Multivariable calculus

Multivariable calculus

I need to do this again to be able to go any farther, I think.  Differential equations, too.  The OCW courses on topology and graph theory sort of offhandedly mention topics that I either never studied or have forgotten.  Anyone want to join me?

It’s not so much that I never took Calc III, it’s that I’m sure the MIT version is more…comprehensive than what I did.

Better School Days

For about 4 months, I’ve thought that there needs to be some kind of balancing force to keep modern ed “reformers” in check.  Even they themselves ought to recognize the importance of a debate where their propositions are not the only ones.  This is the value of competition of ideas.  But it’s all very one-sided right now.

For example, what group is contesting the Common Core State Standards?  There are individuals here and there expressing reservations, but many more groups have gotten on board without, as far as I can tell, any decision making process.  This is not to say that the CCSS are a bad thing, but the debate is important.  What group is contesting the notion that more hours over more days will improve the quality of education for students?

Unions are probably the easiest objection to this post.  But if you follow the links above, or pay any attention at all, you can forget about unions taking any kind of position here.  The exception is the CTU, which is where I got the phrase “better school day.”  Here‘s someone talking about better school days to give you an idea if you’re not familiar.

I’m probably not as informed as I think, but I’m working on it.  Any comments with information are appreciated.

Some things I’ve found since writing this post:

Micro rewards have weird effects on decisions

Micro rewards have weird effects on decisions

How would you implement micro rewards for wrong steps in a classroom?  The idea would be to either have students get savvy enough that they didn’t take the reward automatically or else learn from the experience of receiving a reward that was short-term good and long-term bad.  The easy answer is candy.  Like, “I’ll give you a piece of candy if you eliminate 13x instead of 2x” when 2x is the right thing to do.  The extension and real lesson here is that sometimes people are trying to get you to do something that’s better for them than you by offering you a short term reward.  This article is really about game design, but it has instructional implications.

It’s easy to see the implications of rewarding correct steps.  This is shaping.  But even that is questionable according to the paper in the linked article.  The paper concludes that when someone is rewarded for correct steps, their learning is not transferable to novel problems, only to identical or very similar problems.

Super Teacher Tools

Super Teacher Tools

Maybe the neatest site I’ve seen for whole class math games.

linear combination

At a meeting earlier this week, some of the other math teachers were saying that they didn’t teach their kids about the LCM or how to choose a variable to eliminate when solving systems by linear combination.  I was stunned.  Instead of using the LCM, they said they told the kids to always swap the x coefficients and multiply, being sure to make one equation positive and the other negative.  Let’s leave aside the sloppiness of the language here for a minute.  I can’t really imagine using an approach geared to the lowest level students.  It doesn’t seem fair to everyone else to present material that ignores what ought to be prior knowledge, and after all “Algebra 2” means something. 

But they went on to say that students picked up on the variations on their own.

So, this is a dilemma: should I present a severely dumbed-down approach, hoping that students will make the extensions themselves, or be more comprehensive in my presentation?

If I’m serious about students deciding what and how and how much they will learn, I have to be willing to take the risk of them being lazy.  So I did that today.  The thing is that they’ve seen this before, in Algebra 1.  So, I don’t know how it would have worked with a class that had not seen the technique before.  Luckily (hah) I have an Algebra 1 class too, and can try it there later this year.

curriculum choices

I’m trying to find ways to get my students involved in the curriculum that I teach them.  To that end,  I would like to ask this question:

Is there anything in your daily life that you think has something to do with math, but you don’t know what it might be?  Something that you’re wondering about?

The old way of using a question like this is to try and find matches between their interests and the planned curriculum.  That’s fine as far as it goes.  Teachers can also ask students about this kind of thing as the class progresses, which would be called “making it relevant to their world.”  I don’t think that goes far enough either, but at least it’s not me finding applications.

The danger here (danger!) is that the things they identify are over their heads.  That’s entirely possible.  So what’s the right answer to that?  You don’t know enough math yet.  Investigate that on your own (hah.)

And the challenge, if it’s completely unrelated to the planned curriculum, is to find time to address whatever part of these answers can be brought into their zone of proximal development.  Even assuming that they all have the same proximal zone is a huge error.

What to do…


What difference does it make if I’m more creative as a teacher?  There’s no guarantee that creative lesson planning leads to growth in creativity for my students.  I could use super creative methods to bring students along rails to an objective that I’ve designed, but it’s still me making the hard decisions for them.  If we’re interested in promoting creativity in students, we need to think about curriculum development.

Contrariwise, I could use the same approach every day with open ended tasks that allowed free form solutions, but those wouldn’t always go where I wanted them to.  So that’s not a problem unless you’re determined to control the outcome.  But that’s not how creativity works, right?  You don’t go in looking for a certain and specific outcome.  You don’t need to be creative to get an outcome that you can see in advance.