Mathematical Communication, Conceptual Understanding, and Students' Attitudes Toward Mathematics

This action research study of my 8th grade classroom investigated the use of mathematical communication, through oral homework presentations and written journals entries, and its impact on conceptual understanding of mathematics. This change in expectation and its impact on students’ attitudes towards mathematics was also investigated. Challenging my students to communicate mathematics both orally and in writing deepened the students’ understanding of the mathematics. Levels of understanding deepened when a variety of instructional methods were presented and discussed where students could comprehend the ideas that best suited their learning styles. Increased understanding occurred through probing questions causing students to reflect on their learning and reevaluate their reasoning. This transpired when students were expected to write more than one draft to math journals. By making students aware of their understanding through communicating orally and in writing, students realized that true understanding did not come from mere homework completion, but from evaluating and assessing their own and other’s ideas and reasoning. I discovered that when students were challenged to communicate their reasoning both orally and in writing, students enjoyed math more and thought math was more fun. As a result of this research, I will continue to require students to communicate their thinking and reasoning both orally and in writing.

Improving the Effectiveness of Independent Practice with Corrective Feedback

In this action research study of my 8th grade Algebra class, I investigated the effects of teacher-to-student written corrective feedback on student performance and attitude toward mathematics. The corrective feedback was given on solutions for selected independent practice problems assigned as homework. Each problem being assessed was given a score based on a 3- point rubric and additional comments were written. I discovered that providing teacher-to-student written corrective feedback for independent practice problems was beneficial for both students and teachers. The feedback positively affected the attitudes of students and teacher toward independent practice work resulting in an improved quality of solutions produced by students. I plan to extend my research to explore ways to provide corrective feedback to students in all of my mathematics classes.

Exploring the Influence of Vocabulary Instruction on Students’ Understanding of Mathematical Concepts

In this action research study of my classroom of 8th grade mathematics, I investigated the influence of vocabulary instruction on students’ understanding of the mathematics concepts. I discovered that knowing the meaning of the vocabulary did play a major role in the students’ understanding of the daily lessons and the ability to take tests. Understanding the vocabulary and the concepts allowed the students to be successful on their daily assignments, chapter tests, and standardized achievement tests. I also discovered that using different vocabulary teaching strategies enhanced equity in my classroom among diverse learners. The knowledge of the math vocabulary increased my students’ confidence levels, which in turn increased their daily and test scores. As a result of this research, I plan to find ways to incorporate the vocabulary teaching strategies I have used into current math curriculum. I will start this process at the beginning of the next school year, and will continue looking for new strategies that will promote math vocabulary retention.

EQUIVALENCE OF REAL MILNOR FIBRATIONS FOR QUASI-HOMOGENEOUS SINGULARITIES

We show that for real quasi-homogeneous singularities f : (R-m, 0) -> (R-2, 0) with isolated singular point at the origin, the projection map of the Milnor fibration S-epsilon(m-1) \ K-epsilon -> S-1 is given by f/parallel to f parallel to. Moreover, for these singularities the two versions of the Milnor fibration, on the sphere and on a Milnor tube, are equivalent. In order to prove this, we show that the flow of the Euler vector field plays and important role. In addition, we present, in an easy way, a characterization of the critical points of the projection (f/parallel to f parallel to) : S-epsilon(m-1) \ K-epsilon -> S-1.

The local power of the gradient test

The asymptotic expansion of the distribution of the gradient test statistic is derived for a composite hypothesis under a sequence of Pitman alternative hypotheses converging to the null hypothesis at rate n(-1/2), n being the sample size. Comparisons of the local powers of the gradient, likelihood ratio, Wald and score tests reveal no uniform superiority property. The power performance of all four criteria in one-parameter exponential family is examined.

Geometrical and Spectral Properties of the Orthogonal Projections of the Identity

[EN]We analyze the best approximation

Theoretical and implementation aspects in the mechanization of the metatheory of programming languages

Interactive theorem provers are tools designed for the certification of formal proofs developed by means of man-machine collaboration. Formal proofs obtained in this way cover a large variety of logical theories, ranging from the branches of mainstream mathematics, to the field of software verification. The border between these two worlds is marked by results in theoretical computer science and proofs related to the metatheory of programming languages. This last field, which is an obvious application of interactive theorem proving, poses nonetheless a serious challenge to the users of such tools, due both to the particularly structured way in which these proofs are constructed, and to difficulties related to the management of notions typical of programming languages like variable binding. This thesis is composed of two parts, discussing our experience in the development of the Matita interactive theorem prover and its use in the mechanization of the metatheory of programming languages. More specifically, part I covers: - the results of our effort in providing a better framework for the development of tactics for Matita, in order to make their implementation and debugging easier, also resulting in a much clearer code; - a discussion of the implementation of two tactics, providing infrastructure for the unification of constructor forms and the inversion of inductive predicates; we point out interactions between induction and inversion and provide an advancement over the state of the art. In the second part of the thesis, we focus on aspects related to the formalization of programming languages. We describe two works of ours: - a discussion of basic issues we encountered in our formalizations of part 1A of the Poplmark challenge, where we apply the extended inversion principles we implemented for Matita; - a formalization of an algebraic logical framework, posing more complex challenges, including multiple binding and a form of hereditary substitution; this work adopts, for the encoding of binding, an extension of Masahiko Sato's canonical locally named representation we designed during our visit to the Laboratory for Foundations of Computer Science at the University of Edinburgh, under the supervision of Randy Pollack.

Sensitivity analysis of a parabolic-elliptic problem

We consider the heat flux through a domain with subregions in which the thermal capacity approaches zero. In these subregions the parabolic heat equation degenerates to an elliptic one. We show the well-posedness of such parabolic-elliptic differential equations for general non-negative L-infinity-capacities and study the continuity of the solutions with respect to the capacity, thus giving a rigorous justification for modeling a small thermal capacity by setting it to zero. We also characterize weak directional derivatives of the temperature with respect to capacity as solutions of related parabolic-elliptic problems.

The group of invariants of an inner function with finite spectrum

This paper determines the group of continuous invariants corresponding to an inner function circle dot with finitely many singularities on the unit circle T; that is, the continuous mappings g : T -> T such that circle dot o g = circle dot on T. These mappings form a group under composition.

Shading Silhouettes of Smaller Ones

I am truly honored to have been given the amazing opportunity to create this original piece, this powerful journey through memory and emotive exploration of the loss of childhood. How do we feel about the loss of our child-self? Could we ever get them back? How long, how deep would one have to dig in the graveyards, the playgrounds of memory, to uncover what was buried there... to un-erase what waserased? shading silhouettes of smaller ones will ultimately encourage a reconnection with the Inner Child hidden inside all of us, as well as an intimate awareness of the adult version of the self by looking back to the smaller ones. The main inspiration for this piece is then of course, Inner Child Work. Most people may not be familiar with this therapeutic exploration of childhood... It wasimportant to me then, to present this concept in an imaginative, theatrical way, as a gift to you - a comprehensive and intensely moving gift. Speaking from experience, working on my Inner Child - my little Bianca - has been the most painful, frightening, yetrewarding and powerful experience within my personal life. Some people spend their entire lives trying to love themselves, to prove themselves, or be accepted. Some are too afraid to look back to where it all began. The characters within this piece will face thatfear... in a regression from the complexities of adulthood to the confusion of adolescence, all the way back to the wonder and bliss of childhood. They will reveal memories, of both joy and pain, love and abandonment, journeying backwards through time - through memory - through a playground - back to the beginning... We will enter a world where a push of a merry-go-round spins us to games of Truth or Dare after a high school dance at 16 - or the slam of a metal fence reminds us of the door Dad slammed in our face at 9 - where the sound of chain links swings us back to scrapping our knee by the sandbox at 5 This piece will attempt to connect everyone, both cast and audience, through a universal understanding and discussion of what it means to grow up, as well as a discovery of WHY we are the way we are - how experiences or relationships from our childhood have shaped our adult lives. We will attempt to challenge your honesty and nerve by inviting you to ask questions of yourselves, your past - to remember what it's like to have the innocence and hope of a child, to engage with and discover your Inner Child, to realize when or why you left them behind, and if you want to this magical part of yourself. It is my hope that you will join us in a collective journey - gather the courage to dig up the little kid you buried so long ago...* The creation, design, choreography, and direction for shading silhouettes of smaller ones mark the culminating experience of a year-long independent study in Theatre.

Applications of finite geometries to designs and codes

This dissertation concerns the intersection of three areas of discrete mathematics: finite geometries, design theory, and coding theory. The central theme is the power of finite geometry designs, which are constructed from the points and t-dimensional subspaces of a projective or affine geometry. We use these designs to construct and analyze combinatorial objects which inherit their best properties from these geometric structures. A central question in the study of finite geometry designs is Hamada’s conjecture, which proposes that finite geometry designs are the unique designs with minimum p-rank among all designs with the same parameters. In this dissertation, we will examine several questions related to Hamada’s conjecture, including the existence of counterexamples. We will also study the applicability of certain decoding methods to known counterexamples. We begin by constructing an infinite family of counterexamples to Hamada’s conjecture. These designs are the first infinite class of counterexamples for the affine case of Hamada’s conjecture. We further demonstrate how these designs, along with the projective polarity designs of Jungnickel and Tonchev, admit majority-logic decoding schemes. The codes obtained from these polarity designs attain error-correcting performance which is, in certain cases, equal to that of the finite geometry designs from which they are derived. This further demonstrates the highly geometric structure maintained by these designs. Finite geometries also help us construct several types of quantum error-correcting codes. We use relatives of finite geometry designs to construct infinite families of q-ary quantum stabilizer codes. We also construct entanglement-assisted quantum error-correcting codes (EAQECCs) which admit a particularly efficient and effective error-correcting scheme, while also providing the first general method for constructing these quantum codes with known parameters and desirable properties. Finite geometry designs are used to give exceptional examples of these codes.

Explicit mathematics and operational set theory: some ontological comparisons

We discuss several ontological properties of explicit mathematics and operational set theory: global choice, decidable classes, totality and extensionality of operations, function spaces, class and set formation via formulas that contain the definedness predicate and applications.

A feasible theory of truth over combinatory algebra

We define an applicative theory of truth TPT which proves totality exactly for the polynomial time computable functions. TPT has natural and simple axioms since nearly all its truth axioms are standard for truth theories over an applicative framework. The only exception is the axiom dealing with the word predicate. The truth predicate can only reflect elementhood in the words for terms that have smaller length than a given word. This makes it possible to achieve the very low proof-theoretic strength. Truth induction can be allowed without any constraints. For these reasons the system TPT has the high expressive power one expects from truth theories. It allows embeddings of feasible systems of explicit mathematics and bounded arithmetic. The proof that the theory TPT is feasible is not easy. It is not possible to apply a standard realisation approach. For this reason we develop a new realisation approach whose realisation functions work on directed acyclic graphs. In this way, we can express and manipulate realisation information more efficiently.

Placing Strategy Discourse in Context : Sociomateriality, Sensemaking, and Power

There has been increasing interest in the discursive aspects of strategy over the last two decades. In this editorial we review the existing literature, focusing on six major bodies of discursive scholarship: post-structural, critical discourse analysis, narrative, rhetoric, conversation analysis and metaphor. Our review reveals the significant contributions of research on strategy and discourse, but also the potential to advance research in this area by bringing together research on discursive practices and research on other practices we know to be important in strategy work. We explore the potential of discursive scholarship in integrating between significant theoretical domains (sensemaking, power and sociomateriality), and realms of analysis (institutional, organizational and the episodic), relevant to strategy scholarship. This allows us to place the papers published in the special issue Strategy as Discourse: Its Significance, Challenges and Future Directions among the body of knowledge accumulated thus far, and to suggest a way forward for future scholarship.