Examples, history, food for thought.

Al-Sayyid Bedouin Sign Language

Nicaraguan Sign Language

Nicaraguan Sign Language is a signed language spontaneously developed by deaf children in a number of schools in western Nicaragua in the 1970s and 1980s. It is of particular interest to the linguists who study it, because it offers a unique opportunity to study what they believe to be the birth of a new language.

In 1980, a vocational school for adolescent deaf children was opened in the area of Managua called Villa Libertad. By 1983 there were over 400 deaf students enrolled in the two schools. Initially, the language program emphasized spoken Spanish and lipreading, and the use of signs by teachers was limited to fingerspelling (using simple signs to sign the alphabet). The program achieved little success, with most students failing to grasp the concept of Spanish words. However, while the children remained linguistically disconnected from their teachers, the schoolyard, the street, and the bus to and from school provided fertile ground for them to communicate with each other, and by combining gestures and elements of their home-sign systems, a pidgin-like form, and then a creole-like language rapidly emerged. They were creating their own language.

Staff at the school, unaware of the development of this new language, saw the children’s gesturing as mime, and as a failure to acquire Spanish. Unable to understand what the children were saying to each other, they asked for outside help, and in June 1986, the Nicaraguan Ministry of Education contacted Judy Kegl, an American Sign Language linguist from MIT. As Kegl and other researchers began to analyze the language, they noticed that the young children had taken the pidgin-like form of the older children to a higher level of complexity, with verb agreement and other conventions of grammar.

Nicaraguan Sign Language

• New Nicaraguan sign language shows how language affects thought – Discover Magazine (On this paper)
• Study: Math Skills Rely on Language, Not Just Logic – Wired Science (On this paper)

Plains Indian Sign Language

The Plains Indian sign languages (PISL) are various manually coded languages used, or formerly used, by various Native Americans of the Great Plains of the United States of America and Canada.

Plains Indian Sign Language

[Prior] to the cultural disruption caused by European colonization, it was commonly used across a large swath of North America from the Gulf of Mexico to Calgary, Canada, an area of over 1 million square miles. It spread so far because it was used as a lingua franca between Native American nations speaking at least 40 different languages, but it was also used within native communities as an alternative to their spoken languages and as a primary language for deaf people.

Hand Talk

Poetry by deaf people

The deaf poet is no oxymoron. But one would think so, given the popular understanding that poetry has sound and voice at its heart.

Melodies Unheard – Poetry Magazine

Suppression of sign languages

• Forbidden Signs: American Culture and the Campaign against Sign Language

Forbidden Signs explores American culture from the mid-nineteenth century to 1920 through the lens of one striking episode: the campaign led by Alexander Graham Bell and other prominent Americans to suppress the use of sign language among deaf people.

Review

Sign language and thought

The relationship of language to cognition, especially in development, is an issue that has occupied philosophers, psychologists, and linguists for centuries. In recent years, the scientific study of sign languages and deaf individuals has greatly enhanced our understanding of deafness, language, and cognition. This Counterpoints volume considers the extent to which the use of sign language might affect the course and character of cognitive development, and presents a variety of viewpoints in this debate. This volume brings the language-thought discussion into a clearer focus, both theoretically and practically, by placing it in the context of children growing up deaf and the influences of having sign language as their primary form of communication

Relations of Language and Thought: The View from Sign Language and Deaf Children

Two videos

Various communities of researchers have thought about this class of problems; there are results scattered through journals on image processing, geometric mechanics, numerical analysis, graph theory, machine learning, general relativity (and more?) Besides its use in various domains of application, discrete geometry also provides fresh insights into classical, continuous geometry (e.g. Regge’s discussion of Bianchi identities in his original paper on Regge calculus).

Discrete exterior calculus, cohomology, etc.

• Discrete exterior calculus by Desbrun, Hirani, Leok, Marsden
• Discrete Poincare lemma by Desbrun, Leok, Marsden

Discrete Riemannian geometry, connections, etc.

• A Discrete Theory of Connections on Principal Bundles by Leok, Marsden, Weinstein.
• General Relativity Without Coordinates by Regge (1961) (Introduces Regge calculus.)
• The geometry of classical Regge calculus by Barrett (1987)
• Torsion Degrees of Freedom in the Regge Calculus as Dislocations on the Simplicial Lattice by Schmidt, Kohler (2001)

Discrete spectral geometry

• References on graph Laplacians, etc.

Convergence of discrete geometric quantities to their continuous counterparts.

• Add references.

Discrete geometric mechanics, discrete field theory

• Discrete Euler-Poincaré and Lie-Poisson Equations by Marsden, Pekarsky, Shkoller (1999)
• Multisymplectic geometry, variational integrators, and nonlinear PDEs by Marsden, Patrick, Shkoller.
• Discrete Geometric Mechanics for Variational Time Integrators by Stern, Desburn
• Variational Integrators, a post by Rod Carvalho with relevant references
• Jerrold Marsden on Discrete Mechanics and Optimal Control
• Differential Geometry in Computational Electromagnetics, PhD thesis by Eric Alan Forgy

General links

• The Discrete Differential Geometry Forum at Columbia University
• Applied Geometry Lab at Caltech

People

• Jerold Marsden
• Malvin Leok

I had collected these links a few years ago, and saved them with a note that said “I’d like to think a little about this, before starting to read.”  Still haven’t given much thought, and still haven’t read any of them.

Some issues I keep coming back to: The similarity/relation between the way we parse melodic and rhythmic structure and the way we parse language. The relations between linguistic intonation, emotion, and musical intervals/mini-phrases. (Is this where we will find the roots of melody?) Intonation/melody in other animals.

Some papers, books, talks:

• Music, Language and the Brain, Patel, 2007. Review, review, review. (The last one is behind a pay wall.)
• Musical rhythm, linguistic rhythm, and human evolution, Patel, Musical Perception, 2006.
• Bach Speaks: A Cortical “Language-Network” Serves the Processing of Music, Koelschb, Guntera, Cramona, Zysseta, Lohmanna, Friedericia, Neuroimage, 2002.
• Talk of the Tone, Patel, Nature, 2008.
• Language, Music, Syntax and the Brain, Patel, Nature, 2003.
• , Patel, 2006. (Talk at UCSD)
• Music, Language and Meaning: brain signatures of semantic processing, Koelsch, Kasper, Sammler, Schulze, Gunter, Friederici, Nature Neuroscience, 2004.
• Brain organization for music processing, Peretz, Zattore, Annual Reviews of Psychology, 2005.
• The nature of music from a biological perspective, Peretz, Cognition, 2006.
• The Cognitive Neuroscience of Music, Peretz, Zattore (editors), 2003.

Places, people:

• BRAMS – International Laboratory for Brain, Music and Sound Research (“jointly affiliated with the University of Montreal and McGill University”)
• Isabelle Peretz lab. Isabelle Peretz. Publications.
• Aniruddh D. Patel. Publications.
• Stefan Koelsch

Related:

• Konnakol, bol, kuchi shōga
• Bird vocalization, whale song
• Intonation

Assorted links with tips, tricks, and bits of philosophy.

Suppose you want to teach the “cat” concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractile claws, a distinctive sonic output, etc.? I’ll bet not. You probably show the kid a lot of different cats, saying “kitty” each time, until it gets the idea.

Can we make mathematics intelligible?, by Ralph Philip Boas, Jr.

Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average   colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematicians who are close to the field of the speaker ask a question or two to avoid embarrassment. This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.

On proof and progress in mathematics, by William P. Thurston.

Sometime in my second year in university I suddenly discovered that most of the things I studied really mean something;  that  proofs are not   just  long chains of  logical  deductions discovered by evil math geniuses,   that  behind every worthwhile argument   there’s a beautiful  picture  that   fits   together  with  everything else. Except that my profs tended not to talk about these pictures, and certainly not about  everything else. They didn’t even tell me that these pictures were there.

Teaching philosophy, by Dror Bar-Natan.

This post is about a very simple idea that can dramatically improve the readability of just about anything, though I shall restrict my discussion to the question of how to write clearly about mathematics. The idea is more or less there in the title: present examples before you discuss general concepts. Before I go any further, I want to make very clear what the point is here. It is not the extremely obvious point that it is good to illustrate what you are saying with examples. Rather, it is to do with where those examples should appear in the exposition. So the emphasis is on the word “first” rather than on the word “examples”. If this too seems pretty obvious, I invite you to consider how common it is to do the opposite.

My favourite pedagogical principle: examples first!, by Tim Gowers.

There is also a follow-up post. Also check out this comment by Terry Tao.

Common sense, cognitive psychology, and my personal experience agree that, no matter how interesting a lecture, students’ attention is much keener in minutes 1-10 than in minutes 40-50. I’ve found it makes a big difference to break every lecture up with some kind of active or reflective activity. For example: I write an assertion on the board, like “If a matrix is diagonalizable, it has distinct eigenvalues.” I ask for a show of hands on whether the assertion is true or false. Then I ask each student to turn to the person sitting next to them and try to convince that person of the truth or falsity of the assertion. After one or two minutes, I bring the class back together for another show of hands, followed by a brief discussion. I think the class finds it gratifying that the process tends to converge on the right answer; it seems to me to emphasize the useful lesson that something is mathematically correct because it can be argued, not because I say so. (I first heard about this technique from Eric Mazur at Harvard; I’m not sure whether it’s original to him.)

Notes and links on teaching, by Jordan Ellenberg.

• Mathematical writing, by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts.
• Mathematical Education, by William Thurston.

Updates 5/1/14:

• Real-life module statistics: A happy Harvard experiment, by Kari Lock and Xiao-Li Meng.
• (To read) Teaching labs the Compass way: Engaging students in authentic research practices and regular self-reflection, by Punit R. Gandhi, Jesse Livezey, Anna M. Zaniewski, Daniel L. Reinholz, Dimitri R. Dounas-Frazer.

Updates 7/4/14:

Michael Atiyah has a section devoted to the power of good examples in his Advice to a Young Mathematician. Here is his final paragraph (the whole thing is very much worth reading):

But most of all a good example is a thing of beauty. It shines and convinces. It gives insight and understanding. It provides the bedrock of belief.

George Orwell on writing concretely (from Politics and the English Language, via Faruk):

I am going to translate a passage of good English into modern English of the worst sort. Here is a well-known verse from ECCLESIASTES:

I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favor to men of skill; but time and chance happeneth.

Here it is in modern English:

Objective consideration of contemporary phenomena compels the conclusion that success or failure in competitive activities exhibits no tendency to be commensurate with innate capacity, but that a considerable element of the unpredictable must invariably be taken into account.

This is a parody, but not a very gross one . . . The first sentence contains six vivid images, and only one phrase (“time and chance”) that could be called vague. The second contains not a single fresh, arresting phrase, and in spite of its 90 syllables it gives only a shortened version of the meaning contained in the first.