I would love any thoughts or suggestions. What areas does it seem like there might be some opportunity to discuss further with students?
A while back I posted about a survey that I gave my students during the first week of school. My intent was to get information about their definition of mathematics and their sense of mathematical identity/agency. I am nearing the end of my research period, so I decided to go ahead and give them the same survey to see how/if things had changed for them over the year. I must say, I'm a bit skeptical about this sort of data that attempts to show a change in "x" because of "y." With that said, here are there responses before and after. I found it useful to compare percentages.
The new question I have is how to get more information from the students about this data. What caused certain changes? What caused other areas to remain static?
I would love any thoughts or suggestions. What areas does it seem like there might be some opportunity to discuss further with students?
I've been debating on whether or not to write about this for the past week, but I keep thinking about it so I figure I might as well record it somewhere. My class is in the middle of a unit on probability and last week we were working on this problem from the IMP Year 1 "Pig" unit:
Students played around with it for a bit and we even did some experimental trials before trying to tackle the theoretical probabilities...which is when things got tricky. They started listing all of the possible ways to get a sum of 2, 3, 4, etc. when rolling a pair of dice (pair of die?? not sure? well, you would say "pair of shoes" and not "pair of shoe" so I'm sticking with dice). Anyways, there was a big controversy about whether or not we should count 1+2 AND 2+1 as two different options or if we should just count them together as one option (which, in all honestly, I tried to intentionally bring out by giving students pairs of dice that were different colors).
The group that said we should count them as two different options ended up with 21 total possible outcomes and the following theoretical probabilities:
2 3 4 5 6 7 8 9 10 11 12
4.8% 4.8% 9.5% 9.5% 14.3% 14.3% 14.3% 9.5% 9.5% 4.3% 4.3%
The group that said we should count them together as one option ended up with 36 total possible outcomes and the following theoretical probabilities:
2 3 4 5 6 7 8 9 10 11 12
2.8% 5.6% 8.3% 11.1% 13.9% 16.7% 13.9% 11.1% 8.3% 5.6% 2.8%
Neither side was willing to budge, so I suggested we conduct a HUGE experiment with LOTS of trials across all three of my classes so that we could put the results together and see what conclusions we could draw. We did something like 3,500 trials...and here is what we found (experimental probabilities in brown on the far right):
In the end, there were a few people from the "21 camp" that decided to change their mind and join the "36 camp" but most people stuck with their original idea.
I was reminded of a quote from Les Steffe's work that I read a while back:
"A particular modification of a mathematical concept cannot be caused by a teacher any more than nutriments can cause plants to grow. Nutriments are used by the plants for growth but they do not cause plant growth."
I'm curious what you think and what you would do in this same situation. I let it go. I felt I did my job by helping students test their ways of thinking, not by telling them what to think.
A friend/co-worker, Andrew, and I hosted a workshop at Greater San Diego Math Council's Annual Conference this past weekend. We titled the workshop "Rich Mathematics Through Project-Based Learning," but rather than attempt to give a "here is how you do PBL" workshop (which fails to acknowledge that there are many ways to attempt PBL) we decided to use our own efforts at project-based learning as a lens through which participants, including ourselves, could examine the role of the teacher in fostering mathematical understanding. We each presented one approach to PBL, along with some student work and a dilemma that each approach seemed to pose about the role of the teacher in facilitating (or not?) the direction of student work/learning.
Our Two Approaches
Andrew and I decided to present two very different approaches to planning and implementing projects in our classes. I don't want to try and articulate Andrew's position too much on my own (maybe if he drops by he can leave some thoughts in the comments), but his basic premise was that a "content-based" project might be structured something like this:
Andrew's dilemma here was, given that he is attempting to move students from A to B, "what is the role of the teacher in facilitating that progression?" This led participants to attempt to deconstruct the process and put in steps that might direct student learning. Contrasting that, I have been interested in experimenting around with a more "open-ended" approach to projects. I visualize it like this:
This approach brings an entirely different dilemma, yet still one that largely revolves around the role of the teacher. Specifically, how does a teacher manage such divergent outcomes and how can the teacher facilitate mathematical connections/understandings for students?
A Closer Look at the Open-Ended Approach
We took an in-depth look at a specific example that I have used recently with students. First, I posed the following task for students:
Create as many squares as possible using only 12 lines
Students played with that in groups for a day or so and we made some conclusions as a class. Then, I asked them to brainstorm as many questions as they could that they might be interested in pursuing based on the initial task. Here is what they came up with:
1. Can we create a rule/formula for the maximum number of squares based on the number of lines used?
2. How many triangles/rectangles/etc. can be created using only 12 lines?
3. Is there a difference between even and odd numbers of lines?
4. How many shapes can you create with 12 lines?
5. What would a graph of maximum number of squares vs. lines used look like? Linear? Exponential? Other?
6. What if by "lines" we meant "toothpicks" or "unit lines?"
I passed out some student work samples for participants to take a look at, with the prompt(s):
What do you notice?
What do you wonder about?
What evidence of the Common Core Practice Standards do you see in student work?
Looking for Rule/Formula (Question #1)
Lines as "toothpicks" (Question #6)
Even vs. Odd # Lines (Question #3)
How many triangles (Question #2)
I'm still left with LOTS of questions about what all this means and what my role is in all of this work with students. What are the benefits of this divergence? What are the consequences? How do I help them make connections within and across each others' work? Do I push students reach certain conclusions with their work that I know are still out there? Do I let them just end where they end? If I did push, would they really own the outcome? Would they "know" it? Those last two questions have been nagging at me a lot recently (perhaps more on that in a later post?).
I do think, however, that I am committed to continuing to try figure it out with my students. It is the closest I have come to truly freeing them to think for themselves, follow their own curiosities, make their own conclusions, and be honest with themselves about what is still left unanswered (versus trying to convince me that they know something that they think I want them to know). I mentioned in our workshop that our answers to some of the big questions (what is math? what is the purpose of math education? what is a project? what is the role of the teacher?) determine a lot of the small things we do in our class every day. I'm trying to let myself live in a state of constant re-evaluation with those questions. I think we need to in order to be fully present in the craft of teaching.
I am trying to make a commitment to myself to "write rough" here more often. My hyper-analytic personality leads me to think and rethink things so much that I rarely get to a mental space in which I feel like I can write with any clarity. So, this might be rough.
I attended the Creating Balance conference up in San Francisco last weekend, where Rochelle Gutierrez gave a talk about "Teaching Mathematics as a Subversive Activity." Her talk hit close to home for me but, even given it's closeness, still has left me thinking and rethinking her words this past week.
I appreciated many things about her talk, but mostly the critical way in which she addressed the sort of taken-for-granted discourses, or structures, in mathematics education that politicize the issue. Things like success, proficiency, achievement, and even what counts as mathematics largely have a singular meaning. As she writes,
"What counts as knowledge, how we come to 'know' things, and who is privileged in the process are all part and parcel of issues of power."
Mathematics education deals primarily with the dominant view of mathematics, the "discipline." With such a singular view we fail to recognize that, as Dr. Gutierrez points out, "mathematics needs people as much as people need mathematics." She writes,
"Most often, the goal in mathematics teaching is to try to get the student to become a legitimate participant in the community of mathematicians, thereby subsuming their identity within the currently sanctioned way of communicating in the field."
"Yet, when students offer a different view, they are seen as having deficient, underdeveloped, or misconstrued understandings of mathematics."
Dr. Gutierrez describes this as the "deficit model" which I see as very similar to Friere's "banking model." Attending to dominant mathematics can sometimes mean losing oneself at the expense of adopting another's way of knowing. As Dr. Gutierrez reminded us during her speech, teachers of mathematics are "identity workers." When we define mathematics narrowly and we impose that mathematics, we fail to recognize the mathematics of the individual, the student, the child. These are issues of identity (is my thinking mathematical?), of equity (who gets to participate in mathematics?), and of power (who benefits as a result of this?).
The point where the two axes intersect is a space Dr. Gutierrez has named "Nepantla." Borrowed from the work of Gloria Anzaldua, Nepantla is an Aztec term that refers to "el lugar no lugar" (neither here nor there). It was best described, I think, by Dr. Gutierrez to simultaneously mean "both" and "neither." I think it succinctly captures the paradox and tension I have felt in my research on student agency and identity. The discourses and structures in society require that we attend to dominant mathematics (access and achievement), yet I know there is an alternative in which moving away from dominant mathematics allows us to attend to the mathematics of students/people (identity and power). We can't do both but we also can't do neither. So, as teachers, we live in Nepantla...we live in tension.
It is from this space of Nepantla that new options, new knowledge emerge. To retreat to safety, to settle in the current ways of doing things because no clear alternative is present, is to choose to live in "desconocimiento" (a distancing, ignorant space in which we refuse tension). The alternative is to choose tension, to live in the messy space of Nepantla. Dr. Gutierrez describes curriculum as "both a mirror and a window." A mirror because it allows the student to recognize oneself in the work they do, but a window because it also allows them a new perspective on the world. I don't know that it's attainable, but I think we must choose to pursue the unrealizable philosophical ideal and live in messiness because to settle is to distance ourselves from knowing mathematics with our students.
I have been experimenting recently with different ways of having student curiosities drive our work together. In some ways it has been successful. In some ways it feels like I don't know how to do this well. I thought I'd blog about a few examples from the past two weeks and hopefully you all can help me sort this out a little more.
I put this up on the white board:
x/2 + 5
Me: "Someone give us a number."
Me: "Ok. I heard 7. We are going to put 7 in for x in the expression on the board. Then, whatever we get as the result, we are going to put in for x. And then again...and again....and again. But we aren't EVER going to stop. What do you think is going to happen? Tell your partner."
They had various ideas, we tested them out, and made some cool observations. After that, I encouraged them to experiment with anything they were curious about. What happens if we change the expression? What happens if we change the starting value? What happens if we use two rules instead of one and alternate? There were lots of options.
I posted about this problem before, but this is an extension of my thinking about the launch of that problem. For my first two classes, I gave students the following problem:
They seemed to rely on me to define the task for them and set parameters. So, for the third class, I put this up on the board:
I said, "For the next five minutes, everyone experiment with something you find interesting." They experimented for a bit and then I had them compare their activity with the rest of their group. There were a few different ideas. Most students hovered somewhere around the question of "which numbers can you create" but there was a lot of discrepancy about parameters.
I put the following images up on the projector:
I gave students some time to examine the images and then asked them to brainstorm a list of questions that were raised by the images. We put a list of them up on the board. It was interesting that a lot of the questions were clarifying questions rather than problems to be investigated (Is the first one just a zoomed in portion of the second? What is the dot? Does the line always have to cross diagonally through the small squares?) It seemed like they were so used to asking about parameters, rather than setting them, that it didn't occur to them to just set the parameters and ask a solvable question based on them.
Eventually, we got a few questions with potential. Will it always hit the corner? How many times will it hit the sides before it hits a corner? Does it matter if the side lengths are odd or even? Is it possible to hit every grid line on the side BEFORE it hits the corner? Is it possible to end up in the same corner that you started in?
I suggested that each group: 1) pick a question they were interested in, 2) set their own rules/parameters and 3) get to work. It was interesting. Groups worked for a couple days and then things really stalled out. Because there were only four people (or so) working on a problem, there wasn't the same opportunity for them to bounce ideas off of other groups, for us to work through difficult things together as a class, or the same chance that someone might have an insight that led to progress for the whole class. We eventually proved that 1) it would ALWAYS end in a corner and 2) that corner would NEVER be the starting corner (assuming you launch at a 45 degree angle from a corner). So, from there, I suggested we all work towards finding a way to predict exactly which corner it would land in based on the rectangle size.
Help Me Out...
I'm really interested by idea of using student generated questions but I feel like I need help on how to make it work. Things that I'm thinking about:
Something happened in my class last week that really got me thinking and reflecting. My class was working on this problem...
...when one of the students came up to the board and drew this...
If you haven't already, it might be helpful to think about the prompt and the student's proposal for a bit. Essentially, the question they were considering was "if we started with a perfect square, are all five of the divisions also perfect squares?" As you might have noticed, it is possible to draw it so that it in a way that makes you stop and think. There are a host of interesting things to discuss here including measurement, drawing versus abstraction, and others, but here is what interests me most at the moment...
Another student in the class responded to this student's drawing with a friendly counterargument. In essence, he provided the student with a logical argument about why these five divisions were not squares. The student who had drawn the proposal on the board seemed to recognize the fault in his argument and we kept discussing other cases. Then, on his way out when class was over, I overheard him say, "I wonder what size starting square would make my five case possible?" It seemed to me that this student was still searching for ways to preserve his mental equilibrium; hesitant to make a (perhaps?) necessary accommodation.
All of this got me thinking about the ways our minds work and the role of the teacher in working with students. It's fascinating to me how people work with their ideas. I'm convinced that any lasting change must come from a student. I use the example to make a larger point about our goals in working with students. In the context of this puzzle-like problem, it's pretty easy to suggest that we just keep letting this student play with other size starting squares; to keep letting him play with his own ideas. I know that in my own teaching, it isn't always as easy to preserve this play-like quality with things that we are "supposed" to teach. But to hold students accountable to those things (or what we perceive to be their mastery of that Mathematics) is an oversimplification of the workings of the human mind and a coercive, unjust position to hold in education. Because if we were to force this student to abandon his ideas about that square, to insist that his thinking is wrong, he would inevitably start to reject his own thinking as mathematical. He would inevitably start to surrender his own agency.
I think it brings up some interesting questions about the goals of education. I've been reading The Having of Wonderful Ideas recently (which I highly recommend) and I think Eleanor Duckworth puts it quite nicely:
"Many people subscribe to the goals of the it's-fun (interest) and I-can (confidence) types, but when it comes to detail, almost never does one see a concern with anything other than the-way-things-are beliefs. Lesson-by-lesson objectives are almost without exception of this type, despite the fact that general goals very often mention things like interest, confidence, and resourcefulness. This is, of course, because it is difficult to produce a noticeable change in any of these in the course of one 50-minute lesson. But notice that as a result all the effort is put into attaining the objectives stated for the lesson."
I'm not sure where all of this leaves me other than the realization that when we listen to kids and their ideas, working with students is challenging, perplexing, and a ton of fun.
I recently read 'The Child and the Curriculum' by John Dewey. The essay was written in 1902 and is still relevant today. The article is packed with passages that will make you think...
...but I have tried my best to summarize my reading and interpretation of it through direct quotes from the essay:
The fundamental factors in the educative process are an immature, undeveloped being; and certain social aims, meanings, values incarnate in the matured experience of the adult. The educative process is the due interaction of these two forces.
Instead of seeing these two as an interactive whole, we often view them as conflicting parts, leading to what Dewey views as the "child vs. the curriculum" or "individual nature vs. social culture." Often, he argues, educational movements side with one or the other which leads us to polarized extremism. The two camps, as Dewey describes them:
One school fixes its attention upon the importance of the subject-matter of the curriculum as compared with the contents of the child's own experience...studies introduce a world arranged on the basis of eternal and general truth.
Not so, says the other sect. The child is the starting point, the center, and the end. His development, his growth, is the ideal...Not knowledge, but self-realization is the goal...subject-matter never can be got into the child from without. Learning is active.
Dewey's position is that these two extremes set up a fundamental opposition, left for the theorists, while any settlement on a solution will vibrate back and forth in perpetual compromise. His proposal is that we must refrain from seeing the experience of the child and the subject matter of the curriculum as opposing forces:
From the side of the child, it is a question of seeing how his experience already contains within itself elements - facts and truths - of just the same sort as those entering into the formulated study...
From the side of the studies, it is a question of interpreting them as outgrowths of forces operating in the child's life...
Abandon the notion of subject-matter as something fixed and ready-made in itself, outside the child's experience; cease thinking of the child's experience as also something hard and fast; see it as something fluent, embryonic, vital; and we realize that the child and the curriculum are simply two limits which define a single process. Just as two points define a straight line, so the present standpoint of the child and the facts and truths of studies define instruction. It is continuous reconstruction, moving from the child's present experience out into that represented by the organized bodies of truth that we call studies.
Throughout the essay, Dewey refers to the psychological (of experience and process) vs. the logical (of finality and fulfillment). The two forces are similar to that of the child vs. the curriculum and he argues for "psychologizing" the subject-matter ("restoring it to the experience from which it has been abstracted"):
If the subject-matter of the lessons be such as to have an appropriate place within the expanding consciousness of the child, if it grows out of his own past doings, thinkings, and sufferings, and grows into application in further achievements and receptivities, then no device or trick of method has to be resorted to in order to enlist "interest." The psychologized is of interest - that is, it is placed in the whole conscious life so that it shares the worth of that life. But the externally presented material, conceived and generated in standpoints and attitudes remote from the child, and developed in motives alien to him, has no such place of its own. Hence the recourse to adventitous leverage to push it in, to factitious drill to drive it in, to artificial bribe to lure it in.
And his, perhaps, more action oriented response:
There is no such thing as sheer self-activity possible - because all activity takes place in a medium, in a situation, and with reference to its conditions. But, again, no such thing as imposition of truth from without, as insertion of truth from without, is possible. All depends upon the activity which the mind itself undergoes in responding to what is presented from without. Now, the value of the formulated wealth of knowledge that makes up the course of study is that it may enable the educator to determine the environment of the child, and thus by indirection to direct. Its primary value, its primary indication, is for the teacher, not for the child.
And Dewey's final message to the reader:
The case is of the Child. It is his present powers which are to assert themselves; his present capacities which are to be exercised; his present attitudes which are to be realized. But save as the teacher knows, knows wisely and thoroughly, the race-expression which is embodied in that thing we call the Curriculum, the teacher knows neither what the present power, capacity, or attitude is, nor yet how it is to be asserted, exercised, and realized.
I can't quite decide what that final passage means to me. At present, I take Dewey's words to be a reminder of the subjectivity and intersubjectivity involved with matters of the mind. As a teacher, I make observations of students working and I make inferences about their thinking based on my own ways of knowing the Curriculum. I must remind myself that these are my inferences and that I have no way of knowing the thinking an other because I am not them. At best, my pursuit as a teacher must be to work with them in a constant state of negotiation of meaning; not to direct their thinking until I judge it to be a mirror image of my own.
There are plenty of free downloads of the essay online. I encourage you to read it. Afterwards, come leave your thoughts in the comments.
Through my action research, I have become even more intrigued by task design and the effect it has on student discourse and agency. Of course, there are many other factors influencing agency (teacher expectations/actions, classroom and group norms, etc.) but I think task design might be the most powerful one. Recently, I gave some of my students copies of three tasks that we had done together in class (below) and asked them a variety of questions. They all had incredibly helpful insight and I have featured a few of the key quotes here.
…helping students DO mathematics