Reviewing, Rethinking and Revising Homework Expectations in a Seventh Grade Math Class

In this action research study of my seventh grade mathematics class, I investigated whether de-emphasizing homework assignments as daily grades while stressing them as daily practice encouraged students to focus more on the learning rather than the daily grade. As part of this study, I also looked at how this change in homework expectations affected my daily teaching. I discovered that having students keep notes, examples, practice problems and homework assignments in a notebook helped them concentrate more on the process of getting answers and why they may of had an incorrect answer. Students were more likely to discuss with their peers how answers were found when comparing answers showed differences. When we reviewed the answers, they were more willing to ask questions about why their answer was wrong and then make corrections. As a result of this research, I plan to continue having seventh graders keep using notebooks to organize their notes, examples and assignments.

Summarization in Math Class

This action research study of twenty students in my sixth grade mathematics classroom examines the implementation of summarization strategies. Students were taught how to summarize concepts and how to explain their thinking in different ways to the teacher and their peers. Through analysis of students’ summaries of concepts from lessons that I taught, tests scores, and student journals and interviews, I discovered that summarizing mathematical concepts offers students an engaging opportunity to better understand those concepts and render that understanding more visible to the teacher. This analysis suggests that non-traditional summarization, such as verbal and written strategies, and strategies involving movement and discussions, can be useful in mathematics classrooms to improve student understanding, engagement in learning tasks, and as a form of formative assessment.

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.

Confidence in Communication: Can My Whole Class Achieve This?

In this action research study of my classroom of Algebra 2 students, I investigated the confidence levels and communication skills of these students. I discovered that students who have higher confidence levels are comfortable in their classroom situations. The students with increased levels of confidence also have more open communication with those they respect. As a result of this research, I plan to continue with the implementation of communication skills. I will also look to next school year as a place to start executing a plan to be more available and involved in the active learning process of my students.

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.

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.