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.

Precise Mathematical Language: Exploring the Relationship Between Student Vocabulary Understanding and Student Achievement

In this action research study of my classroom of fifth grade mathematics, I investigate the relationship between student understanding of precise mathematics vocabulary and student achievement in mathematics. Specifically, I focused on students’ understanding of written mathematics problems and on their ability to use precise mathematical language in their written solutions of critical thinking problems. I discovered that students are resistant to change; they prefer to do what comes naturally to them. Since they have not been previously taught to use precise mathematical language in their communication about math, they have great difficulty in adapting to this new requirement. However, with teaching modeling and ample opportunities to use the language of mathematics, students’ understanding and use of specific mathematical vocabulary is increased.

Writing in the Mathematics Classroom: Does It Have an Effect on Students’ Mathematical Reasoning?

In this action research study of my classroom of 5th grade mathematics, I investigate how to improve students’ written explanations to and reasoning of math problems. For this, I look at journal writing, dialogue, and collaborative grouping and its effects on students’ conceptual understanding of the mathematics. In particular, I look at its effects on students’ written explanations to various math problems throughout the semester. Throughout the study students worked on math problems in cooperative groups and then shared their solutions with classmates. Along with this I focus on the dialogue that occurred during these interactions and whether and how it moved students to a deeper level of conceptual understanding. Students also wrote responses about their learning in a weekly math journal. The purpose of this journal is two-fold. One is to have students write out their ideas. Second, is for me to provide the students with feedback on their responses. My research reveals that the integration of collaborative grouping, journaling, and active dialogue between students and teacher helps students develop a deeper understanding of mathematics concepts as well as an increase in their confidence as problem solvers. The use of journaling, dialogue, and collaborative grouping reveals themselves as promising learning tasks that can be integrated in a mathematics curriculum that seeks to cultivate students’ thinking and reasoning.

Student Problem Solving

The purpose of this study is to determine if students solve math problems using addition, subtraction, multiplication, and division consistently and whether students transfer these skills to other mathematical situations and solutions. In this action research study, a classroom of 6th grade mathematics students was used to investigate how students solve word problems and how they determine which mathematical approach to use to solve a problem. It was discovered that many of the students read and re-read a question before they try to find an answer. Most students will check their answer to determine if it is correct and makes sense. Most students agree that mastering basic math facts is very important for problem solving and prefer mathematics that does not focus on problem solving. As a result of this research, it will be emphasized to the building principal and staff the need for a unified and focused curriculum with a scope and sequence for delivery that is consistently followed. The importance of managing basic math skills and making sure each student is challenged to be a mathematical thinker will be stressed.

A Study of Written Communication: Showing Your Steps

In this action research study of my teaching of sixth grade mathematics, I investigated the importance of showing work on daily assignments. I wanted to find out what happens when I ask students to show their work, specifically, whether it would improve students’ grades or not and whether I could help the students to understand the importance of showing their work. I discovered that students need to be shown the proper way to show their work, how to look at a problem and then how to show all of their steps to get to the answer. They need to be encouraged and be held accountable for showing their work when asked. Once they were able to show work, they could start to see the value in showing their work and they tended to show their work more often. Students became more confident in themselves as mathematics students and, in some cases, their grades improved. As a result of this research, I plan to teach and explain to my future classes about how showing their work can benefit them in a variety of ways. They will be able to use the knowledge that they gain in my classroom in their future math classes in middle and high school.

Increasing Student Confidence and Knowledge through Student Presentations

In this action research study of a 9th grade Algebra classroom, I investigated the influence of having students present homework solutions and what effect it had on student learning and student confidence. Students were asked to present solutions to homework problems each day and were rated on how well they did. The students were also surveyed about their confidence and feelings about mathematics. Students were also observed for information about who they asked questions of when presented with a math problem they did not understand. In this classroom, two teachers were involved in instruction and this study examines what affect this had on student learning and who was asked for help. As a result of presentations, students’ confidence increased and students reacted positively to both the presentations and their own mathematical learning. The students felt the presentations were a benefit to the class and watching their peers solve mathematical equations helped them to better understand the mathematics.

Here is the Plan: So What is the Question?

In this action research study of my classroom of 5th grade mathematics students, I investigated their understanding of the mathematical operations by having them write problems to match given equations. I discovered that writing a story to match an equation does provide insight into a student’s understanding of mathematical concepts, however, reading comprehension is a factor in the understanding. Readers who struggle with comprehension do struggle with understanding and writing math story problems. The discussion that follows the writing of a math story problem and the solving of the written problems helps to strengthen the students’ mathematical abilities as well as their communication skills and confidence levels. Through my study, students learned that it was alright to make mistakes because the learning from those mistakes is what is important. As a result of this research, I plan to continue to have students write stories to match given equations as a source of information about student understanding. I will continue to give opportunities to revisit those written problems as a tool to increase students’ comprehension and communication skills, as well as their confidence.