The pictures above hint at the impact the quad problem has had on my family.  My dog dragged one of my quad pages to the middle of the room.   I had cut out my quads and organized and glued them on different pages.  This was the top page of the one additional quad made from a total of sixteen points.  The dog chewed half of it off!  The dog seems to understand what has the focus of my attention, when she thinks it should be her!  She doesn’t usually chew paper but she has chewed the power cord for my laptop and the cord for the vacuum cleaner.  The second is something I found on the ground when my husband and I were going out to breakfast – it’s a quad! with the corners rounded.  It was outside a muffler shop.  I’ve been obsessed!  Driving home from the restaurant I began to notice quad patterns in the brick, roofing, and other tiling on buildings as well as in the wire mesh of a fence. Quads everywhere!

The quad problem became an information problem because it required organizing the data somehow to eliminate duplicates (and hopefully to identify missing quads).  If you are wondering what the quad problem is…  Given a grid of sixteen points in a four by four formation, how many unique quadrilaterals can you form?

I decided to also ask some Brown & Walter inspired questions about the problem.  In particular, what is the significance of this problem (or who cares?).  When I was reading about quads on the Internet I noticed the fact, “any quad tiles the plane” and so I began to wonder about using the problem to identify quads.  Suppose you had a piece of equipment to cut out tile pieces or create a textile pattern and you were limited to at least a one by one inch piece and at most a four by four inch piece, how many unique quads could you use?  When I did an Internet search on quadrilateral tiles, I discovered several technical articles that I didn’t understand but some were about mesh and one was about using quadrilaterals for cell phone areas.  So maybe those folks would be interested in the problem.  We have a piece of turtle shell at home – guess what the scales are quads!  Does nature use quads to tile the surface– maybe reptile scales or crystals.  This time I did an image search for reptile scales – snakes look like diamond scales but other scaly animals seemed to have a combination of quadrilaterals and pentagons.

I wasn’t sure if I should believe the statement that “any quadrilateral tiles the plane” so I tried a little trial and error with a few of my irregular patterns – it wasn’t always easy.  So I looked again on the Internet and found a cool site (Interactive Floor Tiles) that lets you create your quad, choose two colors, and then it creates a tessellation for you!  So now the quad problem had moved me into another area of mathematics.

All of this made me think about how cool the Internet is (A HUGE INFORMATION RESOURCE!) and what it means today for learners that we can almost immediately satisfy our curiosities in this way and make these connections for ourselves.  So the lovely quadrilateral problem became an information problem of another sort that relates in so many ways to my interest in learning, in technology, and in twenty-first century learning.  Using a question about the practical applications of a mathematics problem led me to more of the mathematics.  Questions are wonderful and having access to resources to explore these questions is truly amazing.

Sometimes there are wonderful coincidences.  Or you never know how something you learn in one context will be needed in another context.  Today I had an email from a teacher asking if the library had a video about tessellations.  We didn’t but I offered to look at United Streaming for one.  But I also went to her and asked if I could share my quad problem and what I had learned about tessellations including the cool website with her class.  And I remembered a lesson I had done with tissue paper triangles translated to form a star (like a quilt pattern) and offered to do that with her class as well.  The finished stars are translucent and look beautiful hanging in a window. And they will be so seasonal.  I’ll try to remember to post a picture when we finish them.

This week Lampert gives us another concrete example of the work that is required of teachers to “cover the curriculum” while attending to the different understandings of different students and attempting to move a class forward in their understanding of the “big ideas” in a discipline.  While she does contrast her teaching with the discrete lessons offered in the textbook, I wondered about how testing, essential questions, and principal observations of teachers also constrain the type of teaching that Lampert models.  She addresses the issue of someone coming into her class to observe a single lesson and not being able to see the underlying coherence that she is attempting to bring across a school year and a year’s curriculum.  I am in a school where teachers are subject to periodic visits from other administrators and cringe when they are criticized for not having their schedule posted (or veering from the posted schedule), for what is on their walls, and for whether the essential question is posted and obvious to the observer in the lesson.  At the same time, I really do worry that we teach mathematics from a textbook, or even from a state or district pacing guide without attempting to foster the type of deep understanding and connections promoted by Lampert.  Again and again, I see our students experiencing the difficulties she notes with parts of changing wholes and ratios.

My research interests, like Lampert and those promoted by Duckworth this week are as a practitioner.  As the school librarian, I straddle working with students and working with teachers.  While I am looking specifically at collaborative planning with teachers, I cannot help but zoom out to the bigger picture of our work with students and zoom into my teaching and listening to students.  Lampert’s model of examining her own practice has intriqued me since I encountered the idea in my first doctoral class.  This week I have really been thinking about using her method and presentation of the results as a model.  I have always felt that as a participant in my own research, I have an intimate knowledge that would be inaccessible to any outsider.  Yet, I need a way to access and communicate that knowledge.  I have really found it valuable to see how Lampert uses her journal (before and after a lesson) both to focus in on specifics, to make connections across specific incidents, and to provide a larger picture across a school year.

Math club is this small part of a school week.  It’s about a half an hour on a Friday afternoon.  Yet I am struck by the many topics we have already touched on and the insight I have gained about students and how they think.  Math club exists outside the required curriculum and pacing but because the participants (including me) are impacted by those structures, what we do in math club is both influenced by them and a potential influence on them.  Already in my planning with those teachers, I find myself talking about what we do in math club or discussing particular students and what I have learned about them.  The work I do in the school is part of a whole cloth.  I can’t examine one part without attending to the others.  The more of my context I can capture in my journal, the richer I think my particular analysis of the discourse in collaborative planning will be.

I really like the metaphor of crevices or cracks. If something is too smooth it will slip through our fingers but the more irregular or prickly surfaces are more likely to stick to us. And in the small cracks, we might find an opening or a connection that allows us access. Time and space have a continuum and our social worlds overlap and flow into each other. Why do we treat and teach disciplines as if they were discrete, self-contained and static when clearly the world does not stand still?

And this week I continue to struggle with how I want to connect mathematics and information literacy. There is an obvious connection that can be made with literature and mathematics. Duckworth talked about using a book to offer young students access to the difficult topic of conflict and war and Lampert also talked about using a story (of the Mimi) as a framework for mathematical problems. Numerous articles as well as a few books have been written about using literature in mathematics. There were many literature sessions and NCCTM and I noticed the booth selling children’s books was one of the busiest at the exhibits (just like at our library conference). Literature provides one of those textured contexts that offer numerous hooks or openings for learners to connect with and think about.

The literature connection is obviously one place where school librarians hold legitimacy in mathematical teaching and learning. I continue to want to push beyond that. Recently I have been reading about “quantitative literacy” and wondering what would we expect a literate person to understand about mathematics, or be able to do with mathematics? Make good decisions as consumers? Understand the uses and limitations of statistics? Be critical consumers of data? Enjoy patterns and geometry in nature, in art, in architecture? What should a literate person understand or be able to do with problem solving? Question the given? Notice the attributes of a problem? Organize data? Evaluate the soundness of a solution? consider multiple solutions? Why is there little convergence between my questions about mathematics and my questions about problem solving?

Math club this week was interesting. I did bring a bag of leaves. When I asked what can you do with these leaves related to mathematics, the first answer I got was counting. But when I asked count what, the answer was the points or the lobes on individual leaves which I thought was really interesting. I had crayons and paper and showed the students how to do a leaf rubbing and suggested that they use these rubbings to capture their ideas about using the leaves in math. Right away several did addition problems. One student had three leaves and two leaves and told me it represented three times two. But I said it looks like two plus three – how could you show two times three and someone in the group mentioned what we had been doing with arrays and so some students created arrays of leaves. Another quick answer was sorting and the first way students sorted was by color. Is there another way I asked and someone said shape. That spurred another student to think about congruent shapes and he chose two leaves that were the same shape and size to color. A few students started to create books using the leaves to illustrate math facts. Somewhere in the working, one student said we could read books. In math club, I asked? Books about math she suggested. At the wrap up I asked them what they would like to with math club — do you want me to find some hard problems for you to solve – an enthusiastic yes! I also tried to make the connection between leaves and the fractal patterns we had been cutting out – looking at repetitions that were smaller (and attempting like Lampert to create some coherence between our meetings). I also told them I noticed that several students had started small books – should we keep a scrapbook of math club? They liked this idea. Maybe math club needs a blog…

image by semaphoria http://www.flickr.com/photos/semaphoria/1564809955/

Just last week the AASL Standards for the 21st Century Learner were published by AASL. I am still working to absorb these since they will be important to my research. There are four over-arching goals for learners: inquiry, applying and creating knowledge, participating ethically and productively in a democratic society, and pursuing personal and aesthetic growth. Under each goal are listed skills, dispostions in action, responsibilities, and self-assessment strategies. I need to work each of these into my question about the relationship of mathematics and information literacy. I think there are many connections to make between inquiry and problem-posing. I especially like the inclusion of “dispositions in action”. Doesn’t that almost translate into “identity in practice?”

I had the recent “aha” that literacy (whether information, mathematical, or science) is about identity. When we talk about a literate individual we are talking about someone who takes a certain stance toward learning. I don’t think a literate identity is something that can be attained once and for all; rather it is a fluid relationship with a subject. Literacy is about continuing to learn, to question the given, and to contribute to practice. I read ahead this week in Lampert to see what she had to say about “Teaching Students to be People Who Study in School.” She really talks a lot about identity in this chapter particularly as it relates to the other identities that students have in the classroom that may cause them to find looking smart or good in mathematics as undesirable identities to strive toward.

I find it interesting that most of my math club students are currently girls. They seem to enjoy the social aspect of a math club. In fact when a couple decided to whisper to each other I made the statement, “We don’t whisper in math club. I just don’t want anyone to feel uncomfortable.” The girls seem especially drawn to the more artistic activities. In fact when I asked who wanted to do more with the paper folding and cutting the girls all said yes, and the one boy said no at first but then joined the group. I’ve tried to relate the activity to fractions and so we started with a 3 X 4 array and I asked them to separate it into two parts, then four parts, and then three parts and we talked about one half, one fourth, and one third – which was bigger? Then we went to the fraction number line and added thirds (fourths and halves were already up). We then went back to fold and cut out paper in thirds. I really wonder what the long term effects of this almost casual discussion of fractions will be with these students this year. Will understanding fractions become a part of their identities?

leaf image by antkriz http://www.flickr.com/photos/ananth/58136590/

Cendrillon hopes to go to the ball.  She has to wait for Madame, Vitaline and Papa to leave first.  They leave at 9 p.m.  It takes 30 minutes to find fruit for the carriage, 6 agoutis for horses, and a manichou for the coachman.  Getting dressed takes no time.  The ride to the palace takes 15 minutes.  If she must leave the ball at 12:00 midnight, how long can she dance?

I sent this word problem to third grade classes and asked that the students try to work the problem independently.  When they came to the library, we talked through the problem and I read the book, Cendrillon: A Carribean Cinderella, by Robert D. San Souci.  I kept the student work to look over afterward.  In my first class, most students added 9, 30, 15, and 12, or did some combination of addition and subtraction.  Two unique responses from the first class were the one student who just drew a picture of a clock with no work or answer, but at least he had the idea of a clock.  The other student started at 10:00 and wrote five minute increments starting at ten (so ten was five minutes, then 10:05 was ten minutes and so on until midnight) and got the answer 2 hours and five minutes which was really close and I thought showed a unique strategy.  I’m not sure how he was thinking about the time before ten oclock.  Overall from this first class what I noticed was that minutes and hours were being treated the same and in general, I felt like the students weren’t reading the problem carefully and thinking about their answer.  Answers like 54:00 don’t really make sense, do they?  I asked this teacher later if the problem was too hard and she said yes, there were too many words.

The second class worked in small groups.  One group tried to add 30, 15, 6, and 9.  I am assuming they got the number six from the agoutis needed for Cendrillon’s horses.  When you add these numbers you get 50 which they changed to 5:00. Another group just drew a clock without hands – no work or answer.  A third group added 39 (I assume they added 30 and 9 in their heads) to 15 and got the answer 54 minutes.  One group’s work has two clocks: one shows 9:45 and the other shows 12:00.  They show addition 30 +15 = 45.  Their answer says, “She has 3 hours (a 2 is erased) and 95 minutes (were they thinking 45 minutes??) to dance because if you use the clock.”  The last group shows a picture of a clock and the time seems to be 12:15.  Their answer is 3 hour and 15 minutes.  They wrote “by using math to are time.”  When I worked with this group, more students seemed to have some idea about how to work the problem as I think the work of the last two groups suggests.  What was interesting was the first student I called on told me he used addition to solve the problem.  When I asked him what addition, he said 9 and 12 and I asked about “adding up” from 9 to 12.  I’m not sure how much I prompted or he already had.  Maybe I should begin to ask to see student work before they come to the library so I can see where these misconceptions are.

I was under the impression that these students knew something about elapsed time but their approach to this problem concerns me.  I guess my greatest worry given other observations I have made is that students seem to perform an operation on the numbers without thinking through what the problem is asking, in other words without really reading and comprehending the problem.  Is this a reading problem and not a math problem?

Does my question about the connection between mathematical problem solving and information literacy touch on this issue?  If we want students who are mathematically literate and can solve real-world problems, then we need to contextualize mathematics and approach mathematical problems from a more holistic way.  I really don’t think the problem was too hard, and I certainly don’t think it had too many words.  Don’t we really want students to read and think through a problem like this? What do I know (what does the problem tell me and what do I know about problems like this?), what is the problem asking me?  What do I know about solving problems like this?  What strategy can I try? Does my answer make sense?  Did I answer the question?  These are really information literacy type questions.

Among the obvious links between the library and mathematics is the growing body of literature on mathematical topics including books by Greg Tang, Stuart Murphy, and Marilyn Burns. Many of these books offer novel approaches to mathematical topics and encourage the reader to look at problems from different perspectives. Another clear connection is the data strand in the mathematics curriculum that involves both reading and creating visual presentations of data including tables, charts, and graphs. This strand in the mathematics curriculum has an almost identical counterpart in most information literacy curricula. One could argue that data and information are synonymous terms. Most of the reference books (and internet counterparts) as well as non-fiction titles include information in some sort of graphical format. The Almanac is almost entirely in table, chart or graph format and offers a fantastic resource for mathematical problems. When it comes to classification and organization of information, the library is a clear leader. Arranging books in Dewey Decimal order is an application of ordering decimals. The big idea of Dewey Decimal classification that all knowledge could be organized by assigning numbers with a theoretically infinite number of digits to either side of the decimal is an ingenious application of mathematics.

Beyond these obvious connections between mathematics and the library that could be exploited by every school librarian and teacher, there are several other links to explore. One is the idea of word problems as a particular genre that employs characters, setting, and some problem to be solved. Reading and writing in this genre might be used as extension to other pieces of literature. Take characters and a setting from a favorite story and create a word problem about them. I’ve used this technique many times often sending a word problem to a classroom before they come to the library and using the problem as a hook to introduce the book. A great follow-up is to have students write their own problems based on the book.

A library offers a wealth of resources beyond printed books for mathematics. Computers, digital cameras, scanners and other technologies offer possibilities for presenting mathematical ideas and problems. Our school library houses math manipulatives for teachers to check out. Locating them centrally in the library provides access to professionals including the librarian who might not have access to classroom materials. One answer to the question, “why math club?” is because we house and have access to these material resources and tools. One other answer may be a particular resource the library does not generally include: the textbook.

Equally important in schools are access to the scarce resources of time and place. A library that is flexibly accessed, or available for individuals and classes as needed and relevant provides a space outside of classrooms crowded with desks and schedules crowded with mandates. This space in both time and place provides an opening for activities outside the usual class schedule where students (and teachers) might pursue mathematical ideas that are not bound by textbooks, pacing guides, and correct answers. The library offers a place and a time for the possibility of something like a math club. So I have offered the last thirty minutes of the last day of the week to students who think math might be fun. I have the space for us to spread out at tables or huddle together in a corner and I have the time unfettered by a fixed schedule. Yes, I am thinking about what the goals and objectives are in their curriculum in part because of the clues it provides me of where they are developmentally. I am not totally unbound by curriculum but I am free to look for or invent problems that are imaginative, fun and challenging without being intimidating. I have the luxury of time and space to devote some time to planning as well as implementing such a club. Math club allows me a time and space to listen and observe how students talk and do mathematics when the pressure is off them to find quick answers. Math club also provides a social space for students to listen and observe each other rather than carefully guarding their work.

Finally, I think the most important answer to the question why mathematics and why in the library may not so much be our access to these resources but in the inquiry stance provided by an information literacy perspective. An information literacy stance takes a critical perspective on knowledge and information with a focus more on questions and process than on ready answers. This critical perspective entails identifying assumptions implicit in problems, questioning the given or even asking “what if not” questions, working collaboratively, and with an awareness of the ethical and social dimensions of information. While I think my earlier paragraphs about the material, temporal and spatial resources of a library may help to justify inclusion of the school library in mathematics teaching and learning, I think this final area is the one I want to explore in my paper for this class. I find threads of it in all three of our readings. I hope to familiarize myself with the national mathematical standards and twenty-first century literacy and mine those resources for connections. I sense that reform directions in mathematics education contest the paradigmatic image of mathematics as a solitary endeavor, a static body of facts, and a concern with correct and definitive answers and I think information literacy has much to offer in this area.

What does Duckworth mean when she says, “..no amount of theory can affect children in schools except as it becomes a fundamental part of a teacher’s thinking?” Do you agree? Why or why not? How might this apply to other theories or theoretical information you have learned in teacher education? This quotation from Duckworth actually perplexes me a bit. I think she may be referencing the well worn gap between theory and practice. I guess I find confirmation of this in the continuation of the paragraph where she quotes someone as saying “You aren’t teaching teachers about Piaget, you’re teaching them to be Piaget.” The way Piaget listened and probed into the sense-making of children becomes a model for teachers. In this sense, I really felt like this chapter began to converge with both Lampert and Brown and Walter in a focus on questioning and listening. Throughout each of these writings, I see a focus on posing questions both to understand another and to move that other further in their own thinking. Brown and Walter suggest, “Take an alleged proof that either surprises you or lacks illumination. Then generate new sets of questions which might diminish the surprise or increase the illumination” (p.121). I think this is exactly what Duckworth suggests we do when a child’s response is surprising or seemingly irrelevant: ask more questions in order to understand their sense making. Lampert clearly illustrates using a similar process to understand what meaning her students are making of the mathematics in her classroom. Duckworth goes on to make a tantalizing observation, “The questions the interlocutor asks, in an attempt to clarify for herself what the child is thinking, oblige the child to think a little further also,” (Duckworth, p.97). For me, this is fascinating because in this interplay I believe I see how learning is socially constructed. We make meaning together because in trying to make sense of each other we demand that each makes sense for themselves. So in Lampert’s classroom, the interlocutor is not just the teacher but the classroom norms are established so that students also learn to question and respond to each other.

I guess finally, I would say that no theory has any meaning except as we put it into practice and subject it to probes, what if nots, and the scrutiny of others or, as Brown and Walter say, “we frequently do not appreciate the significance of an alleged solution without generating and analyzing further problems or questions” (p.126). And so I guess I would say that no amount of theory can affect children in school unless teachers put it into practice and subject it to the type of inquiry and reflection that may perhaps best be done in collaboration with other teachers. Because as we ask each other to make sense of both our theories and our practices, we grow in our own learning about teaching.

Problem of the Week

(These are from among the many photos I took on our weekend in Williamsburg – I really had fun looking for geometry in the architecture of Colonial Williamsburg.)

This week I decided to take some liberties with our assignment to do a “problem of the week,” and make my problem a Lampert one of teaching as well as of mathematics. Last week I started a math club with third graders at my school. We started by working on the Quad problem with geoboards and I used the opportunity to talk to them about math and what kinds of math problems they liked to work on. I realized I have created a challenge for myself because I need to identify problems that this group of students will think are “fun” because math club is an activity that occurs during their class “fun Friday.” They each indicated that problems with numbers interested them. I am working under the assumption that a problem that is difficult enough to challenge without completely frustrating them will be “fun.” Like me, they will be attracted to problems that are puzzles.

This week I decided to take the problem of the week that I did for class last week and modify it for this group. Like Lampert, I also thought this problem would provide me with an opportunity to learn about these students: how they interact, how they approach problems, what mathematics they are comfortable with, etc.

This was our class problem:

“An artist has a set of white tiles forming a rectangle twelve tiles across and eight tiles high. She wants to insert down the middle a strip of red tiles that is two tiles wide and a strip of red tiles across the middle that is three tiles high. How many tiles will she need to put in?”

Here are my notes about how I thought about changing this problem for a group of students early in third grade:

I tried to word the problem so they wouldn’t be hung up on the meaning of words like width, length, arrange… One of the things I want to pay attention to is what words they use to talk about the problem. At some point after they have worked through the problem I may talk about area and perimeter. Will some students think of multiplication for this problem? Who draws? Who chooses manipulatives?

Decisions I made
Use even numbers
Use somewhat large, but not too large numbers
Use colors that are reflected in the tiles we have (blue and red)
Use words up and down, and across instead of width and length (will this be more or less confusing?)
Make the cross one tile wide both ways (use two tiles wide as an extension question)
Offer pencil, paper, and color tiles to use to work solution.
Encourage work with a partner (but maybe move to separate tables so one person doesn’t close down everyone with a quick solution – or should we offer it as individual problem first – which will they like best?)

Looking at the tub of tiles – I may have enough for 2 or 3 groups to each have 24… will they understand that it doesn’t matter what color tiles they use? Will we be able to estimate that there are not enough for everyone?? I almost got another tub of tiles but decided I wanted to see how they worked through this problem.

Here is the problem I devised:

An artist has blue tiles placed in a rectangle with six tiles across and four tiles up and down. She wants to put a cross in the middle made out of red tiles. How many blue tiles does she have? How many red tiles does she need? What if she wants the red cross to be 2 tiles thick?

I also decided to offer graph paper to students. When I checked in the office ahead of time to see if we stocked graph paper, I learned that we did not. The secretary indicated she thought graph paper was something they needed in middle school. But we give every student in grades 3-5 a sheet of graph paper to work on end of grade tests. So this has raised the issue of whether we give students enough practice with graph paper before the tests. What affordances might be provided by graph paper?

Here are my notes following the meeting:

I only had three students (2 girls and one boy) so number of tiles was not an issue. When they read the problem they said it was hard. I told them I have paper, graph paper, and tiles. What would you like to use. Everyone said tiles.

Do you want to work alone or together. They said together “because it would be hard” but then each wanted to build with tiles by themselves. Sort of “Together but by myself first”

K. the boy started building the cross first – six by four. Then he built an L shape six by five – not counting the corner twice.

D., one of the girls, started to build a rectangle.

M. the other girl was building a huge rectangle; she seemed to be not counting; and making the same mistake as K – not counting corners. At one point she was building a six by six rectangle.

I told them to stop and look at differences in what others were doing. I asked K. if his shape was a rectangle? No. What do you need to do? So he built a perimeter – not filled in and I told him I thought the problem meant the shape should be filled in.

Finally everyone had rectangles. D’s was four by six. K and M both had five by six – look at how everyone has different sizes… what did you do differently. Again we counted. And K realized what he had done wrong – he took off a row. At some point I asked them if they knew anything about area and K. misheard me and said arrays. They laughed at his mistake… but I said no that’s good how is this like an array… I asked them to answer the first question. How many blue tiles. Korrey counted each tile and answered 24. Another time he counted by sixes – six, twelve, eighteen, twenty-four. Okay I said six rows of four tiles what multiplication problem does that remind you of. Six times four. Great.

When it came to the cross, they wanted to place it on top of the rectangle and I just stopped to show them what I thought the problem meant. It was cool because when we opened the rectangle to fit the cross into it we ended up with four groups of six tiles and I pointed that out.

There were eleven red tiles needed. At one point they weren’t sure if there were eleven or twelve (I think they were counting the center twice. Which is it, I asked and when they double checked it was eleven. Okay, I showed them what I meant by two thick and we build a wider cross. Now how many? K counted by twos – twenty-four. Hmmmm he noticed it wasn’t just 2 times 11 or 22. I wonder why, I asked, where did the extra two come from? We played with it a little but time was up. They wanted to draw their designs on the graph paper so I sent it with them.

Students only came from one class; one teacher was absent and the other must have forgot. I will need to check if that was the case. It was good to have only three students because I really had a chance to interact with them and to observe how they worked. I think they did enjoy the problem – at first they thought it was hard but they were successful. They all chose to work with manipulatives and I think that may be part of what they like about math club. I encouraged them to work together, even though they each wanted to try the problem with the manipulatives for themselves. A couple of times I used their different solutions and encouraged them to look at each others work and think about their differences. When I was reading Lampert this week about small group work I was struck by how she intentionally has students examine and learn from each other’s work, or what in other contexts might be considered copying or even cheating. In this way she lessens her role as the sole expert in the classroom and really allows for legitimate peripheral participation among her students. Math club may be fun in this way because students can work and copy each other.

One thing I noticed about two of the students that I think was interesting. When told to create a rectangle that is six across and four down, they first created a row of six tiles and then created a column of four tiles and put these together which created a shape of six by five tiles. In a similar fashion, when I asked how many red tiles were needed to make a cross in the middle, before they counted, two of the students answered 12. I think these were similar errors. It is as if they have a hard time thinking of a tile as a member simultaneously of two sets (one horizontal and one vertical). In the first case it was the corner tile that was not counted twice and in the second case of the cross it was the center tile that was counted twice – once as a member of the horizontal set and once as a member of the vertical set. In the latter case, they corrected themselves as soon as they actually counted each tile but in both cases they were clearly confused. This very much feels like a Duckworth type of puzzle that the students need to work through to an understanding themselves.

I’m wondering if next week I should give the group a similar problem and let these three students take a leadership role with the students who were not here.

This chapter interests me because this weekend I have been reading and writing about my research into collaboration with teachers. I plan to do a discourse analysis and it’s interesting that what I conceptualize as collaborative meetings are called “grade level planning meetings” at our school which may or may not imply collaboration. This has caused me to think about what teachers mean by planning and how often I have felt we were simply filling in boxes on planning sheets rather than having any substantial conversation or reflection about curriculum or student learning. How might these grade level planning meetings offer more in the way of resources to call on for teaching in the sense evoked by Lampert?

This past week, I decided on the spur of the moment to start a math club with third graders. I told each of the three teachers they could send me three students during their “Fun Friday” time. Yes, I was assured they had students who would think this was a fun way to spend that time. I decided to give them the quad problem to work on since I had introduced it to them before. I got out the geoboards, paper and pencil and while we worked we talked. First, it was interesting how the students told me they like mathematics; how these six girls and three boys clearly enjoyed a mathematics identity. “I’m good at math” they volunteered. Heads nodded when I asked if they would be interested in doing math puzzles and things like that. What kind of math do you like I asked. Many of the students said they were good at addition and just learning their “times tables.” So you would like problems with numbers for you to solve I asked? Like Lampert, I have since found myself wondering what types of problems to present them with. How to find problems that will be open-ended, challenging enough to be interesting but not frustrating, that make use of what they already know but allow them to stretch into some new areas and understandings? The quad problem was a great place to start and it was interesting how much vocabulary entered our conversation. Someone started talking about polygons and so I talked about quadrilaterals as a type of polygon. We talked about flips and slides when trying to eliminate duplicate quads. Trapezoids, squares, triangles.

In general my purpose with this club is to create a space that is unconstrained by curriculum and testing where we can explore numbers and space. To create a culture of inquiry just for the sake of inquiry.

My summaries of the three chapters we read for this week are in the blog entry below. The meaning and importance of these readings deserved a separate entry. First, I was blown away when Brown & Walter opened with Fibonacci. I have been fascinated by this series and it’s relevance to nature and aesthetics for many many years. In fact, this weekend my husband and I were driving to Virginia. I had my books in my backpack in the back seat untouched. Joe and I were talking about my schoolwork and he made the comment out of the blue, “after you get your phd, you need to write that Fibonacci book.” For years, I have thought there should be a children’s book about this fascinating series. Imagine my surprise when a few miles down the road, I decided to start my reading and opened B&W to Fibonacci!! I was fascinated with patterns like this in mathematics and often fought off boredom in high school math classes with exactly the type of mathematical doodling B&W mention (p.80 & 98). I know I played around with the prime numbers some and the reversal of the associative property seemed really familiar. I appreciated his example of the young students who creatively used negative numbers in long division and multiplication. Working with children often provides us with “what if not” questions.

The listing of attributes seems like a really important problem solving (as well as problem posing) exercise as well as questioning the given. Both of these seem like information literacy skills to me as well. In a sense they ask, what do I already know about the problem including what assumptions I am making, and how might I think about these assumptions differently. What if not?

Of the three books we are reading for this class, this one seems to fill my dreams with mathematical problems turned inside out and what if not refrains. And so I come around to Fibonacci and that dream – I really should write that book.

Duckworth also surprised me in the latest chapter by talking about concepts as problematic because they are nouns. Her discussion here very much reminds me of Lave and Wenger’s discussion of situated learning, “Knowing a general rule by itself in no way assures that any generality it may carry is enabled in the specific circumstances in which it is relevant” (Lave & Wenger, p.34). I also found much in this chapter of interest to me, as a school librarian because as she points out, one recognized goal of education is to develop in students an interest in learning and an ability to continue learning independently. Identifying information resources, and judging their relative quality particularly in the case of disconfirming evidence are important lifetime information skills. Much of the goals of information literacy fall into this area of belief, “sharing one’s knowledge and knowing when to call upon other resources” (Duckworth, p.52-3).

The Lampert book has become really important to me as an example of method both in teaching and in research. I was really struck in this chapter by the systematic way that she designs lessons including written reflections not only after a lesson but before the lesson. I found myself wondering if I should include some similar reflections before I meet with a grade level, really thinking deeply about where I am in my relationship with the team, what I hope to contribute to their designs and to the conversations about students and student learning. Lampert is also clearly taking a socio-cultural perspective on her classroom. Sometimes I am frustrated with Duckworth’s occasional references to the social aspects of learning. Duckworth’s discussion of beliefs seemed very centered in individual minds. Contrast this with what Lampert says about routines in her classroom as “largely a matter of guiding student talk and action in such a way as to establish shared understandings among everyone present about what it means to teach and to study and how it is to be done, here with this class and this teacher” (p.93).

I also have a huge WOW! noted in the margins where Lampert says, “The problem I faced in doing the work I set for myself was something like establishing and maintaining a counterculture in the midst of a conventional school environment” (p.65). This reminded me of our classroom discussion about how students don’t seem willing to consider “what if not” type questions, or to make conjectures, or offer evidence to back up their thinking. Are they simply reflecting the conventional school culture? Have we as Duckworth suggests, focused too much on the transmission of nouns or concepts, and not enough on the affective or expansive modes of understanding and belief?

Lastly, I just need to note here that I spent the weekend in Williamsburg, Virginia where I was born fifty years ago. And I spent a good part of “seeing” the landscape again through a mathematical lens: looking for geometric patterns and capturing them with my digital camera. Now, as soon as we find the cord we will upload those pictures to share!