4 ms·

How would you teach mathematics or reading backwards?

gravypod 8y ago
"Backwards" is relative to the original way a subject was taught. My from my experience math is taught formula first rather than by starting with an intuition based on physical representation. Starting with developing an intuition like 3B1B and then deriving a formula would be "backwards" to how I've been taught in the past.
- tomjakubowski 8y ago
  Gilbert Strang's lectures on linear algebra work this way. He typically first gives a simple example (like for least squares, fitting a line to three points in two dimensions) and then over the course of the lecture susses out a broader generalization / formula from the simple example. Unfortunately, OCW seems to be having some kind of copyright issue with YouTube, but here's a link to the video anyway. https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-16-projection-matrices-and-least-squares/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
  - smallnamespace 8y ago
    I generally agree with this approach, but sometimes even the 'simple' example isn't intuitive or motivating to someone; it highly depends on your audience. E.g. a lot of math courses under my school's engineering curriculum would start off with examples from physics, but if you don't have the proper physics background or aren't interested in those applications, it's just another layer of abstraction to slog through. My high school stats textbook would use classic problems from medicine (e.g. x/n people sampled are sick, what bounds can we put on the distribution of how many people are sick in population?), but that might not be particularly motivating to someone who has no interest in medicine or why an epidemiologist might care to know the distribution. The best starting point to teach anything is the point that is closest to what the student understands and motivates them, but unfortunately that can vary a lot from person to person.
    - newsbinator 8y ago
      What you're saying makes sense and is one viewpoint. I could see it the opposite way too: In school, it's good for someone who doesn't know or care about epidemiology to get a slice of what would be useful to an epidemiologist. Likewise in reverse. The reason Macs have always had nice topography is because Steve Jobs happened to audit a calligraphy class taught by a monk (who stopped being a monk because he met a nice girl). I'm not banging the interdisciplinary drum- we all already know learning outside our interests is good. I'm saying it's probably a good educational tool to force students to look at problems based on their utility to someone else. Sometimes the further a problem is away from being directly useful to you, the wider your horizons will be once you understand how to solve it, and (more importantly) why to bother.
ljw1001 8y ago
math for mathematicians excepted, starting with an application that shows how to do something useful and working backwards seems pretty reasonable to me.
newsbinator 8y ago
One way to teach reading backwards is to start by creating a personal alphabet together with the student, from first principles. Make up symbols for various words. This gets cumbersome when you have words like peace, happy, glad, joyful, nice, etc. To solve for that, make up symbols for each sound. Then start writing new words by assembling the custom symbols you made. The problem you run into is that books/articles/internet aren't printed in your symbols, and no one you know can read them. Solve for that by introducing the (English/Spanish/Korean/etc) alphabet and the sounds it makes. Write simple words together. Then read simple words together. Then expand.

"Backwards" is relative to the original way a subject was taught. My from my experience math is taught formula first rather than by starting with an intuition based on physical representation. Starting with developing an intuition like 3B1B and then deriving a formula would be "backwards" to how I've been taught in the past.

tomjakubowski 8y ago
Gilbert Strang's lectures on linear algebra work this way. He typically first gives a simple example (like for least squares, fitting a line to three points in two dimensions) and then over the course of the lecture susses out a broader generalization / formula from the simple example. Unfortunately, OCW seems to be having some kind of copyright issue with YouTube, but here's a link to the video anyway. https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra-spring-2010/video-lectures/lecture-16-projection-matrices-and-least-squares/ https://ocw.mit.edu/courses/mathematics/18-06-linear-algebra...
- smallnamespace 8y ago
  I generally agree with this approach, but sometimes even the 'simple' example isn't intuitive or motivating to someone; it highly depends on your audience. E.g. a lot of math courses under my school's engineering curriculum would start off with examples from physics, but if you don't have the proper physics background or aren't interested in those applications, it's just another layer of abstraction to slog through. My high school stats textbook would use classic problems from medicine (e.g. x/n people sampled are sick, what bounds can we put on the distribution of how many people are sick in population?), but that might not be particularly motivating to someone who has no interest in medicine or why an epidemiologist might care to know the distribution. The best starting point to teach anything is the point that is closest to what the student understands and motivates them, but unfortunately that can vary a lot from person to person.
  - newsbinator 8y ago
    What you're saying makes sense and is one viewpoint. I could see it the opposite way too: In school, it's good for someone who doesn't know or care about epidemiology to get a slice of what would be useful to an epidemiologist. Likewise in reverse. The reason Macs have always had nice topography is because Steve Jobs happened to audit a calligraphy class taught by a monk (who stopped being a monk because he met a nice girl). I'm not banging the interdisciplinary drum- we all already know learning outside our interests is good. I'm saying it's probably a good educational tool to force students to look at problems based on their utility to someone else. Sometimes the further a problem is away from being directly useful to you, the wider your horizons will be once you understand how to solve it, and (more importantly) why to bother.

ljw1001 8y ago

math for mathematicians excepted, starting with an application that shows how to do something useful and working backwards seems pretty reasonable to me.

newsbinator 8y ago

One way to teach reading backwards is to start by creating a personal alphabet together with the student, from first principles. Make up symbols for various words. This gets cumbersome when you have words like peace, happy, glad, joyful, nice, etc. To solve for that, make up symbols for each sound. Then start writing new words by assembling the custom symbols you made. The problem you run into is that books/articles/internet aren't printed in your symbols, and no one you know can read them. Solve for that by introducing the (English/Spanish/Korean/etc) alphabet and the sounds it makes. Write simple words together. Then read simple words together. Then expand.