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."
Student:  "7!"
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.
  • Most students found something to pursue on their own
  • Most gravitated toward something that was appropriately challenging
  • The initial task was relatively narrow and was defined by me, not them
  • It was difficult to have students understand, respond to, and challenge each others' work because they were all working on something different


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.
  • Students were the ones engaged in "finding the task" and setting the parameters
  • It opened up the possibility for students to pursue questions outside of the one that might have been intended or suggested
  • It took a lot longer and was more difficult to facilitate
  • It felt like I was tricking students into asking THE question versus actually giving them freedom to explore their own questions
  • Students had a difficult time accepting the suggested parameters of a different group if they were not the parameters THEY saw as fitting (there might be some implications here about the questions we pose as teachers seeming unnatural to students?)


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.
  • This was the closest I have come to having student curiosities drive the work; felt like students had a genuine opportunity to follow their own question
  • Students were engaged in questioning, setting parameters, exploring, and then adjusting their question or parameters if they needed to
  • Many students really enjoyed the freedom and creativity involved
  • Almost all students were engaged in mathematical activity
  • Students seemed to, initially, search for a more shallow level of depth than we usually accomplish as a class
  • Some students were very turned off by the ambiguity and openness
  • I didn't know how to bring things together or take it further when students were all over the place
  • As a result, I eventually defined a question for the whole class. Even though it evolved out of their work, it was still defined by me

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:
  1. I don't love the idea of tricking students into asking the question you want them to ask, but I also have trouble when students are all working on different things.
  2. I'm curious about the "initial event" that prompts student questions. Should I start small and well-defined and then move to open exploration (example #1) or should I start wide open and leave it wide open (example #3)?
  3. In most cases, I found it hard to facilitate student work. How can I get students to share work, challenge each other, and challenge themselves?
Mostly, I would love to hear about your experiences, questions, advice, or thoughts.
Today a student came in at the end of the day and said, "Mr. Meyer...wanna see why pi equals 4?" I was intrigued. He started me off with this image:
and then said, "The perimeter of that square is 4. But what if we pulled the corners in so that they touch the circle? We wouldn't change the perimeter of the square. Like this..."
"and we could just keep doing that..."
"and keep doing that..."
"until pretty soon it would look exactly like a circle even though its a bunch of really small steps."
"So wouldn't pi be equal to 4?"
"Oh yeah, and ignore the snail I drew."
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 hosted another focus group as part of my action research. In trying to tease apart the pieces of a classrooms experience that foster and promote agency, I asked students questions to try to get a deeper understanding of how they see themselves in the work we do together. Below are the questions I asked, followed by direct quotes from student responses.

Pick one piece of mathematics that you are proud of. Why did you pick it?

“I really like to work with conjecture and testing stuff so I liked when we were able to experiment with trying to get certain numbers using consecutive sums. I like being able to be ‘hands on’ with stuff and try things out to see if they work or not.”
“it made sense”
"We went farther than the original problem and it was really fun doing that."
"at first I wasn’t really sure what to do with it but then when we had time to collaborate with our group, my group member had this really good idea and I was able to build off her idea to create the rule we were looking for. Also, I really like that at the end of class we had the opportunity to keep pursuing it in different ways by looking at challenge options. I thought that was fun because it let us go beyond and I was able to come up with a different rule for a bigger problem."
"I saw patterns going on with it and then I found a rule for it. At first I just based my rule off the pattern that I saw but then I understood where the rule came from and which squares it was talking about."
"I went about finding a rule in a different way than a lot of other people did but I still came to the same conclusion which I thought was really cool and it was good for me to be able to explain the way I do things. Our group also worked on an extension problem and we actually came to part of an equation for that.”
"seemed more like a puzzle than a traditional math problem so I experimented a lot more with it. It was something I could play with."
"testing out what works and what doesn’t and explore."

Pick a piece of mathematics that you were NOT proud of. Why did you pick it?

"we didn’t spend enough time on it in class"
"we had these restrictions that seemed kinda confusing. I think it would have been easier if we had some sort of freedom"
"frustrating and it seemed like this little thing we did for like a day and then it disappeared off the map.”
"I was confused but other people got it"
"It didn’t connect to anything. It just kinda ended…there was nothing to explore"
"it seemed like an enormous problem, like it could go on forever and there was no sure way of knowing."
"a person at my table gave me a rule for the problem but I didn’t do any of the work for it and I didn’t understand it because it was someone else’s work."

We start each day with one big, open question and then use the class period to explore that prompt. How do you experience these questions and the idea of experimenting and playing?"

"I like it because it is usually a question that can be answered in a bunch of different ways so every time you hear from somebody it’s a different answer and it makes you think."
"You have to think about it a bit more on your own and develop your own ideas first."
"It helps you because you can create your own ideas for how to solve that problem. Then we you talk in groups or with the whole class you can bounce ideas off of everyone and they can give you their feedback and then you end up with a rule or a way to solve the problem that is even better than what you could have come up with on your own."
"We have freedom and, in math, you usually don’t think of having freedom. I used to think of math as having a bunch of rules and everything was set and strict. With our open questions, it just kind of blows that concept away. They make you stop and think instead of just memorizing a rule and that will help us outside the classroom or when we have a problem with a rule we don’t know."
"It’s a really nice way to get away from traditional math where it’s like, ‘here’s a worksheet, go figure it out…if you don’t get it, then tough.’ I think experimenting and playing with our questions is a great way to get your mind going."
"I don’t mind them but they frustrate me after a while sometimes because I wish there was just one answer to get to rather than something that just keeps getting bigger and bigger. Sometimes it just keeps going and I lose interest and just want to move on to the next problem. I think it might also be because my previous education was very traditional and so I have become used to just having one answer."

Another student's response to the above quote:
"I have the same problem where I get bored easily, but I like it in here because our problem kinda changes when we look at extensions."

I find these focus groups fascinating. Our students have such incredible insight into the intricacies of the classroom. I was struck by the themes present in work that students were NOT proud of. They noted they didn't connect to work when:

1) there wasn't enough time to let them develop ideas,
2) they didn't see the connection to a larger theme/unit/purpose,
3) there was not an appropriate level of difficulty, and
4) when the problem didn't allow them to think in a way that was natural to them (no room to explore or confusing restrictions).

On the other hand, it seems much easier to boil down their thoughts about work that they DID connect with. As I listened to them and re-read their responses, I just kept thinking, "they are telling me to just let them explore an idea and develop their own ways of thinking and understanding." The discussion about 'real world' comes up a lot in the math ed community. I have seen students be incredibly motivated by exploring block patterns and drawing shapes and I have also seen students be incredibly bored by problems that are based in some real world application. Mostly, I just think students want a problem that does not seem unnatural to them and one that allows them to explore, think, and develop their own 'wonderful ideas.' As Eleanor Duckworth writes:
"Textbooks and standardized tests, as well as many teacher education and curriculum programs, feed into this belief that there is one best way of understanding, and that there is one best, clearest way of explaining this way of understanding."

"...there <are> a vast array of very different adequate ways that people come to their understanding."

"...children can probably be left to their own devices in coming to understand these notions."
I suppose I am not suggesting that we just walk into a classroom with a group of students and say, "explore something" (although I'm also not ready to say that I disagree with that). What I do think could be helpful is to broaden our mathematical goals for students. I'll use the current work with my students as an example. We are currently working on a unit in which we are investigating the area of the Koch Snowflake. In the past, I have used that as an opportunity to engage students in investigations with arithmetic and geometric sequences (part of our Algebra II standards). This year, I broadened that goal to engage students in "looking for patterns and making predictions." That broader goal has allowed me to let students pick multiple paths and explore things that are interesting to them...so long as they are looking for patterns and making predictions from those patterns. I feel confident that the students this year would be able to work with arithmetic and geometric sequences as well as, if not better than, my students in the past even though, this year, we never even discussed them specifically.

There is still a lot to think about, and good task design does not necessarily ensure a rich mathematical experience for students. But, I think it is a good start.
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.
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):
  • 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 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). I am attaching the printouts that I photographed in case anyone wants the details.
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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
  • 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 feel like there are a couple ways to go with this trajectory next:
  1. Stay with resistance, but use a continuous curve
  2. Stay with this discrete model, but switch contexts (maybe speed or rate of growth)
  3. Stay with discrete and resistance, but use a more complicated readout

I'm leaning towards #1. What do you think would be best for the students?
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?

Her response:

"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.

Task A
(click to enlarge)

Task B
(click to enlarge)

Task C
(click to enlarge)

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."