Enhancing Problem Solving Through Math Clubs

This action research paper was about a mandatory math club of seventh graders that met once per week over a 12-week period. The students gathered in the classroom during their regularly scheduled math class. The focus of the math club was to solve challenging math problems, usually cooperatively, and sometimes competitively. The math club activities varied from week to week to offer an element of surprise. Frequently, the students presented their solutions to peers, along with an explanation of the way they solved the problem. Instruments were used to collect information about problem-solving accuracy, student attitudes, and student and teacher behaviors. I discovered a slight improvement in problem solving. Also, on Math Club days, the teaching was less teacher-centered and more student-centered. As a result of this research, I plan to offer my middle school students more problem-solving opportunities and I plan to allow my students to work cooperatively on a regular basis.

Why Are We Writing? This is Math Class!

In this action research study of my classroom of 8th grade mathematics, I investigated writing in the content area. I have realized how important it is for students to be able to communicate mathematical thoughts to help gain a deeper understanding of the content. As a result of this research, I plan to enforce the use of writing thoughts and ideas regarding math problems. Writers develop skills and generate new thoughts and ideas every time they sit down to write. Writing evolves and grows with ongoing practice, and that means thinking skills mature along with it. Writing is a classroom activity which offers the possibility for students to develop a deeper understanding of the mathematics they are learning. Writing encourages students to reflect on and explore their reasoning and to extend their thinking and understanding. Students are often content with manipulating symbols and doing routine math problems, without ever reaching a deep and personal understanding of the material. My goal through this project was to help students understand why they were doing certain operations to solve math problems. Writing is an essential tool for thinking and is fundamental in every class, in every subject, and on every level of thinking; skills in writing must be practiced and refined, and students must have frequent opportunities to write across the curriculum. Communication in mathematics is not a simple and unambiguous activity.

Writing for Understanding in Math Class

In this action research study of my classroom of 10th grade geometry students, I investigated how students learn to communicate mathematics in a written form. The purpose of the study is to encourage students to express their mathematical thinking clearly by developing their communication skills. I discovered that although students struggled with the writing assignments, they were more comfortable with making comments, writing questions and offering suggestions through their journal rather than vocally in class. I have utilized teaching strategies for English Language Learners, but I had never asked the students if these strategies actually improved their learning. I have high expectations, and have not changed that, but I soon learned that I did not want to start the development of students’ written communication skills by having the students write a math solution. I began having my students write after teaching them to take notes and modeling it for them. Through entries in the journals, I learned how taking notes best helped them in their pursuit of mathematical knowledge. As a result of this research, I plan to use journals more in each of my classes, not just a select class. I also better understand the importance of stressing that students take notes, showing them how to do that, and the reasons notes best help English Language Learners.

Writing In Math Class? Written Communication in the Mathematics Classroom

In this action research study of my seventh grade mathematics classroom, I investigated what written communication within the mathematics classroom would look like. I increased vocabulary instruction of specific mathematical terms for my students to use in their writing. I also looked at what I would have to do differently in my teaching in order for my students to be successful in their writing. Although my students said that using writing to explain mathematics helped them to better understand the math, my research revealed that student writing did not necessarily translate to improved scores. After direct instruction and practice on math vocabulary, my students did use the vocabulary words more often in their writing; however, my students used the words more like they would in spelling sentences rather than to show what it meant and how it can be applied within their written explanation in math. In my teaching, I discovered I tried many different strategies to help my students be successful. I was very deliberate in my language and usage of vocabulary words and also in my explanations of various math concepts. As a result of this research, I plan to continue having my students use writing to communicate within the mathematics classroom. I will keep using some of the strategies I found successful. I also will be very deliberate in using vocabulary words and stress the use of vocabulary words with my students in the future.

Educational Investment, Family Context, and Children’s Math and Reading Growth from Kindergarten through the Third Grade

Drawing on longitudinal data from the Early Childhood Longitudinal Study, Kindergarten Class of 1998–1999, this study used IRT modeling to operationalize a measure of parental educational investments based on Lareau’s notion of concerted cultivation. It used multilevel piecewise growth models regressing children’s math and reading achievement from entry into kindergarten through the third grade on concerted cultivation and family context variables. The results indicate that educational investments are an important mediator of socioeconomic and racial/ethnic disparities, completely explaining the black-white reading gap at kindergarten entry and consistently explaining 20 percent to 60 percent and 30 percent to 50 percent of the black-white and Hispanic-white disparities in the growth parameters, respectively, and approximately 20 percent of the socioeconomic gradients. Notably, concerted cultivation played a more significant role in explaining racial/ethnic gaps in achievement than expected from Lareau’s discussion, which suggests that after socioeconomic background is controlled, concerted cultivation should not be implicated in racial/ethnic disparities in learning.

Class Notes for Math 901/902: Abstract Algebra, Instructor Tom Marley

Topics include: Free groups and presentations; Automorphism groups; Semidirect products; Classification of groups of small order; Normal series: composition, derived, and solvable series; Algebraic field extensions, splitting fields, algebraic closures; Separable algebraic extensions, the Primitive Element Theorem; Inseparability, purely inseparable extensions; Finite fields; Cyclotomic field extensions; Galois theory; Norm and trace maps of an algebraic field extension; Solvability by radicals, Galois' theorem; Transcendence degree; Rings and modules: Examples and basic properties; Exact sequences, split short exact sequences; Free modules, projective modules; Localization of (commutative) rings and modules; The prime spectrum of a ring; Nakayama's lemma; Basic category theory; The Hom functors; Tensor products, adjointness; Left/right Noetherian and Artinian modules; Composition series, the Jordan-Holder Theorem; Semisimple rings; The Artin-Wedderburn Theorem; The Density Theorem; The Jacobson radical; Artinian rings; von Neumann regular rings; Wedderburn's theorem on finite division rings; Group representations, character theory; Integral ring extensions; Burnside's paqb Theorem; Injective modules.

Class Notes for Math 921/922: Real Analysis, Instructor Mikil Foss

Topics include: Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon- Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration, Semicontinuity, equicontinuity, absolute continuity, metric spaces, compact spaces, Ascoli’s theorem, Stone Weierstrass theorem, Borel and Lebesque measures, measurable functions, Lebesque integration, convergence theorems, Lp spaces, general measure and integration theory, Radon-Nikodyn theorem, Fubini theorem, Lebesque-Stieltjes integration.

Class Notes for Math 905: Commutative Algebra, Instructor Sylvia Wiegand

Topics include: Rings, ideals, algebraic sets and affine varieties, modules, localizations, tensor products, intersection multiplicities, primary decomposition, the Nullstellensatz

Class Notes for Math 918: Homological Conjectures, Instructor Tom Marley

This course was an overview of what are known as the “Homological Conjectures,” in particular, the Zero Divisor Conjecture, the Rigidity Conjecture, the Intersection Conjectures, Bass’ Conjecture, the Superheight Conjecture, the Direct Summand Conjecture, the Monomial Conjecture, the Syzygy Conjecture, and the big and small Cohen Macaulay Conjectures. Many of these are shown to imply others. This document contains notes for a course taught by Tom Marley during the 2009 spring semester at the University of Nebraska-Lincoln. The notes loosely follow the treatment given in Chapters 8 and 9 of Cohen-Macaulay Rings, by W. Bruns and J. Herzog, although many other sources, including articles and monographs by Peskine, Szpiro, Hochster, Huneke, Grith, Evans, Lyubeznik, and Roberts (to name a few), were used. Special thanks to Laura Lynch for putting these notes into LaTeX.

Class Notes for Math 918: Cohen Macaulay Modules, Instructor Roger Wiegand

Topics covered are: Cohen Macaulay modules, zero-dimensional rings, one-dimensional rings, hypersurfaces of finite Cohen-Macaulay type, complete and henselian rings, Krull-Remak-Schmidt, Canonical modules and duality, AR sequences and quivers, two-dimensional rings, ascent and descent of finite Cohen Macaulay type, bounded Cohen Macaulay type.

Class Notes for Math 918: Local Cohomology, Instructor Tom Marley

Topics include: Injective Module, Basic Properties of Local Cohomology Modules, Local Cohomology as a Cech Complex, Long exact sequences on Local Cohomology, Arithmetic Rank, Change of Rings Principle, Local Cohomology as a direct limit of Ext modules, Local Duality, Chevelley’s Theorem, Hartshorne- Lichtenbaum Vanishing Theorem, Falting’s Theorem.

Class Notes for Math 871: General Topology, Instructor Jamie Radcliffe

Topics include: Topological space and continuous functions (bases, the product topology, the box topology, the subspace topology, the quotient topology, the metric topology), connectedness (path connected, locally connected), compactness, completeness, countability, filters, and the fundamental group.

Activation of 5-HT2C receptors in the dorsal periaqueductal gray increases antinociception in mice exposed to the elevated plus-maze

Several findings have pointed to the role of the dorsal periaqueductal gray (dPAG) serotonin 5-HT1A and 5-HT2(A-C) receptor subtypes in the modulation of defensive behavior in animals exposed to the elevated plus-maze (EPM). Besides displaying anxiety-like behavior, rodents also exhibit antinociception in the EPM. This study investigated the effects of intra-dPAG injections of 5-HT1A and 5-HT2B/2C receptor ligands on EPM-induced antinociception in mice. Male Swiss mice received 0.1 mu l intra-dPAG injections of vehicle, 5.6 and 10 nmol of 8-OHDPAT, a 5-HT1A receptor agonist (Experiment 1), or 0.01, 0.03 and 0.1 nmol of mCPP, a 5-HT2B/2C receptor agonist (Experiment 2). Five minutes later, each mouse received an intraperitoneal injection of 0.6% acetic acid (0.1 ml/10 g body weight; nociceptive stimulus) and was individually confined in the open (OA) or enclosed (EA) arms of the EPM for 5 min, during which the number of abdominal writhes induced by the acetic acid was recorded. While intra-dPAG injection of 8-OHDPAT did not change open-arm antinociception (OAR). mCPP (0.01 nmol) enhanced it. Combined injections of ketanserin (10 nmol/0.1 mu l), a 5-HT2A/2C receptor antagonist, and 0.01 nmol of mCPP (Experiment 3), selectively and completely blocked the OAR enhancement induced by mCPP. Although intra-dPAG injection of mCPP (0.01 nmol) also produced antinociception in EA-confined mice (Experiment 2), this effect was not confirmed in Experiment 3. Moreover, no other compound changed the nociceptive response in EA-confined animals. These results suggest that the 5-HT2C receptors located within the PAG play a role in this type of environmentally induced pain inhibition in mice. (c) 2012 Elsevier B.V. All rights reserved.

Central, but not basolateral, amygdala involvement in the anxiolytic-like effects of midazolam in rats in the elevated plus maze

The role of the amygdala in the mediation of fear and anxiety has been extensively investigated. However, how the amygdala functions during the organization of the anxiety-like behaviors generated in the elevated plus maze (EPM) is still under investigation. The basolateral (BLA) and the central (CeA) nuclei are the main input and output stations of the amygdala. In the present study, we ethopharmacologically analyzed the behavior of rats subjected to the EPM and the tissue content of the monoamines dopamine (DA) and serotonin (5-HT) and their metabolites in the nucleus accumbens (NAc), dorsal hippocampus (DH), and dorsal striatum (DS) of animals injected with saline or midazolam (20 and 30 nmol/0.2 mu L) into the BLA or CeA. Injections of midazolam into the CeA, but not BLA, caused clear anxiolytic-like effects in the EPM. These treatments did not cause significant changes in 5-HT or DA contents in the NAc, DH, or DS of animals tested in the EPM. The data suggest that the anxiolytic-like effects of midazolam in the EPM also appear to rely on GABA-benzodiazepine mechanisms in the CeA, but not BLA, and do not appear to depend on 5-HT and DA mechanisms prevalent in limbic structures.

Both the dorsal hippocampus and the dorsolateral striatum are needed for rat navigation in the Morris water maze

The multiple memory systems theory proposes that the hippocampus and the dorsolateral striatum are the core structures of the spatial/relational and stimulus-response (S-R) memory systems, respectively. This theory is supported by double dissociation studies showing that the spatial and cue (S-R) versions of the Morris water maze are impaired by lesions in the dorsal hippocarnpus and dorsal striatum, respectively. In the present study we further investigated whether adult male Wistar rats bearing double and bilateral electrolytic lesions in the dorsal hippocampus and dorsolateral striatum were as impaired as rats bearing single lesions in just one of these structures in learning both versions of the water maze. Such a prediction, based on the multiple memory systems theory, was not confirmed. Compared to the controls, the animals with double lesions exhibited no improvement at all in the spatial version and learned the cued version very slowly. These results suggest that, instead of independent systems competing for holding control over navigational behaviour, the hippocampus and dorsal striatum both play critical roles in navigation based on spatial or cue-based strategies. (C) 2011 Elsevier B.V. All rights reserved.