6 ms·

For students in higher grades this may very well pay off. Especially in MCQ

knaik94 4y ago
If accuracy is important, students would be allowed to use a calculator. There is no longer practical value of being able to do this without one. So there's no point in testing it in school. Any teacher asking this kind of arithmetic question is wasting the time and energy of their students. I have never come across it in my last 10 years of tutoring high school or college math. If the abstract understanding of root properties is being asked, the question will be presented using variables. The cube root function is effectively equivalent to the square root function execpt the range includes negative values, when looking at it graphically.
- dahart 4y ago
  Are we talking about the practical value of actively learning math, or the practical value for an average high school student who barely tolerates math? Sure, yes, maybe most people can’t be bothered in practice, but isn’t there a lot of practical value in learning number theoretic ideas and seeing relationships between digits, for example, for the person bound for a job or graduate degree in a STEM field? Seems practical for people interested in math history as well, no? I’m a fan of some of the math problems on Project Euler. This isn’t high school math, of course, but many problems are carefully designed such that blindly using a computer to solve them just doesn’t work. This forces you to think about the problem more deeply and use that to come up with tricks that reduce the search space. This is very much along the same lines as this blog post, but generalized to bigger harder problems. And FWIW, stuff like this - a deep interest in math and how there are often many different ways to solve a problem, and the study of digits and numbers and polynomials and roots and efficient ways to manipulate them - is quite core to my job and all the people I work with. I do mostly software at a company that makes hardware and software, and number-theoretic thinking and fun & “impractical” tricks put into practice are literally everywhere. I guess I’m saying I think that there actually is a lot of practical value in this kind of play, for the engaged audience, and once someone goes beyond basic high school & college math, in addition to the benefits of increased motivation that you mentioned. I wouldn’t be so quick to write it off as pointless, and only teach to the test. Maybe if we had more of this in high school, we’d have a better math education system. I’d agree we need to teach all kids how to use calculators, but I also think we need to teach kids how to think deeply about numbers, and have the skills to build tricks and do mental estimation, and send more of them into science and tech jobs with exploration skills emphasized and prioritized over calculator skills.
  - knaik94 4y ago
    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. 1. https://thehacksb.github.io/ https://thehacksb.github.io/
    - dahart 4y ago
      > 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_interpretation_of_the_roots 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.

If accuracy is important, students would be allowed to use a calculator. There is no longer practical value of being able to do this without one. So there's no point in testing it in school. Any teacher asking this kind of arithmetic question is wasting the time and energy of their students. I have never come across it in my last 10 years of tutoring high school or college math. If the abstract understanding of root properties is being asked, the question will be presented using variables. The cube root function is effectively equivalent to the square root function execpt the range includes negative values, when looking at it graphically.

dahart 4y ago
Are we talking about the practical value of actively learning math, or the practical value for an average high school student who barely tolerates math? Sure, yes, maybe most people can’t be bothered in practice, but isn’t there a lot of practical value in learning number theoretic ideas and seeing relationships between digits, for example, for the person bound for a job or graduate degree in a STEM field? Seems practical for people interested in math history as well, no? I’m a fan of some of the math problems on Project Euler. This isn’t high school math, of course, but many problems are carefully designed such that blindly using a computer to solve them just doesn’t work. This forces you to think about the problem more deeply and use that to come up with tricks that reduce the search space. This is very much along the same lines as this blog post, but generalized to bigger harder problems. And FWIW, stuff like this - a deep interest in math and how there are often many different ways to solve a problem, and the study of digits and numbers and polynomials and roots and efficient ways to manipulate them - is quite core to my job and all the people I work with. I do mostly software at a company that makes hardware and software, and number-theoretic thinking and fun & “impractical” tricks put into practice are literally everywhere. I guess I’m saying I think that there actually is a lot of practical value in this kind of play, for the engaged audience, and once someone goes beyond basic high school & college math, in addition to the benefits of increased motivation that you mentioned. I wouldn’t be so quick to write it off as pointless, and only teach to the test. Maybe if we had more of this in high school, we’d have a better math education system. I’d agree we need to teach all kids how to use calculators, but I also think we need to teach kids how to think deeply about numbers, and have the skills to build tricks and do mental estimation, and send more of them into science and tech jobs with exploration skills emphasized and prioritized over calculator skills.
- knaik94 4y ago
  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. 1. https://thehacksb.github.io/ https://thehacksb.github.io/
  - dahart 4y ago
    > 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_interpretation_of_the_roots 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.