I made this lesson planning template based on the Thinking Through a Lesson article and a longer template from Teacher's Development Group. I battled back and forth with myself about making it detailed enough to be helpful in encouraging teachers to think through lesson implementation while also being short enough so that it isn't overwhelming. I decided to post it here just in case anyone found that it might be useful. :)
I wrote a post a while back in which I was trying to make sense of my own constructivist epistemology and what that meant for my classroom. Like a lot of the conversations I hear about constructivism and education, I think at the time I was making an inappropriate association between a constructivist learning theory and pedagogy/classroom practice. You hear a lot of people talk about the "constructivist classroom" and many even implicitly associate that with the idea that lecture is bad and that students need open-ended, exploratory problems in order to "construct knowledge for themselves." All of that is misguided.
The main tenets of constructivism are:
1. knowledge is always actively built by the cognizing subject
2. the function of cognition is adaptive, tending towards fit or viability
This means we are all always constructing...all the time. There is no such thing as a NON-constructivist classroom. In short, constructivism "says" nothing about pedagogy or how to teach. So...lecture, lab, investigation, research, whatever...students are always constructing knowledge for themselves.
Then all of this led me to question, "what are the principles that guide my pedagogy and my work with students?" If constructivism says nothing about how to teach, then why do I teach the way I do? Several things that have happened to me recently led me back to thinking about autonomy as the aim of education. Piaget said that moral and intellectual autonomy should be the goal of education. That means that students/people should be governed by themselves in terms of what is right and wrong (moral) and what is true and untrue (intellectual). My action research over the last two years has made me very convinced that our work with students very much plays a role in their development of autonomy. When not properly nurtured, students lose trust in their ability to think...mostly as a result of our attempt to teach things to students...to get them to think like we think.
I have so much more that I am thinking that I just can't figure out how to put in words right now. So, to be continued...
In the meantime, go read this post...it makes more sense than mine.
I've been having some fun working with teachers recently around brainstorming better questions to ask their students. I've noticed that we have a tendency to want to ask our students computational questions as opposed to conceptual questions. We've been working hard to turn the first type into the second. A few examples from the past week....
Paper Airplane Investigation
Some 6th graders were investigating the best type of paper to use when making a paper airplane...they had decided to compare construction paper, computer paper, and cardstock. After making each, throwing a few trials, and recording data, the plan was to have students fill in the following chart:
We started thinking that by including "average," it implied that this was the best way to make decisions about what was best in this case, forced the teacher to instruct on how to calculate an average before the students had made sense of the idea, and turned the activity into a computational one rather than a conceptual one. We thought about the small change of just giving students this chart...
...and then asking them to decide "which one is best" by inventing a method and justifying their reasoning.
One teacher had been thinking a lot about trying to create discussion in the math classroom. He had been working on scientific notation with his class and was thinking about using this question as a prompt to spark discussion:
We decided that if we wanted kids to talk, they needed to have something rich and complex to talk about and make sense of. After brainstorming some different options, we turned the computational question into a conceptual one by asking:
Of course, this also opens up a host of interesting extensions...
"what about the smallest product?"
"what if we divide instead? biggest result? smallest?
"what if one of the exponents was negative? both?"
"what if we move the decimal point?"
...and many others.
Let's commit to asking different questions and letting students share THEIR ideas with us and each other.
I ran a session with some teachers the other day with a scenario I modified from Thinking Mathematically. They posed some really interesting questions that I wanted to record here.
The Scenario and Their Questions
On a sheet of grid paper, draw a rectangle of any size.
Color in the largest square possible.
In the remaining rectangular region, repeat the process by coloring in the largest square possible.
Keep repeating until you are done.
I asked them to play around with this on their own for a little over five minutes, paying close attention to things they noticed and things they wondered about. After sharing ideas with their groups, we made a list of their wonderings:
1. Can we determine the final number of regions given the dimensions of the original rectangle? (i.e. 5 in the example)
2. Can we determine the number of different sized regions given the dimensions of the original rectangle? (i.e. 2 in the example...3x3 and 1x1)
3. Can we determine the size of the final square to be colored in given the dimensions of the original rectangle? (i.e. 1x1 in the example)
4. Are prime numbers somehow important as dimensions of the original rectangle?
5. What happens with both dimensions of the original rectangle are odd? When both are even? When one is even and one is odd?
Groupwork and Status Treatments
The focus of the session was "facilitating groupwork" with a brief nod to complex instruction. Here are some of the things I tried that seemed to have positive results with this group:
1. Before we did anything else, I started by reviewing group norms...
- Everyone participates and everyone's ideas are valued
- Share and explain your ideas
- Listen to understand the ideas of others
- Ask and respect questions
- Help without telling others how to think
...and we spent five minutes at the end of the session by having participants journal about how they (personally) did in adhering to group norms. They identified things they did well and things they did not.
2. Before launching the scenario, I passed out grid paper with only this on it...
Participants identified at least one way they could contribute for the session and shared that with the group. At the very end of the session, I asked them to identify at least one way each member in their group had been smart and they shared these publicly with each other in their groups.
3. During their group work time, I identified what (from my perspective) seemed to be lower status group members. I looked for ways each of these members was helping the group to function well and highlighted them publicly as we wrapped up the group work phase.
There were two things that stood struck me during the session that I am still thinking about:
1. One participant said that, upon receiving the dilemma, was extremely underwhelmed. Her statement during our meta-debrief was something like, "I was expecting something significant and profound." But, she said during the course of the session she and her group became extremely interested in one of their questions. By the end of the session their group was high-fiving each other over their solution....over rectangles and squares. It left me wondering what sparked this excitement for them. The freedom to pose their own question? To pursue one they found interesting? I'm not sure.
2. One participant commented that she had a very difficult time with the session because she felt like she didn't have enough time to think to herself. We talked a little bit about my decision to not give too much individual time so that, when coming back together as a group, nobody has a finished idea to share...only the beginnings of one. But her point is a good one and something I struggle with as a teacher. It gives me a lot to think about.
All in all, it was a fun investigation.
Thanks to everyone who attended my session on "Rich Mathematics Through Habits of Mind" today at the New Tech Annual Conference (NTAC). As promised, I am including my slides from today that include the notes from your discussions. I have also added a few other bonus resources if you are interested.
I mentioned my research on habits of mind, student agency, and identity. If you are interested in reading more, it is available online or for purchase.
What is the role of the teacher in a more open mathematical learning environment? This article suggests the benefits of listening. If you find that idea interesting, you might also find some usefulness in the book "5 Practices for Orchestrating Productive Mathematical Discussions."
I was very fortunate to get the opportunity to travel to Stockholm recently for a conference on "21st Century Mathematics." The gathering was put together by the Center for Curriculum Redesign and, to oversimplify it, the conference was centered around discussing the question of "what should students learn in the 21st century?" There was an interesting set of presenters and a jam-packed three day schedule. I'm hoping to take some space here to summarize some of the presentations and moments that stood out to me while interweaving some of my thoughts, reactions, and questions. We'll see how that goes....
Charles Fadel (Opening Address)
Also the primary organizer of the conference, Charles started things off by posing one of several main concerns of the event. Namely, he asked us to consider "in the era of technology, what do we teach for?" He suggested that a response to this question might force us to reconsider what knowledge, skills, character traits, and acts of metacognition were important for students. Participants also added that it might be useful to teach for problem posing, creating, synthesizing, and ethics. His opening address ended with what, I think, is truly his central concern: "what do we add? what do we remove?"
Michael Kaplan (Probabilistic Reasoning)
Not surprisingly, Michael Kaplan really has a way with words. His talk was mostly about probabilistic reasoning, but often drifted towards grander themes. I think the best I can do here is loosely quote some of his words (I can't say for certain that I have captures accurate quotes here):
"The less we know about a subject, the more certain we are about our opinions. It's the way our thrifty brains save our precious mental resources. But when we know what we are talking about, we willingly acknowledge complexity and uncertainty."
"Too often we fail to engage students in the uncertainty and complexity of problems and, instead, we treat them as a receptacle."
"Mathematics should be useful, we are told....but useful to whom?"
"We make a sad mistake when we propose that studying mathematics will lead to dream careers."
"So much effort in repetition designed to get some idea in the mind of a student is time wasted."
He said much more, but these were the quotes that caught my attention. Lots to think about here.
Jon Star (Cognition and Neuroscience)
Jon Star set out to investigate the claim that teaching math leads to the development of "higher order thinking skills." He defined higher order thinking as problem solving, critical thinking, reasoning, deduction, and logic. The basic message of his talk is that he was unable to find any evidence that supported the claim that teaching math produces an increase in higher order thinking. This didn't surprise me, although it seemed that it made many people in the crowd particularly uncomfortable or upset. Coupled with Michael Kaplan's talk, it had me thinking early on about the reasons we give ourselves for teaching math...of assigning purpose to our work.
Keith Devlin (Mathematical Thinking)
The most stunning thing about Devlin's talk was that he started it out by saying something along the lines of, "everything technique and method I learned in obtaining my bachelor's and doctorate in mathematics in math can now be outsourced. What makes me still marketable is mathematical thinking." He mentioned a particular project he did for the Department of Defense in which, upon reflection, he realized that "it wasn't the mathematics <he> knew, it was the way he approached the problem or framed the question" that made him valuable and marketable. Lastly, he recognized that innovative mathematical thinkers need to think outside the box, adapt or create methods and techniques, collaborate, and communicate.
Sverker Lundin (The Drift Towards Purity)
Sverker gave a really fascinating talk about the history of mathematics education and how many progressive movements were started through the work of Pestalozzi and his rejection of Enlightenment ideals. His main conclusions were:
1. be careful about higher goals in education (democracy, thinking, etc)
2. think about how what students do in schools could lead to these goals...expect no magic from the subject matter of mathematics
3. open up for other ways of approaching mathematics than through tool-free pure problem solving
4. we have inherited a problem AND its solution...we should rethink both
Conrad Wolfram (Stop Teaching Calculating, Start Teaching Math)
Conrad's talk was very similar to his TED Talk. His basic premise is that computers have changed the way we experience our world and, as a result, they should change what schools are valuing. He repeatedly asked us to "stop turning humans into third rate computers and, instead, turn them into first rate problem solvers." He pushes hard for a problem-centered curriculum in which the use of computers as engines of calculation allows a teacher to ratchet up the level of complexity that they can be carried out in the classroom.
Merrilea Mayo (What Math Do People Use in the Workplace)
Some really interesting tidbits from this session:
- correlation between education and career readiness/success is 0.1 (almost none)
- more than 90% of people will never use more than 6th grade level mathematics
- in practice, even engineers were not using differential equations...the software was
- found that people were learning their "applied math" outside of school
- evidence to suggest that taking math does not mean you know/can apply it
Bryan Meyer <-- That's ME! (Interdisciplinary Education)
I was on a panel with two other teachers and we were asked to talk about our experience in schools that valued interdisciplinary education. I talked a bit about how the school I work in is specifically structured to make an alternative form of education possible (strategic scheduling, teaching teams, math/science paired, etc.) but that many factors continue to pull us back to the norm (testing and accountability, definitions of what math is, perceptions about what it looks like to learn math, the difficulty in doing it well). I also talked about a "tale of two classes" in which I compared my work with my 10th grade class to the work with my 12th grade class. Some of the specifics of these experiences have lead me to question:
1. How/Who decides what counts as mathematical knowledge and/or mathematical activity? What are the benefits and consequences of how we answer that question?
2. Is it "inter"disciplinary education that we are after or is it "anti/un"disciplinary education? What are the benefits and consequences of defining a discipline of mathematics?
3. Related to both above, should we operate under the goal of "mathematics for all" or "all people are mathematical?" How will our actions be affected by the place that we choose to operate from?
So....What Did I Learn?
I'm not exactly sure what I learned, but I do know that I've been thinking about a few things since I have returned from Stockholm.
1. I have been thinking A LOT about the reasons we hear/accept/give ourselves for why we teach math. Many of the common ones were subtly brought into question by a few speakers. I'm not going to give my thoughts on that here and now, but I think it is worth considering. I also think it is worth considering how our answer to that question might impact our work with students...for better or worse? When we push towards something, what are the effects of that pushing?
2. The conference was premised on "what should students learn in the 21st century?" That's a big question and I just never felt settled on the idea that changing what students should learn will somehow change anything at all. I guess I'm left wondering and thinking about what it is we want to change in math education? Enjoyment? "Success?" Something else?
Not sure....still lots to think about I guess.
p.s. Conference session slides and notes are available here if you are interested.
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.