Making Sense of Word Problems

In this action research study of my classroom of sixth grade mathematics, I investigated word problems. I discovered that my students did not like to try word problems because they did not understand what was being asked of them. My students also saw no reason for solving word problems or in having the ability to solve them. I used word problems that covered topics that were familiar to the students and that covered the skills necessary at the sixth grade level. I wanted to deepen their understanding of math and its importance. By having my students journal to me about the steps that they had taken along the way to solve the word problem I was able to see where confusion occurred. Consequently I was able to help clarify where my students made mistakes. Also, through writing down the steps taken, students did see more clearly where their errors took place. Each time that my students wrote their explanations to the steps that they used in solving the word problems they did solved them more easily. As I observed my students they took more time in writing their explanations and did not look at it as such a difficult task anymore.

How to Better Prepare for Assessment and Create a More Technologically Advanced Classroom

In this action research study of my classroom of 8th grade mathematics, I investigated how to better prepare these students for quizzes and how technology can be used in the classroom. I discovered that there are many different ways to challenge students and help them prepare for assessments. There are also many ways to use technology in the classroom if one has the opportunities to use some of the tools, such as Power Point and Algebra Tiles. As a result of this research, I plan to increase the scores on state standards while also allowing the students to enjoy technology during this process.

Student Understanding and Achievement When Focusing on Peer-led Reviews

Math in the Middle Institute Partnership, Action Research Project Report, In partial fulfillment of the MAT Degree. Department of Mathematics. University of Nebraska-Lincoln. July 2009.

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.

Understanding the Mathematical Language

In this action research study of my calculus classroom consisting of only 12th grade students, I investigated activities that would affect a student’s understanding of mathematical language. The goal in examining these activities in a systematic way was to see if a student’s deeper understanding of math terms and symbols resulted in a better understanding of the mathematical concepts being taught. I discovered that some students will rise to the challenge of understanding mathematics more deeply, and some will not. In the process of expecting more from students, the frustration level of both the students and the teacher increased. As a result of this research, I plan to see what other activities will enhance the understanding of mathematical language.

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.

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.

Can homework become more meaningful with the inclusion of oral presentations?

In this action research study of my 7th grade math class, I investigated the inclusion of homework presentations to see if they would improve students’ attitude toward mathematics, participation, and understanding. I discovered that although the implementations of presentations into our homework routine did not drastically influence grades, or even improve attitudes (according to test grades and student surveys), a multitude of other changes surfaced. These changes consisted of an increase in discussion, a team effort among students in my class, and an overall “learning community” effect. I plan to continue to pursue presentations as a major part of my homework routine, and also incorporate presentations into review sessions.

Reduce Late Assignments through Classroom Presentations

In this action research study of my 7th grade math class, I investigated homework presentations, to see if they would reduce the amount of late homework assignments. I did not find any significant results that weekly presentations given by students were beneficial to reduce the amount of late assignments, but found many other positive things that happened because of presentations. As a result of this research, I plan to use classroom presentations because they foster listening skills and student interaction, and promote deeper thinking.

“Let’s Review.” A Look at the Effects of Re-teaching Basic Mathematic Skills

In this action research study of my classroom of 8th grade mathematics, I investigated the effect of reviewing basic fraction and decimal skills on student achievement and student readiness for freshman Algebra. I also investigated the effect on the quality of student work, with regards to legibility by having students grade each other’s work anonymously. I discovered that students need basic skill review with fractions and decimals, and by the end of the research their scores improved. However, their handwriting had not. At the end of the research, a majority of the students felt the review was important, and they were ready to take math next year in high school. As a result of this research, I plan to implement weekly fraction and decimal review assignments in all middle school grades: 6th, 7th, and 8th. In addition, fraction and decimals must be incorporated into daily assignments, where appropriate, in order to encourage students to retain these skills.

Reasonable or Not? A Study of the Use of Teacher Questioning to Promote Reasonable Mathematical Answers from Sixth Grade Students

In this action research study of my sixth grade mathematics class, I investigated the influence a change in my questioning tactics would have on students’ ability to determine answer reasonability to mathematics problems. During the course of my research, students were asked to explain their problem solving and solutions. Students, amongst themselves, discussed solutions given by their peers and the reasonability of those solutions. They also completed daily questionnaires that inquired about my questioning practices, and 10 students were randomly chosen to be interviewed regarding their problem solving strategies. I discovered that by placing more emphasis on the process rather than the product, students became used to questioning problem solving strategies and explaining their reasoning. I plan to maintain this practice in the future while incorporating more visual and textual explanations to support verbal explanations.

Student Transition to College

In this action research study of recent graduates from my district, I investigated their level of readiness for college-level mathematics courses. I discovered that the students have a wide variety of experiences in college. There are many factors that determine success in college mathematics courses. These factors include size of college, private or public, university or community college. Other factors include students’ choice of major, maturity level, and work ethic. As a result of this research, I plan to raise the individual expectations in my classroom. It is our duty as high school educators to prepare the students for a wide variety of experiences in college. We cannot control where the students attend college or what they study. High schools need to prepare the students for all possibilities and ensure that they have a solid knowledge of the baseline mathematics skills.

Holding Students Accountable

This action research study of approximately 90 high school algebra students investigates how frequent quizzing benefits them during the course of a semester. The intent of the research was to see how well students kept up with the material and if frequent quizzing helped them on the chapter tests. It was also designed to help me gain a better understanding of what students know and how I need to adjust daily routines so that all students stay caught up. I discovered that although frequent quizzes are not the students’ favorite activity to take part in, they learn to accept the quizzes and benefit greatly because of the amount of information students learn from them. Holding students accountable with frequent quizzes forces students to stay caught up and pushes them to excel as many found the tests to be much easier because of the practice they received. My research revealed many advantages to holding students accountable through frequent quizzes and although it can be somewhat time consuming, it is definitely a practice that will be continued in my classroom for years to come.

Daily Problem-Solving Warm-Ups: Harboring Mathematical Thinking In The Middle School Classroom

In this action research study of my classroom of 8th grade mathematics, I investigated the use of daily warm-ups written in problem-solving format. Data was collected to determine if use of such warm-ups would have an effect on students’ abilities to problem solve, their overall attitudes regarding problem solving and whether such an activity could also enhance their readiness each day to learn new mathematics concepts. It was also my hope that this practice would have some positive impact on maximizing the amount of time I have with my students for math instruction. I discovered that daily exposure to problem-solving practices did impact the students’ overall abilities and achievement (though sometimes not positively) and similarly the students’ attitudes showed slight changes as well. It certainly seemed to improve their readiness for the day’s lesson as class started in a more timely manner and students were more actively involved in learning mathematics (or perhaps working on mathematics) than other classes not involved in the research. As a result of this study, I plan to continue using daily warm-ups and problem-solving (perhaps on a less formal or regimented level) and continue gathering data to further determine if this methodology can be useful in improving students’ overall mathematical skills, abilities and achievement.

Vanishing of Ext and Tor over complete intersections

Let (R,m) be a local complete intersection, that is, a local ring whose m-adic completion is the quotient of a complete regular local ring by a regular sequence. Let M and N be finitely generated R-modules. This dissertation concerns the vanishing of Tor(M, N) and Ext(M, N). In this context, M satisfies Serre's condition (S_{n}) if and only if M is an nth syzygy. The complexity of M is the least nonnegative integer r such that the nth Betti number of M is bounded by a polynomial of degree r-1 for all sufficiently large n. We use this notion of Serre's condition and complexity to study the vanishing of Tor_{i}(M, N). In particular, building on results of C. Huneke, D. Jorgensen and R. Wiegand [32], and H. Dao [21], we obtain new results showing that good depth properties on the R-modules M, N and MtensorN force the vanishing of Tor_{i}(M, N) for all i>0. We give examples showing that our results are sharp. We also show that if R is a one-dimensional domain and M and MtensorHom(M,R) are torsion-free, then M is free if and only if M has complexity at most one. If R is a hypersurface and Ext^{i}(M, N) has finite length for all i>>0, then the Herbrand difference [18] is defined as length(Ext^{2n}(M, N))-(Ext^{2n-1}(M, N)) for some (equivalently, every) sufficiently large integer n. In joint work with Hailong Dao, we generalize and study the Herbrand difference. Using the Grothendieck group of finitely generated R-modules, we also examined the number of consecutive vanishing of Ext^{i}(M, N) needed to ensure that Ext^{i}(M, N) = 0 for all i>>0. Our results recover and improve on most of the known bounds in the literature, especially when R has dimension two.