Interpreting: March 2008 Archives

Push of Chang, Pull of Cronen

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A vigorous debate between two faculty members dominated conversation about Marc Crépon's "What We Demand of Languages," an extended footnote to Derrida's Monolingualism of the Other.

I had been worried about arriving late to the Center for Communication Studies event, however Briankle Chang and Vernon Cronen were deep in discourse, ranging from the mistake of theology (not a feature of all religions), the influence of the Platonic opening, Aquinas' linkage of physics with the New Testament, to structuralism as the antidote to transcendentalism, and whether "topos" is a place that contains all topoi and all vocabularies or a place that can be talked about in infinitely many ways.

I always learn more from faculty interactions with each other than from monologistic pedagogy!


A colleague translated Crépon's article from French. Srinivas Lankala explains:

"Crépon summarizes Derrida’s argument, provides references to the argument that Derrida did not provide, and extends the argument to new areas:
  • the question between what language is and what language means in terms of politics of nationalism or politics of identity
  • the definition of identity
  • the definition of the self

"One important thing called into question is the notion of a singular cultural identity: identity is formed in advance by language – the whole question of identity which cultural studies depends on, what post-colonial studies depends on, is nonexistent in that sense, it does not exist before language. Crépon extends Derrida’s proposition that the monolingualism of the other is not just his unique case of (to put it too simply) a French-speaking Jew in Algeria who is speaking French as the language of colonizer, this is one kind. Derrida goes beyond the particular to show that the idea of monolingualism is not simplistic. Crépon builds on the understanding that the colonized has no other language than that of the colonizer, but that all cultures are always colonized, because a culture comes into being through the question of naming, giving names, which is a function of language and calls language into being."


Naming sets Chang and Cronen off again (providing me descriptive data for "saying something," according to Chang). [Note: the provided link is not particular to the discussion, it merely invokes the complexity.]

It was suggested that "The point behind this extension of monolingualism is so that it is not understood as the empirical problem cultural studies tries to make it but rather a broader problem that applies to all of us: we all only ever speak one language and we never speak only that language." I am not familiar enough with cultural studies to know the (attempted?) formulation of this "empirical problem" - and I certainly won't speculate (although I am curious!)

Meanwhile, Lankala continues:

"What Crépon is doing to extend Derrida’s notion is to explore: how do we go beyond this situation, what do we do to go beyond this restrictive monolinuguaism that we all share? Derrida suggests the way beyond is to invent one’s language as one is speaking it. This is something Derrida associates with translation as a radical way to call language into question, to call identitiy into question. Not in the simple sense of from one source language to another, but a translation without sources, which only has a target language, which only has arrival; in its arrival it creates its own sources. This radical idea is what Crépon extends. How to invent new language to go beyond the monolingualism of the other that is a common situation for all."


The subsequent exchange between Cronen and Chang was much too quick to transcribe adequately, here are the main points that I think I can parse from the words I managed to capture.

Cronen questions the privileging of the speaker, the one who speaks, i.e., the one who names over "the responsiveness of the other." His argument is that there is no stability of language - any language - without a correspondence of action/response between the speech of one and the responsiveness of another. Cronen goes so far as to say that "the emphasis on naming is fundamentally misplaced" and poses "joint action" instead.

"Where," Chang asks, "does that joint come from" Joint, you already presuppose jointed, being joined. That is la langue, the package." Cronen illustrates by describing how a child learns language only through interaction. Chang concedes "two facts: we all have a father and a mother, and we speak," agree also with Cronen's emphasis on vision. Later, Cronen will characterize this vigorous exchange between them as a horror to those with a strict or narrow conception of dialogue, and Chang will call it "quotidian. We do this every day."

Hmmm. Yes. I am getting ready to "say something" (but certainly not everything! and guaranteed not yet well enough) by building on the use I made of their exchange to illustrate a distinction between representation and symbolism. After some more discussion on Derrida's emphasis on language, Lankala asserts, "The whole question of naming comes up because he’s talking of language as the force which calls culture into being, and culture is nothing but this whole process of naming."


  • Lankala: Leaving Derrida’s book aside, how can one disagree that culture is nothing but naming? Culture means singular. The point exists because someone is naming someone else…. It exists because it is named as a culture….. [this is] "naming" properly understood; it is not making things ostensive.

  • Cronen: Then what is [language]?

  • Chang: [Language] allows us to call things out from their natural state, again. Not fixing a lexicon, it is about establishing presence.

  • Cronen: [The] complexitiy [is] in the interactive process.

  • Chang: No. Naming – we found the point of disagreement. Interaction makes no sense without naming having already taken place.

It is this "point of disagreement" that I will take up, eventually. First, here is the rest of my re-construction of the conversation. Lankala moves to another interesting question:

"the relationship that Crépon makes between language and how language is appropriated in movements seeking nationalism or defining identity. [Crépon uses a] completely opposite definition of what language should mean from the way language is generally used in more mainstream cultural studies tradition, [which is] as the language of the colonizer or language of the oppressed without calling into question or breaking apart what language actually is, what its function is in defining that movement or culture, where is it from. [Derrida and Crépon’s] move goes one step beyond the relationship between language-culture to discuss the functional role of language in creating a culture….."

What ensues is a discussion of how forced multilingualism can lead to monolingualism (e.g., the case of India), and problems of language being misconceived as a possession - the "mother tongue," as if language exists outside of/beyond the "me," which returns us to the beginning assumptions of appropriation (for identity construction: of "self", "culture," "nation," etc.)

"Language,"
says Chang, building on 35-40 years of Derrida reading Nietzsche,
"is always a promised language."

  • Cronen adds that promise is "linked to the notion that meaning is not just a presence but a pointing-into the future" (drawing on William James).
  • Chang: A promise never promises anything, nothing but another promise. This is why it is linked to time, the future; that’s why they smuggle in Kant (law) and Kafka, how can they assemble these ideas together?!
  • Cronen: Dewey [also, with his notion of] ends in view, not fixed. Using my vocabulary, [there is] always a punctuality, not a destination, [there is] always opening up, even when we think we’ve fulfilled ... still [there] opens up a new horizon of possibility
  • Chang: If fulfilled, [a promise] is not a promise any more, it is not promising.


Finally the giants (!) relented enough (!) to let us peons into the fray. :-)

I mentioned the ideas I'm working on regarding interpretation as a way to keep promising, to keep language and meaning in motion. George asked about the use of the term, "political." Ellen brought in the notion of "power."

The entire 90 minutes rocked!

"2 hours talking about poop"

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Pete said it, summing up the party.

2 bouquets, combined SMALL.jpg

We started at the UMass Sunwheel circa 6:15 pm. The clouds cooperated, beginning to clear an hour in advance of sunset. The wind was bitter, though: fortitude was required to make it through until the moon cleared the 7 degrees of forest obscuring the horizon in the East.

full moon rises.jpg
Dr. Judith Young from the Astronomy Department at UMass regaled the crowd (52 brave souls who stayed) with enlarged photos, anecdotes, history, and education. I was struck by the range of nuance embedded in the careful alignment of static stone with the motions of our solar system. In particular, I learned of the Callanish Stones for the first time. Dr. Young showed some pictures and explained the presence of an "extra" stone that - if one stands just right - creates a visual notch with the stone next to it that outlines the precise location on the horizon where the summer solstice sunrise occurs. "They found," she said, "a way to let us know."

Hmmm, a way to know - what? If there is a message in these stone circles, what might it be? Was there an active intent to leave a sign that would invite us to wonder? What would people from four or five millennia ago want to convey to us, their descendants in a future as dim to them as their present is removed in a distant past? I considered these questions: they want us to know there is another mode of perception. They want us to remember that scientific measurement with all its technical specificity is not the only way to apprehend life. (My conviction was profound in the moment. Some hours later, I imagine that the possibilities of their intended meanings range beyond imagination, yet in this time - our time, now - this meaningfulness jumped into consciousness.)
I missed parts of the lecture walking around taking photographs and whispering with friends, still - the qualities of the equinox that I did learn struck me as propitious. Were there four? Equal lengths of night and day, sun (and moon?) rising and setting due east and west, sun directly above the equator, and . . . well. At the moment I heard the list I thought, this is exactly how I need to go about my upcoming research. (When I told Anne the good news she described it as "impossibly cool!" "I know!" I hollered.)

Just-in-Time and I spoke about the need for evenness in one's emotional life as we drove from the Sunwheel to the apartment for soup. Suppose 80% of your emotional experience is "okay," 10% is elation, and 10% is all the other stuff? That 80% takes in a lot, eh? It's good! Is such a spread worth the highs of the high and the lows of the low? "Hey, maybe I've already done my ten percent? Five years or so of the lows....finished! It's out of my system! Been there, done that!" :-)


Option A: Tissue + Plastic Wrap/Newspaper

or

Option B: Clean, Dry Container


The Béguine Cream Soup was a hit. (Yah!) I confess I doctored it a bit. (Who, me?) Check out this description from Twelve Months of Monastery Soups:


"This recipe is a version of a soup from Flanders in northern Belgium. Its name suggests it originated among the Béguines. Béguinage was a medieval institution that allowed pious laywomen to lead a form of religious life in common, without becoming actual nuns. It was one of the few alternatives to either marriage or the cloister...In general, they were a progressive group of women who wished to assert, as much as the times allowed, their independence from men. They were women of great culture, and some of them became renowned mystics."



"Open flap of Collection Card"



The recipe calls for chervil, which I could not locate. Having received an email from one of my teachers about the Apache New Year (which, like many other cultures, recognizes the spring equinox as the beginning of the year), sage seemed an ideal replacement. We were cold coming in from our hour in the wind; it took a few minutes to settle in and get the soup warming. Soon enough, the Wanokip put on The Doors and the party started. :-) Pete and Sinead got me going on my research question, so much so that I had to take notes! They gave me an absolutely crucial framing, later clarified even further by The Ever-Smiling Evil Indian and Ambarish, who asked, "Isn't it obvious [why certain people use or don't use the interpreters]?" Aha! The fact that they are making a choice is obvious, but the reasons for the choice are not! I have no idea what their reasons are, and (to be honest, gulp) I'm not sure (?) they have thought (?) very much (?) about it themselves. This is what I need to find out!

"Collect a pea-size sample with provided Applicator Stick."



Searching human behavior for patterns is not so far removed from searching the stars for meaning, is it? I mean, come on, Renee found her way to the event by approximating a time in memory and correlating that temporal position with its internal references to other times (if she received the invitation two days ago and the event was specified as "tomorrow" then that meant "tonight" not Friday). A skill she has improved, apparently, after reading Longitude by Dava Sobel. (What role does the chronometer now play as "control" in a cybernetic civilization?!) Then there was the long convo with Anuj about inattention blindness and the basic fact that our brain must select - and therefore also de-select - where to aim one's focus. (We also conjured the amazingly cool idea of eye tracking deaf people as they watch sign language.)

"Apply sample to top half of window."



Obviously, with so much goin' on in this puny brain, I had to take notes and remind everyone about the blog. "Is this informed consent?" I was challenged. I responded with the options. "Shut the F*** Up" has been duly noted.


sunset in the west.jpg



"Reuse Applicator Stick... spread samples over entire window..."



mathematical thinking

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I'm closer to Brian Butterworth than Stanislas Dehaene, as this comparative review describes:

"Butterworth is a neuropsychologist who came to studying mathematical ability via his work on natural languages....Dehaene, on the other hand, started off as a mathematician, but became fascinated by the abstractness of his subject. He began to wonder where mathematical ability came from, and why some people are so bad at it, and others so good."

The Mathematical Brain appeals to me from the start, with the author's writing style being compared with Oliver Sacks (Seeing Voices: A Journey into the Land of the Deaf). The Number Sense reminds me of Barry Mazur's, Imagining Numbers (which I started and now want to finish).

The reviewer argues, "cognitive science tells us that it is possible to teach mathematics in a way that fits with our psyche, a way that minimises maths-induced fear and boredom." Lots of "sideways" exposure is doing it for me....all that three-dimensional American Sign Language interpreting has (seriously!) re-wired my conceptual circuits for math.

Just last week, the New Yorker's "Numbers Guy" wrote about whether our brains are actually wired for math, featuring Stanislas Dehaene.

One tidbit: in addition to a certain kind of math perception, the language you use also influences cogniive processing:

English is cumbersome. There are special words for the numbers from 11 to 19, and for the decades from 20 to 90. This makes counting a challenge for English-speaking children, who are prone to such errors as “twenty-eight, twenty-nine, twenty-ten, twenty-eleven.” French is just as bad, with vestigial base-twenty monstrosities, like quatre-vingt-dix-neuf (“four twenty ten nine”) for 99. Chinese, by contrast, is simplicity itself; its number syntax perfectly mirrors the base-ten form of Arabic numerals, with a minimum of terms. Consequently, the average Chinese four-year-old can count up to forty, whereas American children of the same age struggle to get to fifteen. And the advantages extend to adults. Because Chinese number words are so brief—they take less than a quarter of a second to say, on average, compared with a third of a second for English—the average Chinese speaker has a memory span of nine digits, versus seven digits for English speakers.

I'm excused from interpreting this talk, Nanometers, Femtoseconds, and Yoctomoles: Molecular-Dynamics Simulations of Diffusion in Garnet, which means I can take notes and play!

The professor is highly billed: Dr. Bill Carlson from UT at Austin. You think I'm kidding about "play"? No way, Jose!
Scale: plates, rocks in the field, mineral grains, atoms....
Geologic Time:
Sizes from macro to nano.....

Diffusion gives direct qualitative information on rates and duration of metamorphic processes. Garnet is present in a wide range of bulk compositions, is stable, and has a wide array of diffusive behaviors that can be monitored to help us understand rates of diffusion and the mechanisms behind them. You know my parallel? Groups (of people) and knowledge/understanding (disseminated via language).

Main topic: Molecular dynamics simulations.... (microdynamic intergroup relations?)

Problem: existing theories for diffusion at atomic scale don't explain the phenomena we observe...(sounds like social science to me!)

Novel systematics emerge from recent synthesis...

Elastic Strain Theory (EST) - diffusion by vacancy mechanism: work is required to move atoms apart and squeeze this atom in-between them....larger atom = more strain which slows down diffusion. Like all theory (!) "sometimes it works...sometimes it doesn't."

There's a "misfit parameter" (!) = "how badly an atom fits in its new site." If a good fit, then the number is small; if the atom is too big you get a positive misfit parameter, if the atom is too small you get a negative misfit parameter. (No speculation, thanks, on the size or charge of my misfits!)

Observation: a fundamental gap in our knowledge, sometimes smaller sizes diffuse more slowly (instead of faster, which is what theory predicts).

How else can observable systematics be explained if EST doesn't do it? Perhaps - molecular dynamics (MD) ...EST relies on a visualization based on Hooke's Law ;MD takes into account all of the binary potential fields (imagine: all 756 (?) potential dyads we calculated as the total combination of interpersonal pairings (28 individuals, each with 27 unique relationships - except I don't know how to do this math!) in the current course on Group Dynamics).

Comparison of Potential Barriers for Atoms of Different Size: take potential energy, over time, and compare it to optimal diffusion (and yield (?) energy barriers to diffusion). EST predicts well for larger atoms.....for smaller atoms....start with lower energy well because more tightly bound....then a smaller atom has a larger energy barrier to cross than the optimum size....

MD: Newtonian mechanics at the atomic scale.... with forces as sum of pairwise interactions: interatomic potential, interatomic distance.

Interaction potentials:


  1. ionic charges (same = repel; different = attract)

  2. Born-Mayer repulsion - atoms can't get too close to each other, will begin to push each other away = gives an indication of how hard the atom is (large value = billiard ball, small value = nerf ball)

  3. dipole attraction (van der Waal) - an induced dipole, if the force is strong it leads to a large value, if the force is small then it leads to small value.


Interaction parameters are determined by fitting MD models to data on static properties, eg...molar volumes, expansivities, compressivities (ah, no static properties in human relations - although social science (and basic prejudice) TRIES to make "identity" static/stereotypical...)

You have to select time steps that are a function of atomic motion...durations long enough to obtain many diffusive jumps... (time...always time! not to mention timing!)

Assign initial positions ("groups" never simply "begin" they are a convergence in time of dynamics already in motion, already historical), throw in random velocity (intensity/emphasis of attention to the storming phase of group development?)....

Diffusive Jump - Dr. Carlson shows an animation of atoms in motion.....cute!!! I wanna link to it! Could we model interpersonal relations in some kind of analogue? I've envisioned forever - do we have the technology?


Einstein relates diffusivity to time using a mean-square displacement.....average over all atoms, average over all possible times....get tau....then see how it changes, the slope is the diffusion quotient...

Vacancy concentrations are crucial - but how do we figure this out? One method comes up with a physically impossible result (100% vacancy) which indicates some of the physics is still being missed in the calculations. The standard MD simulations.... tend to significantly underestimate.... (something crucial. Kinda like social science, language, social construction of reality, you know what I mean).

Tracer diffusion simulations: replace 10% of the atoms with some other elements and examine the rate of diffusion of that element. (Can I just say, as if anything only ever goes in one way?!!!!!?)

Failure to generate (via simulation) the relationships that match measured behavior in strain relationships but the gaps/discrepancies point us to what we're missing... STATIC properties all MATCH up But the DYNAMICS do not!!! (Same as with social science?!)
Failing to account for what's happening to atoms when there are other atoms in the vicinity. Different cases pending varying polarizability. (I swear this is group relations jargon!)


This semester I've had the opportunity to work at the extremes: one professor who has seamlessly blended me into her extensive use of the chalkboard, and another whose gaze apparently registers only empty space whenever she accidentally happens to glance my direction. The contrast heightens my belief that attempts to make ourselves "invisible" (while interpreting) are worse than "professional": they are downright counterproductive.

For instance, the day I knew I could not possibly reconstruct the meaning for something uttered very quickly yet of obvious conceptual importance for the subject matter, and had to ask professor #2 (the one for whom language accommodation is nonexistent) for a repetition or clarification (I forget which), an expression crossed her face as if a voice had come from the woodwork. Her answer was curt (to say the least). However, I had established my presence (albeit momentarily). An interesting consequence of actually "being there" was that a non-deaf student requested my attention to a sight-line that I thought I was not blocking but, in fact, was. This had been going on for some time; the students had suffered simply because the non-verbal behavior of the professor indicated that I was to be ignored. I am always annoyed by teachers who assume that they have no responsibility for accommodating a bilingual situation - as if the interpreter wields some special magic that automatically transforms the laws of physics such that what you say is signed by me in fully comprehensible fashion before you're even done saying it, so that you can pick up and go on without pause, without any need to confirm meaningful understanding!

It is this matter of time that seems to drive much of the refusal to recognize the presence, and unavoidable effect, of having a language interpreter involved in the interaction. Not that everyone does this!

But here's the rub:

the time we experience bears little relation to time as read on a clock. The brain creates its own time, and it is this inner time, not clock time, that guides our actions.

This dilemma with time came sharply into focus for me while interpreting some one act plays. Occasionally a dialogue would be paced in such a way by the verbal turn-taking between the actors that the interpreters could replicate it in its entirety, building in the proper visual cues, creating pauses - fully mimicking the action instead of being in a constant rush to catch up to it. Yes, it may have been obvious that the interpreters were a turn "behind" the actors in the dialogue, but I can imagine that this would not appear so except at the beginning and end of the scene... if you were bilingual you could follow the auditory-and-visual languages in the echo or prelude, if not, you could focus exclusively on the accessible language and tune is as you wish to the additional layer of communicative spectrum. A sophisticated troupe could craft storylines that complement and replicate each other in ASL and English - neither audience would miss out nor find the process of interpretation distracting because the languages would be part of the weft and weave of the action.

It may sound abstract, but I'm closer to an articulation of such a bi-linguistic merger than I've ever been because of the experience working with professor #1 this semester. She was as surprised as most teachers are when I walk in on the first day of class and introduce myself, "Hi, I'm the interpreter for your class." Usually there is not much time to talk about the how (and, in my experience, those conversations don't usually mean much until we've actually gotten through a class or three). On the first day of class I'm usually a bit more concerned about where the Deaf person chooses to sit, and how that affects my range of choices for creating good sightlines between them, me, and the action. If a teacher uses powerpoint, I have to be near the screen, if they use the blackboard, I've got to be near where they are writing , if they lecture without visual aids, then I need to stick close by. Most teachers move, they pace, point things out on the board, may even drift among student's chairs....I have to judge whether the talk coming from the person is most important, or the visual matter they are talking about (on the board or screen), and position myself accordingly.

Most interpreters plant themselves in one place, and deaf people are accustomed to this - they know the interpreter is always going to be in that location. But the ACTION is happening ALL OVER! The meanings move, the important stuff is not only the conceptual message that an interpreter repackages in another language - it is the relationship between those words and the person saying them, to their audience, with particular intentions in mind, and - especially in the case of teaching - with relevant symbols or concepts, a diagram or an equation represented in visual form. I don't have to know how to do calculus, for instance. But I sure as heck need to comprehend the relationships between numbers, the rationale for procedures, and the ways different applications relate to each other! If I do not, then I cannot sign them in a meaningful way.

No one has ever said, "That looks like babble," but it must. I know - plenty of interpreters say, "I just sign the words and they get it because they do know the subject"....I have even had this experience myself. One deaf student in an upper level math course on matrices used to love teasing that she could tell the difference between when I understood what I was signing and when I didn't. How many deaf people just put up with us when we don't understand because it's better than nothing? ugh.

I know that it matters, now, for two reasons. One is that when I get a concept it enables me to sign it differently and I watch the deaf person's mental lightbulb go off simultaneously. This is wicked cool. Obviously, there's an argument to be made for specialist interpreters in the maths and other deeply-developed sciences. But here's the thing, if the teacher understands that the interpreter must understand, and if the teacher realizes that there is no one:one correspondence between English words and ASL signs, and if the teacher is able to incorporate the added dimension of this extra mediator between them and one (or more) students in the class, then magic can happen.

I'm not kidding! I enter a math classroom this spring, introduce myself per usual, and the teacher and I have the usual brief conversation:


You'll use the board a lot?
Yea.
Ok, so I'll try to stay out of your way but I need to be able to be in the same sightline with the student and whatever you're writing.
Ok.


After the first day, immediate debrief: "How did that work?" the teacher asked me! I can tell you I almost fell over. Very well, I responded, I think I stayed out of your way, eh? A few times I get into my head working on the puzzle of expressing a concept and can't simultaneously move out of her path as she fills the boards from right to left....but no problem, she simply moves around me and we continue. Oh, I remember: can you explain this concept of differentiation? Because the sign in ASL for "different" implies that there is no relationship between the objects being described, but I figured out that in the math usage, "differentiation" is a process that can be reversed: in other words, there is a very definite relationship, which seems more like a transfer between terms rather than a categorical distinction. I showed her the sign, "BUT" which is also "DIFFERENT," as we spoke. She began to wrap her mind around the challenge of translation.

We get easier over the next few classes as I find the positions that work best depending on whether she's working the right, center, or left sides of the board. We pace back and forth, with me shifting from right-to-left of her, and sometimes lagging behind as I make the connection between what I'm interpreting and the part of the procedure she wrote on the board several seconds before. I keep myself within her field of vision as much as possible, as well as keeping her, me, and the board in the visual field of the deaf student. It's a kind of choreography. She offers to meet with me to explain the trajectory of the course, help me get a handle on the conceptual flow. My questions are so basic it's embarrassing.

But the meeting helps, and a rapport continues to develop. She knows she's not teaching me, per se, but she is trying to provide the conceptual framework so that the particular lessons make sense. She has grasped that - in order for me to interpret well - I need to know not only the details of the immediate lesson, but especially how that lesson relates to the larger principles this course is all about.

The day I figured out how to sign anti-differentiation conceptually accurately, the deaf student's eyes lit up. I could almost SEE her thinking, "Oh, THAT's how those two processes are related to each other!" Meanwhile, it isn't just my own comprehension that is improving. The teacher has been listening when I've spoken about sightlines, she now notices how far behind her I am in the interpretation and takes advantage of those moments to pause. "I'm sure the other students appreciate it too," she said to me one day, "we are covering this material very quickly." She's gotten so good at observing the visual information that most non-deaf people never even register, that she is aware of when I am waiting for the deaf student to look at me so that I can interpret! I've never had this occur before; I think it is quite extraordinary. She'll see that I'm not signing, look at the deaf student, and then make a judgment how long to wait until she continues. Sometimes she does start before I've delivered all the info to date, and of course I get lost somewhere and have to ask a clarifying question. No problem: immediate and precise repetition is provided.

Last week, some new concepts came up: convergence and divergence. I had convergence down but the way I was understanding divergence it was like the line never catches up to the limit, the limit is constantly receding. But I wasn't sure and I kept listening for some hint about how to convey the concept within the three-dimensionality of American Sign Language. Finally, I asked for clarification. "For the interpreter," I said, "I need to understand the concept of divergence so I can sign it accurately." I showed her what I was doing for convergence, then, as I adapted to show the way I thought I was grasping divergence she jumped in, explaining that the line always crosses the limit, there IS no limit, not because the line can't catch it, but because the line always exceeds it. Wow! I was excited. :-) Ok, I confess, some of it was selfish (I understand how mathematical divergence contrasts with mathematical convergence!) but the neatest thing was the simple 20 seconds it took, the natural flow we've developed where a question from the interpreter is obviously not a reflection on the deaf student, and all that is required is a precise, focused response: a genuine collaboration in guaranteeing conceptual understanding across two perceptually- and grammatically-distinctive languages.

Again, the deaf student's nod of comprehension provided the icing on the cake.

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