We actually believe in a lot of the same things. I was speaking much more specifically to the case of learning the cubic function. The method posted here doesn't generalize to all cubes, only perfect cubes, which is not an assumption you can infer from looking at a number. On a similar thread about an article applying heron's formula to squares, I was arguing for the opposite of what I am arguing here.
I tutor so I get a lot more freedom compared to typical teachers. Instead of cubes, I think time is better spent teaching students logarithms. I personally love teaching how to estimate log_2 values and helping build an intuition about mapping non-linear relationships. Log_2 and logarithms in general allows tracing math from the the first origins of compution ( and how "back in the day" they used log tables for everything) to modern technology storing data in binary form. On my high school's PUMaC A team, I found properties of logarithms way more useful than roots.
I completely agree that teaching number sense is critical but I think excel/spreadsheets and Fermi questions is more useful approach at a high school level. The math education system is broken because using a calculator is still seen as "cheating", and so excel is very rarely a part of the math curriculum. In my opinion, the average student who doesn't puruse a hard math or related field will appreciate the value of math more with an intuition built on logarithms. And those who do pursue math will have an easier time with number theory after having learned how to abstract from the decimal system to numeral systems in general. Fermi questions force students to think deeply about numbers without needing to understand high level math concepts.
I value your perspective, and I have shown project euler to certain high school students in the past. My high school has a very strong CS department, they have been running a 12 hour hackathon since 2017, that also MLH sanctioned/affiliated [1]. We ultimately care about achieving the same goal. I am very intrested to know some keywords I can look up related to your work, how you use roots and polynomials, and how/when you use cubes specifically.
> I am very interested to know some keywords I can look up related to your work, how you use roots and polynomials, and how/when you use cubes specifically.
Sure, happy to geek out a little. (FWIW, I’m not suggesting that this specific cube root trick is super useful, but I think this trick is generalizable in an abstract way, and representative of a lot of different techniques including any exponent, and also has elements that you can use for multiplication and division).
I work in computer graphics, ray tracing specifically, currently using GPUs, and perfect cubes come often up in volume rendering, for example. One little expression that’s reminiscent of this cube root article is how to calculate the number of total nodes in an octree (A cube subdivided into 8 children, and each child subdivided into 8, and so on.) Counting the first cube, and the next 8, then 64, down to N subdivisions, the total has some interesting variations. In binary, it’s a 1 every third digit (the article mentioned how the cube root trick is repeatable every 3 digits). The formula for number of octree nodes is ((2^N)^3 - 1)/7. The 7 in there can be unintuitive at fist, and I think it’s really instructive to see what the divide by 7 does in binary.
Another couple of applications of cube roots I’ve used recently are solving for the distance between a point and a quadratic spline curve, and solving for ray intersections with trilinear surfaces. Obviously these are continuous cube roots, not perfect cubes, but there’s a useful variation of Viete’s formula that made a few rounds on demo scene web sites. The idea that cube roots can be thought of as equally spaced points on the circle blew my mind a little. https://en.wikipedia.org/wiki/Cubic_equation#Geometric_inter...
BTW, is it still common to disallow calculators? My kids are generally allowed to use calculators for math homework and tests. I just assumed that was true most places now.
Thank you for those topics, they are interesting. Graphics programming has been something I've avoided because it's always felt too intimidating, but this feels like a nice place to start. Dynamic detail rendering built on octrees feels like a good way to handle the demands VR. I am not sure what benefits of ordering it as (2^N)^3 rather than (2^3)^N or just thinking of it as 8 all the time. Intuitively the inverse function will use log_8. I haven't come across Viete's formula before, I love these kinds of graphical solutions.
The use of calculators depends on two things, the kind of test and how accelerated a curriculum is. Standardized tests like the SAT have specific calculator and non-calculator sections. As of this year AP Chemistry will allow them during both the MC and FRQ, but that wasn't always the case. Macro and Micro both only allow a four fucntion calculator.
Whether calculators are allowed in classes depends on the teacher, but generally advanced/honors/ap classes emphasize learning to solve problems without one. Advanced classes focus on encouraging a more bottom up approach to learning how to solve problems, but in a class like Calculus, the CAS system in mainstream calculators make assessments trivial. On the AP and SAT exam, the NumWorks calculator is allowed and that's capable of running MicroPython 1.12. For my finals in AP Physics C, AP Chem, multi-var calc (in high school), I wasn't allowed a calculator. Similarly for all of my college math and cs courses, I wasn't allowed anything more than a four function calculator, despite taking the Matlab section of linear algebra and numerical analysis.
I feel frustrated that most schools fail at teaching students how to use calculators effectively. I get genuinely annoyed when students tell me they prefer a Ti-84 over a Ti-89, because it's "less confusing". Some teachers in my school district even encourage a Ti-84 for 90% of Calc BC and only talk about the Ti-89 for those handful of topics that require it. Very few teachers, let alone students, use the CAS system. I tutor in those classes every year, it hasn't changed.
I remeber I was asked by a teacher I had a good rapport with to give a presentation/demo for a handful of the other teachers during lunch. Having students graduate not knowing the "solve( f(x) = 0, x)" or how the calculator can do both symbolic and numerical integration is a failure of the education system. The TI Nspire II CAS not only has python, it has a color screen and can display animations, like turtle. If you jailbreak the Nspire with Ndless, you can even get micropython and there's an sdk that supports c and assembly to make apps. The newest Nspire OS doesn't allow the jailbreak though, and there's no way to downgrade. The only thing we used excel for in HS was making a graph or chart. I make sure my students leave with some profiency in excel, function nesting, cross sheet references, conditionals, and sorting at a minimum.
There's a difference in knowing how to use a calculator and being able to use the calculator to do what ever you want. It's the same as students knowing the steps to solve a specific math problem but not knowing why.
Oh hey my oldest got the Nspire, and did jailbreak it with Ndless! ;) She is geeking out at me while I type this with all the games she installed on it (DooM at FULL SPEED, on a calculator!!), and says the games on the calculator are the only reason she passed her math class. She agrees with you, they didn’t teach her how to use the calculator, and feels they should have.
I highly recommend diving into some graphics programming. For me, graphics is opposite of intimidating, and easily one of the best ways ever to motivate math. I love love love writing code that makes pictures & interactive graphics. It helps so much develop a more complete and intuitive understanding compared to the abstract symbolic pedagogy we got in math class. ‘Ray Tracing in One Weekend’ is a really gentle/easy introduction to ray tracing that can lead you down interesting paths depending on your motivation and interest. https://raytracing.github.io/ (Full disclosure, the book was written by a professor of mine, and I helped draw the whiteboard illustrations for the book.) One of the fun bits about ray tracing is it only requires high school math at most to get started, and you can go a long way on that alone, but if you want, you can go super deep on harder stuff, it’s both open-ended with research problems and forgiving & beginner-friendly at the same time. Graphics programming is a fabulous way to motivate matrices too. ShaderToy.com is another pretty easy way to get started, and has endless examples of neat tricks.
I tutor so I get a lot more freedom compared to typical teachers. Instead of cubes, I think time is better spent teaching students logarithms. I personally love teaching how to estimate log_2 values and helping build an intuition about mapping non-linear relationships. Log_2 and logarithms in general allows tracing math from the the first origins of compution ( and how "back in the day" they used log tables for everything) to modern technology storing data in binary form. On my high school's PUMaC A team, I found properties of logarithms way more useful than roots.
I completely agree that teaching number sense is critical but I think excel/spreadsheets and Fermi questions is more useful approach at a high school level. The math education system is broken because using a calculator is still seen as "cheating", and so excel is very rarely a part of the math curriculum. In my opinion, the average student who doesn't puruse a hard math or related field will appreciate the value of math more with an intuition built on logarithms. And those who do pursue math will have an easier time with number theory after having learned how to abstract from the decimal system to numeral systems in general. Fermi questions force students to think deeply about numbers without needing to understand high level math concepts.
I value your perspective, and I have shown project euler to certain high school students in the past. My high school has a very strong CS department, they have been running a 12 hour hackathon since 2017, that also MLH sanctioned/affiliated [1]. We ultimately care about achieving the same goal. I am very intrested to know some keywords I can look up related to your work, how you use roots and polynomials, and how/when you use cubes specifically.
1. https://thehacksb.github.io/