Geometric Landscape

I got shifted from my usual (of late) secondary student teacher supervision to elementary preservice teacher prep this fall. (We have an unusually low number of student teachers this fall.) I love this teaching, too, so it will be a treat. Pam Wells, David Coffey and Jon Hasenbank were already coplanning a revision to the course, so it gave me a chance to dive in and collaborate. And gave me my first chance to look in detail at the K-5 Geometry common core. So I thought I'd share what I saw:

First I collated them, then tried to look for a way to organize them more sensibly. They are pretty unevenly written. From vague generalities to hyper-specifics. The best common threads I saw were the action verbs about what the students were supposed to be able to do.

Our assignment was to sort them into a concept map or landscape of learning.  I'm very fond of the landscape of learning model for teachers. I first saw the idea in Fosnot and Dolk.  In addition to those Young Mathematicians at Work books, they are involved in the great Mathematics in the City project and the excellent curriculum Contexts for Learning Mathematics. Here's a sample chapter from the YMAW: Algebra book. This sample chapter from Contexts for Learning has a Multiplication Landscape of Learning (page 16).

A landscape emphasizes the many paths through understanding that students might take, and are loosely organized from bottom to top in terms of students development. (Read also Christopher Danielson on landscapes. Here's a landscape from years ago I developed with novice teachers for teaching money.)

Here's what I came up with. I'd love feedback on ordering from top to bottom, what you would add, and classification into strategies, concepts and models.

(Here it is as a PDF.)

There's things that are quite sophisticated present (hierarchical structuring, Van Hiele level 2 and level 3 reasoning) and very accessible things missing (motions, congruence and similarity). Even though they are not included, of course, you can still teach them; use those ideas to help students access the ideas that are required.

As I develop and revise activities for the course I'll be sure to share them. Again, if you have feedback about the landscape, shout it out!

Projects

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.

When They Work

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.