### © 2009 Teodora CoxRURAL HIGH SCHOOL MATHEMATICS TEACHERS ‘ RESPONSE TO MATHEMATICS REFORM CURRICULUM INTEGRATION AND PROFESSIONAL DEVELOPMENT BY

"... The purpose of this qualitative study was to examine rural high school mathematics teachers ‘ responses to the initial implementation of Louisiana‘s Comprehensive Curriculum during their second year of involvement in a professional development program. The curriculum changes were the culmination of ..."

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The purpose of this qualitative study was to examine rural high school mathematics teachers ‘ responses to the initial implementation of Louisiana‘s Comprehensive Curriculum during their second year of involvement in a professional development program. The curriculum changes were the culmination of an alignment between standards, curriculum, assessments and instruction which exemplified the shift to standards-based accountability and high-stakes testing characteristic of post-NCLB systemic reform efforts. I further investigated some of the discrepancies between the teachers ‘ professed beliefs about mathematics and their classroom practices. The research questions probed the responses of forty-seven teachers to the implementation of the Comprehensive Curriculum, their impressions of the impact of the professional development program, and the nature of mathematics as portrayed in the new curriculum. The study was framed in symbolic interactionism and grounded theory. The concerns and interests of the rural mathematics teachers guided the interview discussions and some of the observations. Data sources included surveys, participant-observations, interviews and other documents. Predetermined and constant comparative coding themes contributed to the constant

### Descartes Numbers

"... We call n a Descartes number if n is odd and n = km for two integers k, m> 1 such that σ(k)(m + 1) = 2n, where σ is the sum of divisors function. In this paper, we show that the only cube-free Descartes number with fewer than seven distinct prime divisors is the number 3 2 7 2 11 2 13 2 19 2 61, wh ..."

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We call n a Descartes number if n is odd and n = km for two integers k, m> 1 such that σ(k)(m + 1) = 2n, where σ is the sum of divisors function. In this paper, we show that the only cube-free Descartes number with fewer than seven distinct prime divisors is the number 3 2 7 2 11 2 13 2 19 2 61, which was discovered by Réne Descartes. We also show that if n is a cube-free Descartes number not divisible by 3, then n has over a million distinct prime divisors. 1

### Logic and Computerisation in mathematics?

, 2009

"... – If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ ..."

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– If you give me an algorithm to solve Π, I can check whether this algorithm really solves Π. – But, if you ask me to find an algorithm to solve Π, I may go on forever trying but without success. • But, this result was already found by Aristotle: Assume a proposition Φ. – If you give me a proof of Φ, I can check whether this proof really proves Φ. – But, if you ask me to find a proof of Φ, I may go on forever trying but without success. • In fact, programs are proofs and much of computer science in the early part of the 20th century was built by mathematicians and logicians. • There were also important inventions in computer science made by physicists (e.g., von Neumann) and others, but we ignore these in this talk. ISR 2009, Brasiliá, Brasil 1An example of a computable function/solvable problem • E.g., 1.5 chicken lay down 1.5 eggs in 1.5 days. • How many eggs does 1 chicken lay in 1 day? • 1.5 chicken lay 1.5 eggs in 1.5 days. • Hence, 1 chicken lay 1 egg in 1.5 days. • Hence, 1 chicken lay 2/3 egg in 1 day. ISR 2009, Brasiliá, Brasil 2Unsolvability of the Barber problem • which man barber in the village shaves all and only those men who do not shave themselves? • If John was the barber then – John shaves Bill ⇐ ⇒ Bill does not shave Bill – John shaves x ⇐ ⇒ x does not shave x – John shaves John ⇐ ⇒ John does not shave John • Contradiction. ISR 2009, Brasiliá, Brasil 3Unsolvability of the Russell set problem

### Embodied Cognition and the Origins of Geometry: A Model Approach of Embodied Mathematics Through Geometric Considerations

, 2003

"... In this paper, we propose that ’embodied mathematics ’ should be studied not only by reduction to the present individual bodily experience but in an historical context as well, as far as the origins of mathematics are concerned. Some early mathematical results are the Theorems of Geometry and arose ..."

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In this paper, we propose that ’embodied mathematics ’ should be studied not only by reduction to the present individual bodily experience but in an historical context as well, as far as the origins of mathematics are concerned. Some early mathematical results are the Theorems of Geometry and arose as attempts to objectively render the main perceptual categories such as verticality, horizontality, similarity (or its varieties). Inasmuch as these are of a qualitative nature, it was required that they be expressed in a quantitative way in order to be objectified. The first form of this objectification occurred in the case of ’archetypal results’, namely the Pythagorean triads and the internal ratio of the legs in the right triangles. In the next stage, a ’scientific ’ treatment would come from a shift of objectification and descriptions inside an abstract theory, which would constitute the first logicomathematical knowledge. In this theory, the ’archetypal results ’ were incorporated, generalized and acquired their unquestionable, supertemporal validity. The study presents a particular epistemological analysis of some of the main terms used in the beginnings of Geometrical Thought and Euclid’s Elements, utilizing the theoretical apparatus of the theory of ’embodied mathematics’. It also traces models of objectification for the ’archetypal results ’ and indicates their diffusion in later mathematical developments.

### NON-EUCLIDEAN PYTHAGOREAN TRIPLES, A PROBLEM OF EULER, AND RATIONAL POINTS ON K3 SURFACES

, 2006

"... Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an alg ..."

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Abstract. We discover suprising connections between three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geometry, finding three integers such that the difference of the squares of any two is a square, and the problem of finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler’s work on the second problem with a modern point of view. 1. Problem I: Pythagorean triples An ordinary Pythagorean triple is a triple (a, b, c) of positive integers satisfying a 2 + b 2 = c 2. Finding these is equivalent, by the Pythagorean theorem, to finding right triangles with integral sides. Since the equation is homogeneous, the problem for rational numbers is the same, up to a scale factor. Some Pythagorean triples, such as (3, 4, 5), have been known since antiquity. Euclid [6, X.28, Lemma 1] gives a method for finding such triples, which leads to a complete solution of the problem. The primitive Pythagorean triples are exactly the triples of integers (m 2 − n 2, 2mn, m 2 + n 2) for various choices of m, n (up to

### Heriot-Watt University Edinburgh, Scotland

, 2005

"... The evolution of types and logic in the 20th century ∗ ..."

### Um Ceclo de Computeraçao

"... Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible ..."

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Brasiliá 2010Welcome to the fastest developing and most influential subject: Computer Science • Computer Science is by nature highly applied and needs much precision, foundation and theory. • Computer Science is highly interdisciplinary bringing many subjects together in ways that were not possible before. • Many recent scientific results (e.g., in chemistry) would not have been possible without computers. • The Kepler Conjecture: no packing of congruent balls in Euclidean space has density greater than the density of the face-centered cubic packing. • Sam Ferguson and Tom Hales proved the Kepler Conjecture in 1998, but it was not published until 2006. • The Flyspeck project aims to give a formal proof of the Kepler Conjecture.

### MathLang, a framework for computerising

, 2007

"... Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected) ..."

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Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) University of West of England, Bristol 1A brief history • There are two influencing questions:

### MathLang, a framework for computerising

, 2007

"... Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the cont ..."

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Saarbruecken, GermanyWhat is the aim for MathLang? Can we formalise a mathematical text, avoiding as much as possible the ambiguities of natural language, while still guaranteeing the following four goals? 1. The formalised text looks very much like the original mathematical text (and hence the content of the original mathematical text is respected). 2. The formalised text can be fully manipulated and searched in ways that respect its mathematical structure and meaning. 3. Steps can be made to do computation (via computer algebra systems) and proof checking (via proof checkers) on the formalised text. 4. This formalisation of text is not much harder for the ordinary mathematician than L ATEX. Full formalization down to a foundation of mathematics is not required, although allowing and supporting this is one goal. (No theorem prover’s language satisfies these goals.) Saarbruecken, Germany 1A brief history • There are two influencing questions: