Physics and Biology: Helping students understand their ‘connections’

Topic: Helping students understand ‘connections’ between physics and biology
Discipline(s) or Field(s): Physics, Biology
Authors: Shusaku Horibe, Bret Underwood, Peter Timbie, University of Wisconsin – Madison.
Submission Date: June 17, 2008

Executive Summary: The goal of the lesson is for students to develop an understanding of how physics is connected to biology through the building of physics models of biological phenomena.

We developed three versions of the lesson, evaluating Versions 1 and 2 and making changes based on those evaluations. In Version 1 students engaged in model building activities and were asked to develop physics-based models for a variety of biological and physiological facts. In Version 2 significant modifications were made to address difficulties students had in meeting the learning goals of Version 1. In particular, the number of different biological facts students were asked to model was reduced significantly, and more attention was paid to developing students’ model building skills. Only minor modifications were made in Version 3 to help provide more feedback and a clearer framework for model building to students.

We found that students suffered from several difficulties that prevented them from achieving the learning goals: a lack of conceptual understanding; underdeveloped models; and a lack of reflection on the models that they built. The revisions in the lesson were designed to address these difficulties, resulting in a lesson, which provides ample opportunities for feedback to students on the model building process and how it helps to make connections between physics and biology

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Mathematics: Arc Length of a Curve as an Application of Integration

Topic: Arc length of a curve as an Application of Integration
Discipline(s) or Field(s): Mathematics
Authors: Kavita Bhatia, University of Wisconsin-Marshfield/Wood County & Kirthi Premadasa, UW-Marathon County
Submission Date: March 30, 2010

Lesson Goals:

  1. Students will learn how to make a manual calculation of a Riemann sum for the arc length of a givensample curve using a few subdivisions.
  2. Students will use the knowledge obtained through the Riemann Sum ‐> Integrations models that they have seen before, to “discover” an integration formula for the arc length of any continuous curve.
  3. Students will use the formula that they “discovered”, together with the integration techniques, taught in the course to evaluate the actual arc length of the curve.
  4. Students will understand the underlying theme behind all the Riemann sums that they have encountered.

Mathematics: Intro to Partial Derivatives in a Business Calculus Course

Title: Introduction to Partial Derivatives in a Business Calculus Course
Discipline(s) or Field(s): Mathematics
Authors: Erick Hofacker, Ioana Ghenciu, Don Leake, Alexandru Tupan, University of Wisconsin-River Falls
Submission Date: March 2, 2009

Executive Summary: This introductory lesson to partial derivatives to a class of business and social science majors focuses on conceptual understanding in several different ways. It opens with a couple of questions on car loans aimed at assessing the experience and intuition of the class concerning changes in multivariable functions. Then with the help of a computer applet borrowed from MIT the lesson introduces the concept of partial derivatives through its geometrical meaning. TI-89 calculators provide a way for students to easily compute partial derivatives algebraically for a simple polynomial function. Through these two technological tools students explore the relationship between the 3-D graph of a two-variable function and its partials. The 75 minute lesson ends with a couple of partial derivative applications from the fields of business and economics.

The lesson is based on a laboratory/guided discovery approach. Technology is used as a tool for exploration. The learning activities were ordered to achieve understanding first geometrically, then algebraically, and finally through application. Lower-level computational skills were placed in support of higher-level conceptual understanding. Some later questions were directed toward giving students the opportunity to discover connections with previously-learned material. The application portion of the lesson is designed to help students see connections between the mathematics curriculum and other disciplines.

This lesson study reinforced the notion that discovery learning, supported by technology that helps students visualize and compute, is very helpful in the introduction of a conceptually difficult topic such as partial derivatives. The lesson also highlighted the importance of constant and immediate assessment in the classroom. The gulf between an instructor’s perception of student understanding and what is actually the case can be tremendously broad, especially toward the end of a long semester. A third revelation is that usually simpler is better. It is preferable to focus on understanding a few concepts well in the classroom. Finally, the importance of personal contact, student-to-student or student-to-teacher, cannot be overemphasized. While working in a computer lab, the information is right there in the face of the student on the computer screen. In a lengthy classroom or lecture hall, it is far too easy for the weaker student to disengage. In addition every learning environment needs to provide a way for instructors to get within every student’s “sphere of learning.”. Students that are not easily accessed in the classroom, whether in the back of a long classroom or against the wall in a computer lab are in danger of being lost.

Mathematics: Introduction to Partial Derivatives in a Business Calculus Course (Final Report)

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Mathematics: Related Rates in Calculus

Title: The “Perfect” Related Rates Lesson: A lesson study in calculus
Discipline(s) or Field(s): Mathematics
Authors: Joy Becker, Christopher Bendel, Petre Ghenciu, Laura Schmidt, Radi Teleb, University of Wisconsin – Stout
Submission Date: February 28, 2007

Executive Summary

The lesson topic is related rates in Calculus I or Calculus & Analytic Geometry I. Related rates problems tend to be difficult for students since they are generally word problems that require setting up equations before solving. This topic is important as one common example of an application of derivatives.

Learning Goals:  There are two immediate goals for this lesson: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. A longer-term goal is that students’ problem-solving and critical thinking skills will be improved.

Lesson Design: The lesson is designed to span two class days. On the first day, students start by working through an introductory worksheet, which extends what they have previously learned to introduce the concept of related rates. Since word problems are often a stumbling block for students, the lesson includes an overview of problem-solving strategies, somewhat specific to related rates, although they can be generalized. A warm-up worksheet reviews necessary material and gives students a chance to set up equations, an essential part of the problem-solving process. On the second day of the lesson, the instructor works through two examples with the class to model the problem-solving process, and students are given a chance to solve problems on their own or in small groups. The examples and worksheet problems were chosen to show students a variety of different types of related rates problems, starting with more straightforward problems and ending with more difficult problems.

Major Findings about Student Learning:  In terms of our specific lesson goals, by looking at the data we collected, the first two were achieved by most students: 1) Students will understand that related rates problems are applications of implicit differentiation and 2) Students will be able to translate, compile, model, and solve a related rates problem and interpret the meaning of the answer. Since the third goal, “Students’ problem-solving and critical thinking skills will be improved,” is more general, there will need to be a series of lesson studies in order for it to be assessed properly. Is this the “perfect” lesson? The answer is probably no. However, the planned activities did visibly increase student engagement and responsiveness. The lesson developed will help instructors to assemble an excellent lesson, depending on the classroom settings and other institutional factors.

Below are links related to the lesson plan and the materials used to teach it.

Basketball Example
Warm-Up to Related Rates Worksheet
In-Class Related Rates Examples
Rales Rates Worksheet

Below are links related to the study of the lesson.

Observation Forrm
Student Survey
Related Rates Homework Spring 2006
Related Rates Homework Fall 2006
Student Data

 

 

Mathematics: Rate of Change in Context

Topic: Rate of Change in Context: a Lesson Study in Calculus at the University of Wisconsin at River Falls
Discipline(s) or Field(s): Mathematics
Authors: Laurel Langford, Alexandru Tupan, Ioana Ghenciu, Don Leake, University of Wisconsin – River Falls
Submission Date: May 5, 2007

Executive Summary: Our goal is for students to better understand rate of change in context, including the skills of moving flexibly between algebraic and graphical representations and analyzing behaviors given information about the rate of change. In this lesson, students practice these skills in concrete examples using average rate of change, as a preparation for doing similar work with derivatives. These activities are at an appropriate level, with some review, and some critical thinking work, and they prompt valuable discussion among students about rates of change.

Below are links to materials used to teach the lesson.