I have the week off for Thanksgiving, so some friends and I have spent the last few days in Napa hanging out and doing some wine tasting. We got a recommendation to visit Freemark Abbey Winery
, which we did, where I came across what seems like a pretty good problem-based investigation. Here is the scenario as they describe it (based on actual events):
"In September 1976 William Jaeger, a member of the partnership that owned Freemark Abbey Winery, had to make a decision: should he harvest the Riesling grapes immediately or leave them on the vines despite the approaching storm? A storm just before the harvest is usually detrimental, often ruining the crop. A warm, light rain, however, will sometimes cause a beneficial mold, botrytis cinerea, to form on the grape skins. The result is a luscious, complex sweet wine, highly valued by connoisseurs."
Basically, they had to decide: harvest the grapes now and guarantee the production of a moderately priced wine, OR wait for the incoming storm and hope that it encourages the mold growth necessary for production of high end wine. There are some accompanying details that the winery used in the decision making process (which I have somewhat shortened here):
I can imagine several different versions of the problem depending on the group of students you were working with (by adding/removing details and alternative scenarios).
- There was a 50% chance the storm would hit Napa Valley
- There was a 40% chance that, if the storm did strike, it would lead to the development of botrytis mold
- If Jaeger pulled grapes before the storm, he could produce wine that would sell for $2.85 per bottle
- If he didn't pull and the storm DID strike (but didn't produce the mold), he could produce wine that would sell for $2.00 per bottle
- If he didn't pull and the storm DID strike (and did produce the mold), he could produce wine that would sell for $8.00 per bottle (although at 30% less volume because of the process to produce this wine)
- If he didn't pull and the storm DIDN'T strike, he could produce wine that would sell for $3.50 per bottle
I am attaching the printouts that I photographed in case anyone wants the details.
The 12th graders and I started a new unit/project/problem/whatever last week. It started with this image:
What is this?
We took some guesses before someone correctly identified the image as the readout from a stationary bike machine. If this was the readout, I nudged, then:
What did the course look like?
I let them play with it for a while. They identified some important questions they had and information they wanted:
- What do the dots represent?
- How can we quantify this?
- Does speed matter?
- Did the bike have gears?
- What do we mean when we say, "what does the course look like?
Ultimately, they agreed upon some answers and, as a result, gave themselves some space to work within. This part takes time and I really think we need to give it the time it deserves. If we don't let students pose and answer those questions, the rest of the unit suffers. I let them know that we would examine some simpler cases first and return to the unit question somewhere down the road.So, the next day they came back and and I
had them map the course for these two:
This took another class period for them to sort out. By the end, though, they started to feel pretty good about the fact that the dots were a measure of resistance which they connected to the slope of the course at that point. Towards the end of class, someone asked, "What would it look like if you were going downhill?' They talked for a bit and devised a system to handle that, so the next day I gave them this:
This allowed them to have conversations about what it meant when the resistance was positive, what it means when resistance is zero, and what it means when resistance is negative.It's fun to watch the students construct an understanding of antiderivatives. The three-day span reminded me of a few things:
- the importance of listening to how students are thinking about and making sense of problems
I feel like there are a couple ways to go with this trajectory next:
- our role as teachers in responding to that way of knowing by bringing what might be the "next good problem" for them
- allowing them the time to sort things out together
I'm leaning towards #1. What do you think would be best for the students?
- Stay with resistance, but use a continuous curve
- Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
- Stay with discrete and resistance, but use a more complicated readout
I recently asked a student of mine the question:
Do students care about 'real world' connections and problems or is there something else that motivates and drives them?
"To be perfectly honest, as a student I do not know really know what I need for the real world. The term real world confuses me sometimes. While in the process of learning something I have never thought to myself "This will really help me later on in life. Especially while I work on so and so in the future" because it's irrelevant. The helpfulness for the "real world" comes later and I think that we can never really realize the significance until we naturally and effortlessly apply that knowledge throughout our daily lives.
When I am bored or I don't want to try or the assignments seems super difficult and tedious, I subconsciously think to myself "all this effort will contribute to nothing for me later on in life. When will I ever need this? Never." And therefor justify my actions for either not doing it or not caring or not doing the work to my best ability.
extreme challenges with no personal connection or interest, or very confusing requirements = I won't need this!
I believe that most student's are not motivated by the real world. What students want is something that they can personally connect with on a deeper more intimate level. Something exciting, current, and easily relatable.
I think relatable can translate to most students definition of "real world." When we argue for something real world and applicable, I think what we really mean is that we want something relatable and personal."
Students have a lot of interesting things to say about education. Maybe we just don't ask them for their opinion often enough?
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.
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What task did you feel most successful with? Why?
"Problem A...I connected with the most because it was extremely hands-on and I got to just go with the problem in any way that I chose and I didn't feel limited."
"Problem A...The way this problem is designed, there are several different ways you can approach it."
"Problem A...we definitely tore this problem apart. We looked at all the different possibilities, all the restrictions, we looked at a variety of theories and hypothesis, and we proved or disproved them. I felt successful because we created rules that we could use for any given triangle and it really helped me to understand the problem more. B and C were good, but they were more, like, content-based I would have to say. But with problem A, there was no real content idea that you had to learn."
"Problem B...my main problem is I never know where to start and with this problem we were kinda given a place to start."
"Problem A...Unlike B and C, I was able to get started on this problem right away. With A, you can explore in whatever way you want to, so you are almost, like, being successful the whole time because there isn't really a right or wrong, you are just testing.
Which task did you enjoy the most? Why?
"Problem A...you're not just following some previous knowledge to solve the problem, you are going into this new, unknown problem that you have never seen before and tackling it in the best possible way. We just played around with it and I think that is the best way to really enjoy math....just play around with it."
"Problem C...I enjoyed this because it was really visual and you could clearly see the diagonal lengths."
"Problem C...it was outside of my comfort zone. Even though we used Pythagorean Theorem it was still challenging and a new learning experience."
"Problem A...It bothered me that I didn't actually know how to start. It was fun to play around and see what you could do but it was like 'frustrating fun.'"
"Problem C...before this problem we were working on the Geoboard and playing with squares and Mr. Meyer came over and showed me a diagonal square. I had never even thought of that and it just opened my eyes and I started to enjoy it and I started just looking for all these crazy squares." Here he is actually referring to a problem we did before Task C.
"Problem B...I enjoy structure a lot. With Task A we were just drawing a bunch of triangles and my mind was, like, 'what?!?' were are the instructions here. I enjoyed B most because it was more straightforward and it was just, like, do this, do this do this...and that may seem boring to some people but to me it helps me understand it."
What is 'frustrating fun?' When does something become too frustrating to be fun anymore?
"It becomes too frustrating when I have tried everything I can do and there is no way to move forward. But, ever sense last year working with you, I have learned to not get too frustrated, to step back and think it over. There are still episodes of frustration, but nothing too bad."
"Problem B is way more frustrating to me than Problem A. In B, you have to find specifics like slope of the tangent line and average rate of change and, for me, if I don't know that one thing I get frustrated and just shut down. In Problem A there is no specific thing you need to find right here and right now, so it's less frustrating because I can feel confident in myself because every individual student is finding, like, their own personal thing compared to B where everyone is trying to find one thing."
"With Problem A, the frustration is with how far can we get. With Problem B, I got frustrated because when I got stuck, I didn't know how to go forward but in Problem A when I got stuck, I just tried something else because there were so many different ways to approach it."
"I think we are all mentioning when you feel like you are falling behind. We are really interactive with our groups and when you talk to someone else and they seem to be getting it and you don't you get frustrated with yourself. With math, there seems to be invisible pressure. Like, nobody is saying 'do it as fast as that person is' but you still feel it. But it depends on the problem because, like you guys are saying, with A you don't feel pressure at all. But with C, I'm not the person to just draw a bunch of triangles so after I drew like 5 triangles and I didn't find anything I kinda gave up...I'm not gonna draw like 6 pages of triangles."
A while back we took a vote and most of the class said they would be happy doing ONLY "POWs." Why is that? What is it about POWs that students like so much?
"I was kinda stuck between what I think would benefit me more for college and what I enjoy more. Like, Problem A is not gonna show up on the SAT. Personally, I just wanna do POWs but I don't know that we would be as prepared as if we also focused on content."
"I personally love POWs. There are no repetitive formulas that you have to do over and over and over again. I go more in depth with POWs than I would with other types of math problems that just use a formula over and over again."
"I like POWs because I feel extremely intelligent when I'm working on them. When you feel insignificant compared to others around you, you have a tendency to not feel important to the conversation but with POWs I feel like my opinion matters. In terms of college, I think it depends because I don't really feel like I need math for what I want to do in college. As a creative person, I enjoy POWs because they force you to think in creative ways and challenge your brain in ways you have never thought of before. When you have that 'a-ha' moment, there is really nothing like that and I never really experience that moment with Problem B. I mean, I eventually 'got it' and I was, like, now it makes sense but with Problem A I felt like I was doing something bigger than just math. It's almost like 'therapy math' in a way because you just feel really, really good about yourself."
"I really love POWs but I also love the structure and questions of the projects in class. But, like you said, I also love that 'a-ha' moment."
"I also don't really need math for my plans in college. Yes, there is some content that is helpful but with POWs we learn so much more than content. We learn to think outside the box and we have all these 'habits.' Other kids...when they're stuck, they're stuck. With us, we're, like, 'hey, we have other ideas.' We are learning things that, in my opinion, are more powerful than the content."
"There are some kids that really enjoy their content. So, if we were to create a math class like this for all schools I don't know if everyone would like it."
"What if we could have POWs based around content?"
"Well, we already kinda do that. Like the penny problem for instance. It is like one big POW with content built into it. I think all schools should have that."
"It boosts your critical thinking PLUS it gets that math content. And it boosts your confidence and independence."
In your minds, when are we doing 'content' and when are we not?
"I feel like content is when you are learning something new. With POWs, I don't feel like I am learning something new, I feel like I am using what I already know to explore a problem."
"I feel like we are always doing content, POWs or not. Like, in C for example, you had to use Pythagorean Theorem to solve for the diagonal. So, you kinda use content in POWs...just not as much as in the Penny of Death problem."
"I went to a traditional middle school and content, for me, was sitting in a class and taking down notes. Then studying for a test and trying to get good grades. But when I came here in 9th grade I had pretty much forgotten all of that content that I learned in middle school. I got a little sad and was, like, 'what did I work for?' and 'what did I gain?' Pretty much just note taking skills...and that's it."
"I feel like we are always learning things, but in here sometimes the content is kinda, like, hidden. Now that we are thinking back on it, I realize I did learn more about the Pythagorean Theorem. I guess sometimes I just don't have an exact name for what I'm learning. You learn something, but then you don't learn it's name....so you might think, I guess we just drew some triangles.
"Sometimes we are just doing work and until points it out you don't realize 'oh yeah, I did content.'
ME: "This is all so interesting to me. To me, content doesn't have to be something that is coming from a textbook or from some source. Anytime we are doing math together and we create a rule...that is content. Doing math together IS the content. It's not like we need to create something that already exists in order for it to be considered a good use of our time."
To me, that is the difference between A and B. In A, because this isn't in a standards list somewhere, I feel OK about just giving you this problem and saying 'let's see what happens.' You all created some interesting rules and those rules are pieces of mathematics that you have created. With B, this is designed to get at a specific thing that does exist. From my perspective, that is more limiting. There is one right thing and, like, if we don't get that one right thing we haven't accomplished the goal."
Do you notice a difference in how your group functions with POWs versus other types of problems?
"It depends. In both scenarios, sometimes a person will latch onto something quickly and have a piece figured out."
"I can see a clear shift when we are doing content-based problems versus POWs. There are people who get it and people who don't. Another student and I will still be figuring something out and the other part of our table is like, 'done...we got it.'"
"With content, our table is usually split. But when we are working on POWs, we all work really cohesively. We have all these different ideas that everyone is throwing into the mix and from that 'idea throwing' you come up with this great new idea that we have all created together. And with content it's more like 'this goes to this.' With POWs like Problem A, we had all these different approaches and they were all correct."
"With POWs, even someone is struggling with math will put their idea out there and then we will combine them all together to create this cool hypothesis that we can just go and test. It's really cool to see everyone putting their ideas into this giant pot and we just see what happens."
"I think group work is a lot different when we are doing POWs than content. With content their is one answer and one solution....there isn't really a lot of in between. With POWs everyone usually has something different that we just pile together to create a central answer. I think POWs are a lot better for group work."
I use "Problems of the Week" (or POWs) with my class every week. As far as I know, the Interactive Mathematics Program
authors coined the term, but the problems themselves are usually well-known (?) puzzles from the history of mathematics. Students have always responded well to these types of problems...even students that don't always respond well to other types of tasks in the classroom. This has always been interesting to me, but it has really come onto my radar since I started my action research on habits of mind and agency. It has led me to believe that the tasks we use, teacher expectations about student outcomes, and what we value as teachers all have pretty powerful effects on the preservation of student agency.
I have become really interested by the difference between the two. I have included two tasks below that I used in class recently. The first is modified from an IMP textbook and is, in my opinion, "procedures with connections." The second is a "Problem of the Week" borrowed from 'Thinking Mathematically'
which, in my opinion, is "doing mathematics."
(Procedures with Connections)
Context: Students have been working on a unit problem about whether or not a penny dropped from the Empire State Building would kill someone if it hit them on the head. We dropped a ball from the roof of our school, modeled it, and found that this equation was useful.
Students are pretty interested in the unit question about the penny, but the second task was by far more popular and definitely engaged more students in a variety of ways. Today I asked my Seniors, "How many of you would be happy if the class was only POWs?" Out of about 25, all but two said that would like that. The two that said they wouldn't like it cited "preparation for college" as their reason for not wanting that.
I think there is a lot here about the set up and structure of tasks in relation to student agency, but I haven't unraveled that yet. I would love to hear your thoughts.
In 'Task 1' above, I think it is clear to the student that this is a typical math problem designed to get them to understand a particular topic or concept. In other words, it is clear from the outset to them that they are expected to get to a certain place by the end of the task. It has given them something external to value themselves against. Now, all of a sudden, if they understand it they are smart and if they don't they are not smart (I'm hypothesizing here about what the task is implicitly saying to students).
In 'Task 2," the situation is much different. The problem is pitched like a puzzle. There is a clear question, but the solution to that question is not the end of the problem. The problem ends (potentially) when students create a piece of mathematics to describe what they are noticing. I think this makes things a lot different. This task isn't designed to get students to understand something specific. It is designed to get the DOING mathematics and thinking mathematically. There is no pressure here. I don't feel like I need to hold everyone accountable to something. We can just think together and wherever we end up....that is where we will be.
What We Value
Standards and teacher expectations put pressure on students to 'know' certain things on a specific schedule. When we (teachers/curriculum/whomever) set this finish line in advance, it changes a learning environment. It becomes about measurement and judgement. I would just much rather put the priority on student thinking, student confidence, and student agency.
I think we should ask ourselves..."Why teach math?" I can't help but think that a lot of the things students 'learn' in school will soon be forgotten. What they might remember, though, is what our classes and teaching taught them about themselves. I want students emerge from high school trusting their own thinking and having confidence in their ability to figure things out. I'm beginning to think that sometimes the tasks we choose and our expectations for students might get in the way of that.
I am about a month into the school year and, consequently, a month into my action research project. Very briefly, my action research project involves looking at how a classroom that is centered in mathematical habits of mind can support students in realizing (or regaining) a sense of agency/authority. Early on the year, my classes worked on a couple different tasks ("Checkerboard Squares" and "Consecutive Sums"). Along the way, I asked them to be particularly meta-cognitive about the mental actions that were helpful in making sense of each task. They ended up developing this list of the "habits of a mathematician:"
Look for Patterns and Regularity
Experiment and Play
Be Confident and Persistent
Conjecture and Test
Solve a Simpler/Related Problem
Be Systematic and Organized
Justify and Support
Create a Rule (Generalize)
We posted these habits in the room and refer to them often. Last week, I was hoping to get some preliminary/baseline data about how students perceived themselves as mathematicians. I asked them simply, "Do you feel like you have strengths as a mathematician? If yes, what are they? If not, why do you feel this way?" I was expecting many students to respond "no." The results were pretty interesting:
Out of the 68 students polled:
54 replied yes
10 replied no
4 replied yes and no
BUT, of the 54 students that said yes, HALF of them cited one or more of the habits as their mathematical strengths and they used the exact verbiage of that/those particular habit(s). These habits are new to them this year, so I was surprised to see that so many of them are already internalizing them as part of their mathematical identity. I'm optimistic to see where this leads because I plan on having all students do a "snapshot of a mathematician" soon where they identify personal strengths and weaknesses based around these habits.
Each student has a blog where they post weekly about one piece of work that they are proud of and how it is representative of one (or more) of these habits. There have also been some interesting things happening on their blogs:
One student wrote about how she finds herself "experimenting and playing" in singing:
"This is an important step for any singer, weather they know they are doing it or not. The more a tune is played with, the more original it will be, even if it has been sung 1,000 times before. A song can be sped up, slowed down, the pitch can be changed, the melody can be altered...there are endless possibilities, but progress can't be made without trying new ideas, even if they aren't all golden."
One student wrote about how she used "systematic organization" in rearranging her bookshelf:
"When I was organizing my bookshelves, I needed to find a system that would work for the books."
One student wrote about how he "looked for patterns and regularities" in the stock market (not an assignment):
"The task at hand was to search for trends and patterns within the graphs of the DOW Jones industrial that I had made. Essentially I was looking for ways to predict stock market behavior and how to ensure some safe investment."
One student took a task about group norms that we did and turned it into a task about systematic list making by "experimenting and playing:"
"The interesting part came when my group member tried to pair them into couples. We had an interesting discussion as to whether monogamy was possible in a closed environment such as this one. I didn't think so, and my group member insisted it was. So, I attempted to create a genetic graph."
The really interesting thing to me about all of this is that students are taking mathematical ways of thinking and recognizing these same mental actions in other parts of their daily lives. I am excited to see where this leads!
My grad school advisor keeps telling me that I need to write more about my thoughts and observations in the classroom for my action research project. I figure I might as well multi-task here and just blog about my research along the way.
We are now a month into the new school year, but I collected some data from them during the first week that I never had time to analyze and write about. There are some interesting (but not all that surprising) things that I found. First, I gave them a journal prompt that asked:
In math class, what is the role of the teacher and what is the role of the student?
Of my 72 students, 79% identified the teacher as the authority (status and epistemic), the student as passive recipient, and/or the role of school as knowledge transmission. Here are some of their responses:
Role of Teacher
All of the following are pieces of direct quotes from students:
Grade the work
Impart knowledge on the students
Teach the material
Share their knowledge
Help solve problems and do them over and over again
Pass on knowledge
Teach math concepts so they are simple and easy to understand
Show examples of a problem
Teach students how to do the assigned work
Give and deliver information
Role of Student
All of the following are pieces of direct quotes from students:
Listen to the teacher
Take all the knowledge the teachers have to offer
Learn from the teacher
Absorb the knowledge
Learns what the teacher teaches
To be honest, this wasn't surprising...but it is alarming. It's alarming because the way in which we teach math inevitably (and implicitly) simultaneously teaches students things about themselves as mathematicians. Here is the evidence (responses from beginning of the year survey):
80% of my students think they can't do a math problem unless I tell them how to do it first...
85% think they need to memorize things...
and about half of them don't think they can create mathematical ideas, formulas, and rules.
All of this is further support that, as I cited in my research proposal (bold added for discussion here):
1. "our classrooms are the primary experiences from which students abstract both their definition of mathematics (Schoenfeld, 1994) and their sense of self as an active participant in the authoring of mathematics (Lawler, 2010)."
2. "Identity is a model for self-direction and, as a result, a possibility for mediating agency (Holland et al., 1998). Many students have established their identity as receivers of knowledge, with no active role in creating or critiquing mathematical claims. As a result, their sense of agency is surrendered. Research supports the view that such environments cause students to surrender their sense of thought and agency in order to comply with the procedural routines outlined by the teacher/authority figure (Boaler, 2000). Signs of this include negative attitudes towards math, lack of connected knowing, and the belief that mathematics is absorbed rather than created."
I'm interested in the idea of agency (mathematical and otherwise). I'm interested in the hidden curriculum in our classes and how it impacts students' definition of math, students' formation of self, the mediation (or perpetuation) of status/race/economic/power issues, and the recognition of their own ways of thinking and being mathematical in the world.
As I have posted about before
, I really want to do a problem-based unit this year in which students attempt to answer the question, "How far away is the horizon line?" I'm getting ready to start the unit in about a week, so I have been thinking about it a lot lately. Mostly, I have been thinking a lot about the idea of a line that is tangent to a circle and how students might conceptualize that. The "visual" that I get in my head when I think about this horizon line question is this:
This morning I was sitting around the house with my girlfriend, and I decided to see what visual she might come up with and what ideas she might have about tangent lines. First, I asked her to draw the visual that comes to mind when she thinks about the horizon line problem. This is what she drew:
Pretty fantastic, right?!?! Certainly more artistic than my picture! It was interesting to me how differently two people might be thinking about the same scenario.
Then I asked her to draw a circle. Then I asked her to draw a line that touched that circle in only one point. She drew line #1 below (I added the numbering to make some distinctions here). Our conversation went something like this:
ME: Tell me why you decided to draw it that way.
HER: Well, cause it would only touch the circle in one point.
ME: What would happen if you continued your line?
HER: It would cross the circle on the other side.
ME: So would that work?
HER: I guess not....Well, I was thinking about this (draws line #2).
ME: Why did you decide not to draw that one?
HER: It just seems like it would touch the circle in more than one point.
ME: What if we zoomed in? (I drew the picture on the right)
HER: Hmmm...not sure. I still feel more certain that line #1 would only touch in one spot.
To me, this was really interesting. I wonder about how students think. I wonder about their mental models. And, mostly, I wonder how much we actually listen to them and respond to how THEY think. It can be tempting to tell students about a tangent line in the context of this problem, but that would be a missed opportunity for rich discussion. Perhaps more importantly, it would be imposing a way of thinking on them that is incompatible with how they are currently thinking. You hear a lot of people say that they don't like math. I wonder how much of that is due to the fact that they have learned that math doesn't care about their ideas, that math is always right, and that they need to learn to think more like math.
For one hour of my school day, I work with an amazing group of Seniors in a class I have called "Mathematical Thinking" (more about that in a later post, perhaps). Mostly, the class is a mixture of problem-based units and other miscellaneous open-ended puzzles, problems, and mathematical games. Yesterday, we worked on a puzzle/game called Cartesian Chase. I played a few games against students to demonstrate the rules (we confined ourselves to a 3x7 rectangle) and then just let them play for while. Then, I had them stop and record anything they were noticing in terms of a strategy that seemed to be working. Then, they switched and played with new partners for a while longer. I stopped them again after a few games and had them record updated strategies. We ended with a class a few "undefeated" people playing each other. It quickly became apparent that there was a winning strategy at play. In the process of all this, here is what I noticed:- nearly ALL of the students were engaged and playing for the whole time- students were having fun with each other- we had a few early conjectures in place about what strategy might be best- students uncovered structure in the problem, used it to win every time, & were able to clearly explain it- after the game was "solved," a few students were curious:
"what if we added another column?" and "what happens with other board sizes?"I work hard to bring this same spirit of playfulness to other lessons. I work hard to make every day feel like a puzzle in our class. For some reason, I can never quite bridge that gap in the way I would like. I think I get pretty close most days, but for some reason "how many burger combinations are possible?" still feels more like a math problem and less like a puzzle to students. Maybe it has to do with our intent as teachers? Do we place too much emphasis on students "knowing" something specific by the end of the lesson? Could we set up the task better (slower?) so that it emerges as a puzzle? I have a lot of questions, but I do know that I value
what students are learning about themselves as mathematicians and thinkers from a lesson just as much, if not more than, I value students knowing some piece of the thing we call "mathematics."
I should start this post by saying that I work in a school with "inclusive classrooms." Students are in the same class based on grade level alone, no tracking by "ability level" or any other metric. I suppose I should also say that I am doing an action research project on mathematical agency. Agency is a slippery word to define, but for the sake of simplicity for now let's say mathematical agency is defined by a positive self-concept as a mathematician (as one who believes in their ability to make sense of mathematical tasks and situations and to judge the validity of those responses). Lately, I have been thinking most closely about how the set-up and discussion of tasks in the classroom can affect student agency and power - for good and bad.This all came about the other day when some of Brian Lawler's credential students were in observing my 10th grade class working on "Consecutive Sums." Basically, the prompt is: "explore consecutive sums and see what you discover" (where equations such as 1+2+3=6 and 7+8=15 are considered consecutive sums). Eventually, the conversation turned into trying to figure out which numbers can (or cannot) be written as consecutive sums. I put a table up on the board with the numbers 1-25 on it and asked groups to send people up to fill in the chart once they had found one. What ended up happening is that about 5 students dominated this part of the lesson while others sat and watched. I know there are plenty of suggestions about better ways to handle this particular part of the lesson, but I think the implications are greater than that.I have read several articles and books for my action research (a couple good ones if you are interested) that outline an amazing vision for a classroom community in which students present ideas, challenge each other, and construct meaning together.
Most times when I try this, one of two things happens:1. I select and sequence student share outs so that certain voices are heard that are usually silenced. Mostly, because I have created the conversation, there isn't much to talk about and students seem disinterested. They aren't debating anything; they aren't solving things collaboratively.
OR...2. I'll select one or two pieces of work to get a conversation started and then step out of the way. This usually gets students talking and debating. The only problem is, it's usually no more than 10 students out of a class of 20-30.I'm not sure I have any answers to this yet, or even that an answer exists that will work for all groups of students. But, I am really interested by the intricacies of teaching...by the tasks we choose, by how we set up those tasks, by how we get students talking about those tasks, by how we conclude those tasks, and, especially, how ALL of those moves inevitably make a difference in what students are learning about themselves as capable mathematicians.
This last bit, to me, is far more important than the "mathematics" that they learn.