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.