Most people who learn it have a very hard time doing so, and they’re already well above average in mathematical ability.
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud. You multiply by the number up to the top right of the letter then reduce that number by 1. Or you do the reverse in the reverse order. You know, like you put on your socks then your shoes but have to take off your shoes then take off your socks.
Sometimes drawing a picture helps prime an intuitive understanding of the physics. You start with a graph of velocity vs time. That is the ‘acceleration’. See… it is getting faster each second. Now, use a pencil and progressively color in under the line. that’s the distance that is getting covered. See how later on more when it is going faster more distance is being traveled at one time and we have to shade in more area? Now, remember how we can find the area of a triangle? Well, will you look at that… the maths came out the same!
has to do with the derivative as a rate of change or as a slope of a tangent line. And from the perspective of a calculus student who has gone through the standard run of American school math, I can understand. It does require a level up in mathematical sophistication.
from the perspective of a calculus student who has gone through the standard run of American school math,
That’s the problem. See that bunch of symbols? That isn’t the best way to teach stuff. It is like trying to teach them math while speaking a foreign language (even if technically we are saving the greek till next month). To teach that concept you start with the kind of picture I was previously describing, have them practice that till they get it then progress to diagrams that change once in the middle, etc.
Perhaps the students here were prepared differently but the average student started getting problems with calculus when it reached a point slightly beyond what you require for the basic physics we were talking about here. ie. they would be able to do 1. and but have no chance at all with 2:
I’m not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly″.
Doing derivatives without the definition is just calling on a black box.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan “brought to you by apostrophe, the letters LIM and an arrow” but you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn’t easy.
you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it
That is close to saying “this stuff is hard”. How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?
(Actually, many would struggle with 1. due to difficulty with comprehension and abstract problem solving. They could handle the calculus but need someone to hold their hand with the actual thinking part. That’s what we really fail to teach effectively.)
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud.
People get the simple concepts mixed together with a bunch of mathy-looking symbols and equations, and it all congeals into an undifferentiated mass of confusing math. Yes, I know calculus is actually pretty straightforward, but we’re probably not a representative sample. Talk with random bewildered college freshmen to combat sample bias. I did this, and what I learned is that most people have serious trouble learning calculus.
Now, if you want to be able to partially understand a bunch of physics stuff but you don’t necessarily need to be able to do the math, you could probably get away with a small subset of what people learn in calculus classes. If you learned about integration and differentiation (but not how to do them symbolically), as well as vectors, vector fields, and divergence and curl, then you could probably get more benefit-per-hour-of-study than if you went and learned calculus properly. It leaves a bad taste in my mouth, though.
what I learned is that most people have serious trouble learning calculus.
When taught well the calculus required for the sort of applications you mentioned is not something that causes significant trouble, certainly not compared to vector fields, divergence or curl. By taught well, if you will excuse my lack of seemly modesty, is how I taught it in my (extremely brief—don’t let me get started on what I think of western school systems) stint teaching high school physics. The biggest problem for people learning basic calculus is that people teaching it try to convey that it is hard.
I’m only talking here about the level of stuff required for everyday physics. Definitely not for the vast majority of calculus that we try to teach them.
It has been said that democracy is the worst form of government except all the others that have been tried.
--Sir Winston Churchill
I’m not quite going to make that analogy but I will hasten to assert that there are far worse systems of education than ours. Including some that are ’like ours but magnified”.
In terms of healthy psychological development and practical skill acquisition the apprenticeship systems of various cultures have been better. Right now I can refer to the school system on one of the Solomon Islands. The culture is that of a primitive coastal village but with western influences. Western teaching materials and a teacher are provided but occurs in the morning for 4 hours a day. No breaks are needed and nor is any pointless time wasting. The children then spend their time surfing. But they surf carrying spears and catch fish while they are doing it.
What appeals to me about that system is:
The shorter time period.
Most of the time kids spend at school is a blatant waste. in particular, in the youngest years a lot of what the kids are doing is ‘growing older’. That is what is required for their brains to handle the next critical learning skills.
Much more than 4 hours a day of learning is squandered on diminishing returns. The Cambridge Handbook of Expertise and Expert Performance suggests that 4 hours per day of deliberate practice (7 days a week for 10 years) is a good approximate guide for how to gain world-class expert level performance in a field. It is remarkably stable across many domains.
The children’s social lives are not dominated by playground politics and are not essentially limited to same age peers.
Not only are the extracurricular activities physically healthier than more time wasted in classes they are better for brain development too. What is the formula for increased release of Neurotrophic Growth Factors, consolidation into stable Neurogenesis and optimized attention control and cognitive performance? Aerobic exercise + activities requiring extensive coordination + a healthy diet including adequate Omega-3 intake. That’s right. Spending hours swimming, surfing and catching fish with spears is just about perfect.
I agree, and this is a tragedy in that it makes it so students don’t have marketable skill by 14 as they would in an apprentice system, and so are dependent on mommy and daddy. This “age of genuine independence from parents” is increasing all the time, and there’s no excuse for it. It disenfrachises children more than any legal age restrictions on this or that.
Well above average mathematical ability and cannot do calculus to the extent of understanding rates of change? For crying out loud. You multiply by the number up to the top right of the letter then reduce that number by 1. Or you do the reverse in the reverse order. You know, like you put on your socks then your shoes but have to take off your shoes then take off your socks.
Sometimes drawing a picture helps prime an intuitive understanding of the physics. You start with a graph of velocity vs time. That is the ‘acceleration’. See… it is getting faster each second. Now, use a pencil and progressively color in under the line. that’s the distance that is getting covered. See how later on more when it is going faster more distance is being traveled at one time and we have to shade in more area? Now, remember how we can find the area of a triangle? Well, will you look at that… the maths came out the same!
I teach calculus often. Students don’t get hung up on mechanical things like (x^3)′ = 3x^2. They instead get hung up on what
has to do with the derivative as a rate of change or as a slope of a tangent line. And from the perspective of a calculus student who has gone through the standard run of American school math, I can understand. It does require a level up in mathematical sophistication.
That’s the problem. See that bunch of symbols? That isn’t the best way to teach stuff. It is like trying to teach them math while speaking a foreign language (even if technically we are saving the greek till next month). To teach that concept you start with the kind of picture I was previously describing, have them practice that till they get it then progress to diagrams that change once in the middle, etc.
Perhaps the students here were prepared differently but the average student started getting problems with calculus when it reached a point slightly beyond what you require for the basic physics we were talking about here. ie. they would be able to do 1. and but have no chance at all with 2:
I’m not claiming that working from the definition of derivative is the best way to present the topic. But it is certainly necessary to present the definition if the calculus is being taught in math course. Part of doing math is being rigorous. Doing derivatives without the definition is just calling on a black box.
On the other hand, once one has the intuition for the concept in hand through more tangible things like pictures, graphs, velociraptors, etc., the definition falls out so naturally that it ceases to be something which is memorized and is something that can be produced ``on the fly″.
A definition is a black box (that happens to have official status). The process I describe above leads, when managed with foresight, to an intuitive way to produce a definition. Sure, it may not include the slogan “brought to you by apostrophe, the letters LIM and an arrow” but you can go on to tell them “this is how impressive mathematcians say you should write this stuff that you already understand” and they’ll get it.
I note that some people do learn best by having a black box definition shoved down their throats while others learn best by building from a solid foundation of understanding. Juggling both types isn’t easy.
That is close to saying “this stuff is hard”. How about first showing the students the diagram that that definition is a direct transcription of, and then getting the formula from it?
(Actually, many would struggle with 1. due to difficulty with comprehension and abstract problem solving. They could handle the calculus but need someone to hold their hand with the actual thinking part. That’s what we really fail to teach effectively.)
People get the simple concepts mixed together with a bunch of mathy-looking symbols and equations, and it all congeals into an undifferentiated mass of confusing math. Yes, I know calculus is actually pretty straightforward, but we’re probably not a representative sample. Talk with random bewildered college freshmen to combat sample bias. I did this, and what I learned is that most people have serious trouble learning calculus.
Now, if you want to be able to partially understand a bunch of physics stuff but you don’t necessarily need to be able to do the math, you could probably get away with a small subset of what people learn in calculus classes. If you learned about integration and differentiation (but not how to do them symbolically), as well as vectors, vector fields, and divergence and curl, then you could probably get more benefit-per-hour-of-study than if you went and learned calculus properly. It leaves a bad taste in my mouth, though.
When taught well the calculus required for the sort of applications you mentioned is not something that causes significant trouble, certainly not compared to vector fields, divergence or curl. By taught well, if you will excuse my lack of seemly modesty, is how I taught it in my (extremely brief—don’t let me get started on what I think of western school systems) stint teaching high school physics. The biggest problem for people learning basic calculus is that people teaching it try to convey that it is hard.
I’m only talking here about the level of stuff required for everyday physics. Definitely not for the vast majority of calculus that we try to teach them.
Aw, please ? I’d be interested in hearing about the differences with other systems :)
It has been said that democracy is the worst form of government except all the others that have been tried. --Sir Winston Churchill
I’m not quite going to make that analogy but I will hasten to assert that there are far worse systems of education than ours. Including some that are ’like ours but magnified”.
In terms of healthy psychological development and practical skill acquisition the apprenticeship systems of various cultures have been better. Right now I can refer to the school system on one of the Solomon Islands. The culture is that of a primitive coastal village but with western influences. Western teaching materials and a teacher are provided but occurs in the morning for 4 hours a day. No breaks are needed and nor is any pointless time wasting. The children then spend their time surfing. But they surf carrying spears and catch fish while they are doing it.
What appeals to me about that system is:
The shorter time period.
Most of the time kids spend at school is a blatant waste. in particular, in the youngest years a lot of what the kids are doing is ‘growing older’. That is what is required for their brains to handle the next critical learning skills.
Much more than 4 hours a day of learning is squandered on diminishing returns. The Cambridge Handbook of Expertise and Expert Performance suggests that 4 hours per day of deliberate practice (7 days a week for 10 years) is a good approximate guide for how to gain world-class expert level performance in a field. It is remarkably stable across many domains.
The children’s social lives are not dominated by playground politics and are not essentially limited to same age peers.
Not only are the extracurricular activities physically healthier than more time wasted in classes they are better for brain development too. What is the formula for increased release of Neurotrophic Growth Factors, consolidation into stable Neurogenesis and optimized attention control and cognitive performance? Aerobic exercise + activities requiring extensive coordination + a healthy diet including adequate Omega-3 intake. That’s right. Spending hours swimming, surfing and catching fish with spears is just about perfect.
I agree, and this is a tragedy in that it makes it so students don’t have marketable skill by 14 as they would in an apprentice system, and so are dependent on mommy and daddy. This “age of genuine independence from parents” is increasing all the time, and there’s no excuse for it. It disenfrachises children more than any legal age restrictions on this or that.