Sharing Teaching Ideas Research Articles

Physics meets mathematics: a dam model *

This paper demonstrates an experimental integrated lesson of physics and calculus in a topic of fluid force applying on different shapes of dams. This lesson was designed for the first year students in engineering program in an attempt to show them the connection between these two disciplines, as well as to introduce more advanced engineering-physics problems to them. Experimental results present students’ learning achievement from pre-test, homework and post-test scores, students’ ability in solving problems using an appropriate method, and students’ satisfactory on the integrated lesson. The authors hope that our work could provide other lecturers in different fields with an idea to conduct crossover lessons or courses and encourage them to do so. This practice will not only enable students to apply their knowledge to practical applications, but also is a way to create collaborative atmosphere among lecturers, broadening their interdisciplinary knowledge and sharing teaching ideas to one another.

Earth Core: Enhancing Delivery of Geoscience Content in a Diverse School System During Times of Changing State Standards

The Earth Core program provided a one week earth science content workshop to urban public school sixth grade teachers who lacked geoscience training with the goal of increasing participants' pedagogical content knowledge (PCK) and confidence in teaching earth science. The workshop evolved from a concern that teachers would not have the training or experience required to engage students with earth science material due to new state standards switching geoscience instruction from 8th to 6th grade. The program, aligned to Cochran's (1992) five suggestions for enhancing teachers' PCK, was designed to encourage participants to reflect by sharing teaching ideas, discuss what we found to be often naïve earth science conceptions, share lesson plan ideas, and conduct action research. Field trips facilitated a bonding experience that encouraged collaboration by participants, and interactive lesson activities helped participants recognize their limited knowledge of earth science. Both qualitative and quantitative data collection methods were used in a mixed methods approach to this study. Participant confidence in teaching earth science was slightly elevated after the workshop. The study suggested that teachers realized how little earth science understanding they have after being confronted with advanced level geoscience concepts.

Sharing Teaching Ideas: Finding a Parking Spot for the Binomial Theorem

Oftentimes when learning the binomial theorem, students can apply the theorem's formula, but it has little meaning for them. Introducing the binomial theorem using an engaging application gives the theorem life and meaning. This activity relates the coefficients in the formula to finding a parking spot. First, the students discover Pascal's triangle using a parking spot simulation.

Sharing Teaching Ideas: Demystifying Systems

It has been twenty-five years since I taught my first beginning algebra course. For many years, classes in the private school at which I taught typically comprised highly motivated, mathematically gifted students. However, my class rosters changed dramatically six years ago when I first started teaching at a public, urban high school with a focus on the performing arts. It did not take long for me to realize that this was not going to be my father's algebra class, to paraphrase a well-known advertising slogan. Many of my students lacked fundamental mathematical skills. Furthermore, the overwhelming majority of my students were far more interested in dance, sculpture, and dramatic readings than in algorithms, graphs, and linear equations. For the first time in nearly two decades, I was truly challenged to create classes that would appeal to the nontraditional algebra student. After considerable trial and error, I discovered that my students responded well to lessons that included hands-on activities, incorporated hand-held technology, and connected to their experiential base. The following multiday lesson on the topic of solving linear systems of equations illustrates these three strategies.

Sharing Teaching Ideas: A Deeper Look at Related Rates in Calculus

There are numerous places in the calculus curriculum where students can discover important concepts. Some of these ideas are counterintuitive, while many others are very reasonable. Whenever possible, our students should go a little further than just settling for an answer. All too often they obtain an answer and simply leave it at that. or they might check to see if their answer matches the back of the book and then move on to the next problem. Instead, we must encourage them to analyze the result, checking for reasonableness. Is the answer in line with their expectations, or is it so counterintuitive that further exploration is required to determine why? In calculus, where many of the concepts are truly profound, this extra step is necessary for real understanding. Moreover, sometimes a secondary lesson can be learned, or at least verified, by looking at the results of a problem from several different perspectives.

Sharing Teaching Ideas: Classroom Voting in Mathematics

How do you get your students to take an active role during a mathematics class? For instance, how do you get them to form opinions and to participate in discussions about difficult concepts? There is a large body of education research demonstrating how active learning methods can be very effective, especially in comparison to traditional lectures (e.g., Bonwell and Eison 1991; Davidson 1990; Dees 1991; Hagelgans et al. 1995; Norwood 1995; Springer, Stanne, and Donovan 1999). Even when supplemented with demonstrations and PowerPoint, lectures encourage students to be passive observers, and passive students rarely learn. Classroom voting is a powerful technique promoting active learning. It engages every student in the material, and it can easily be incorporated into an otherwise traditional class. This technique breaks students out of the passive-receptive mode and requires them to participate, creating a more effective learning environment.

Sharing Teaching Ideas: Multiplying Polynomials

I find that the method demonstrated below works very well in my algebra classes, as well as for students in other mathematics classes. Several of these students who have been diagnosed with a learning disability receive individual instruction in a resource room setting and most likely are on an Individualized Educational Plan (IEP).

Sharing Teaching Ideas: Did You Code?

If you were to observe one of my algebra classes on a day that the students were solving equations, you would probably hear me ask, “Did you code the equation?” Whether we are solving simple equations, multistep equations, multivariable equations, systems of equations using substitution, or radical equations, I have taught my students that the task will be easier if they “code” the equation.

Sharing Teaching Ideas: Say What You Mean and Mean What You Say

Communication, whether verbal or written, is an essential tool in our society. Yet we find students in mathematics classes resisting efforts aimed at improving their skills in this area. They consider the usual writing assignments boring. As requirements in the mathematics curriculum shifted to call for better communication abilities from students, many teachers discovered that the mathematics classroom is an ideal setting for the development of a more precise use of language that leads to better communication.

Sharing Teaching Ideas: Understanding Functions without Using the Vertical Line Test

Since I started teaching precalculus eighteen years ago, I have struggled with how to convey a function's definition so that its importance and usefulness comes across to students. This is especially the case with the Vertical Line Test (VLT), a mechanism for testing whether the graph of a relation is a function. (Vertical lines are drawn through the graph. If every vertical line intersects the graph in exactly one point, the relation is a function.) In this era of “learning mathematics with understanding” (NCTM 2000, p. 20), the Vertical (and Horizontal) Line Tests present a challenge. I find that once students are introduced to these devices, they tend to disregard the concepts that the devices are testing and instead mechanically apply the tests. This rote application limits not only the students' understanding of the concepts but also their ability to work within and across representations to solve problems related to identifying functions.

Sharing Teaching Ideas: Solving Absolute Value Equations Algebraically and Geometrically

Understanding the geometrical meaning of algebraic equations or solving algebraic equations geometrically can help students improve their comprehension.

Sharing Teaching Ideas: Teaching Logarithms Day One

Students being taught logarithms for the first time often memorize the exponent and logarithm properties and never understand the material. Teachers who have taught this concept have repeatedly told the students that logarithms are exponents. For some reason students hear the terms exponents and logarithms but often do not understand the relationship between them. I have had a great deal of success in helping students understand the relationship between exponents and logarithms in high school and college courses using the following procedure.

Sharing Teaching Ideas: Identifying Features of Functions Stated in Graph Form or Equation Form

This article describes an activity that emphasizes identifying the features of functions. Such features include rates of change, intercepts, zeros, asymptotes, and local and global behavior. I designed this activity to enhance my students' understanding of functions and to detect any misconceptions they might have.

Sharing Teaching Ideas: Why Is the square root of 25 Not ±5?

You might want to try this brief experiment in one of your mathematics courses.

Sharing Teaching Ideas: Algebra 1's Speed Trap

When will i ever use this in life”? This question evoked many childhood memories of mathematics classes in which I struggled to learn the real-life applications of many of the concepts that I was learning: The question has also acted as a driving force to impel me to connect mathematical concepts with real-life situations. As a result, I devised a hands-on activity designed to engage learners. It used time-rate-distance problems to reinforce many mathematics concepts in an algebra 1 class. As a student teacher at Loveland Hurst Middle School in Loveland, Ohio, I developed this activity with my cooperating teacher. In this lesson, students measure the speed of moving cars, on the basis of the time spent traveling a given distance, and relay the information back to the drivers.

Sharing Teaching Ideas: Discovering New Trigonometric Identities

For most of our students, trigonometric identities appear as if by magic. A recent class activity gave me the rare opportunity to lead my students to discover what appears to be a new trigonometric identity by using a combination of graphical, numerical, and algebraic tools. In this article, I demonstrate this result and then show how it can be extended to a wider class of apparently new identities that are all extensions of the Pythagorean identity.

Sharing Teaching Ideas: Linear Equations: High School Algebra 1

Teaching linear equations is an important part of algebra 1—and the topic involves so much information that students can become overwhelmed and frustrated with the sheer volume of notes, formulas, and other “stuff” to remember. Algebra 1 students, like students in any entry-level course, need help organizing notes, discriminating important information, and finding a way to remember formulas, calculator steps, and relationships. Teachers new to teaching entry-level courses frequently assume that special education students need more concrete learning experiences or that English Speakers of Other Languages (ESOL, or limited English) students have more vocabulary needs. These assumptions are often correct, but I have discovered that most high school algebra 1 students have the very same needs. Many have not been successful in previous mathematics classes, and few know how to pick out important concepts and relate them to other ideas. To help students keep track of the information needed in studying lines, we make a flip chart that the student can keep and use throughout the year, as well as in succeeding years.

Sharing Teaching Ideas: The Area under a Curve: Conjecturing the Fundamental Theorem of Calculus

The NCTM standards for courses designed to prepare students for the “new calculus” include using “all appropriate calculator and computer technologies,” as well as promoting “the use of experimentation and conjecturing” (NCTM 2000, p. 289). Both these goals can be met in the following experiment to find the area under a curve. The experiment is a modification of a classroom exercise that I conducted in an introductory college calculus course for nonmajors.

Sharing Teaching Ideas: The Triangle Technique for Solving Direct Variation Problems

Students encounter and have difficulty with problems that involve direct variation and joint variation. Many of these problems are called work problems, rate problems, or percent problems. However, many real-life situations involve quantities that change (vary) as a result of another quantity or quantities.

Sharing Teaching Ideas: Problem Solving the Problems of Society

By learning problem solving in mathematics, students should acquire ways of thinking, habits of persistence and curiosity, and confidence in unfamiliar situations that will serve them well outside the mathematics classroom. In everyday life and in the workplace, being a good problem solver can lead to great advantages.