Math Hombre

What makes a good project? Teachers argue over how much they should be predetermined or up to student direction, the difference between problem-based learning and project-based learning and other aspects.  The summer capstone course I just finished teaching had an opportunity for maximum openness. It had the context of the history of mathematics, so any mathematical topic is fair game. One of my weaknesses as a teacher is not giving students enough structure - I'm so interested in what they'll do with freedom that I provide more than many want.

The condition of learning that this connects to the most, for me, is employment. Brian Cambourne explains employment:

Employment. This condition refers to the opportunities for use and practice that are pro- vided by children’s caregivers. Young learner-talkers need both time and opportunity to employ their immature, developing language skills. They seem to need two kinds of opportunity, namely those that require social interaction with other language users,  and those that are done alone.
    Parents and other caregivers continually provide opportunities of the first kind by en- gaging young learners in all kinds of linguistic give-and-take, subtly setting up situations in which they are forced to use their underdeveloped language for real and authentic pur- poses. Ruth Weir’s (1962) classic study of the presleep monologues of very young children is an example of the second kind of opportunity. Her work suggests that young learner-talkers need time away from others to practice and employ (perhaps reflect upon) what they’ve been learning.
    As a consequence of both kinds of employment, children seem to gain increasing control of the conventional forms of language toward which they’re working. It’s as if in order to learn language they must first use it.

Brian Cambourne, Towards an Educationally Relevant Theory of Literacy Learning, Reading Teacher, v 49 n3, Nov 1995.

The project directions were minimal - instead I tried to communicate the idea in discussion, having the whole class talk about the kinds of things into which they might look, and who might be interested in that also. This worked pretty well

Project Possibilities: a project should show an investment of 16 or more hours. You might want to keep a log.

• developed mathematical writing on content of your own working
• historical profile of period in mathematics or of significant mathematician
• series of lessons that includes historical connection or context or connects significant math content to the Common Core.
• video or video series on any of the above
• mathematical art that explores any of the above

Since this capstone class had an emphasis on writing for an audience and sharing work, more of this is available to share than in a usual semester. So... here's some student work! Hope you enjoy it, and that it gives an idea of what the exemplars are about.

Several of the teachers in the class put together lesson plans or a unit. For example, Erika, Kyndra and Kelsey made a website, the 3rd Grade Brigade, with lessons and resources for the 3rd grade common core in mathematics.

Bre Zielinski and Jessica Bracey went the farthest out there. One got interested in the platonic solids and the other in tessellations so they tried to combine the two to make tessellated polyhedraa. Lots of neat photos of their results in what was definitely the most artistic project.

Jeff Holt investigated something near and dear to my heart as he tried to make a new statistic for studying Magic: the Gathering. I may have egged him on, but he was genuinely interested in studying this or World of Warcraft.  (He also did a history of the mathematician who invented Magic, Richard Garfield, for a weekly assignment.)

The project that had the most impact on their colleagues was this dandy from Ryan Garman and Joe Kargula. Ryan is a baseball coach at Grand Valley, and a former star player. He had the idea to dig into Sabermetrics and got some fascinating results:

• Final Paper 
• Runs-Created
• Pitching-Runs 
• All-Sabermetrics Team

If you enjoyed these you might be interested in the student-chosen exemplars from this same course.

One of the conditions of learning traditionally not well represented in math classrooms is approximation.  Not as a math practice (also traditionally under-utilized), but as set forward by Brian Cambourne:

"When learning to talk, learner-talkers are not expected to wait until they have language fully under control before they’re allowed to use it. Rather they are expected to “have a go” (i.e., to attempt to emulate what is being demonstrated). Their childish attempts are enthusiastically, warmly, and joyously received. Baby talk is treated as a legitimate, relevant, meaningful, and useful contribution to the context. There is no anxiety about these unconventional forms becoming permanent fixtures in the learner’s repertoire. Those who support the learner’s language development expect these immature forms to drop out and be replaced by conventional forms. And they do."

Brian Cambourne, Towards an Educationally Relevant Theory of Literacy Learning, Reading Teacher, v 49 n3, Nov 1995.

One of the reasons that article made such a huge impact on me when Dave Coffey first shared it was this idea of approximation.  It both supported some of the things I was trying to do in my classtime, and convicted me of many of my grading practices.  The grading structure in many of my classes involves the choice of exemplars.For example, the summer capstone course on math history I just finished teaching had daily work that was just to be done. Their choice, ungraded, noted for attempt. From that they chose, expanded or made anew some weekly work, which they submitted for feedback. Then at the end of the semester, they submit their choice for exemplars, with a short description of what makes it exemplary. We had four main themes in the class, and they submitted an exemplar for each:

• Doing Math
• Communicating Math
• History of Math
• Nature of Math

Mathy, no? In the end of term evaluations the students felt like we were strongest in class on history,  and weakest on the nature of mathematics. People were divided on whether some of what we did counted as doing math or not. 

Since this capstone class had an emphasis on writing for an audience and sharing work, more of this is available to share than in a usual semester. So... here's some student work! Hope you enjoy it, and that it gives an idea of what the exemplars are about.

Calvin needs some choice in his school work.

Jamie Paolino is probably more of the inspiration for this blogpost than any other, as she did a nice job presenting her work all semester. Here's two of her exemplars:

• Doing Math - Hypocycloids (Google doc)
  "What makes this piece exemplary is it displays my thought process and inquiry while working with hypocycloids and the student worksheet created with geogebra. I spent a substantial amount of time working on this weekly writing and discovered a lot about their creation with the use of a combination of variables. I think this type of work is often overlooked in the school setting because there is sometimes more of a focus on the finished product as opposed to the route that was taken to reach that final product and really having an understanding of something means more than simply being able to do it."
• Nature of Mathematics - What is an Axiom?
  What makes this work exemplary is my understanding of an axiom and the many roles they play in various proofs. I had a big misconception of axioms prior to investigating them further  and was able to clarify my misunderstanding. After researching more on this topic and looking at different examples the meaning of the term “axiom” started becmoming more clear and didn’t seem so scary as it once had.

Ros Rhodes - Desmos (All her exemplars are in one Google doc; this is the first.) Totally new tool to her, and she really got into exploring with it. HT to Daily Desmos for the class activity that helped engage students in exploring with Desmos online graphing calculator.

• Doing Math - "Why is this considered my best work?- A lot of mathematics is done through the use of observations. My experience with working with Desmos was an incredible experience to encounter and taught me a lot about how powerful hands-on computer programs are. With the hands-on experience that I have encountered with this program advanced my understanding of the relationships of how various functions operate with each other. I think with this work, not only was my work creative, but I was able to articulate how I created such a powerful piece of art using mathematics."

Erin Jurek - Rascal's Triangle (Google doc) One downside to having so much to cover is all the stuff we don't get to talk about in class. I like this as an example of a learner following up independently on something she found interesting.

• Doing Math - "I am using this piece of work as an Exemplar because I feel as though I explored this topic very deeply and I was able to bring myself into the work by actually doing the math that these students did in order to determine the next rows of Rascal’s Triangle. I really enjoyed reading about these students and the hard work they did in order to come with the diamond formula."

Luan Huynh - Chinese Numbers (Google doc) After discussing the development of Hindu-Arabic numbers pretty extensively in class, Luan got interested in Chinese numeration and I learned a lot from his work. Several students chose number systems explorations for a communicating exemplar, mostly about Mayan numerals.

• Communicating Math: "this can be an exemplar for math communication since it gives us an introduction to the Chinese number system, which allow us to understand how the Chinese learning and doing math."

Bri Zielinski - Modernizing Euclid. (Google folder of all her exemplars; this is the 1st.) Brianna took the proof from one of my all time favorite pieces of mathematics - Oliver Byrne's Euclid's Elements, 19th century full color visualization of  - and wrote it up as a modern written proof. (Her 4th exemplar is a quite nice essay on math as a language, also worth a read.)

• Communicating Math: "I consider myself pretty good at writing proofs, so this Weekly Writing kept my attention and focused my ideas."

 Milli Brown - What is Doing Mathematics? (Google doc)

• Nature of Mathematics: What is Doing Mathematics? "What makes my work exemplary is the way I described my journey to deciding what “doing math” means to me. I included my research, past experiences, and a summary of what I have arrived at for a definition of what “doing mathematics” is to me." 

Erika Bidlingmaier - What is Elchataym?   I developed a new appreciation for Leonardo of Pisa while preparing this course. Reading some of the Liber Abaci convinces you of his great place in mathematics.

• History of Mathematics: "In this writing I let a simple curiosity lead into a full-out study of the historical method of elchataym used by Fibonacci. Although I left it open at the end (and would have built a stronger piece if time permitted), I still exhibited my understanding of a very influential part of math's history."

Alyssa Boike - The House of Wisdom (Google doc) Our time spent studying Islamic mathematics seemed to make a big impression on students.

• History of Mathematics: "Week 3 is a good exemplar because I was able to concisely note a few of the most important people who worked at the House of Wonders and include their significance in the development of mathematics as a field. "

Anna Krivsky - Tessellations
I like her personal connections here and the nature of mathematics. Is making a tessellation a mathematical act? (Hard to choose between this and her Magic Square.)

• History of Mathematics:The following is exemplary of my learning about the history of mathematics during this semester because it discusses the historic development of a branch of mathematics that was new to me this semester: TESSELLATIONS. Furthermore, this work shows my ability to research about the history of mathematics.

If you enjoyed these, you might be interested in some of the student-directed projects from this same course.

I've been loving the posts people are writing for this year's virtual conference responding to the prompt: What is at the center of your classroom?. The conference is an amazing collection of writing put together by Riley Lark, blooger at Point of Inflection.  (Which I've always liked because it sounds - and reads - like he's synthesizing intention and reflection.)

This is an attempt to emulate my betters, which is actually not a bad strategy for improvement.

I want the center of my classroom to be empowerment. As a bad beginning teacher, emulating David Letterman of all people, I realized that I loved teaching math. I'd tell people that there weren't many things you could teach where the student would literally be able to do something at the end of the class that they couldn't do at the beginning.  While I still like that, I now think it can happen in  many more places than math class, and have a much different idea of what I want the students to be able to do.

On my own I got to realizing that problem solving was what I really wanted to teach, and my friend Sue Feeley introduced me to Polya (figuratively) and the other NCTM process standards.  Both helped give me language to describe things I had realized, and both indicated a path to set out on. Vygotsky helped me understand why students responded so differently to the same task.

Mosaic of Thought, Cambourne's Conditions of Learning and other literacy education work helped me start to understand how to teach processes, and I don't know that I would ever have found that were it not for Kathy and Dave Coffey. The conditions are the heart of what I want for my students, and creating or nurturing those conditions is what I see as the main responsibility of my work.

Engagement is first and foremost. Cambourne notes that engagement requires learners to believe:

• they are potential doers of what is being taught. This fits with and explains the idea that the students need to be the ones working in the classroom. (See also expectations)
• what they are doing matters to them. This is why teaching the processes is so important to me. Polynomial division does not matter to 99% of students. Problem solving will make 100% of my students' lives better.
• they are safe to try. This is why classroom community is so important. (See also approximation.)

Cambourne links engagement with two conditions:

• Immersion. Learners need to experience real and rich mathematics of all kinds.  Still one of my measures of how rich a question is is to consider how many processes it invites learners to engage in.
• Demonstration. Learners need many and authentic demonstrations of what they are learning. This was huge for me. I had become a hardcore discovery-based teacher. I was proud of my students saying things like "Why ask him, you know he won't tell us." (Oh, that is painful to me now.) I took students' feedback about their frustration and decided that it was because they weren't used to this mode of learning.  But really, I was asking them to do things they'd never seen.  Asking them to learn how to dance when they'd never seen one. Coupled with a lack of good feedback, it's a real testament to those learners that they got as much out of it as they did. Adding demonstration let me back into the discussion. Not to tell the learners how to do it, but to share with them authentically how I think about it.  I make space for them first, as I prefer if they're demonstrating to each other, but I watch and assess for when they need demonstration as a support.

The conditions that make engagement more likely are:

• Expectations. This goes hand in hand with the Equity Principle from the NCTM, which is near and dear to me. I believe all people can do significant, important mathematics. I really do. I try hard to communicate this to my learners.
• Responsibility. Learners make their own decisions about when, how and what to learn. THIS IS ANARCHY. I know. It's dangerous, especially when our learners have been trained for dependence and helplessness. Most students are not ready for full freedom immediately, but it is my goal for all of them. It's also my ongoing struggle to get students to understand that I both mean this and it doesn't mean that work is optional.
• Employment.  Learners need time to try out their learning in authentic situations. This connects with real life mathematics, with project based learning, with discovery learning and more. The students need to be the ones working if they are to be the ones learning.
• Approximation.  Learners need to be able to make mistakes without fear of punishment. If there's one area that math has completely screwed up on in the past, it is this. I do it, now you do it perfectly. This is crazy. We all know that no one learns anything important this way. And that the mistakes people make are crucial for learning in the first place.
• Response. Learners need real and meaningful feedback about what they are interested in working on. I now have my students put stickies on any work they turn in with what questions they have for me. Cambourne: "response must be relevant, appropriate, timely, readily available and nonthreatening." Grades are not feedback in this sense, and can only strive to be timely and readily available. (Although that does make grades better so far as it goes.)

Though I have made progress in creating the conditions, I have a long way to go.  (Although that implies an end to the journey and there may not be one.) They help determine what I teach, what my classroom is like, what I want to know about the students and their learning and how I evaluate their work.  That I would not have found them without colleagues in my professional learning community speaks to the heart of why that is important. So thank you, to Sue, Dave and Kathy, for helping me get this far.

I had the good fortune to win a bet recently (well, best 2 out of 3) by the performance of the Yukon Huskies (jk) in the 2011 NCAA Division I men's Basketball Tournament.  My prize? A guestpost from Dave Coffey, @delta_dc.  (I was actually rooting for Butler, but that's a quality consolation prize!) This is Dave with Juneau.  Juneau is asking, "How could you bet on bulldogs?  Have I taught you nothing?  Haw!"

A few weeks back, one of our teacher assistants said, “You just started on Twitter this semester. I never would have guessed.” I wasn’t sure if she was talking about my quantity or quality. I chose quality and explained that it could be traced to Cambourne’s Conditions of Learning (a Foundational Framework of our Teacher Assisting Seminar).

This reminded me that John, my co-teacher, had asked me about blogging about the Conditions. I turned to him and said, “I’m thinking about writing about how the Conditions of Learning helped me to communicate using Twitter.” I thought this would be a good example of authentic learning in action.

The teacher assistant chimed in, “Maybe you could describe each condition in a Tweet.” John laughed, understanding that she had issued me a challenge without knowing it. Well, “challenge” accepted…


[Note from John - t was very tempting to put this in twitter-typical reverse order... but that would make it less readable.  Please forgive the lack of verisimilitude.]







Thanks, Dave and Jim Calhoun! And Brian Cambourne, of course.  The Reading Teacher has put the article introducing the Conditions online for download.  Or you can read the whole story in his book The Whole Story.  Also, I put the date on the cartoon at '95, the date of the RT article, but 1988 would be more accurate.

Photo Credit: Kathy Coffey, Rosaura Ochoa @ Flickr