I think being able to understand how different people understand the same idea is essential to deep learning. That's why having mentors work with my students has been such a critical part of how I orchestrate their learning spaces; their blogs.

My classes have been enriched beyond measure by the thoughtful, and thought provoking, comments left my students by visitors to their blogs. Lani Ritter Hall (Ohio), Roland O'Daniel (Kentucky), and Emina Alibegovic (Michigan) have each pushed my students thinking in different ways. Lani, as a non-math educational consultant, has always asked them to dig deep into their learning styles and motivation for success. Roland, as a former math teacher, has focused on both the motivation and the specific content of what they write. And Emina, as a professor of mathematics at the University of Michigan has pushed them to dig deep and explore their mathematical (mis)understandings to better understand what they are learning.

Last semester Alec Couros and I collaborated to have two of his classes of student teachers (Regina, Saskatchewan) mentor my grade 10 Consumer Math class. The experience helped my kids grow as learners and Alecs' students grow as teachers.

Some of my students have been mentored for three years now, having taken all their high school mathematics courses with me. And they're starting to pay it all forward ... and backward.

I've blogged before about the excellent work two of my students, Grey-M and Mr SiWwy, are doing with Clarence and Barbara's classes in the Thin Walled Classroom. Now Mark has begun another unique mentorship project with my current Pre-Cal 40S (grade 12 pre-calculus) class.) (I'd be grateful towards anyone who left m@rk a comment over there.)

Mark took Pre-Cal 40S and has already earned his grade 12 credit. He did really well and wants to upgrade his mark so he's taking it again. All he really wants to do is challenge the final exam. We discussed it. He decided it would be a good idea to take the class again. He's taking Advanced Placement courses that conflict with the timetable for my class. We've worked out an arrangement whereby he can stay on top of his work without attending every one of my classes (all a repeat for him) so he can pursue his Advanced Placement courses concurrently. So Mark is mentoring his classmates, who he doesn't see every day, and he's doing some really amazing work!! Like Grey-M and Mr SiWwy, he's leaving comments on the class blog and archiving them with some reflections on a separate blog in order to aggregate all the content he's producing in one place. This allows me (and anyone else) to easily follow his work. Also, it will be a concrete archive of his learning and teaching that he can take with him when he leaves high school.

Recently, I've begun talking with Chris Lehmann (the principal at SLA) about having one or more of my students mentor one of the students at the Science Leadership Academy in Philadelphia who wants to pursue some advanced math on her own. Mark is the first student to volunteer to do this.

All these connections criss cross in all sorts of unexpected ways. It starts to look like network spaghetti:

Lani mentored my kids for almost three years. They pay it back by mentoring Clarence's kids and Lani starts a conversation with Grey-M and Mr SiWwy (my two student mentors) about effective mentoring in the course of their work with Clarence's and Barbara's kids. When Alec's student teachers come online and ask: "What does a good mentor's comment look like?" I point them to a comment Grey-M (my student) left one of Clarence's kids. So Alec's students learn the art of mentoring from my older students and pass that on to my younger (mathematically wounded) students all the while Lani and my two older students (Grey-M and Mr SiWwy) are engaged in this high level discussion (call it PD) about how to be effective mentors the results of which model good mentorship for the student teachers in Regina. Full circle.

And now m@rk is continuing this thread with his classmates on their class blog.

I suspect Al Upton will be having similar experiences; if he hasn't already. ;-)

Well this just made my day ...

Linger is a student in my AP Calculus class. She does some of the most exceptional work when she is scribe. She recently published her Developing Expert Voices project. Lani left her this comment:

Hi Linger!

What a great introduction!

"The questions that I chose are of interest to me, and I chose them to challenge myself. My goal for this project is to learn something new and to be able to teach you, the reader as well. Yes, becase "together, we can learn.""

I really appreciate your wanting to move out of your comfort zone. And from your reflection, "Now I am twice amazed, because I got to make the problems on my own and I got to understand it in a different point of view.", it was definitely worth the effort!

Your problems were of interest and your explanations clear and valuable! I personally really liked the music to accompany my learning; just the right touch -- helping to move calculus out of "gross" as was your intent! :-)

You mentioned that you learned math and about yourself from this project. Do you find one more valuable than the other and why?

Kudos to you on your learning and teaching!

My very best wishes,


Lani

Here is her reply ...

Hi Lani, thank you for your comment!

I find learning about math and learning about myself equally valuable. I'm a very competitive person and I saw this project as a competition where I was against myself.. and that is the best kind. Because it's not like being in a team, where people have to pull their own weight to keep things balanced. This is the type where you're all in or you're not. It's the kind where you know how well you did, and you've got no one else to blame.

Mr. K told us the other day that he made us do all these things (blog, project, other assignments) so that we learn in many different ways. It doesn't matter what you're learning as long as you know how. He wants us to become life-long learners, and I think, on my behalf anyway, he succeeded. I really value this project, and our AP calc blog. Mr. K's job is to teach, and he did it well.

This project taught me a lot.. yes, math was the main point but learning about myself was a double bonus ;)

<smile>

I've been slow to blog for lots of reasons. One of them is a feeling (self imposed and, as a result, particularly paralyzing -- does anyone else ever feel this way?) that there are things that I have to write about; but I just read something that I really want to share.

On Monday the first two Developing Expert Voices projects were published to the project blog. Two more will be published next week and the week following will see a steady stream of projects published until June 4.

Ivy was the second student to publish her project. If you look at it you'll see she worked hard on her presentation but struggled with the content. This led to her talking with me about ways she could go back and review the content she had difficulty with. Many students who struggle in math are happy to wait for a difficult unit to pass, put it behind them and forget about it. I think Ivy's renewed interest in relearning the content she had trouble with is a direct result of this project. The idea is to "teach other people and learn at the same time." They do this by creating their own problems and explaining the solutions as clearly as possible. The students are learning that making up good problems is no easy task. Writing well written solutions is also hard work.

One student, Graeme, told me he learned more doing this project than anything he's ever done before. Graeme published his project first. It uses text, graphics, a SlideShare slideshow, and several explanatory podcasts. Lani was struck by what he wrote in the Reflection section of his project and asked him about it:

Hi Grey-M,

The podcasts really help to contribute to understanding the content. Your voice, tone, and easy speak were a very good listen!

I am very interested in your reflection especially when you say:
"When I did the problems I knew exactly what I was doing but when explaining them the words escaped me. I almost think that it would have been worth doing journeyman level questions for the boost in the mark of annotation because I don't know if my answers are going to get through to everyone. I know what I mean but others may not. With an easier question that you have a stronger foundation in you'll be able to explain it much better."

This first question may be unfair-- Do you think you would have learned as much had you chosen journeyman level questions? Is this really all about the marks?

Do you think, now that you have experienced difficulty in explaining what you feel you really know (that has happened to me alot!) that perhaps you didn't know it as well as you thought? And that in composing an explanation you've really extended your understanding?

Best,

Lani

We're so fortunate to have Lani "in" our class. Not being a math specialist she focuses on helping the students stay focused on their learning. And she asks just the right questions in just the right way. ;-) (Emina, a professor of mathematics, has also been doing some great mentoring. In Vincent's scribe post the comments hold a great discussion pushing Vince to clarify his thinking again and again and again.) I was particularly struck by Graeme's reply to Lani, for several reasons ...

Hi Lani,

I must say that doing the harder level questions did by far extend my understanding but as the project description says it is not just our understanding, it is everyone who wants to learn the materials understanding that counts as the point of this is to assume the role of the teacher while teaching ourselves. Doing the expert questions taught me more but in doing so may have given the other people less in comparison to a journeyman question that would have given me less but with their explanations given others more. Then again if people get my explanations completely then the expert questions are the way to go because they'd be teaching yourself as much as the other person.

Graeme really groks the purpose and intent of the assignment. He goes on to contrast the value of learning with the value, validity and fairness associated with grading student work ... (my emphasis added)

As for your other question the marks are important to me but it is merely a byproduct of the hard work invested into the teaching of ourselves and others in this case as that is the main goal. If you don't achieve that goal you won't get the marks as it is based on how well you teach and apply concepts. (and a high mark in Mr.K's class is a badge of honor as he has the most wicked hard tests you can imagine as most questions aren't routine and truly test your understanding. A mark of say 70 in Mr.K's class is probably and 80 or 85 in other math teachers classes.). But I must say that marks do unfortunately matter. A University is going to choose the person with the higher marks if all the rest is equal (even though certain teachers are easier to pass then others so I don't see it as a fair process). In the end though I think that understanding trumps marks. I failed an identities test but I understand that concept completely, foolish errors caused by going to fast and the stress of the test. In the end that will get me farther than if I had fluked the questions and got a high mark without understanding what I did completely.

[I really didn't know I was considered such a "tough" teacher. ;-)] Here's a student who has gotten everything out of this assignment that I could have hoped and dreamed for all my students. He articulates the tension between learning and grading so well and concludes "that understanding trumps marks." He finishes by talking about a particular moment of clarity he had that came about not from creating a problem but from the act of explaining (teaching) the solution ...

Explaining the work in words and diagrams so that others can understand it made my understanding of it become rock solid. I may have known methods to merely complete them before but to be able to explain them is to be able to fully grasp problem and see the nuances and intricacies that you may not have seen before leading to a better way. Like the last slide on the velodrome question. I never saw that until I had explained everything.

As Mr.K would say... cheers!

Grey-M

I think we're going to do this sort of work again in my class. ;-)