4 ms·

> Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and

> Ah, sounds like the common trend in the hard sciences of putting "difficulty" on a pedestal. Can't make it too easy to learn, need to make things generic and obtuse so only the most dedicated can master the material! I'm not sure that this kind of snark is helpful if you want to teach something. My point is precisely to make things as easy as possible, not harder! Variables are just slots. How would you rewrite the following using "meaningful" variable names? The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ? Any "meaningful" names that you could assign to the terms X, a, b, c would depend heavily in the context. You would convert this equation into an unrecognizable monstrosity. But this equation appears in many different, completely unrelated contexts! The main point of math, I'd say, is that no matter what is the context, the equation is always one and the same and it is solved in exactly the same way. Thus it pays off to have variable names as meaningless as possible.

adam_arthur 5y ago
Apologies for the snark, definitely not warranted. I do see sentiments along those lines a lot, so it's a bit of a pet peeve of mine! My suggestion is to first teach the generic concept wrapped in context that's more relatable to the student. Similar to code, it's easier to understand the purpose of a variable when you understand the surrounding context as well. For your formula above, you could define the axes as something meaningful. Say, e.g. instead of A, B and C you have Supply, Demand and Price. If I read a formula such as: Demand = 100 + 50Price Supply = 50 + 20Price It's easier for somebody to relate to utility of the math than something like: A = 100 + 50C B = 50 + 20C Of course, the math is exactly the same, but you give the student something to relate to. Now, Supply/Demand is not something the average person can relate to. Obviously context has something of a cultural and familiarity bias. But there are examples that are likely to be close to universal. I use economics here, just because it's a discipline I'm familiar with :) I don't advocate teaching strictly in contextual examples, but to use them first to provide ELI5 level introduction to concepts. The utility behind the math becomes more clear. Of course, this is just my take/experience. I spent the majority of time in college skipping class and self teaching, and I discovered that the fastest approach to learning something new is to start at highest level of abstraction and work down. Usually I would learn concepts from resources other than the textbook, as they conveyed information more effectively. Trying to learn concepts "bottom up" rather than "top down" works, but is less time efficient.
- username90 5y ago
  > Of course, the math is exactly the same, but you give the student something to relate to. No, you give the student an extra problem they need to solve: Figure out what the hell supply and demand has to do with price. Understanding that isn't expected of high schoolers and teaching it isn't a part of mathematics. History shows there are no intuitive equations outside of basic counting and summing. Even the concept of rate of change weren't explored before the 17th century, before then people had very ridiculous views on what equations physics uses, that is what you get if you try to invoke students intuition here.
  - adam_arthur 5y ago
    > Figure out what the hell supply and demand has to do with price. Are more people going to relate more to a problem framed that way, or framed as what A has to do with B? If you're comparing the two, it's self evident that the real world example is easier for the average person to understand. People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this domain is not the most widely understood... just off the cuff to make a point. The solution to both problems is the same, does not require specialized domain knowledge at all, so I don't get your point. There are plenty of more culturally widely understood domains we could use, rather than economics. Physics is of course the easy go to. Not sure what you mean by ridiculous views on physics, clarify?
    - username90 5y ago
      > People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this example is not the most widely recognized... just off the cuff to make a point. Have you ever tried to argue with an average person? Those deductions aren't simple to make at all. > Are more people going to relate more to a problem framed that way, or framed as what A has to do with B? All students have already seen countless of examples of what A and B can be. So when they see an equation with A and B they can relate it to apples, to money, to distances, to mass, to temperature etc, at once. If you use special names for the variables most students assumes that the equation only works for those variables and either memorises it or tries hard to understand what makes those input kinds special. Edit: > Physics is of course the easy go to. If you use physics then your class is no longer a math class but a physics class. It takes years for students to build intuition for even basic physics. > Not sure what you mean by ridiculous views on physics, clarify? Before newton people didn't understand rate of change or movement properly, instead they came up with shit explanations like this: > Based on Aristotelian physics, Scholastic physics described things as moving according to their essential nature. Celestial objects were described as moving in circles, because perfect circular motion was considered an innate property of objects that existed in the uncorrupted realm of the celestial spheres. The theory of impetus, the ancestor to the concepts of inertia and momentum, was developed along similar lines by medieval philosophers such as John Philoponus and Jean Buridan. Motions below the lunar sphere were seen as imperfect, and thus could not be expected to exhibit consistent motion. More idealized motion in the "sublunary" realm could only be achieved through artifice, and prior to the 17th century, many did not view artificial experiments as a valid means of learning about the natural world. Physical explanations in the sublunary realm revolved around tendencies. Stones contained the element earth, and earthly objects tended to move in a straight line toward the centre of the earth (and the universe in the Aristotelian geocentric view) unless otherwise prevented from doing so.[25] https://en.wikipedia.org/wiki/History_of_physics#Medieval_Europe https://en.wikipedia.org/wiki/History_of_physics#Medieval_Eu... That was what the best of humanity could come up with over 2 millennia. You can't expect an average modern high schooler to do better than that without first being taught physics.
hutzlibu 5y ago
" The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ? You would convert this equation into an unrecognizable monostrosity." For any beginner this is a unrecognisable monstrosity. Having real names for the variables can give them something to hold on. Something they can relate to. So it makes sense to their brain. And later on, you can abstract it - and they are fine with it and see the benefits. (I still want programming variable names to be as descriptive as possible) You (and probably everyone else around here) are used to abstraction - a beginner to math is not! (and sadly, most people are even after 12 years of schoolmath still beginners in math) And the focus on abstraction is a huge wall, making everything harder than it needs to be. " Thus it pays off to have variable names as meaningless as possible. " I agree that this is very helpful to achieve, when you are dealing with plain math only. But I think this thinking actually lead to lots of real world harm - when variables are not meaningless factors in math - but representing real persons. Which is forgotten quite often, if some buerocrat for example just sees them as a "meaningless variable" and treats them like this. But in reality this can mean, someones existence just got crushed.
- enriquto 5y ago
  > For any beginner this is a unrecognisable monstrosity. Having real names for the variables can give them something to hold on. Can you rewrite this same equation using "real names" so that it becomes less monstrous? I don't think that's possible, but maybe you have something in mind.
  - hutzlibu 5y ago
    You equation is abstract. I recognize it exactly as this: as some abstract equations - but nothing more. (am I supposed to recognize them?) There is of course no real point in adding names later on. At least not without context. I would start with a real world example. And then - yes have the equations with long names. And then after that - have the equation as abstract plain math - in the shape you posted it. But not starting with it.
    - username90 5y ago

adam_arthur 5y ago

Apologies for the snark, definitely not warranted. I do see sentiments along those lines a lot, so it's a bit of a pet peeve of mine! My suggestion is to first teach the generic concept wrapped in context that's more relatable to the student. Similar to code, it's easier to understand the purpose of a variable when you understand the surrounding context as well. For your formula above, you could define the axes as something meaningful. Say, e.g. instead of A, B and C you have Supply, Demand and Price. If I read a formula such as: Demand = 100 + 50Price Supply = 50 + 20Price It's easier for somebody to relate to utility of the math than something like: A = 100 + 50C B = 50 + 20C Of course, the math is exactly the same, but you give the student something to relate to. Now, Supply/Demand is not something the average person can relate to. Obviously context has something of a cultural and familiarity bias. But there are examples that are likely to be close to universal. I use economics here, just because it's a discipline I'm familiar with :) I don't advocate teaching strictly in contextual examples, but to use them first to provide ELI5 level introduction to concepts. The utility behind the math becomes more clear. Of course, this is just my take/experience. I spent the majority of time in college skipping class and self teaching, and I discovered that the fastest approach to learning something new is to start at highest level of abstraction and work down. Usually I would learn concepts from resources other than the textbook, as they conveyed information more effectively. Trying to learn concepts "bottom up" rather than "top down" works, but is less time efficient.

username90 5y ago
> Of course, the math is exactly the same, but you give the student something to relate to. No, you give the student an extra problem they need to solve: Figure out what the hell supply and demand has to do with price. Understanding that isn't expected of high schoolers and teaching it isn't a part of mathematics. History shows there are no intuitive equations outside of basic counting and summing. Even the concept of rate of change weren't explored before the 17th century, before then people had very ridiculous views on what equations physics uses, that is what you get if you try to invoke students intuition here.
- adam_arthur 5y ago
  > Figure out what the hell supply and demand has to do with price. Are more people going to relate more to a problem framed that way, or framed as what A has to do with B? If you're comparing the two, it's self evident that the real world example is easier for the average person to understand. People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this domain is not the most widely understood... just off the cuff to make a point. The solution to both problems is the same, does not require specialized domain knowledge at all, so I don't get your point. There are plenty of more culturally widely understood domains we could use, rather than economics. Physics is of course the easy go to. Not sure what you mean by ridiculous views on physics, clarify?
  - username90 5y ago
    > People know what price of a good is. People know that factories make goods, so there must be some supply. Demand is maybe more foreign, but I admit this example is not the most widely recognized... just off the cuff to make a point. Have you ever tried to argue with an average person? Those deductions aren't simple to make at all. > Are more people going to relate more to a problem framed that way, or framed as what A has to do with B? All students have already seen countless of examples of what A and B can be. So when they see an equation with A and B they can relate it to apples, to money, to distances, to mass, to temperature etc, at once. If you use special names for the variables most students assumes that the equation only works for those variables and either memorises it or tries hard to understand what makes those input kinds special. Edit: > Physics is of course the easy go to. If you use physics then your class is no longer a math class but a physics class. It takes years for students to build intuition for even basic physics. > Not sure what you mean by ridiculous views on physics, clarify? Before newton people didn't understand rate of change or movement properly, instead they came up with shit explanations like this: > Based on Aristotelian physics, Scholastic physics described things as moving according to their essential nature. Celestial objects were described as moving in circles, because perfect circular motion was considered an innate property of objects that existed in the uncorrupted realm of the celestial spheres. The theory of impetus, the ancestor to the concepts of inertia and momentum, was developed along similar lines by medieval philosophers such as John Philoponus and Jean Buridan. Motions below the lunar sphere were seen as imperfect, and thus could not be expected to exhibit consistent motion. More idealized motion in the "sublunary" realm could only be achieved through artifice, and prior to the 17th century, many did not view artificial experiments as a valid means of learning about the natural world. Physical explanations in the sublunary realm revolved around tendencies. Stones contained the element earth, and earthly objects tended to move in a straight line toward the centre of the earth (and the universe in the Aristotelian geocentric view) unless otherwise prevented from doing so.[25] https://en.wikipedia.org/wiki/History_of_physics#Medieval_Europe https://en.wikipedia.org/wiki/History_of_physics#Medieval_Eu... That was what the best of humanity could come up with over 2 millennia. You can't expect an average modern high schooler to do better than that without first being taught physics.

hutzlibu 5y ago

" The two solutions of aXX+bX+c = 0 are X=(-b+√(bb-4ac))/2a and X=(-b-√(bb-4ac))/2a ? You would convert this equation into an unrecognizable monostrosity." For any beginner this is a unrecognisable monstrosity. Having real names for the variables can give them something to hold on. Something they can relate to. So it makes sense to their brain. And later on, you can abstract it - and they are fine with it and see the benefits. (I still want programming variable names to be as descriptive as possible) You (and probably everyone else around here) are used to abstraction - a beginner to math is not! (and sadly, most people are even after 12 years of schoolmath still beginners in math) And the focus on abstraction is a huge wall, making everything harder than it needs to be. " Thus it pays off to have variable names as meaningless as possible. " I agree that this is very helpful to achieve, when you are dealing with plain math only. But I think this thinking actually lead to lots of real world harm - when variables are not meaningless factors in math - but representing real persons. Which is forgotten quite often, if some buerocrat for example just sees them as a "meaningless variable" and treats them like this. But in reality this can mean, someones existence just got crushed.

enriquto 5y ago
> For any beginner this is a unrecognisable monstrosity. Having real names for the variables can give them something to hold on. Can you rewrite this same equation using "real names" so that it becomes less monstrous? I don't think that's possible, but maybe you have something in mind.
- hutzlibu 5y ago
  You equation is abstract. I recognize it exactly as this: as some abstract equations - but nothing more. (am I supposed to recognize them?) There is of course no real point in adding names later on. At least not without context. I would start with a real world example. And then - yes have the equations with long names. And then after that - have the equation as abstract plain math - in the shape you posted it. But not starting with it.
  - username90 5y ago