8 ms·

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different sh

DiffEq seemed like black magic to me when I took it as a freshman in college. They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. When I started a gamified music discovery company, I actually ended up using DiffEq to define a scoring algorithm that would produce continuously varying point values based on time series input data. I had to relearn how to do it, but the concepts made far more sense with a real application.

craig_asp 8y ago
Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".
- llamaz 8y ago
  I think the calculus of variations might be a better approach to introducing ODEs in first year. You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation). This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system. Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you've motivated ODEs and given the students are reason to solve ODEs with constant coefficients.
  - whatshisface 8y ago
    You don't need a Lagrangian to invent mechanical ODEs. You could talk about mixing tanks, objects under complicated forces, and so on with a lot less background information.
- cgriswald 8y ago
  We had a great professor and this was one of the most enjoyable classes I've ever taken. One particular assignment was a group paper where we were supposed to essentially explain and use the SIR model. We extended the model to an SIRZ model and effectively argued that zombie apocalypses in fiction are essentially impossible unless they include some supernatural elements. Under a wide range of assumptions (e.g., zombies rot/zombies don't rot) the infection always stopped before it spread significantly. (We got an A.)
  - Fomite 8y ago
    I use zombies in my epidemic modeling class - one of my favorite results is, with a semi-complex model, you can replicate the script of most zombie movies mathematically (lots of people die, the survivors take shelter somewhere and are safe for awhile, attrition starts to take hold, things collapse and then you're left with a small surviving fraction of protagonists at the end).
- jshowa1 8y ago
  You need to learn the math before applying it. You don't apply things that you don't understand.
  - acjohnson55 8y ago
    There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.
    - jshowa1 8y ago
      The problem is math is entirely abstract. So you must learn the abstraction before applying it. Otherwise, you learn how to add 2 + 3, but you don't learn how to add n + m.
      - acjohnson55 8y ago
        That strikes me as a strange way of looking at it. Most math taught to people who aren't pure mathematicians is taught precisely because of application to concrete situations. We value pure math largely because of the potential for future concrete applications. In case you aren't aware (and I wasn't until I trained to be a teacher, so this isn't meant to be condescending), there are alternative methods of teaching besides abstraction-first. See https://en.m.wikipedia.org/wiki/Inquiry-based_learning https://en.m.wikipedia.org/wiki/Inquiry-based_learning
        jshowa1 8y ago
        I'm aware. Examples are given to demonstrate abstract concepts. However, that doesn't preclude you from needing to learn the abstract concepts eventually. Otherwise, you go back to knowing times tables for 2's and 3's, but not n's and m's.
  - ruang 8y ago
    I need the concrete before the abstract just so my mind knows that what I’m seeing is not BS. Because in finance, 95% of the math is BS formulas that have no connection to reality.
freetime2 8y ago
"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks that they taught us to solve them. 10. TEACH CONCEPTS, NOT TRICKS What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality, and the justifications I have heard of it, citing poor preparation of the students, their unwillingness to learn, and the possibility of assigning clever problem sets, are lazy ways out. In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach. We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations; such information can nowadays be gotten in much better ways than sitting in a classroom. A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course. https://web.williams.edu/Mathematics/lg5/Rota.pdf https://web.williams.edu/Mathematics/lg5/Rota.pdf
amelius 8y ago
Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.
- hackandtrip 8y ago
  One of my professor used to say that you can do derivatives of every functions, it's mechanical, but you will only be able to find exact solution of an infinitesimal fraction of diffeq, and thinking about that, the same apply with integrals.
  - amelius 8y ago
    Yes the same applies to integrals. But when you go from 1d to 2d or 3d problems, the analytical approach quickly becomes too difficult.
- Cobord 8y ago
  Or you learn how to make the toy problem that has the same behavior as what you are trying to model. If you don't understand how to simplify problems, you end up just chucking the computation at it and calling it a day. If you have a solution to the basic problem you can give yourself a warm start.
- gh02t 8y ago
  You can do quite a lot simplifying to get equations that are solvable. A lot of engineering techniques do this, using simplifications that capture the essential behavior of a system (in fact I'd argue that knowing when such approximations are appropriate is one of the core skills of an engineer). That said, even if you need a numerical solution it will still often require a lot of simplifications in order to be tractable. Multiphase fluid flow, for instance, relies on tons of physics simplifications and empirical correlations in order to make numerical techniques viable.
- jshowa1 8y ago
  If you consider analyzing an RL/RC circuit as a "toy problem" then I guess you're right.
Cobord 8y ago
The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?
- _jal 8y ago
  Not the OP, but I've had good and bad math teachers. The bad ones tend to teach rote steps, "do this, do this, do this, done," without any attempt to explain why things work the way they do, without drawing parallels to already-learned things, without trying to teach any _why_. Then you hear students asking, "when am I ever going to use this?" My good teachers, on the other hand, always tied what we were doing into a larger scheme. If there were similarities or other relations between concepts, they'd be pointed out. If someone wasn't 'getting it', they had other ways of looking at it at hand, would sometimes give alternate methods of doing the same thing, etc. In short, one tells you to memorize in a vacuum for no good reason. The other helps you learn.
  - Fomite 8y ago
    This has a huge impact for me. I didn't get calculus. It was a disaster for me. Now I'm a mathematical epidemiologist. Why? Because someone introduced me to the grander scheme of things.
  - Cobord 8y ago
    But I would think the same applies to your algorithms teacher. There should be just as many good and bad ones there. Yet it seems different level of what is counted as too rote and unusable.
- acjohnson55 8y ago
  Depends on how they're taught, but to be honest, at a certain point in algorithms class, the subject matter also became tedious and abstract beyond recognition.
agumonkey 8y ago
I'd like to write a more algebraic understanding of differential equation, e, derivative and integrals. Something like Maxwell equations (pardon the pompous aspect) but for all things differentials. There's a webpage about differentiation that starts that (and is the inspiration behind my quest).
Sharlin 8y ago
> They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. This is why I just dropped off my DiffEq class. It was optional anyway, but when I go to a university level math class I expect insight, not rote memorization.

craig_asp 8y ago

Yeah, it's relatively dry material which is hard to grasp without the context of why we need it and how it can be applied in the real world. While the intro is good, it still has the same problem - just shows (in a good way) "some math".

llamaz 8y ago
I think the calculus of variations might be a better approach to introducing ODEs in first year. You can show that by generalizing calculus so the values are functions rather than real numbers, then trying to find a max/min using the functional version of dy/dx = 0, you end up with an ODE (viz. the Euler-Lagrange equation). This also motivates Lagrange multipliers which are usually taught around the same time as ODEs. They are similar to the Hamiltonian, which is a synonym for energy and is derived from the Euler-Lagrange equations of a system. Of course you would brush over most of this mechanics stuff in a single lecture (60 min). But now you've motivated ODEs and given the students are reason to solve ODEs with constant coefficients.
- whatshisface 8y ago
  You don't need a Lagrangian to invent mechanical ODEs. You could talk about mixing tanks, objects under complicated forces, and so on with a lot less background information.
cgriswald 8y ago
We had a great professor and this was one of the most enjoyable classes I've ever taken. One particular assignment was a group paper where we were supposed to essentially explain and use the SIR model. We extended the model to an SIRZ model and effectively argued that zombie apocalypses in fiction are essentially impossible unless they include some supernatural elements. Under a wide range of assumptions (e.g., zombies rot/zombies don't rot) the infection always stopped before it spread significantly. (We got an A.)
- Fomite 8y ago
  I use zombies in my epidemic modeling class - one of my favorite results is, with a semi-complex model, you can replicate the script of most zombie movies mathematically (lots of people die, the survivors take shelter somewhere and are safe for awhile, attrition starts to take hold, things collapse and then you're left with a small surviving fraction of protagonists at the end).
jshowa1 8y ago
You need to learn the math before applying it. You don't apply things that you don't understand.
- acjohnson55 8y ago
  There are certainly people who can learn that way, but it's not effective for most people. Most people have a limit to the amount of abstraction they can operate under before they need a tangible connection. Once that connection is made, most people can continue on to higher levels of abstraction.
  - jshowa1 8y ago
    The problem is math is entirely abstract. So you must learn the abstraction before applying it. Otherwise, you learn how to add 2 + 3, but you don't learn how to add n + m.
    - acjohnson55 8y ago
      That strikes me as a strange way of looking at it. Most math taught to people who aren't pure mathematicians is taught precisely because of application to concrete situations. We value pure math largely because of the potential for future concrete applications. In case you aren't aware (and I wasn't until I trained to be a teacher, so this isn't meant to be condescending), there are alternative methods of teaching besides abstraction-first. See https://en.m.wikipedia.org/wiki/Inquiry-based_learning https://en.m.wikipedia.org/wiki/Inquiry-based_learning
      - jshowa1 8y ago
        I'm aware. Examples are given to demonstrate abstract concepts. However, that doesn't preclude you from needing to learn the abstract concepts eventually. Otherwise, you go back to knowing times tables for 2's and 3's, but not n's and m's.
- ruang 8y ago
  I need the concrete before the abstract just so my mind knows that what I’m seeing is not BS. Because in finance, 95% of the math is BS formulas that have no connection to reality.

freetime2 8y ago

"Ten lessons I wish I had learned before I started teaching differential equations" is relevant here. I feel that DiffEq was the most useless undergraduate course that I took for my comp sci degree. They really didn't spend enough time going into the fundamental concepts so that I am not even sure I could recognize a differential equation if it were staring me in the face at this point... much less any of the tricks that they taught us to solve them. 10. TEACH CONCEPTS, NOT TRICKS What can we expect students to get out of an elementary course in differential equations? I reject the “bag of tricks” answer to this question. A course taught as a bag of tricks is devoid of educational value. One year later, the students will forget the tricks, most of which are useless anyway. The bag of tricks mentality is, in my opinion, a defeatist mentality, and the justifications I have heard of it, citing poor preparation of the students, their unwillingness to learn, and the possibility of assigning clever problem sets, are lazy ways out. In an elementary course in differential equations, students should learn a few basic concepts that they will remember for the rest of their lives, such as the universal occurrence of the exponential function, stability, the relationship between trajectories and integrals of systems, phase plane analysis, the manipulation of the Laplace transform, perhaps even the fascinating relationship between partial fraction decompositions and convolutions via Laplace transforms. Who cares whether the students become skilled at working out tricky problems? What matters is their getting a feeling for the importance of the subject, their coming out of the course with the conviction of the inevitability of differential equations, and with enhanced faith in the power of mathematics. These objectives are better achieved by stretching the students’ minds to the utmost limits of cultural breadth of which they are capable, and by pitching the material at a level that is just a little higher than they can reach. We are kidding ourselves if we believe that the purpose of undergraduate teaching is the transmission of information. Information is an accidental feature of an elementary course in differential equations; such information can nowadays be gotten in much better ways than sitting in a classroom. A teacher of undergraduate courses belongs in a class with P.R. men, with entertainers, with propagandists, with preachers, with magicians, with gurus. Such a teacher will be successful if at the end of the course every one of his or her students feels they have taken “a good course,” even though they may not quite be able to pin down anything specific they have learned in the course. https://web.williams.edu/Mathematics/lg5/Rota.pdf https://web.williams.edu/Mathematics/lg5/Rota.pdf

amelius 8y ago

Also, what they usually don't tell you is that the recipe bag only works for toy problems. For real applications you most often need numerical approximations.

hackandtrip 8y ago
One of my professor used to say that you can do derivatives of every functions, it's mechanical, but you will only be able to find exact solution of an infinitesimal fraction of diffeq, and thinking about that, the same apply with integrals.
- amelius 8y ago
  Yes the same applies to integrals. But when you go from 1d to 2d or 3d problems, the analytical approach quickly becomes too difficult.
Cobord 8y ago
Or you learn how to make the toy problem that has the same behavior as what you are trying to model. If you don't understand how to simplify problems, you end up just chucking the computation at it and calling it a day. If you have a solution to the basic problem you can give yourself a warm start.
gh02t 8y ago
You can do quite a lot simplifying to get equations that are solvable. A lot of engineering techniques do this, using simplifications that capture the essential behavior of a system (in fact I'd argue that knowing when such approximations are appropriate is one of the core skills of an engineer). That said, even if you need a numerical solution it will still often require a lot of simplifications in order to be tractable. Multiphase fluid flow, for instance, relies on tons of physics simplifications and empirical correlations in order to make numerical techniques viable.
jshowa1 8y ago
If you consider analyzing an RL/RC circuit as a "toy problem" then I guess you're right.

Cobord 8y ago

The algorithms you would learn for graphs etc are also just math and equally abstract. What makes the difference in how real you treat the two concepts that are both abstract but generally applicable?

_jal 8y ago
Not the OP, but I've had good and bad math teachers. The bad ones tend to teach rote steps, "do this, do this, do this, done," without any attempt to explain why things work the way they do, without drawing parallels to already-learned things, without trying to teach any _why_. Then you hear students asking, "when am I ever going to use this?" My good teachers, on the other hand, always tied what we were doing into a larger scheme. If there were similarities or other relations between concepts, they'd be pointed out. If someone wasn't 'getting it', they had other ways of looking at it at hand, would sometimes give alternate methods of doing the same thing, etc. In short, one tells you to memorize in a vacuum for no good reason. The other helps you learn.
- Fomite 8y ago
  This has a huge impact for me. I didn't get calculus. It was a disaster for me. Now I'm a mathematical epidemiologist. Why? Because someone introduced me to the grander scheme of things.
- Cobord 8y ago
  But I would think the same applies to your algorithms teacher. There should be just as many good and bad ones there. Yet it seems different level of what is counted as too rote and unusable.
acjohnson55 8y ago
Depends on how they're taught, but to be honest, at a certain point in algorithms class, the subject matter also became tedious and abstract beyond recognition.

agumonkey 8y ago

I'd like to write a more algebraic understanding of differential equation, e, derivative and integrals. Something like Maxwell equations (pardon the pompous aspect) but for all things differentials. There's a webpage about differentiation that starts that (and is the inspiration behind my quest).

Sharlin 8y ago

> They basically just taught us the recipe bag for solving equations in different shapes, but very little insight. This is why I just dropped off my DiffEq class. It was optional anyway, but when I go to a university level math class I expect insight, not rote memorization.