Perspectives - the story so far.

I think I am learning more in this Perspectives class than anyone else. Probably okay.

My sense of our emergent narrative over the past month is:

(1) what happens in school is often "schooly" and (with Steve's insight that that's a term worth pursuing) we've worked on what we think this means and have a few ways of thinking about it, but my favorite is the idea: "I'm learning what someone else wants me to know." And, I'd add, providing answers that someone else finds reasonable. It's a kind of performance of someone else's goals. And, with some caveats, this is not what we want school to be.

(2) we thought then about the Bakhtin quote and what words feel like a "performance" of sorts - not really "our" words. (natural log, e, curl, entropy, genus, etc.) - and there were a range of categories (which we represented as emojis), but most interesting to me: these are words we can use for a test and in class, but they don't feel like they are our own. (related): there's a disconnect between the word and any ability to use it in another setting. ('why would i use natural log?')

(3) I suggest that one of the ways we get students out of this "schooly" game is to engage students in puzzles. (I used the analogy to the infield fly rule, and those in our class who understood it were all the same as those people who play baseball; I argue that playing the game is part of being able to understand its concepts.) That is, we don't teach students about ln and practice using it, we get students involved in a puzzle that has, perhaps, ln as its solution. In class, I tried to set up puzzles that get at those ideas. (draw all the kinds of currents that might make a paddle spin; sketch a function where the derivative of the function is also the value of the function (and if you're stuck, start with a function that crosses (1,1)); and then "if 2 has one 2 in it, and 4 has two 2s and 8 has three 2s..., sketch the function for the number of twos in a number...")

(4) from this came the question of "what's a puzzle" (posed by Steve, who always finds the right question) - and the answer (posed by Steve after a lot of discussion) is kind of genius I think - a puzzle is anything that you find worth solving. Don't care about solving? Not a puzzle. (a little aside: I want to push against the idea that - say - "for inner city students to learn about physics you should have them write rap songs, b/c writing rap songs is a fun puzzle to those kinds of kids." -- While I am aware of how complex rap can be, I want it to be the kind of puzzle that has, in its solution, some piece of fascinating content. So I said something like "it's tempting to think this means...")

(5) but from there we became intrigued by "worth solving" -- with competing claims that might not be so different -- With crosswords as the context (something I love to do) some said "but you don't actually care about the solution" and "you wouldn't accept $5 for someone to give you the answer" (what I like is solving, I don't care about the answer.) Others argued that I *do* care about the answer -- it's the pursuit of the answer that drives me. Steve suggested that what you want is to succeed in some kind of rule-governed activity ( I think "rule - governed" just means that 'success' is somehow defined?) ... but I think it's something other than / in addition to / maybe the same as that. It's the idea of vexation. I think vexation implies care about solving / finding resolution -- but I'm not sure that it means "I like playing the game required to reach that solution." but maybe it does mean "I'm willing to work to figure out the solution."



(I remember a student, Dee, saying something about how frustrating our class was at times - because, I would say, she deeply cared about getting the answer; and I remember the students in a different semester really wanting to know what the rule was for how light would bend when it entered a material and I said something like "they're doing a lab on just this thing today in physics" and they were like "we have to run over there and ask them! - TELL US WHAT YOU KNOW!" and I said "you know they don't even think what they are learning is interesting??")

three favorite passages

These three quotes -- I can almost quote them from memory -- keep coming to mind.

"It’s very difficult to avoid, the student being lost in the beginning and the school set up to emphasize short-term performance.  So they tend to imitate what you do as a way of associating with what you say.  But what you’re trying to do is develop their sensitivities and not your own.  I have strong philosophic reservations about what it is we are actually talking about when we use the world morality, but as that word is most commonly used, I would think that the most immoral thing one can do is have ambitions for someone else’s mind.  That’s the crux and the challenge and the responsibility of having the opportunity to deal with young people at such a crucial time in their formation.  One of the hardest things to do is not to give them clues—‘Here, do it this way, it’s a lot easier’—and instead to keep them on the edge of the question." - Robert Irwin

“You fight your superficiality, your shallowness, so as to try to come at people without unreal expectations, without an overload of bias or hope or arrogance, as untanklike as you can be, sans cannon and machine guns and steel plating half a foot thick; you come at them unmenacingly on your own ten toes instead of tearing up the turf with your caterpillar treads, take them on with an open mind, as equals, man to man, as we used to say, and yet you never fail to get them wrong. You might as well have the brain of a tank. You get them wrong before you meet them, while you're anticipating meeting them; you get them wrong while you're with them; and then you go home to tell somebody else about the meeting and you get them all wrong again. Since the same generally goes for them with you, the whole thing is really a dazzling illusion. ... The fact remains that getting people right is not what living is all about anyway. It's getting them wrong that is living, getting them wrong and wrong and wrong and then, on careful reconsideration, getting them wrong again. That's how we know we're alive: we're wrong. Maybe the best thing would be to forget being right or wrong about people and just go along for the ride. But if you can do that -- well, lucky you.”  - Philip Roth

‘Of course, I see your point of view, Archie, I do. But my point is, and has always been, from the very first time we discussed the subject; my point is that this is not the full story. And, yes, I realize that we have several times thoroughly investigated the matter, but the fact remains: full stories are as rare as honesty, precious as diamonds. If you are lucky enough to uncover one, a full story will sit on your brain like lead. They are difficult. They are long-winded. They are epic. They are like the stories God tells: full of impossibly particular information. You don’t find them in the dictionary.’  - Zadie Smith

one last comment

My philosopher friend explained to me, in discussing who gets to be a "real philosopher:" "If you study free will, you get to play against the avatar of Leibniz."

He also described teaching college as: "we’re gathering up probationary citizens en masse and they want to know 'how do I become a full member of this tribe?'" (In discussing rites of passage.)

I have an idea.

Let's all (* all = anyone who reads this blog and anyone else who wants to come and especially anyone with a currently-three-year-old kid) take a year-long sabbatical at the same time, and rent a bunch of houses in a small village in Switzerland together and do lots of fun work together and eat well every night. And ski in the winter, and take trips all over the place.

This will be in 2021 or maybe 2022, when I am eligible for a sabbatical.  START PLANNING.

More thoughts on what we mean by "Schooly"

We continued to talk in class about what we mean when we say school-y. There was some (extensive) conversation around: "we can't keep using this word until we define it" and (from faculty) "we can't define it until we use it a lot more!"

And someone (B) suggested that "schooly" things can be really amazing and exciting - they really engage you in the discipline: she followed a step-by-step procedure to embed some DNA (?) in a seed (?) that then made the plant purple when it grew up. She was so excited just to talk about it. (I love this example b/c I think I would do things in school to make it so that it wasn't 'schooly' -- that you can approach a task as 'schooly' or not... but some tasks are harder than others to find ways in.)

So we gathered their ideas of what it might mean:

• you can't apply your learning to "reality" 
• someone elsewhere wants me to know this
• hand-holding through the process (this was B's example)
• anything at school
• looking for the 'school-expected answer'
• robotically/mechanically, no fun or engagement
• "book smart"
• "I go to work so I don't have to 'do school'"

What I pulled out of the conversation was that there are some themes: there's an affective part -- things where we're really engaged and that "spark" us are not "schooly," things that are authentic and have real-worldness about them are not "schooly," and having ownership in the activity (not someone else's demands on you) is not "schooly."

Another objection that came up - from S.N. - was "but I like school - this makes it sound like school is bad." I tried the analogy that saying something "feels homey" or "feels like home" has a certain meaning -- and we can kind of agree on that meaning even if not everyone likes home very much.

"school problems are designed to imagine someone else's thinking"

Students read some brief vignettes about students doing wacky things in math:

Episode 1

At the end of the 1970s, French mathematics education researchers gave the following assignment to elementary students:

“There are 26 sheep and 10 goats on a boat. How old is the captain?”

76 of 97 students calculated the captain’s age by combining the given numbers by some operation like addition or subtraction. This has since been repeated in a number of countries and the findings have been the same.

Episode 2

The story took place in a class of 9-10 year old students.

The teacher taught the following algorithm facilitating calculation of the difference between two numbers:

328 - 47  = [add three to each number to make the ones place a zero]

331 - 50  = [add 50 to each number to make the tens place a zero]

381 -100 =  281 [... and now that it looks simple, subtract!]

Several weeks later, the students were assigned the following task: How would you carry out the following calculations?

999 - 111 =

Most students (16 out of 19) applied the algorithm:

999  -  111 = [add nine to each number to make the ones place a zero]

1008 - 120 = [add 80 to each number to make the tens place a zero]

1088 - 200 =  888 [... and now that it looks simple, subtract!]

Episode 3

The story takes place in a class of 15-16 year old students.

On a math test, the students were asked to solve the following problem:

Find x ∈ R such that: a) sin x = π/3, b) cos x = π/2.

Only 25 % of the students give the correct answer to a) and 29 % to b).

And then as a survey question (for quick feedback/snapshot before class):

Imagine you give students a test that asks the following: "If we know that one biker covers the distance between the towns A and B in 6 h, how long will it take three bikers to cover the same distance if they set off together?" (given on a math test on multiplication.)

One student (T.) mentioned that this was a trick question. So I began with that idea - is this a trick question? Why would someone say "yes" and why would someone say "no."

Profound ideas that came up:

1 - when given a hammer, everything looks like a nail. (and a student called it "heuristic exhaustion")

2 - there's a difference between what a question asks and what a question means. (love this)

3 - testing multiplication isn't the same thing as testing comprehension

4 - it's not "really a problem" - it's "schooly" - by which we mean (and we discussed this a bit) "school problems are designed to imagine someone else's thinking" - that is: a student's job on a test is not to answer the problem honestly, but to imagine what the teacher wants you to write. (LOVE THIS) (they compared it to this - I love thinking about why this "works" as a joke in light of their ideas: https://static.fjcdn.com/pictures/Find+x_30c251_5488082.jpg)

5 - there's a difference between a "trick" question and an "unfair" question (would be curious to hear what that line is)

More quotes/summary of the DC

From Didactical contract and responsiveness to didactical contract: a theoretical framework for enquiry into students’ creativity in mathematics - by Bernard Sarrazy and Jarmila Novotná

It is our conviction that mathematics already discovered is ‘dead’ mathematics and must be brought to life by teachers. To achieve this revival, teachers create situations in which they can show to students the use, the interest, and other aspects of the mathematics that they are planning to teach. But the teachers cannot place themselves in their students’ position in order to teach them—just as we cannot walk instead of a small child, although we do everything to help the child do so. This is what didactics of mathematics calls the didactical contract (Brousseau 1997; Sarrazy 1995; Novotná and Hospesová 2009); it is based on the fact that what the pupil has to learn is already known in the culture but cannot be taught. This contract is paradoxical as it cannot be recognized unless it is broken: the student can learn only when he/she accepts that he/she will not to be taught everything; when he/she accepts he/she will engage in an activity in which he/she can learn mathematics. 

Therefore, to teach means to create conditions in which something new may emerge. This creation is central to the teacher’s work: to create problems and situations which will enable the student to look for new ways of solving the problems. The student’s creation is not based on reiteration of the taught algorithms but on a unique and innovative way of using them! Therefore it is less creation itself that is to be examined, but the social, pedagogical and didactical conditions (characteristic for mathematics) of this creation. 

It was introduced by Guy Brousseau in 1978 as one of the possible causes of specific failures in mathematics: students answer to comply with what they think is expected of them by the teacher, not by the assigned situation. The most-cited of Brousseau’s definitions is that the didactical contract corresponds to ‘‘the set of the teacher’s behaviours (specific [to the taught knowledge]) expected by the student and the set of the student’s behaviour expected by the teacher’’ (1980, p. 127). Although this definition is very concise, if it is to be understood correctly it requires further clarification. Only then does it escape the danger of psychologizing interpretations such as: students misinterpret what they are asked by the teacher, they think that..., they believe that...; this interpretation results in the conviction that if the problem or question were better formulated or explained, there would be no difficulty. One of the manifestations of learning is students’ ability to propose original solutions to new problems. This ability is one of the criteria by which the teacher may find out whether the student has grasped the taught mathematics (Novotná and Sarrazy 2009; Brousseau and Novotná 2008). It is obvious that the teacher cannot teach (at least directly) this ability of creating new solutions: he/she can demand it, expect it, motivate it, but cannot require it. This is one of the fundamental paradoxes of the whole didactical relationship, which Brousseau (1997) modelled by the didactical contract. As stated above, the teacher cannot teach what he/she expects and hence the student has to re-create it on the basis of what he/she already knows (Sarrazy 1995, 2010; Sarrazy and Novotná 2005).

... It is clear why the traditional mechanistic epistemology of a rule and its application is so persistent. The purpose of knowing an algorithm is selection of uses appropriate for and effective in a particular situation.

One thing that strikes me as I read this is how this is not synonymous with framing, but has elements of framing:

Goal 1 of our paper is to pinpoint the variables influencing the quality of solving of the problem and to show that what influences this quality is not to be sought in the pupil’s psychology but in the situation which, defined by its limitations, imposes certain types of attitudes. 

And later (reminding me of Jasmine Ma's paper ... and mine):

Strong variability in teaching makes these strategies impossible as it brings disruptions to the routines: the student cannot rely only on superficial indices any more, and can neither anticipate, nor master (or if so, then in a very difficult way), the chain of sequences that allow him/her to decipher what behaviour is expected by the teacher. 

Some things they examined:

• Situation 1 (Evaluation by the experimenter—explicit frame ‘a researcher’s practice’). It is the experimenter who proposes the assignment; the students were informed in advance that the test would not be marked; intentionally, other information about the test was not disclosed.
• Situation 2 (Mathematical competition—explicit frame ‘a mathematical competition between classes’). This test was presented to students as a competition between classes in which each class chooses which of the posed problems will be assigned to the other classes. The test was divided into two phases: in the first phase the rules of the competition were presented and then each student posed a problem and then submitted it to the experimenter. The test itself was carried out in the second phase, a few days later, and the researcher posed his own problems. Here, the nature of the task is not individual as in the other situation, but collective.
• Situation 3 (Evaluation by the teacher—explicit frame ‘summative evaluation at the end of term’). Each teacher was asked to carry out such an evaluation as they would normally use at the end of the term. This evaluation should in addition include the target problem. Here, the nature of the activity is clear to students: individual and marked by their teacher.
• Situation 4 (Warning—frame ‘experience of a researcher’). The aim of this situation is to verify that students are able to correct a ‘defective’ problem assignment. That is why the students were informed of presence of both non-calculable and calculable (classical) problems in the given set of problems. 

The results of factor correspondence analysis (Fig. 2) clearly show that production of answers to the equivalent problem is, regardless of the students’ school level, more strongly linked to the situation than to their mathematical abilities (in the set of students v2 = 88.01; s.; p\.001).

more... later.

Didactic contract

I am working today on a lit review of Brousseau's didactic contract. -- My hope is that this will be a productive way of describing teachers' (and preservice teachers') concerns with / dilemmas around responsive teaching. This is for a project with Amy.

The Didactic Contract (DC) is: "a relationship which determines—explicitly to some extent, but mainly implicitly—what each partner, the teacher and the student, will have the responsibility for managing and, in some way or other, be responsible to the other person for. This system of reciprocal obligation resembles a contract."

So we're interested in seeing what the LAs/teachers seem to suggest is their implicit didactic contract -- I've been thinking that there are a range of possible "contracts" (obligations) that each side (teacher // student) could be assuming are in place. Things like:

•  it's my job to make sure they have the right answer
•  it's my job to correct their misconceptions
•  it's my job to ensure they are reasoning correctly

... and so on.

After a little more reading, I think that what Brousseau means by the DC is more of a set notion (teaching involves THE didactic contract, not A didactic contract): it is the paradox of teaching (unless you're teaching "skills") - your job is to cause students to (my words) "understand," their job is to understand, but that "understanding" is not something you can give them.  And the more you try to compel them to understand, the less likely it is that they will.  Brousseau's analysis of Topaze is the example he uses in many writings, and I'll summarize here because I LOVE IT. -- but first a French lesson: the plural of sheep (moutons) is pronounced (more or less) the same as the singular of sheep (mouton), although they are spelled differently.

Topaze (a teacher) is having a student spell out what he says — “Unable to accept errors that are too gross and too numerous, and not being able to give the required spelling more directly, he ‘suggets’ the right answer by hiding it under increasingly transparent didactic encoding: [my translation— ] “the sheep are gathering in the field.”  [student writes "mouton" instead of "moutons"] He tries again. And again. Finally says, “the sheepSSS ARE gathering…”

We imagine that he could continue to requiring the chanting of the rule, and then require it to be copied out a certain number of times. The complete collapse of the act of teaching is represented by a simple order: put an "s" on "mouton:" the teacher has in the end taken over what was the essential part of work.

(roughly from Brousseau, G., & Otte, M. (1991). The Fragility of Knowledge. In Mathematical Knowledge: Its Growth Through Teaching (Vol. 10, pp. 11–36). Dordrecht: Springer Netherlands. ) 

That is, you can ask yourself: why doesn't the teacher just say "it has an 's' on the end?" -- Because, of course, that would be cheating. We have an implicit contract in place around what the teacher's job is. Of course, in the teacher's steps to help the student get the answer, (as another author notes, not Brousseau) he changes a spelling problem (a problem of recognizing and responding to context) into a phonetics problem. The teacher needs to cause the student to understand and produce the correct thing out of that understanding :

If the teacher says or indicates what he/she wants the student to do, he/she can only obtain it as the execution of an order, and not by means of the exercise of the students’ knowledge and judgment (this is one of several didactical paradoxes brought about by the DC). But the student is also confronted with a paradoxical injunction: The student is aware that the teacher knows the correct solving procedure and answer; hence, according to the DC, the teacher will teach him/her the solutions and the answers, he/she does not establish them for himself/herself and thus does not engage the necessary (mathematical) knowledge and cannot appropriate it.

Wanting to learn thus involves the student in refusing the DC in order to take charge of the problem in an autonomous way. Learning thus results not from the smooth functioning of the DC, but from breaking it and making adjustments. When there is a rupture (failure of the student or the teacher), the partners behave as if they had had a contract with each other. 

That above quote comes from a lovely and easy intro to DC - I like it because it begins with the things that need explanation (seemingly weird things students do in math classes), and then offers the didactic contract as a construct that helps to explain those weird things.

And it highlights the paradox of the contract: "everything that they do in order to produce in the learners the behavior they want, tends toward diminishing the students’ uncertainty, and hence toward depriving them of the conditions necessary for the comprehension and the learning of the notion aimed at." (This has parallels to what I'm thinking about/wondering about in terms of 'scaffolding' - when is a scaffold a break in the contract?)

The things written by Brousseau tend to be more dense (for me) -- at least they're not the place I start. Working on this one now... will hopefully flesh this out more tomorrow before my meeting with Amy!

Two thoughts on my mind

1 - I've been taking courses on the Catechesis of the Good Shepherd.- a Montessori-influenced "religious formation" (sunday school) program for ages 3- 12 (and maybe beyond?) that Kate goes to. One thing that has struck me is the emphasis on God as mystery and child as mystery -- you, the teacher, don't have the answers -- it is impossible to know "answers" because it is a mystery. This seems like such a fantastic way to approach teaching physics. How the universe works is and always will be a mystery, and what your students "understand/know" is and always will be a mystery, and how do you support a productive relationship between the students and the workings of the universe? How do you use text/experience/dialog to support that? 

2 - I'm loving Brousseau - particularly this idea that the instructor can be so focused on student's success that they achieve it at the expense of student understanding. This has me reading and thinking a lot about "scaffolds." We can think of Pasco equipment, I would argue, as a scaffold that emphasizes success at the expense of understanding. I was observing in a high school classroom last week where the trajectory problem was algorithmized by the teacher ... and the students would, if they followed the algorithm, be successful. So it's a successful scaffold for answering trajectory problems, but is it a successful scaffold for promoting understanding? For understanding something that students could not otherwise understand? can 'understanding' always be operationalized as 'doing'? can it ever?

I wish I knew more Vygotsky.

doing my best

To anyone to whom I owe something, collaborate with or similar: I feel like things are coming across my desk so fast I can barely acknowledge them, let alone respond to and deal with them in a reasonable time. Sorry. I'm doing my best!

Engineering

From Cunningham & Kelly:

"Importantly, the NGSS recognizes the disciplinary differences across the sciences and engineering fields—science aims to produce knowledge, whereas engineering aims to provide solutions. One goal of science education is help students understand core knowledge and practices; educational standards reflect this by including core concepts that K–12 students might develop as they engage in scientific processes. Engineering asks students to create innovative solutions that reflect particular contexts. The underlying knowledge that students need will differ depending on their stated challenge; therefore, it is not productive to try to prespecify a set of concepts (i.e., core disciplinary knowledge) that underlie the myriad of possible engineering problems."

I think that is a misunderstanding of the products of scientific inquiry: inquiry doesn't lead to disciplinary core ideas any more than engineering practices will lead students to design the Brooklyn Bridge or a cam. I mean, carefully structured, scaffolded engagement in scientific practices can foster ideas that are consistent with scientific knowledge (the SSRP for example) - but almost never exactly replicate.

And I think it undervalues the power of particular design solutions: the electric motor; the combustion engine; gears?; capacitors? siphons? valves? arches? ... I don't even know what the most critical/significant core products of engineering design are, but I feel like many many engineers don't invent new vocabulary, just new sentences. And I think it's also pretty reasonable to think that students should be introduced to how an engine or motor or valve or speaker works as part of a K-12 STEM education.

I'll be doing a toy dissection again this summer, but I want to shift to focus to things that have particularly useful/common design elements. Surely there is work on this question (what are the biggest/most fundamental engineering design ideas)... where to look?

* PS: related to this idea is the idea of "naming" -- giving a thing a term helps us to notice it (Angie). If we take apart toys and name the parts - even giving them our own names - I think it will support a fun way to "notice" in the world.

My summary!

The goal of this paper is to articulate a set of design principles that support transfer. In doing so, I will first describe a perspective on transfer in which an idea is not so much abstracted from its original context and applied to a new context (as transfer is often conceptualized), but one in which the learning context is invoked in a new context — for example, when opportunities to extend a classroom investigation on light are recognized and pursued outside of class — in what I will refer to as intercontextuality (cf., Wagner, 2006; Engle, 2012). After characterizing a range of context domains that may be positioned intercontextually, I will then argue that such intercontextuality is fostered in classrooms that are themselves intercontextual: where out-of-class contexts (e.g., physical, social, functional, and temporal contexts) are invoked by students in scientifically consequential ways as they develop and vet ideas. I describe episodes of such intercontextuality in detail, with transcripts, field notes and artifacts from a course, Scientific Inquiry (Atkins Elliott, Jaxon, Salter, 2016), that shows evidence of high transfer (Atkins and Frank, 2015). Finally, I will argue that intercontextuality in class is supported by disruptions (Ma, 2016) to traditional instructional practices that confer power on students, i.e., a devolution (Brousseau, 1997). In this final section, I contrast structures and expectations in a traditional physics course with those from Scientific Inquiry, describing how those structures represent a disruption and devolution that sponsor intercontextuality and, therefore, transfer.

Paper is getting there...

I'm aiming this behemoth of a paper at JLS, hopefully ready to send off this summer. Feels awfully ambitious for me, and I'm nervous! If any readers would like to give feedback on this paper at any point, let me know and I'll send you what I've got so far.

The outline of my claims:

FIRST CHUNK: describing transfer as intercontextuality

1 - We have two classes teaching optics - a traditional physics class and a weird Scientific Inquiry class. They show very different responses to a survey on transfer of ideas about optics, see?
2 - Let's unpack one student's rich response from the SI class (the Walgreens story) and think about how this might count as transfer using Barnett & Ceci's coding...
3 - Turns out that transfer from one context to another might not be as productive a way of defining this, but instead we should call it intercontextuality: she invokes the context of our class in an out-of-class context. (This is consistent with ideas about transfer from Wagner, Greeno, Wilensky, Goldstone, Hammer, Engle, etc.)

SECOND CHUNK: a coding scheme of IC and seeing IC in class

4 - So here's my claim: "transfer" to out of class contexts is promoted by rich moments of scientifically consequential intercontextuality IN class. I'll use the coding scheme from transfer (6 different contexts) to see if we see those contexts showing up IN class.
5 - First let me flesh out this coding scheme of intercontextuality using brief examples from class.

Intercontextuality domains: What non-classroom context is invoked in the development of scientific ideas?
brief description	example
	Not IC	Weakly IC	Richly IC
Knowledge domain: Ideas from another knowledge domain are positioned as relevant in class.	Data generated in class is used as evidence for a model of color mixing.	Knowledge from art class (another academic context) is used used to dispute our model for color mixing.	Knowledge from bartending (a non-academic context) is used to justify a three-color model for colors.
Physical context: Physical objects and spaces (present or not) that are not typically part of class are positioned as relevant.	A lens and laser are provided as lab equipment and used to measure the index of refraction of glass.	A dry erase marker, intended for the whiteboard, is used to color half of a lens to trace where light rays go.	A student uses a jell-o cup to view the path of laser light before and after a lens.
Temporal context: Prior ideas are held accountable to current and future knowledge.	A model of reflection is established and built on over time, but not challenged or modified.	An established model of reflection is challenged in class the next day when a student disputes that objects don’t “glow.” Our model is modified but not discarded.	An model of reflection students had established - “secondary rays” - is refuted and discarded in light of the “koosh model” of reflection. *cite*
Functional context: Ideas and objects used to explain/inform/perform X are now used to explain/inform/perform Y.	Students are asked to use luxmeters to measure the amount of reflection from a mylar surface provided for this purpose.	Unable to find a flashlight, a student uses her smartphone light to project a shadow as part of a lab.	A hot playground slide is mentioned to justify the idea that light is absorbed by mirror-like surfaces.
Social context: Relationships, identities and roles not usually relevant are invoked and relevant to developing scientific ideas.	Students refer to ideas from the teacher, the book and other students in class.	Student mentions she showed this experiment to her daughter, who had the same ideas and questions that we did.	Student mentions his wife’s ideas in class, and notes he does not understand this until he can explain it to her.
Modality: Multimodal: an idea is expressed using multiple modes.	Students sit quietly and use written text to learn about and communicate ideas about reflection.	Students use and interpret variety of written representations to describe reflection: graphs, diagrams, and text.	Diffuse reflection is drawn as a “shattered” ray, demon-strated with a mirror and flashlight, compared to a cartoon, and sung about.

And with that done, let's use this scheme to examine some longer class episodes...

6 - *lack of methodological rigor warning??* I pull fascinating vignettes from the class and describe how they are richly intercontextual using this coding scheme. I summarize: the high-transfer class shows rich moments of IC in these contextual domains identified by B&C.


THIRD CHUNK: teaching for transfer - disruption and devolution
7 - here I begin by returning to the traditional physics lab (the one with limited transfer happening) - describing the activity and noting that this activity is driven by an implicit contract: “the set of reciprocal obligations and sanctions which each partner in the didactic situation imposes, or believes to impose, explicitly or implicitly, on others, and those which are imposed upon him/her, or s/he believes which are imposed on him/her.”

8 - [METHODOLOGY ALERT!] I simply claim this as the teacher's side of the contract (I'd love to find someone else who has claimed this b/c I don't want to do the qualitative research to back up this point):

• the teacher provides only correct scientific knowledge; 
• the teacher engages students in producing only correct scientific knowledge;
• the teacher assesses whether or not students have “acquired” that knowledge. 

and I show in detail how this is at play in the (Pasco-structured) lab that students perform, and that this limits opportunities for bringing other contexts to bear.

9 - to change this contract, I propose what Ma calls "disruption" and what Brousseau calls "devolution" (this brings this work into contact with work on hybridity and third space):

• disruption: disruptions to typical classroom mathematics in order to provide opportunities for students to recruit a variety of funds of knowledge and other resources for problem solving
• devolution: “the activity of the teacher in attempting to induce the student to take on responsibility for a Situation.” (the term is literally “an act by which the king, by divine right, gave up power in order to confer it on a Chamber.”)

10 - So what are the ways in which this course offers disruptions and devolution? (AGAIN: NO METHODOLOGY. I JUST SAY IT!):

• Disruption/devolution via the absence of traditional text-based resources.
• Disruption/devolution via the lack of traditional lab equipment. 
• Disruption/devolution via student authoring of questions. 
• Disruption/devolution via student authorship, assessment and iteration of scientific models.

11 - For each of those, I unpack how these are both a disruption and a devolution: how do these structures change the traditional structure (disruption) in ways that give students authority/power over the content (devolution)? - And see how students recruit other contexts because of it?  These are four long sections. 

FOURTH CHUNK: implications

I haven't fleshed this out yet, but the general ideas I want to bring in are ideas that are related to my claims:

- Engle, PDE, Framing.

- distinguishing "scientifically consequential" from other examples of hybridity- connected to PDE and Ma's paper

- Responsive teaching.

- Chief Justice Roberts  ("When physics is taught in such a way that students do not introduce other contexts in class, as is often the case, then the “unique perspectives” the Chief Justice asks about are not leveraged. But when this happens - when classes are taught in such a way that there are no “benefits to diversity” - this is a statement of pedagogy and not a statement about the importance of diversity when constructing and vetting scientific ideas.")

- epistemic framing and implicit contracts as related ideas. 

- Brousseau's work.

Equipotential lines

... are places where a KE converts to a PE (or vice versa).

(Assuming those lines are like a terrain map and spaced at equal intervals.)

Overview of IC Contexts

An outline of types of intercontextuality: what it would mean to have no IC, to what it means to be richly IC, in all six contextual domains from Barnett & Ceci.

These frame the rest of the paper (illustrating what IC looks like and why it supports transfer) -- so I want to make sure these are clear. I have longer paragraphs supporting each idea, but this summary I think should stand on its own, too. (If the statement on "temporal context" doesn't make sense as being like the others, that's okay, it's not - but it should still make sense!)

If anyone has feedback, I'd love to hear it!

Intercontextuality domains: What non-classroom context is invoked in the development of scientific ideas?
brief description	No IC	Weakly IC	Richly IC
Knowledge domain: Ideas from another knowledge domain are positioned as relevant in class.	An idea mentioned earlier in class is used in a new conversation.	Knowledge from another science class is used in class.	Knowledge from bartending is used to justify a model in class.
Physical context: physical objects and spaces (present or not) that are not typically part of class are positioned as relevant.	Lab equipment is used in addressing lab questions.	A dry erase marker is used to color half of a lens to trace where light rays go.	A student brings in jell-o to view the path of laser light before and after a lens.
Temporal context: Prior ideas are held accountable to current and future knowledge.	Students and teacher draw on prior ideas from class, but do not modify those.	Prior ideas are reinterpreted in light of new ideas.	Prior ideas confirmed in class are discarded in light of new ideas.
Functional context: Ideas and objects used to explain/inform/perform X are now used to explain/inform/perform Y.	Students are asked to use luxmeters to measure the amount of reflection from a mylar surface.	Unable to find a flashlight, a student uses her smartphone light to project a shadow as part of a lab.	A hot playground slide is used to justify the idea that light is absorbed by mirror-like surfaces.
Social context: Relationships, identities and social roles not usually relevant in class are invoked and relevant to developing scientific ideas.	Students refer to ideas from the teacher, the book and other students in class.	Student mentions she showed this experiment to her daughter, who had the same ideas and questions that we did.	Student mentions his wife’s ideas in class, and does not feel like he understands something unless he can explain it to her.
Modality: Multimodal: an idea is expressed using multiple modes.	Students sit quietly and only use written text to learn about and share ideas.	Students use and interpret variety of written representations: graphs, diagrams, and text.	Specular reflection is demonstrated with song, drawn as a single ray, and demonstrated with paper and flashlight.

Temporal Intercontextuality

I continue to creep along at a snail's pace on my TE/IC (transfer/transformative experiences/ intercontextuality) paper. One element I'm not sure how to work in is to talk about "temporal context" - there are some things I think I could relate, and I want to work out these ideas here...

B&C describe "temporal contexts" of transfer thusly:

This dimension reflects the elapsed time between training and testing phases (e.g., a few minutes, a week, or years later). … to justify the effort invested in education, ideally one would hope for transfer to last for several years after training. 

But what does it mean for me to argue that a different time is invoked during class? And not just invoked (like, "yesterday I was thinking about this...") but consequentially invoked - so that the time itself is relevant? 

What I want to argue is that our evolving narratives - the story we can tell of the construction of our scientific ideas - hopscotch across time. And part of this is because our ideas are iterative - we develop, challenge, build, critique and abandon ideas over months of instruction. -  We do not finish the day's lab and turn our attentions to new ideas, or treat those day's ideas as 'settled.'  We don't have a book that tells us "this is how to understand this idea." (And the book might build on but does not modify that idea.) And since we are never "done" with an idea, it keeps us on the edge of our questions. So I think what I mean is that other times are positioned as 'continuous' with our current narrative... does that make sense? ... that (a) there is an evolving narrative - forward and backward looking (unlike a textbook that builds from right-idea to right-idea) and (b) other, new elements to our story can show up at any time (and any place - which I establish in other 'context' stories). It's a little like a book we can open and close (and never ends!). When we open it, we "pick up" the story from a few hours or days ago to thread together discontinuous times into a continuous one. (This reminds me of Flat Stanley  how carrying him around meant sometimes I would be pulled into "seeing" like a 6-year-old boy for a moment.) So the "temporal" contexts are relevant / consequential to our evolving narrative because it frames our activity in a particular way and promotes/builds on our iteration in a way that makes the other contexts possible. (? do i buy this ?) 

Examples: Maddy completes a lab while in Walgreens. Andy answers his wife's questions during classtime. Wendi extends her critique of primary colors at home. Our "seconds" theory in class is later dismantled by the "koosh" theory (an example of backwards-looking revision).

G&W describe similarity as a tesseract:

"Our hope, then, is not to have students transfer by connecting remotely related situations, but rather to have students warp their psychological spaces so that formerly remote situations are similar."

I would reword this to say, of time: 

"Our hope, then, is not to have students transfer across time by connecting a distant idea to a current situation, but rather to position a current situation as a continuation of prior classroom situations; that is, to warp psychological time so that otherwise remote times are seen as continuous."

And, I think, this is fostered by those prior situations being viewed as never-quite-finished, open, always, to revision.

New website

Inspired by colleagues (though I'll admit to still feeling sheepish about this!) I made myself a website.

atkinselliottlab.com

After leaving Chico, my website disappeared and I've been bereft. So maintaining this through wordpress (for $20/year) means I don't have to use university templates and it goes with me if I ever leave. (BUT: I'M NEVER LEAVING.)

It includes a password-protected blog where I will write about my teaching. If you'd like to read along as I teach an energy-project version of the inquiry course (co-teaching with an engineer!), just send me an email and I'll send you a password.