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?