K-8 Pre-service Teachers’ Algebraic Thinking: Exploring the Habit of Mind Building Rules to Represent Functions

In this study, through the lens of the algebraic habit of mind Building Rules to Represent Functions, we examined 18 pre-service middle school teachers\u27 ability to use algebraic thinking to solve problems. The data revealed that pre-service teachers\u27 ability to use different features of the habit of mind Building Rules to Represent Functions varied across the features. Significant correlations existed between 8 pairs of the features. The ability to justify a rule was the weakest of the seven features and it was correlated with the ability to chunk information. Implications for mathematics teacher education are discussed

An Exploratory Study of Pre-service Middle School Teachers’ Knowledge of Algebraic Thinking

Using algebraic habits of mind as a framework, and focusing on thinking about functions and how they work, we examined the relationship between 18 pre-service middle school teachers’ ability to use the features of the algebraic thinking (AT) habit of mind “Building Rules to Represent Functions” and their ability to recognize and interpret the features of the same AT habit of mind in middle school students. We assessed the pre-service teachers’ own ability to use the AT habit of mind Building Rules to Represent Functions by examining their solutions to algebra-based tasks in a semester-long mathematics content course. We assessed the pre-service teachers’ ability to recognize and interpret students’ facility with the AT habit of mind Building Rules to Represent Functions by analyzing their interpretations of students’ written solutions to algebra-based tasks and their ability to plan and analyze AT interviews of middle school students during a concurrent field-based education course. The data revealed that the pre-service teachers had a limited ability to recognize the full richness of algebra-based tasks’ potential to elicit the features of Building Rules to Represent Functions in students. The pre-service teachers’ own overall AT ability to Build Rules to Represent Functions was related to their ability to recognize the overall ability of students to Build Rules to Represent Functions, as exhibited during one-on-one interviews, but not to their ability to recognize the overall ability to Build Rules to Represent Functions exhibited exclusively in students’ written work. Implications for mathematics teacher education are discussed

Oddness from Rigidness

We revisit the problem of constructing type IIA orientifolds on T^6/(Z2 x Z2) which admit (non)-factorisable lattices. More concretely, we consider a (Z2 x Z2') orientifold with torsion, where D6-branes wrap rigid 3-cycles. We derive the model building rules and consistency conditions in the case where the compactification lattice is non-factorisable. We show that in this class of configurations, (semi) realistic models with an odd number of families can be easily constructed, in contrast to compactifications where the D6-branes wrap non-rigid cycles. We also show that an odd number of families can be obtained in the factorisable case, without the need of tilted tori. We illustrate the discussion by presenting three family Pati-Salam models with no chiral exotics in both factorisable and non-factorisable toroidal compactifications.Comment: 20 page

Progress in D-brane model building

The state of the art in D-brane model building is briefly reviewed, focusing on recent achievements in the construction of D=4 N = 1 type II string vacua with semi-realistic gauge sectors. Such progress relies on a better understanding of the spectrum of BPS D-branes, the effective field theory obtained from them and the explicit construction of vacua. We first consider D-branes in standard Calabi-Yau compactifications, and then the more involved case of compactifications with fluxes. We discuss how the non-trivial interplay between D-branes and fluxes modifies the previous model-building rules, as well as provides new possibilities to connect string theory to particle physics.Comment: Using w-art.cls, 27 pages, 6 figures. Based on a talk given at the RTN `Constituents, Fundamental Forces and Symmetries of the Universe' Workshop and Midterm meeting in Napoli, October 2006. v2: typos corrected and references adde

Принцип экономичного формирования сепарационной характеристики технологического разделительного блока

На основании закона построения технологических схем найдена зависимость сепарационной характеристики схемы от порядка следования сепараторов с различными разделительными свойствами.Scheme separating characteristic dependence on arrangement of separations with different characteristics based on technological schemes building rules is founded

Chiral D-brane Models with Frozen Open String Moduli

Most intersecting D-brane vacua in the literature contain additional massless adjoint fields in their low energy spectrum. The existence of these additional fields make it difficult to obtain negative beta functions and, eventually, asymptotic freedom. We address this important issue for N=1 intersecting D-brane models, rephrasing the problems in terms of (open string) moduli stabilization. In particular, we consider a Z2 x Z2 orientifold construction where D6-branes wrap rigid 3-cycles and such extra adjoint fields do not arise. We derive the model building rules and consistency conditions for intersecting branes in this background, and provide N=1 chiral vacua free of adjoint fields. More precisely, we construct a Pati-Salam-like model whose SU(4) gauge group is asymptotically free. We also comment on the application of these results for obtaining gaugino condensation in chiral D-brane models. Finally, we embed our constructions in the framework of flux compactification, and construct new classes of N=1 and N=0 chiral flux vacua.Comment: 55 pages, 4 figures. Bibtex forma