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This is a sensitive subject for me. I was in a Linear Algebra class where I was asked on an exam to do, among lots of other things, a (nonzero) determinant of

This is a sensitive subject for me. I was in a Linear Algebra class where I was asked on an exam to do, among lots of other things, a (nonzero) determinant of a 6x6 matrix by hand. I got no credit for making an arithmetic mistake. I'm a CS major. I will never in my life (outside of school) be asked to solve a difficult math problem without having a ridiculously overpowered tool like Mathematica in front of me. I'm all for teaching hard math to students. But we need to be teaching them how to do things their calculator can't (or better yet) teaching them how to build the calculator. Not teaching them how to add 600 numbers by hand.

hansen 17y ago
You use the same formula to explicitly calculate the determinant in abstract proof. How can a student understand that, for example, the volume form on a riemannian manifold is invariant under a change of the chart, w/o knowing how to explicitly calculate a determinant? We all hate cumbersome and (at a first sight) useless calculations, but it's the easiest way to become familiar with the math behind it.
- tome 17y ago
  You really ought not to use the formula for anything other than calculation. You should characterise the determinant as the highest order form in the Exterior Algebra[1] and then use the fact that it's a group homomorphism so det(UAU^{-1}) = det(A) That's not to say that I agree with the GP. There's a lot to be said for testing the ability to perform abstract calculations with no practial use. A dimension 6 determinant seems completely unreasonable though. [1] http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality
  - ghurlman 17y ago
    It's posts like this that remind me how now, 10 years out of college in my software dev career, just how much math I've completely forgotten. I like to think I'd pick it up again quickly if need be - but that's the thing... I've forgotten it, because I've never needed it.
  - yummyfajitas 17y ago
    In an intro linear algebra class, I think a 6x6 determinant is legit. I asked one on my midterm. It was actually a very easy problem since the matrix was about 2 row reductions steps away from being upper triangular. Unless the professor is evil, "compute this 6x6 determinant" is really asking "do you know the properties of determinants which will simplify this problem?"
- drewcrawford 17y ago
  My point was not to say that familiarity with the determinant formula was worthless. Potentially, with an infinite amount of time and scratch paper, etc., doing the determinant of one 6x6 in a homework setting might be a worthwhile activity. However, presenting a nonzero nonsparse 6x6 as a "break" between two formal proofs on a large timed exam is not my idea of the level of arithmetic agility that should be expected of an undergrad CS student. No doubt there are great insights to be gained from application of the determinant formula. One that occurred to me at the time was "Computationally, what is the best order to perform these operations to get a numerically exact result? What is the time complexity of a perfect algorithm to determine the best-exact order? Is there a faster approximation that still provides good results?" etc. Unfortunately, I never pursued any of these arguably-more-relevant questions, because I was busy getting my rote multiplication up to speed. I never even thought of some of your applications, because I was too busy with nonsense to make connections. While this is a point example, it in many ways is representative of my math education.
babycakes 17y ago
I also find this a sensitive subject. I always thought that the learning process was held up by the need to constantly reinforce things I learned a long time ago. For example, electrical circuits classes at Purdue do not allow you to use calculators and must therefore make numbers that work out nicely to keep the math manageable. However, this means that your problems are not particularly realistic; 4.7 uF capacitors, 33 mH inductors, and 2.2 kohm resistors are far more common than 1/2 F, 1 H, and 1 ohm resistors. When I taught senior design, I was stunned to find students looking for super-huge capacitors, but that's all they had seen on their exams up until that point. Likewise, complicated FET sizing problems may require a complex system of equations. Students are short-changed by having to design circuits with only one or two unknowns so the math can be done by hand. On the other hand, mathematics courses are supposed to teach you how to solve these problems by hand. A TI-89 or Mathematica renders my first three semesters of calculus useless. However, those skills are necessary for understanding probability, electromagnetics, and all sorts of advanced material. Calculators with symbolic manipulation cannot possibly be allowed in those classes. Otherwise, students will have superficial understanding of all the dependent courses. IMHO, the only answer to this question is "it depends." We can explore much deeper levels of mathematics, science, and engineering if students aren't burdened with basic arithmetic, but we must ensure that students are only able to take shortcuts after they have demonstrated mastery of the material using manual computation.

hansen 17y ago

You use the same formula to explicitly calculate the determinant in abstract proof. How can a student understand that, for example, the volume form on a riemannian manifold is invariant under a change of the chart, w/o knowing how to explicitly calculate a determinant? We all hate cumbersome and (at a first sight) useless calculations, but it's the easiest way to become familiar with the math behind it.

tome 17y ago
You really ought not to use the formula for anything other than calculation. You should characterise the determinant as the highest order form in the Exterior Algebra[1] and then use the fact that it's a group homomorphism so det(UAU^{-1}) = det(A) That's not to say that I agree with the GP. There's a lot to be said for testing the ability to perform abstract calculations with no practial use. A dimension 6 determinant seems completely unreasonable though. [1] http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality http://en.wikipedia.org/wiki/Exterior_algebra#Functoriality
- ghurlman 17y ago
  It's posts like this that remind me how now, 10 years out of college in my software dev career, just how much math I've completely forgotten. I like to think I'd pick it up again quickly if need be - but that's the thing... I've forgotten it, because I've never needed it.
- yummyfajitas 17y ago
  In an intro linear algebra class, I think a 6x6 determinant is legit. I asked one on my midterm. It was actually a very easy problem since the matrix was about 2 row reductions steps away from being upper triangular. Unless the professor is evil, "compute this 6x6 determinant" is really asking "do you know the properties of determinants which will simplify this problem?"
drewcrawford 17y ago
My point was not to say that familiarity with the determinant formula was worthless. Potentially, with an infinite amount of time and scratch paper, etc., doing the determinant of one 6x6 in a homework setting might be a worthwhile activity. However, presenting a nonzero nonsparse 6x6 as a "break" between two formal proofs on a large timed exam is not my idea of the level of arithmetic agility that should be expected of an undergrad CS student. No doubt there are great insights to be gained from application of the determinant formula. One that occurred to me at the time was "Computationally, what is the best order to perform these operations to get a numerically exact result? What is the time complexity of a perfect algorithm to determine the best-exact order? Is there a faster approximation that still provides good results?" etc. Unfortunately, I never pursued any of these arguably-more-relevant questions, because I was busy getting my rote multiplication up to speed. I never even thought of some of your applications, because I was too busy with nonsense to make connections. While this is a point example, it in many ways is representative of my math education.

babycakes 17y ago

I also find this a sensitive subject. I always thought that the learning process was held up by the need to constantly reinforce things I learned a long time ago. For example, electrical circuits classes at Purdue do not allow you to use calculators and must therefore make numbers that work out nicely to keep the math manageable. However, this means that your problems are not particularly realistic; 4.7 uF capacitors, 33 mH inductors, and 2.2 kohm resistors are far more common than 1/2 F, 1 H, and 1 ohm resistors. When I taught senior design, I was stunned to find students looking for super-huge capacitors, but that's all they had seen on their exams up until that point. Likewise, complicated FET sizing problems may require a complex system of equations. Students are short-changed by having to design circuits with only one or two unknowns so the math can be done by hand. On the other hand, mathematics courses are supposed to teach you how to solve these problems by hand. A TI-89 or Mathematica renders my first three semesters of calculus useless. However, those skills are necessary for understanding probability, electromagnetics, and all sorts of advanced material. Calculators with symbolic manipulation cannot possibly be allowed in those classes. Otherwise, students will have superficial understanding of all the dependent courses. IMHO, the only answer to this question is "it depends." We can explore much deeper levels of mathematics, science, and engineering if students aren't burdened with basic arithmetic, but we must ensure that students are only able to take shortcuts after they have demonstrated mastery of the material using manual computation.