Comments for Continuous Everywhere but Differentiable Nowhere

Comment on Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me) by Jim Doherty

Jim Doherty — Thu, 31 May 2012 12:38:58 +0000

Actually, what I do when we first re-encounter the dividing by a negative rule I play kind of dumb and talk about how I am always bothered by negative coefficients. We take a few extra steps early on and add or subtract appropriately until my variable has a positive leading coefficient. I would always write 3 < x rather than x < 3 and this leads to some interesting discussions about how we read statements and what the number line tells us. In my first statement (3 < x) I tell them to look at the fact that x is to the right of 3 so the solutions correspond to numbers to the right of 3 on the number line. Conversely, if I write x < 3 then I note that the variable is to the left of 3 and the corresponding solution set is the numbers on the number line to the left of 3. They naturally tire of the manipulations and remember that the negative multiplier simply changes the order of the inequality statement. I don't know if this is any more lasting than remembering to switch the sign, but for some reason(s) it feels like a better explanation to me to tie in the order of the number line when discussing this. Thanks for pushing me into what I hope is a more coherent statement. I had been awake for less than 30 minutes as of my last post to you.

Also – thanks for offering to share. As we get closer to August I'll get in touch.

Comment on Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me) by samjshah

samjshah — Thu, 31 May 2012 11:13:24 +0000

Hi Jim,
Thanks for your thoughts. If your colleague doesn’t have material and it is a nonAP class, I’m always happy to share my materials (some good, some bad).

With regards to your random string of words (hahaha), I don’t actually get what you’re saying: “I always write my inequalities from left to right and refer back to the number line for them. So, I switched the old rule of changing the inequality sign to a more universal change the order of everything I write rule.” I think you’re saying that if you have x' title='3>x' class='latex' /> you always rewrite it as , and I FORBID THE USE OF MULTIPLICATION OR DIVISION, how would we get the solution. Some come up with a graphing way (test points on a number line), but many actually finally (after some thinking) decide to add x to both sides and subtract 2 from both sides, to end up with , instead of using that whole “trick,” all they should do is rearrange the inequality so that the x does not have a negative coefficient… so 5' title='4x>5' class='latex' />. Then they can go from there.

That may be what you were talking about… but I’m not sure…

Sam

Comment on Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me) by Jim Doherty

Jim Doherty — Thu, 31 May 2012 10:47:38 +0000

Sam
I love the question you posed with the student response. I had a colleague in NJ who had a series of problems regarding a character she called ‘Careless Carl’. She would present Careless Carl’s work on a problem and the students had to identify if he got the problem correct (rarely) or where his mistake in his work was. I think that what you were asking was a bit higher level but it feels similar to me. I have a new colleague coming in here in PA next year who will be teaching Calculus for the first time and I will be sharing with her many of the ideas I have been picking up from you and others in the blogging community. Regarding the last inequality. This might just be a hangup of mine, but I always write my inequalities from left to right and refer back to the number line for them. So, I switched the old rule of changing the inequality sign to a more universal change the order of everything I write rule. By referring to the negative sign as more of a negating sign I think that I have been able to help some kids see this rule as a bit more meaningful. Does that sentence make sense, or is it a random string of words that just makes sense to me…?

Comment on Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me) by Fawn Nguyen

Fawn Nguyen — Thu, 31 May 2012 05:03:42 +0000

This post is so good, Sam. I feel ripples of shame because I say I want my kids to write more and I say I hate the tests I give, yet I’m not really doing anything to change how I test. I work hard to bring more discovery-based learning to my kids (so I’m trying to be kind to self and not feel the tidal waves of shame), but boy you’re dead on about kids giving us the correct answer and not know squat what that answer means.

Ahhh, it’s like you read my mind when you added at the end about -3x < 6 ! I was thinking this while reading your post about procedural and algebraic regurgitation. And also I just learned last week a great way to teach this using the number line, so no more of this "rule" of switch the inequality sign when blah blah blah.

Thank you!! I'm going to be better next year by following your lead here.

Comment on Students communicating mathematics has opened my eyes to mathematical ugliness (and what that means to me) by Joshua Zucker

Joshua Zucker — Thu, 31 May 2012 03:58:36 +0000

I always ask on the first day of Alg 2 class “What does the quadratic formula tell you about the graph of a parabola?”

I learn a lot every time.

Comment on Senior Letter 2012 by cheesemonkeysf

cheesemonkeysf — Thu, 31 May 2012 03:49:25 +0000

Another year, another wonderful letter to your seniors. Bravo!

- Elizabeth (aka @cheesemonkeysf on Twitter)

Comment on Academic Integrity as part of Personal Integrity by Cheating « agktmte

Cheating « agktmte — Tue, 15 May 2012 22:49:27 +0000

[...] we push our kids to take Algebra 1 super young so they have nothing better to do later). I took Sam’s idea of putting an integrity statement at the end of the [...]

Comment on Make it Better: Drawing with GeoGebra by Integration Drawing Projects ’12 « Bowman in Arabia

Integration Drawing Projects ’12 « Bowman in Arabia — Tue, 15 May 2012 18:55:17 +0000

[...] wrote about this project back on Sam’s blog this summer when Sam gave me the reigns of his kingdom for a month or so, but I wanted to [...]

Comment on About by Daniel Minsky

Daniel Minsky — Tue, 08 May 2012 06:24:40 +0000

Hi Sameer,

Awesome blog! I am a graduate student in mathematics at UCSD and my roommate and I built an app for the iPhone that helps introduce binary numbers by way of a cute magic trick. We were hoping that you might be interested to share it with some of your colleagues, students and possibly readers. I’ve tried it out on college students and gotten some great responses. I also posted an entry about introducing it as part of a math lesson on my blog at bluetrapezoid.wordpress.com.

The app itself is available on the iTunes App Store at http://itunes.apple.com/us/app/birthday-genie-math-trick/id516139520?mt=8.

If you would like a free promo code just email me and I’d be happy to send you one.

Thanks for taking the time to read this!

Best,
Daniel

Comment on Reprise on Integration by Honing My Skills Instruction « Bowman in Arabia

Honing My Skills Instruction « Bowman in Arabia — Mon, 07 May 2012 21:16:00 +0000

[...] inspiration from a great worksheet from Sam, I wanted students to rely on their intuition at first to find andiderivatives, instead of relying [...]