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Advanced Language Technologies for Mathematical Markup
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Advanced Language Technologies for Mathematical Markup



IMA Hot Topic Workshop on The Evolution of Mathematical Communication in the Age of Digital Libraries, December 8-9, 2006.

IMA Hot Topic Workshop on The Evolution of Mathematical Communication in the Age of Digital Libraries, December 8-9, 2006.



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  • Mathematical markup languages like OpenMath and MathML offer the possibility to represent mathematical content in a level of abstraction that is not dependent on localized information. This representation typically focuses on the semantics of the mathematical object and postpones localization aspects of mathematics, such as those influenced by notation and by culture, to the rendering process of the markup. While typesetting of mathematical markup has been the object of a numerous efforts, from MathML-presentation to SVG converters, the rendering of mathematics in a "verbalized" jargon has not yet received similar attention. In this talk, I will present the results of the WebALT EU eContent project concerning the application of language technologies to the automatic generation of text from mathematical markup. Mathematical jargon is an important aspect of the education of students. Not only does a teacher train pupils in problem solving skills, but she also makes sure that they acquire a proper way of expressing mathematical concepts. To our knowledge, digital eLearning resources have used a representation in which text is intermixed with mathematical expressions even in situations where the actual abstract representation, for instance of the statement of a theorem, can be reduced to a single mathematical object. One reason for this representation choice is that the rendering process would otherwise produce a symbolic, typeset mathematical formula that might prove too difficult to understand for the students or simply just too hard to read. However, by representing this kind of mathematical text in a language-independent format such as the one provided by markup languages, it is possible to apply language technologies that generate the same text in a variety of languages including English, Spanish, Finnish, Swedish, French and Italian. The project results include editors for mathematical multilingual markup, a web service for generating multiple languages versions and a digital repository of multilingual interactive mathematical exercises and drill questions.

Advanced Language Technologies for Mathematical Markup Advanced Language Technologies for Mathematical Markup Presentation Transcript

  • Olga Caprotti, Mika Seppälä Jordi Saludes, Gloria Casanellas, Wanjiku Ng'ang'a Work funded by EDC-22253-WEBALT Advanced Language Technologies for Mathematical Markup
  • Vision in the Spirit of Jules Verne Professor oversees the instruction, graduate student propels the machine. Students learn by looking to a screen and listening
  • Mobile math
  • The future is here today
    • Students may take online quizzes, anywhere, anytime, on their phones
    • Answers are automatically graded and recorded
    • Feedback is personalized (LeActiveMath)
    • Lectures are available online, a given. (G. Strang videos)
    • Content for online delivery is expensive :
    • It combines expertise in several areas like
    • web technologies, user interfaces, education, subject area.
  • Reusable Online Math Content
    • Mathematical exercises are
    • naturally reusable by
    • changing the parameters of the problem
    Such reusable content is many times over more valuable translate content for use across border
  • Multilingual mathematics
    • Multilingual mathematics is mathematics expressed in a variety of languages
    • Mathematics is abstract yet it can be verbalized using specialized jargon in natural language
    • Verbal mathematics is syntactically and symbolically different in the different languages
      • 93, “ninenty three”, “quatre vingt treize”, “drei und neunzig”
      • GGT, GCD, MCD
  • Why multilingual math education
    • Students need to learn how to express themselves, also mathematically (thinking, speaking, writing)
    • Linguistic differences must be preserved for cultural heritage
    • Linguistic minorities must not be discriminated
  • Linguistic Minorieties
    • 55 millions speakers in EU
  • Linguistic Minorieties
    • regional/minority languages are spoken in all European countries, except for Iceland.
  • Mathematics: a universal language
    • Due to its abstract and exact nature,
    • one can expect to be able to obtain verbalizations of math in natural language, without loss of information,
    • provided one generates them from rich mathematical content
  • WHY content markup?
  • Language Generation in WebALT
    • Short problems (numerical, drill)
      • “ compute the determinant of M”
      • “ find n such that P(n)”
      • “ let x be positive, show that A(x)”
    • Long problems, feedbacks and hints
    Generate multilingual verbalizations for mathematical problems Study of the state of the art in multilingual and multicultural creation of digital mathematical content . L. Carlson, J. Saludes, A. Strotmann. WebALT Project Deliverable D1.2., April 2005.
  • No essays yet
  • MathDox Player in Moodle
  • Automatic Multilingual Exercise
  • Representation of problems
    • Conceptually layered:
    • Mathematics, OpenMath
    • Sentence, WebALT extended OpenMath
    • Problem + algorithmic flow, Math-QTI
    for interactivity and multilinguality Problem Sentence Mathematics
  • Multilingual Short Problems
    • Math is represented in OpenMath
      • extensible: e.g. by private own NLG primitives
      • language independent
      • embeds in Math-QTI and renders with MathML-P
    • Multilingual Generation is done on
      • well-formed
      • language-independent rich encoding
  • a ttrib([nlg:mood nlg:imperative nlg:tense nlg:present, nlg:directive nlg:determine],plangeo1:are_on_line(A,B,C))
      • Determine if A, B and C are collinear.
      • Määritä ovatko A, B ja C suoralla.
      • Determina si A, B y C son colineales.
      • Déterminer si A, B et C sont sur une droite.
      • Determina se A, B e C sono su una linea.
      • Bestäm om A, B och C är på en linje.
    Note the linguistic differences: Imperative vs. Infinitive Adjectives vs. Adverbial phrases
  • Grammatical Framework
    • Grammar formalism, based on type theory, and supporting:
      • Multilinguality by abstract grammar + concrete grammars for parallel languages
      • Semantics, like well formedness of expressions
      • Modular grammar engineering: abstract grammar reflects mathematics
      • Reusable grammars as software components
    • Resource Grammar Library: Danish, English , Finnish , French , German, Italian , Norwegian, Russian, Spanish , Swedish . (Catalan, Swahili)
    Aarne Ranta, Chalmers
  • Mathematical Problem Grammars
    • Operations: sentences
    • OpenMath: math
    • Ground: variables, literals, integers
    Abstract Grammars 175 OpenMath symbols 36 categories 12,000 lines of code 158 source files (Not including the catalan resource grammar. This part itself contains a bit less than 10,000 lines). Concrete Grammars
  • NL_Library Coverage
    • WP3, June 2005
      • Old API GF 0.9
      • Stricter OpenMath
      • linearize but no parse
      • All CDs!
    • WP7, December 2006
      • GF API 1.0
      • Looser OpenMath
      • linearize and parse
      • modular
      • CDs: alg1, , arith1, calculus1, complex1, fns1, integer1, interval1, limit1, linalg1, linalg2, logic1, minmax1, nums1, relation1, rounding1, setname1, transc1
  • Text and Math Editor
  • Good news and bad news
    • NL Library is work in progress
      • FP7 IST call in April 2007: good research topic!
        • Extend grammar coverage for short problems
        • Extend types of sentences to cover more feedback
        • Extend content markup
    • Swahili GF grammars, in progress
    • Catalan GF grammars
    • GF summer school for EU official languages
  • Tools for Mathematical Markup W3C-Math WG survey You will get an email from us
  • Linguists