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"Mathematical notation"
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Math words and symbols
Introduces young children to the meaning and use of math words and symbols.
Book
Numerical Notation
This book is a cross-cultural reference volume of all attested numerical notation systems (graphic, non-phonetic systems for representing numbers), encompassing more than 100 such systems used over the past 5,500 years. Using a typology that defies progressive, unilinear evolutionary models of change, Stephen Chrisomalis identifies five basic types of numerical notation systems, using a cultural phylogenetic framework to show relationships between systems and to create a general theory of change in numerical systems. Numerical notation systems are primarily representational systems, not computational technologies. Cognitive factors that help explain how numerical systems change relate to general principles, such as conciseness or avoidance of ambiguity, which apply also to writing systems. The transformation and replacement of numerical notation systems relates to specific social, economic, and technological changes, such as the development of the printing press or the expansion of the global world-system.
eBook
The Development of Children's Algebraic Thinking: The Impact of a Comprehensive Early Algebra Intervention in Third Grade
A study investigated the impact of a sustained, comprehensive early algebra intervention in third grade. The authors share and discuss students' responses to a written pre- and post-assessment that addressed their understanding of several big ideas in the area of early algebra, including mathematical equivalence and equations, generalizing arithmetic, and functional thinking.
Journal Article
The language of mathematics : the stories behind the symbols
\"Galileo famously wrote that the book of nature is written in mathematical language. The Language of Mathematics is a wide-ranging and beautifully illustrated collection of short, colorful histories of the most commonly used symbols in mathematics, providing readers with an engaging introduction to the origins, evolution, and conceptual meaning of each one. In dozens of lively and informative entries, Raúl Rojas shows how today's mathematics stands on the shoulders of giants, mathematicians from around the world who developed mathematical notation through centuries of collective effort. He tells the stories of such figures as al-Khwarizmi, René Descartes, Joseph-Louis Lagrange, Carl Friedrich Gauss, Augustin-Louis Cauchy, Karl Weierstrass, Sofia Kovalevskaya, David Hilbert, and Kenneth Iverson. Topics range from numbers and variables to sets and functions, constants, and combinatorics. Rojas describes the mathematical problems associated with different symbols and reveals how mathematical notation has sometimes been an accidental process. The entries are self-contained and can be read in any order, each one examining one or two symbols, their history, and the variants they may have had over time. An essential companion for math enthusiasts, The Language of Mathematics shows how mathematics is a living and evolving entity, forever searching for the best symbolism to express relationships between abstract concepts and to convey meaning\"-- Provided by publisher.
BOOK
A Learning Trajectory in 6-Year-Olds' Thinking About Generalizing Functional Relationships
The study of functions is a critical route into teaching and learning algebra in the elementary grades, yet important questions remain regarding the nature of young children's understanding of functions. This article reports an empirically developed learning trajectory in first-grade children's (6-year-olds') thinking about generalizing functional relationships. Findings suggest that children can learn to think in quite sophisticated and generalized ways about relationships in function data, thus challenging the typical curricular approach in the lower elementary grades in which children consider only variation in a single sequence of values.
Journal Article
Language and the rise of the algorithm
\"A wide-ranging history of the intellectual developments that produced the modern idea of the algorithm. Bringing together the histories of mathematics, computer science, and linguistic thought, Language and the Rise of the Algorithm reveals how recent developments in artificial intelligence are reopening an issue that troubled mathematicians long before the computer age. How do you draw the line between computational rules and the complexities of making systems comprehensible to people? Here Jeffrey M. Binder offers a compelling tour of four visions of universal computation that addressed this issue in very different ways: G. W. Leibniz's calculus ratiocinator; a universal algebra scheme Nicolas de Condorcet designed during the French Revolution; George Boole's nineteenth-century logic system; and the early programming language ALGOL, whose name is short for algorithmic language. These episodes show that symbolic computation has repeatedly become entangled in debates about the nature of communication. To what extent can meaning be controlled by individuals, like the values of a and b in algebra, and to what extent is meaning inevitably social? By attending to this long-neglected question, we come to see that the modern idea of the algorithm is implicated in a long history of attempts to maintain a disciplinary boundary separating technical knowledge from the languages people speak day to day. Machine learning, in its increasing dependence on words, now places this boundary in jeopardy, making its stakes all the more urgent to understand. The idea of the algorithm is a levee holding back the social complexity of language, and it is about to break. This book is about the flood that inspired its construction. \"-- Provided by publisher.
BOOK
THE GENEALOGY OF ‘’
The use of the symbol$\\mathbin {\\boldsymbol {\\vee }}$for disjunction in formal logic is ubiquitous. Where did it come from? The paper details the evolution of the symbol$\\mathbin {\\boldsymbol {\\vee }}$in its historical and logical context. Some sources say that disjunction in its use as connecting propositions or formulas was introduced by Peano; others suggest that it originated as an abbreviation of the Latin word for “or,” vel . We show that the origin of the symbol$\\mathbin {\\boldsymbol {\\vee }}$for disjunction can be traced to Whitehead and Russell’s pre- Principia work in formal logic. Because of Principia ’s influence, its notation was widely adopted by philosophers working in logic (the logical empiricists in the 1920s and 1930s, especially Carnap and early Quine). Hilbert’s adoption of$\\mathbin {\\boldsymbol {\\vee }}$in his Grundzüge der theoretischen Logik guaranteed its widespread use by mathematical logicians. The origins of other logical symbols are also discussed.
Journal Article
The operator tensor formulation of quantum theory
In this paper, we provide what might be regarded as a manifestly covariant presentation of discrete quantum theory. A typical quantum experiment has a bunch of apparatuses placed so that quantum systems can pass between them. We regard each use of an apparatus, along with some given outcome on the apparatus (a certain detector click or a certain meter reading for example), as an operation. An operation (e.g. Bb2a3a1) can have zero or more quantum systems inputted into it and zero or more quantum systems outputted from it. The operation Bb2a3a1 has one system of type a inputted, and one system of type b and one system of type a outputted. We can wire together operations to form circuits, for example, . Each repeated integer label here denotes a wire connecting an output to an input of the same type. As each operation in a circuit has an outcome associated with it, a circuit represents a set of outcomes that can happen in a run of the experiment. In the operator tensor formulation of quantum theory, each operation corresponds to an operator tensor. For example, the operation Bb2a3a1 corresponds to the operator tensor . Further, the probability for a general circuit is given by replacing operations with corresponding operator tensors as in 1 Repeated integer labels indicate that we multiply in the associated subspace and then take the partial trace over that subspace. Operator tensors must be physical (namely, they must have positive input transpose and satisfy a certain normalization condition).
Journal Article
Causal Inference Using Potential Outcomes
Causal effects are defined as comparisons of potential outcomes under different treatments on a common set of units. Observed values of the potential outcomes are revealed by the assignment mechanism-a probabilistic model for the treatment each unit receives as a function of covariates and potential outcomes. Fisher made tremendous contributions to causal inference through his work on the design of randomized experiments, but the potential outcomes perspective applies to other complex experiments and nonrandomized studies as well. As noted by Kempthorne in his 1976 discussion of Savage's Fisher lecture, Fisher never bridged his work on experimental design and his work on parametric modeling, a bridge that appears nearly automatic with an appropriate view of the potential outcomes framework, where the potential outcomes and covariates are given a Bayesian distribution to complete the model specification. Also, this framework crisply separates scientific inference for causal effects and decisions based on such inference, a distinction evident in Fisher's discussion of tests of significance versus tests in an accept/reject framework. But Fisher never used the potential outcomes framework, originally proposed by Neyman in the context of randomized experiments, and as a result he provided generally flawed advice concerning the use of the analysis of covariance to adjust for posttreatment concomitants in randomized trials.
Journal Article