What Mathematicians Think

Earlier this month Jan Nordgreen at Caymath posted about an interview of a couple of professional mathematicians talking about their work. Here's one quote.

Isadore Singer: ... when I try out my ideas, I’m wrong 99% of the time. I learn from that and from studying the ideas, techniques, and procedures of successful methods. My stubbornness wastes lots of time and energy. But on the rare occasion when my internal sense of mathematics is right, I’ve done something different.

Another quote:

Michael Atiyah: My fundamental approach to doing research is always to ask questions. You ask “Why is this true?” when there is something mysterious or if a proof seems very complicated. I used to say — as a kind of joke — that the best ideas come to you during a bad lecture. If somebody gives a terrible lecture — it may be a beautiful result but with terrible proofs — you spend your time trying to find better ones; you do not listen to the lecture. It is all about asking questions — you simply have to have an inquisitive mind! Out of ten questions, nine will lead nowhere, and one leads to something productive. You constantly have to be inquisitive and be prepared to go in any direction. If you go in new directions, then you have to learn new material.

The full interview is right here.

These are two things I find myself constantly belabouring in my classes when teaching problem solving:

  • »Take risks! Experiment, play, try something out and see where it takes you. Good math isn't knowing what to do with any problem -- good math is knowing what to do when you don't know what to do. ;-)
  • »Ask questions! If you don't ask questions then I can't tell whether you understand or not. I'll either go on to something new, leaving you confused in the dust, or go over and over something you already understand trying to help you but really just wasting our time.

I'm going to cross post this one on all my classroom blogs. ;-)

Blogging Assessment

I'm working on adding another piece to my Rip, Mix, Learn blog on assessment. I found that I had to associate some kind of mark with blogging in order to really get my students to buy in. The thing was, I didn't want to make the parameters too rigid so as to encourage reflection and creative thought on the students part. So in the end there are two ways students get blogging marks in my classes.

(1) They have to write their own post at least once per unit. The post should be some kind of reflection on what they are learning. The parameters for this post are on the class blog (feel free to copy it verbatim if you wish).

The first question on every unit test they write is:
Did you blog at least once before this test as outlined in my post "Blogging on Blogging?"      Yes    No

It's easy to verify because every post is time and date stamped -- it's either in the web log or not. I give them the mark regardless of what they write as long as they have made at least some small effort to reply to one of the prompts in my post.

(2) One mark for each scribe post. The scribe works like this: Each day (except test days) one student writes a summary of what we did in class that day. That's it. They end their post by choosing who will be the next day's scribe. (The first student scribe was a volunteer.)

I update a post I wrote called The Scribe List each day to make sure that the same person isn't chosen twice before everyone has had a chance to be scribe first. (It takes about a minute each day to do this.) I also leave a permanent link to The Scribe List in the links menu in the sidebar.

Their final scribe mark will be the number of scribe posts they wrote divided by the number of times they were supposed to write such a post. They write some pretty amazing stuff! As I've written earlier each scribe has been trying to outdo the previous scribe; particularly in the grade 11 class.

I've recently begun posting thought provoking games on Sundays on each class blog. An idea I got from reading Jan Nordgreen's blog; Think Again! The idea is too expose them to a variety of problems in a low pressure environment that encourages mathematical thinking. They don't even realize they're learning as they play these games. I think one of the reasons this type of learning is so powerful is related to the instant feedback they get while playing; something that is difficult to do in a classroom setting.

Bud explored blogging assessment last year with the kids in his Blogging Experiment. One of the students, Elle, wrote a very thoughtful post on how blogs should be assessed.

I wonder how other teachers get their students to buy into the process of blogging. Do you also have to use marks to goad them into it? Do you assess their blogging? If so how? If not how do you get them to take advantage of the opportunities blogging represents? How do you assess student blogging?

Blog Genealogy

The Politburo Diktat has a post where he's building a family tree of bloggers. It's called "Who's Your Daddy?" It's got me thinking about the people who most influenced my foray into blogging and to whom I'm most grateful.

My blogmother is easy! It's Anne Davis. I can still remember the overwhelming feeling I had reading her post about my first blog. This later evolved into an exchange of email. Without a doubt, Anne was the most nurturing and encouraging blogger I've "met" in my blog lifetime. She still is; may she live to 120!

My blogchildren are easy too. Mrs. German started her own classroom blog following a workshop I gave. And one of my AP Calculus students, Sarah S, started her (most excellent) blog after being in my class for about a month. If I have any other blogchildren I don't know about them yet.

My blogfather is a little harder. I remember having my eyes opened wide by reading Alan Levine's Blogshop tutorial on educational blogs. But my blogfather, the male blogger who offered encouragement and commented on my work on his own blog, spurring me on to greater efforts .... that would have to be Will Richardson. So I guess I have two blogfathers, or maybe a blogfather and a bloguncle. ;-)

How could I forget my first blogchild?!? Terry Kaminski from the GCHS Math Blog. Sorry about that Terry. ;-)

RML After the Show -- Sheryl and Bud, Wow!

I just got home from my inservice. My part, Rip, Mix, Learn was in the morning only, from 9 to 11:30 am. I went to another session on project based learning in applied math in the afternoon. But my head has been reeling from the morning's session.

Sheryl and Bud, I can't thank you enough for your incredible contributions! I was only able to read your comments, on the blog and in the chatbox, now. I was so rushed during the workshop I just couldn't participate in or observe your part of things. I've got to tell you that I think the comments you made and the simple fact of your presence made it a success beyond my wildest imagination! You both made such insightful and helpful comments to the teachers in the workshop. And to answer one of Sheryl's questions -- yes, they were all math teachers. ;-)

Ok, so here are some lessons learned:

  • »It has to be a full day workshop next time -- it was way too rushed!
  • »Sheryl emailed me last night and suggested I skype the presentation. A brilliant idea! I definitely will next time.
  • »I talked too much and went through some of the links too quickly. I've got to slow down and give them more time to absorb what we're talking about and allow more time for discussion.
  • »"Visitors from afar" are another definite must! I'm so appreciative of the contributions made by Sheryl and Bud! They addressed some of the issues I just didn't have time for and, in a very concrete way, illustrated how blogs "bring the world into the classroom on a daily basis."
  • »I think some teachers were left wondering: "Where's the math!" Although I used several examples of concrete mathematics teaching and learning on the blogs later on we became so immersed in the technology that we lost sight of the pedagogy. It's hard to reconcile this one because, IMHO, to do this effectively teachers need a new kind of technological literacy. I need to be more explicit about that.
  • »I found myself saying, a few times, "I don't really care about the marks anymore. It's all about the learning." I'm growing -- feels good.
  • »Will has a post that I read last night where he talks about the difference between "push and pull learning." I found myself talking about that an awful lot today. It had a real impact on my thinking.

One of the people who attended my morning session, John, was in the applied math session with me in the afternoon. John was really excited by the things he had learned. We're now talking about forming a Manitoba Math Teacher Bloggers blog where we can share and discuss ways to assimilate and incorporate this technology into our teaching. We also talked about how to use del.icio.us and other social bookmarking tools to share resources. -- Very exciting stuff!! -- I'm also going to add a follow-up post to the workshop asking for teachers feedback whether or not we should have another session like this next year -- either something similar to this morning or a number of breakout sessions that run all day where we each get a chance to share our work and struggles with each other and try to figure out ways to improve teaching and learning. I'll try to have it up this weekend and I'll cross post it here and at Rip, Mix, Learn.

Lastly, for you "easter egg hunters" out there, I'm going to make the hidden links explicit if not today, before I go to sleep tomorrow. ;-)

Andrew Jackson and the number e

I just read this over at the brightMystery blog. A cool way to learn about e; especially if you're American. ;-)

He who wonders discovers that this in itself is wonder. -- M. C. Escher

Andrew Jackson and the number e

By robert on math

I'm going into isolation for the next couple of days to work on my CV and tenure portfolio, all of which is due on November 11. But I thought I'd give you something extremely clever and cool to think about, which I learned over the weekend. This is due to my colleague Bill Murphy, who teaches precalculus for us here. (And he attributes it to his wife, who is a high school social studies teacher.)

In mathematics, the number e is a constant used a lot in calculus and related fields. It is an irrational number, which means it has an infinite, non-repeating decimal expansion. (Here's a proof.) A 15-place approximation to this number is

e = 2.718281828459045

Here's how to memorize that 15-place approximation. First of all, memorize by rote the fact that the ones digit for e is 2. To memorize the 15 decimal places, we invoke the following perfectly square picture of Andrew Jackson:

Andrew Jackson was our seventh president. And he was elected in 1828. Using these facts, you can memorize the "7" and the "1828" in the decimal expansion. And if you remember that this is a square photo, you can remember to label both the height and the width of the photo with "1828", so you get two copies of "1828" in the decimal expansion. So now we have a way of remembering e = 2.718281828.

For the remaining six digits, remember that this is a SQUARE picture. So draw a diagonal:

The diagonal splits the square into two triangles. Think back on basic geometry. What are the angle measures in one of those triangles? You guessed it: 45-90-45. And those are the remaining six digits of the decimal expansion: 2.718281828459045.

So now you can amaze/impress/alienate your friends by spouting off the value of e to 15 places at will. And you learn a little about one of our Presidents in the process. Cool geek stuff, no?

[Update: If you start with the picture first, you don't have to memorize the ones digit of 2 by rote. Just remember that if you "square" something it means raising it to the power of 2, and there you go.]

Don't Throw Out the Powerpoint with the Blogwater

Since I posted about my upcoming presentation this Friday I've noticed people commenting how excited they are about using a blog as a presentation tool. I am too. One of the best posts I read about this is here. (If you don't read French copy the text and paste it here.)

People seem to be excited about having another way to give a presentation other than using powerpoint. Alan Levine and others have often commented that powerpoint presentations often degenerate into the presenter reading the slides -- we can read, thank you, no need to read it for us. But look at the way Laurence Lessig uses powerpoint in his lectures. This one has become a classic.

Laurence's presentation style is having a real impact on his audience. Watch this presentation on digital identity by Dick Hardt.

These dynamic examples show how to effectively use powerpoint to deliver a powerful message. They also make the prospect of doing a powerpoint presentation exciting again.

A Rip, Mix, Learn Invitation ....

I'm working on the Rip, Mix, Learn workshop I'll be giving this Friday. It occurred to me that when the teachers see the chatboxes on my classroom blogs they might like to play too. Well, this led to two more thoughts...

(1) I'll install the same chatbox on both the Rip, Mix, Learn blog and The Playground. That way, no matter where they're working in the course of the workshop they can talk to and help each other across the two blogs.

I thought this was a really cool idea. Really give them a taste of using the technology and learning from each other collaboratively.

(2) As I thought about this idea some more, I thought, "Darren, you're thinking too small. Think BIGGER. Collaboration is no longer dependent on space. Anyone anywhere can participate online if they're near a computer at the same time the workshop is going on.

It's scheduled to start at 9:00 am here in Winnipeg; that's central time. Is there anyone out there who would like to join in our fun? I mean, this'll really blow them away! Imagine, if even just a few people from different places join in the conversation by participating in the chatbox we'll have a real Rip, Mix, Learn Event with lots of Small Pieces, Loosely Joined. ;-)

By 9:30 am the participants will be into the blogging part of the workshop. The chatbox will be invisible on the Rip, Mix, Learn site until then. Then or shortly after would be a good time for anyone who's available to jump in on the chatbox. Of course, you're ALL invited! If you could let me know whether or not to expect you by leaving a comment on this post it will make the facilitating a little easier for me -- of course, you could just surprise me. ;-)

PS Has anyone found any of the "easter eggs" yet?

Bloglines - Visuals: Keep It Simple

I found this article at the "Eide Neurolearning Blog." I'm storing it here and I'll send it to all the teachers in my department on Monday.

This is also the first time I've used the "email to your blog and post it" feature of Blogger -- let's see how it works. ;-)

Eide Neurolearning Blog
Daily blog articles related to brain-based learning and learning styles, problem-solving and creativity, gifted and visual learners, dyslexia, attention deficit disorders, autism, and more.

Visuals: Keep It Simple

By Drs. Fernette and Brock Eide

Here's a study with practical implications - at Ohio State, researchers found that students were better able to learn a new mathematical system when symbols were plain and not visually distracting.

Said Dr. Vladmir Sloutsky, "Many teachers believe that concrete materials make learning more fun for students, and that will increase their motivation and help them understand the concepts,” he said. “While this may be true, in many cases, the concrete materials also interfere with what they are trying to learn.” A reduced version of the stimuli are shown below. The actual figures were apparently in color - to accentuate possible distracting factors.

The press release of the article is titled "Students Learn Better When the Numbers Don't Talk and Dance". The truth is really not really so simple, but still this study is a good reminder to avoid unnecessary distractions in the visual presentation of lessons. In some cases, movement or color can enhance the memory of the items presented, but its true, excessive detail to surface or other insignificant aspects of objects can lead to more confusion.

In the Sloutsky's study, the task demands were fairly high (artificial mathematical task for college students)so that simple symbols simplified the perception of categories and relationships, reducing distraction and demands on attention and memory.

This may be why many designers and inventors prefer to sketch generally when brainstorming or working with new ideas.

Simple Visuals Study
Students Learn Better When The Numbers Don't Talk And Dance

Man, This Technology is Empowering!

I found this in the chatbox on the AP Calculus AB blog tonight:

11:31 PM Oct 14, 2005 IP 77:203
hey guys. I just created a blog [link] Tell me your thoughts on it.

Follow the link -- I had nothing to do with this. It's all her idea.

It would be really motivating for Sarah and the rest of the class if someone outside our little group left her an encouraging comment. Hint. Hint. ;-)

Rip Mix Learn

It's been quiet here for the last little while because I've been working on a Rip, Mix, Learn presentation I'm giving to math teachers here in Manitoba next Friday, October 21. I've tried to make the presentation easily naviagable and mostly self-explanatory for those teachers that wish to take a non-linear approach to learning the material at their own pace. For those that wish to follow a guided tour I've tried to make it as interactive as possible by including lots of play time.

I'll be touching on four main web 2.0 technologies -- blogs, RSS, social bookmarking and flickr. At the end we take a look at the future and discuss the role technology can play in education. In the section on blogs I'm trying to give the the experience my students have when they first sign up as contributors to our group blogs. I've got a number of email accounts to give them and I've already sent invitations to our blogging playground. They'll sign up, respond to a Blogging Prompt, and write their first blog.

Here is the Rip, Mix, Learn workshop.

Here is the Rip, Mix, Learn Playground. The participants can only find the playground by picking up the email I've sent them.

For those of you that like treasure hunts I've included a few "easter eggs" in the workshop. These are invisable links to illustrative sites I'll be refering to. Can you find them all? I'll let you know by Friday how many there are. ;-)

You'll notice that I'm trying to squeeze a lot into about a 2 hour presentation.

The comments will be closed until the day of the workshop, Friday, October 21; but I'd love to get some feedback from you in the comments to this post. I've got a week to make any improvements. Any suggestions? ;-)

Update: read more here.