow and making money. But then they make choices, and design a system, that makes money for everyone but themselves. Do they really demonstrate the concepts they talk about?
The possibility of saying something that is true, and following it up with actions that are false, can escalate to the point of a tragic flaw. The problem is that we often believe we are acting on our own words, on our promises, on what we say are our beliefs. But in fact, our actions may be decoupled from our words. Even from the words we think privately to ourselves.
Because of this potential flaw, I think one of the greatest gifts we can receive from another person is to have them point out one of these misconceptions, a discrepancy between our words and our actions. What could be more powerful, more potentially life-changing, than that?
But do I really believe that? If you point out a misconception to me, according to my own words, I should be grateful. If I get defensive instead, I'm demonstrating that I don't truly believe the concept behind those words. How complex we are!
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