Functions of Number Theory in Music

2001 ◽  
Vol 94 (8) ◽  
pp. 700-707
Author(s):  
Craig J. Johnson
Keyword(s):  
Number Theory ◽  
General Education ◽  
High School Teachers ◽  
Music Theory ◽  
Liberal Arts ◽  
School Teachers ◽  
Science And Engineering ◽  
Mathematics Courses ◽  
Universities And Colleges ◽  
Liberal Arts Mathematics

Connecting mathematics with other disciplines is one of the major goals of many general-education mathematics courses taught at universities and colleges across the nation. Of course, from kindergarten to calculus, good teachers demonstrate and explore these connections as a means of making clear the relevance and importance of mathematics to every person. In particular, I have cheerfully noticed a renewed interest in the relationships between mathematics and music. This article examines several fundamental notions of number theory as they arise in music theory, and in the process, develops several music-related functions. I see an added pedagogical benefit in emphasizing the importance of functions in mathematics with examples from a different arena than the usual function pool. Middle school and high school teachers might find that a subset of these topics are handy nuggets of material that can show examples of applications of mathematics outside the mainstream uses in science and engineering. I created this article as notes for the liberal-arts mathematics course that I teach at my college and have consistently found this topic to be one of the students' favorites.

Teacher Praise and Reprimand: Examining the Generalizability and Dependability of Observational Estimates

2021 ◽  
pp. 019874292110120
Author(s):  
Margaret T. Floress ◽  
Amy M. Briesch ◽  
Lyndsay N. Jenkins ◽  
Kaylee A. Hampton
Keyword(s):  
General Education ◽  
High School Teachers ◽  
Generalizability Theory ◽  
Teacher Behavior ◽  
School Teachers ◽  
Future Directions ◽  
Teacher Praise ◽  
Behavior Specific Praise ◽  
General Education Classrooms ◽  
Specific Praise

This study examined the generalizability and dependability of observational estimates of middle and high school teachers’ use of praise and reprimand. Frequency of behavior-specific praise, general praise, and total reprimand were collected across 67, 20-min observations that took place during class-wide instruction in general education classrooms. Generalizability theory was used to determine the number of observations needed to obtain dependable estimates of teacher behavior. Behavior-specific praise rates were consistently low. General praise rates were slightly higher and reprimand rates were notably higher and slightly more variable. Total reprimands had the strongest generalizability results and general praise had the weakest. Behavior-specific praise reached an acceptable level of dependability after 15 min, whereas general praise required a 35-min observation, and reprimand only required 5 min. Implications and future directions are discussed.

scholarly journals Elementary and Junior High School Teachers’ Promotion of Self-Determination in Taiwan

2017 ◽  
Vol 10 (9) ◽  
pp. 141
Author(s):  
Pen-Chiang Chao ◽  
Yu-Chi Chou
Keyword(s):  
High School ◽  
General Education ◽  
High School Teachers ◽  
Junior High School ◽  
Junior High ◽  
Teaching Experience ◽  
Self Determination ◽  
School Teachers ◽  
Problem Solving Skills ◽  
Junior High School Teachers

instruction is implemented by elementary and junior high school teachers; (b)examine the frequency with which the components of self-determination are taught; and (c)investigate whether teachers’ gender, class setting, and teaching experience affect their classroom practices regarding the promotion of self-determination. The participants were 1,039 teachers recruited from elementary and junior high schools nationwide in Taiwan using a random sampling method. The Teaching Self-Determination Scale (TSDS) was used to gauge the extent to which educators teach knowledge and skills related to self-determination. Descriptive statistics, analyses of variance (ANOVAs) and multivariate analyses of variance (MANOVAs) were employed to analyze data collected. Findings showed that more than half of the teachers surveyed reported having often or always provided instruction to promote students’ self-determination. The most frequently taught skills are related to Psychological Empowerment (self-advocacy skills, expecting positive outcomes), while the least frequently taught skills were primarily located in the domain of Self-Regulation (goal setting and problem solving skills). Furthermore, our findings showed that teachers’ gender, class setting, and teaching experience were factors attecting the extent to which teachers delivered instruction to promote self-determination. Female teachers exhibited higher levels of implementation with respect to self-determination instruction. Teachers in general education classrooms showed significantly higher levels of applied self-determination instruction, followed by resource room teachers and self-contained classroom teachers. Additionally, teachers with more teaching experiences more frequently employed instructional activities promoting self-determination. Suggestions and implications are provided.

Teaching Theory on the High School Level

2020 ◽  
pp. 90-124
Author(s):  
Jennifer Snodgrass
Keyword(s):  
High School ◽  
High School Teachers ◽  
Music Theory ◽  
Aural Skills ◽  
Management Skills ◽  
School Teachers ◽  
High School Level ◽  
School Senior ◽  
School Music ◽  
And Performance

Only 90 days separate a high school senior from a college freshman sitting in an 8:00 a.m. music theory course. However, in most cases, there is very little interaction between high school teachers and college-level music theory and aural skills instructors. High school music teachers are reaching hundreds of students each day in ensembles and classes and may be one of the greatest resources in better understanding freshmen music majors. These outstanding high school teachers are also extremely effective educators who place emphasis on material that goes well beyond music making and performance, instead focusing on literacy and listening and teaching to the masses. There is much to be learned from their time management skills, emphasis on aural skills and listening, methods of assessment, and integration of ensemble literature that can immediately be transferred to the undergraduate music core.

Delving Deeper: The Chebyshev Polynomials: Patterns and Derivation

2004 ◽  
Vol 98 (1) ◽  
pp. 20-25 ◽  
Author(s):  
Benjamin Sinwell
Keyword(s):  
High School ◽  
Number Theory ◽  
Graph Theory ◽  
Fourier Series ◽  
High School Teachers ◽  
Chebyshev Polynomials ◽  
Mathematical Community ◽  
Russian Mathematician ◽  
School Teachers ◽  
Mathematical Exploration

Pafnuty Lvovich Chebyshev, a Russian mathematician, is famous for his work in the area of number theory and for his work on a sequence of polynomials that now bears his name. These Chebyshev polynomials have applications in the fields of polynomial approximation, numerical analysis, graph theory, Fourier series, and many other areas. They can be derived directly from the multiple-angle formulas for sine and cosine. They are relevant in high school and in the broader mathematical community. For this reason, the Chebyshev polynomials were chosen as one of the topics for study at the 2003 High School Teachers Program at the Park City Mathematics Institute (PCMI). The following is a derivation of the Chebyshev polynomials and a mathematical exploration of the patterns that they produce.

Differentiating Study Guides

2017 ◽  
Vol 53 (1) ◽  
pp. 19-27
Author(s):  
Greg Conderman ◽  
Laura Hedin
Keyword(s):  
High School ◽  
General Education ◽  
High School Teachers ◽  
Student Outcomes ◽  
Instructional Support ◽  
Learning Preferences ◽  
Upper Elementary ◽  
School Teachers ◽  
Study Guides ◽  
General Education Classrooms

Study guides are a popular and frequently student-requested instructional support developed primarily by upper elementary, middle, and high school teachers to help students navigate through a course of study. With the increase of the number of learners with diverse needs in general education classrooms, teachers need to be mindful of various ways to differentiate study guides, that is, make adjustments that maintain the integrity of student outcomes while offering choices and respecting each learner’s background knowledge, skill-level, interests, and learning preferences. Specifically, teachers can differentiate study guides by adjusting questions and content, providing different kinds of study guides, allowing students to demonstrate knowledge in different ways, reinforcing student efforts, and allowing students to use classroom materials and space in various ways. This article presents ways teachers can apply the components of differentiation to maximize the use of study guides in inclusive classrooms.

Correlation and Structural Analysis for Determining the Connection of Universal State Exam (USE) Results with the Ratings of First-Year Students

2018 ◽  
Vol 6 (3) ◽  
pp. 18-27
Author(s):  
Р. Габдрахманова ◽  
R. Gabdrahmanova ◽  
Г. Гарнаева ◽  
G. Garnaeva ◽  
Л. Нефедьев ◽  
...  
Keyword(s):  
Structural Analysis ◽  
General Education ◽  
High School Teachers ◽  
First Year ◽  
School Teachers ◽  
Educational Organization ◽  
First Year Students ◽  
Student Groups ◽  
Education School ◽  
Pedagogical Analysis

One of the problems that is of interest to researchers is the adaptation of fi rst-year students. One of the indicators of their successful adaptation, many researchers believe performance. With the help of correlation and structural analysis, the interrelation between the results of fi nal examinations in the general education school and the results of training in the fi rst half of the year during the adaptation of fi rst-year students of the Institute of Physics of the Kazan (Volga) Federal University (KFU) was investigated. At the fi rst stage of the pedagogical analysis, we came to the conclusion that between the results of the fi nal examinations in the general education school and the results of training in the fi rst half of the year, during the adaptation of the fi rst-year students of the Institute of Physics, a slight diff erence was revealed, which demonstrates their interdependence. The article also presents an algorithm for working with a correlation matrix. The materials of the article can be used by high school teachers, curators of student groups, as well as by researchers in organizing and conducting an experiment on their problem in the higher school educational organization.

Ways That Engineers Can Get Involved In Recruiting Future Materials Scientists and Engineers

2001 ◽  
Vol 684 ◽  
Author(s):  
Stacy H. Gleixner
Keyword(s):  
High School ◽  
High School Teachers ◽  
Materials Science ◽  
School Teachers ◽  
Science And Engineering ◽  
Materials Engineering ◽  
Materials Science And Engineering ◽  
Web Videos ◽  
Scientists And Engineers ◽  
The Web

ABSTRACTFor materials science and engineering departments across the country, recruitment of students is a critical concern. To increase enrollment, students' interest in materials science needs to be sparked early, preferably before they begin college. However, many entering college students have never heard of the major and/or are unsure of what a career in materials science would be like. There are many excellent resources that have been developed to target this problem directly including comprehensive career resources on the web, videos describing careers in materials engineering, and teaching materials developed for middle and high school teachers.

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