Comments on: Teaching math – oops! http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/ Education, Math, Teaching, New York, Bronx, Union, Language, Travel Mon, 16 Nov 2009 16:03:43 +0000 http://wordpress.com/ hourly 1 By: Carnival of Mathematics 1000 « JD2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-36363 Carnival of Mathematics 1000 « JD2718 Sun, 02 Mar 2008 04:01:45 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-36363 [...] 22 - And Vlorbik rails against publishers and their sloppy use of the √ (sqrt) symbol. Trust me, he takes this (-: seriously. 22 - Is Jackie’s class data normally distributed? Read about it at [...] [...] 22 – And Vlorbik rails against publishers and their sloppy use of the √ (sqrt) symbol. Trust me, he takes this (-: seriously. 22 – Is Jackie’s class data normally distributed? Read about it at [...]

]]> By: More on the roots of i « JD2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7444 More on the roots of i « JD2718 Sat, 17 Mar 2007 17:04:38 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7444 [...] pm Posted by jd2718 in Math Education, High School, mathematics, Math, Education. trackback In this post I mentioned looking for , and in this comment mentioned searching for . In this comment, reader [...] [...] pm Posted by jd2718 in Math Education, High School, mathematics, Math, Education. trackback In this post I mentioned looking for , and in this comment mentioned searching for . In this comment, reader [...]

]]> By: JBL http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7442 JBL Sat, 17 Mar 2007 01:52:06 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7442 The problem he's pointing out is that there are four fourth roots of 1, and no nice convention to distinguish between them. In fact, though, I think the complaint about the square root (or any rational power) on the complexes is silly: each number has two square roots, but it is absolutely possible to define a square-root function; the only "problem" is that it can't be continuous everywhere. The problem he’s pointing out is that there are four fourth roots of 1, and no nice convention to distinguish between them. In fact, though, I think the complaint about the square root (or any rational power) on the complexes is silly: each number has two square roots, but it is absolutely possible to define a square-root function; the only “problem” is that it can’t be continuous everywhere.

]]> By: jd2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7429 jd2718 Tue, 13 Mar 2007 23:19:31 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7429 Vlorbik, your comments are (to me) a bit cryptic. Can you explain the problem with the fourth root of i ? Vlorbik,

your comments are (to me) a bit cryptic. Can you explain the problem with the fourth root of i ?

]]> By: vlorbik http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7426 vlorbik Tue, 13 Mar 2007 22:18:04 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7426 please don't use the square root symbol so freely. \sqrt{-1} is slang; if it's to be made at all formal it should always have \pm ("plus or minus")-- for example, as in the quadratic formula. there simply isn't a function on {\Bbb C} that does what "\sqrt" does for the postitive reals. and it's about time we admitted it. \root4\of{i} ? unask the question (ask instead for solutions to x^4 = i --yes, i really do think the difference matters). please don’t use the square root symbol so freely.
\sqrt{-1} is slang; if it’s to be made at all formal
it should always have \pm (“plus or minus”)–
for example, as in the quadratic formula.

there simply isn’t a function on {\Bbb C}
that does what “\sqrt” does for the postitive reals.
and it’s about time we admitted it.
\root4\of{i} ? unask the question
(ask instead for solutions to x^4 = i
–yes, i really do think the difference matters).

]]> By: jd2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7408 jd2718 Sat, 10 Mar 2007 22:32:48 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7408 Nick, no, I had not considered mentioning the history of imaginary numbers, because I neither knew it, nor Cardano's method. I have printed your link, will study it a bit, and share appropriate parts with the kidlets. I will also, btw, share how I learned it. It is important for them to understand the collaborative nature of mathematical learning, and to keep in mind that I can't really be a teacher of mathematics unless I continue to be a student. Thank you. Nick,

no, I had not considered mentioning the history of imaginary numbers, because I neither knew it, nor Cardano’s method.

I have printed your link, will study it a bit, and share appropriate parts with the kidlets.

I will also, btw, share how I learned it. It is important for them to understand the collaborative nature of mathematical learning, and to keep in mind that I can’t really be a teacher of mathematics unless I continue to be a student.

Thank you.

]]> By: Nick http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7407 Nick Sat, 10 Mar 2007 22:13:48 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7407 jd, have you considered mentioning that, historically, imaginary numbers first came to mathematicians' attention through study not of quadratic, but of cubic equations? This arises when using Cardano's method to solve a cubic with three real roots: even though the roots are real, the formula expresses them in terms of imaginary numbers! This meant that mathematicians were forced to work with "imaginary" numbers, in contrast to an equation such as x^2 + 1 = 0, where they could instead declare that there was no solution. This situation is known as Casus Irreducibilis; see e.g. http://www.sosmath.com/algebra/factor/fac111/fac111.html jd, have you considered mentioning that, historically, imaginary numbers first came to mathematicians’ attention through study not of quadratic, but of cubic equations? This arises when using Cardano’s method to solve a cubic with three real roots: even though the roots are real, the formula expresses them in terms of imaginary numbers! This meant that mathematicians were forced to work with “imaginary” numbers, in contrast to an equation such as x^2 + 1 = 0, where they could instead declare that there was no solution.

This situation is known as Casus Irreducibilis; see e.g. http://www.sosmath.com/algebra/factor/fac111/fac111.html

]]> By: jd2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7395 jd2718 Sat, 10 Mar 2007 14:07:20 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7395 Karl, I love this approach. I use something similar in the introduction (which I abbreviated with this group, not knowing they hadn't seen it before) $latex 2 \times \square = 7 $ I write on the board. Before anyone can answer seven halves or three-point-five, I ask what a first grader would say. The idea that fractions extend the number system makes sense. Then I write $latex 4 + \square = 1 $. We have the same discussion. For a young enough student, this equation has no solution <i>in their number system</i>. So we extend the number system again. The game continues. $latex x^2 = 11 $ (which reminds me I owe the class a proof of the irrationality of $latex \sqrt{2}$) By this point a student can jump in without prompting that we used to think there was no solution, but we extended our number system to include square roots (side discussion of other irrational numbers). And then when I write $latex x^2 = -1 $ I can almost hear the little voice now: "Are we going to extend the number system again?" We forget, each one of these extensions is a minor trauma for the most thoughtful kids. I guess part of my overall concern here is that this class accepted the algebra of complex numbers without challenging their existence. That still worries me, even after seeing that most of the students easily manipulate a + bi. Karl,

I love this approach. I use something similar in the introduction (which I abbreviated with this group, not knowing they hadn’t seen it before)
2 \times \square = 7 I write on the board. Before anyone can answer seven halves or three-point-five, I ask what a first grader would say. The idea that fractions extend the number system makes sense.

Then I write 4 + \square = 1 . We have the same discussion. For a young enough student, this equation has no solution in their number system. So we extend the number system again.

The game continues. x^2 = 11 (which reminds me I owe the class a proof of the irrationality of \sqrt{2}) By this point a student can jump in without prompting that we used to think there was no solution, but we extended our number system to include square roots (side discussion of other irrational numbers).

And then when I write x^2 = -1 I can almost hear the little voice now: “Are we going to extend the number system again?”

We forget, each one of these extensions is a minor trauma for the most thoughtful kids. I guess part of my overall concern here is that this class accepted the algebra of complex numbers without challenging their existence. That still worries me, even after seeing that most of the students easily manipulate a + bi.

]]> By: Xanthir, FCD http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7394 Xanthir, FCD Sat, 10 Mar 2007 13:59:18 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7394 I came up with something else that might be useful for introducing complex numbers. Start by introducing the evil numbers. The basic evil number is e, non-evil numbers are called good numbers, and mixed numbers are called moral. They use all the same rules as complex numbers. The only difference? e^2 = 1. Once they've gotten all of this, reveal that they were just learning negative numbers, and imaginary numbers work in the exact same fashion except that i^2 = -1. I imagine being able to just draw a single line on the board to magically transform the evil into the imaginary. This should help overcome the mystique of complexes as something unreal or crazy. Definitely integrate what Karl is saying as well, where you emphasize that numbers are used to model things; they aren't real in and of themselves. Focus on the non-reality of negatives if you go this route to link up with the above. This should be easy enough, since major mathematicians didn't accept negatives as real until the 19th century! I came up with something else that might be useful for introducing complex numbers. Start by introducing the evil numbers. The basic evil number is e, non-evil numbers are called good numbers, and mixed numbers are called moral. They use all the same rules as complex numbers. The only difference? e^2 = 1. Once they’ve gotten all of this, reveal that they were just learning negative numbers, and imaginary numbers work in the exact same fashion except that i^2 = -1. I imagine being able to just draw a single line on the board to magically transform the evil into the imaginary.

This should help overcome the mystique of complexes as something unreal or crazy. Definitely integrate what Karl is saying as well, where you emphasize that numbers are used to model things; they aren’t real in and of themselves. Focus on the non-reality of negatives if you go this route to link up with the above. This should be easy enough, since major mathematicians didn’t accept negatives as real until the 19th century!

]]> By: Karl http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7392 Karl Sat, 10 Mar 2007 05:35:57 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7392 Haven't taught math for 30 years. But, my approach was - before introducing complex numbers, take them back to natural numbers, show that integers are artificial, created for the purpose of solving certain equations, give an example of such an equation, give a real life example of why they are useful, that the symbolism is arbitrary, and what these new numbers do to the number line Then do the same with rationals. when they are comfortable with the concept of inventing numbers to solve certain equations to arise normally, and that symbols are arbitrary, and that the number line can be expanded, give them an equation that can't be solved with a rational number (x^2=-1). INVENT a new number, propose some possible symbols for it, explain who first thought of it and why the particular synbol was used. Where does this fit on the number line? What uses does it have ? Then go to complex. If they UNDERSTAND, they will have a lot less trouble, will not be just manipulating symbols. Haven’t taught math for 30 years. But, my approach was – before introducing complex numbers, take them back to natural numbers, show that integers are artificial, created for the purpose of solving certain equations, give an example of such an equation, give a real life example of why they are useful, that the symbolism is arbitrary, and what these new numbers do to the number line Then do the same with rationals. when they are comfortable with the concept of inventing numbers to solve certain equations to arise normally, and that symbols are arbitrary, and that the number line can be expanded, give them an equation that can’t be solved with a rational number (x^2=-1). INVENT a new number, propose some possible symbols for it, explain who first thought of it and why the particular synbol was used. Where does this fit on the number line? What uses does it have ? Then go to complex. If they UNDERSTAND, they will have a lot less trouble, will not be just manipulating symbols.

]]> By: jd2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7390 jd2718 Sat, 10 Mar 2007 01:35:15 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7390 Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding $latex \sqrt{i} $ was not possible without significant help. I gave them but one problem this weekend: try to find $latex \sqrt[3]{i} $. Thanks e. Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding \sqrt{i} was not possible without significant help. I gave them but one problem this weekend: try to find \sqrt[3]{i} .

Thanks e.

]]> By: e http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7389 e Sat, 10 Mar 2007 01:22:49 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7389 \sqrt[3]{i}? \sqrt[3]{i}?

]]> By: jd2718 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7388 jd2718 Sat, 10 Mar 2007 00:45:41 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7388 Alon, LSquared, there are things I cannot prove at this level - the Fundamental Theorem of Algebra being one. Further, while some of my proofs look like detailed high school proofs, I drill that proof is "that which convinces," lowering or raising the bar, appropriate to the kids' experience with mathematics. For the FTA there will be no proof. There will be some observations that this seems to make sense, a number of examples, etc. Oh, and when I don't prove, I do so explicitly. They will know that they have gotten something without proof. I wish this class was more like my precalc from a few years ago, though. They weren't as strong overall, but one time I tried to introduce a formula (tangent of a sum, I seem to recall), and they absolutely rebelled. The claim was that they could look up the formula, and apply it by copying the models from the text. Class was for me to prove and for them to ask question. I can still see the girl scolding me. That's a proud, memorable, moment. Nick, these are high school kids, so they are only getting a taste of complex numbers here. My question about $latex \sqrt{i} $ was way out of curriculum. For them complex is a bit novel, and will give meaning to the FTA (which they will learn about, but not prove!) Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding $latex \sqrt{i} $ was not possible without significant help. I gave them but one problem this weekend: try to find [wanted cube root of i, but couldn't latex it. Tried latex \sqrt[\3]{i} .] Alon, LSquared,

there are things I cannot prove at this level – the Fundamental Theorem of Algebra being one. Further, while some of my proofs look like detailed high school proofs, I drill that proof is “that which convinces,” lowering or raising the bar, appropriate to the kids’ experience with mathematics. For the FTA there will be no proof. There will be some observations that this seems to make sense, a number of examples, etc. Oh, and when I don’t prove, I do so explicitly. They will know that they have gotten something without proof.

I wish this class was more like my precalc from a few years ago, though. They weren’t as strong overall, but one time I tried to introduce a formula (tangent of a sum, I seem to recall), and they absolutely rebelled. The claim was that they could look up the formula, and apply it by copying the models from the text. Class was for me to prove and for them to ask question. I can still see the girl scolding me. That’s a proud, memorable, moment.

Nick, these are high school kids, so they are only getting a taste of complex numbers here. My question about \sqrt{i} was way out of curriculum. For them complex is a bit novel, and will give meaning to the FTA (which they will learn about, but not prove!)

Finally, for all: I gave a practice lesson today, to make sure that all of this was solid. They had good control of basic operations, though finding \sqrt{i} was not possible without significant help. I gave them but one problem this weekend: try to find [wanted cube root of i, but couldn't latex it. Tried latex \sqrt[\3]{i} .]

]]> By: Alon Levy http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7386 Alon Levy Fri, 09 Mar 2007 19:42:57 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7386 There are purely algebraic proofs that only assume very elementary analysis (i.e. the intermediate value theorem), but require you to know various things about Galois theory. There are purely algebraic proofs that only assume very elementary analysis (i.e. the intermediate value theorem), but require you to know various things about Galois theory.

]]> By: Lsquared http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7383 Lsquared Fri, 09 Mar 2007 17:35:49 +0000 http://jd2718.wordpress.com/2007/03/08/teaching-math-oops/#comment-7383 What proof of FTA do you use? The one I like uses rather a lot of the geometry of complex numbers, and you need a feel for the re^(i theta) version of complex numbers to appreciate it. If you have a more elementary proof, I'd love to see it. What proof of FTA do you use? The one I like uses rather a lot of the geometry of complex numbers, and you need a feel for the re^(i theta) version of complex numbers to appreciate it. If you have a more elementary proof, I’d love to see it.

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