To get back on topic:
Jarvis323 said:
I think that a lot of people in this thread are missing the point of the OP's idea.
It's not like the idea is to teach them numerical methods for the sake of gaining a practical skill that they take with them along their education and career.
The idea is to give them a simple intuition about what differential equations are and what we do with them. The other side is that a simple hands on approach might introduce them to the subject in a way that is less scary, less abstract, and more fun. The possibility is that for some students this could inspire and motivate them to want to and not be afraid to get into physics, because they can wrap their heads around it to some extent to begin with. So the point is that it is an early course, rather than a later one. And the measure of success is more about the potential students subsequent confidence and interest.
Whether subsequent courses in numerical methods are redundant or will replace what was learned, is only of concern if the students end up deciding they want to be physicists.
Quoted for relevance. One cannot start early enough with exposing children to mathematical ideas; there is a joke often made about teaching a course in analysis in kindergarten but I cannot find the image. The key point to be made is that one cannot underestimate the value of having a true understanding of something compared to just being able to recall lifeless facts; mathematics is one of the best ways to spark such an understanding. This feeling of understanding is empowering for individuals and naturaly spreads across all domains of an individuals life: this is the true goal of any education.
atyy said:
George Jones and Dr Courtney have both taught Euler's method in high school. See post #3 and post #8. I agree that's a good idea. And you can see it's pretty standard in the introductory differential equations course that many physics and engineering majors take at university (it's Chapter 2 of Edwards and Penney, one of the standard texts; Boyce and DiPrima, another standard text, have it later, but mention early in the text that elements of the chapter can be taught early). If Euler's method is already commonplace in university and at least sometimes taught in high school (as George Jones and Dr Courtney relate), why is the OP's proposal a "coming revolution"?
The difference is that teaching this at university is already after having done a postselection, i.e. only the STEM students - those who are able to navigate the current education system successfully - will learn this, while teaching in high school all students learn this. The issue of what teaching method is best for which specific aims at a certain age range is essentially a classic epidemiologic problem of how to evaluate a new intervention versus a control intervention within the context of high school education; i.e. the problem is best decided by using a double blind placebo controlled randomized controlled trial, or a less ideal variant if the DBPCRCT is not realistic.
Having mentored hundreds of students, from high school level up to masters level, across dozens of fields, there is one and only one conclusion I have come to: mathematics and physics education in high school is typically quite abysmal. The paradox is that this is not necessarily because the textbooks or teachers are bad, but because both the textbooks as well as the teachers almost universally reliably fail to engage with all the students, i.e. this is to a large extent a marketing problem.
Take note that I am not considering the students who are already interested in mathematics or physics from a young age, but all students. For the students already interested in mathematics and physics, typically about 10% of any given classroom, the current system works quite well; for the other 90% things aren't usually so optimistic. The numbers may not be completely accurate, but we are literally talking about almost the entire population, apart from the few that end up in STEM jobs.
For most, dare I say for all, of the students for which the current system doesn’t work, they do not usually comprehend the impact not grasping elementary mathematics will have on their life; this is after having factored out those who do grasp this through their teachers, mentors or parents upplaying it and making clear that it will be necessary down the line in their careers, obviously if they want to choose a STEM career but actually pretty much independent of which career they end up choosing.
In very expensive private schools we somehow see this problem far less, i.e. somehow they have figured out a way how to engage all students, not just the ones who go onto do a STEM degree. How have they done this? Easy: by spending enormous amounts of money to let hugely influential charismatic public intellectuals - i.e. guest teachers such as Leonard Susskind, Hannah Fry or Brian Cox - come and convince their kids that STEM is awesome.
This educational strategy seems to work with these students, in multiple ways, namely they understand why science is important, they understand how it benefits them and society, they understand what mathematical literacy can do for them, they even understand the beauty of mathematics. This is revolutionary especially for those not going into STEM in that they typically end up having or wanting a strong mathematics background.
A digression: when I was in university, there were two girls majoring in psychology and law which I came across during my undergrad physics classes. Both of them came from two such private schools and both of them were taking complete course in calculus with the one a course in differential equations and the other mathematical methods for physicists. Coming from medicine, and therefore being an intruder in physics, I was extremely interested in their reasons for taking these courses; their answer was simple, like was mine: they understood mathematical beauty and wanted to expose the mathematical beauty within thrir own disciplines.
Can you imagine what an edge a psychologist - or literally any social scientist or scholar within the humanities for that matter - can have within their own discipline if they can create mathematical models at the same level of sophistication as a physicist, instead of relying purely on less than stellar undergrad statistics course to do quantitative research? Based on this experience, while still a student myself, I started mentoring students in all areas, teaching them not to be afraid of mathematics, but I digress.
To come back to the main question of how to engage high school students in mathematics, without necessarily spending millions of dollars on celebrity scientists to personally come dazzle the kids, it is almost obviously that a change to the curriculum is necessary. There have been large educational experiments in the past who have tried this and there are as I illustrated many smaller experiments which try this, but nothing as of yet in a truly integrative manner.
Garnishing an understanding for the average student - be it by demystifying something as relatively simple as differential equations from physics using something as simple as Euler's method - is a step in the right direction. Suffice to say the advocation of teaching black boxes is failing to see the forest for the trees; this is similar to arguing that all that is needed for one to be able to produce literature such as the works of Shakespeare is an understanding of English grammar.
! I suspect an upper-division course would be more useful for those students going to physics graduate school as well. The pedagogical benefits would have to be significant to prefer the freshman version. This likely means that the syllabi of the subsequent physics courses would need to change. I wonder what topics you propose to eliminate from each course to make room for this new numerical work? Or do you think it can be added without removing anything at all? I doubt it...