Presentation on theme: "Revitalizing College Algebra A workshop for the MMATYC Conference at Howard CC. NOTE: This version of the original PowerPoint was edited to go on the RS."— Presentation transcript:
Revitalizing College Algebra A workshop for the MMATYC Conference at Howard CC. NOTE: This version of the original PowerPoint was edited to go on the RS website. If the PPT links do not work use the links listed at the site.
Agenda for the Workshop 1.A sample module prepared by AMATYC members that exemplifies the type of activity that promotes best practices: engaging students in an activity that requires they use mathematics in a meaningful way, with the aid of appropriate technology, to make decisions and/or draw conclusions. 2.Some background on the reform movement for College Algebra. 3.Sample lessons that promote the “rule of four” in instruction. 4.Q & A: content, pedagogy, technology, and assessment
Part 1 Module 7 Hurricanes
Hurricanes – this will blow you away Module 7 This project is sponsored, in part, by a grant from the National Science Foundation: NSF DUE Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.
Goal of the Module Students will construct a graph and a regression model which shows the relationship between wind speed and the force of the wind. The students will analyze the math model that fits the data to make predictions from the model.
MAA Competencies Problem Solving ◘ solving problems presented in the context of real world situations with emphasis on model creation and interpretation ◘ creating, interpreting, and revising models and solutions of problems
MAA Competencies Functions and Equations ◘ effectively using multiple perspectives (symbolic, numeric, graphic, and verbal) to explore elementary functions ◘ investigating linear, exponential, power, polynomial, logarithmic, and periodic functions, as appropriate
MAA Competencies Data Analysis ◘ collecting (in scientific discovery or activities, or from the Internet, textbooks, or periodicals), displaying, summarizing, and interpreting data in various forms ◘ fitting an appropriate curve to a scatter plot and use the resulting function for prediction and analysis
CROSSROADS Standards Intellectual Development I – 1 Problem Solving I – 2 Modeling I – 3 Reasoning I – 4Connecting with other Disciplines I – 5Using Technology I – 6Developing Mathematical Power I – 7 Linking Multiple Representations
CROSSROADS Standards Content C – 1 Number Sense C – 2Symbolism and Algebra C – 3Geometry and Measurement C – 4Function Sense C – 5Continuous and Discrete Models C – 6Data Analysis, Statistics, Prob. C – 7Deductive Proof
CROSSROADS Standards Pedagogy P – 1Teaching with Technology P – 2Active and Interactive Learning P – 3Making Connections P – 4Using Multiple Strategies P – 5Experiencing Mathematics
Introduction Hurricanes make the news every fall. In recent years, the interest in hurricanes has escalated because of the devastation done by Katrina, and other powerful storms. This activity will attempt to quantify the damage winds are likely to cause. Video: Video:
Preparing for the Module: QL skills 1. Atmospheric pressure is measured in force per unit area (click here to learn more). One atmosphere is 14.7 pounds per square inch. Convert one atmosphere to kilograms per square centimeter.click here to learn more atmosphericpressure.pdf
Preparing for the Module: QL skills 2.If one atmosphere of pressure exerts 14.7 pounds per square inch, how much pressure is exerted over one square foot?
Preparing for the Module: QL skills 3.A man weighing 200 pounds wears size 12 shoes. The surface area of the shoes is approximately 100 square inches. A woman wearing high heals who weighs 120 pounds comes in contact with only 12 square inches of surface. Compute the psf exerted by each person. If each stepped on your back, which one would hurt more? If each stood on a 2x4” plank, which one is more likely to break it?
Preparing for the Module: QL skills 4.According to USA Today, a wind speed of 39 miles per hour produces a force of 6.1 pounds per square foot. How much force would be exerted on a door (6 ft 8 inches by 32 inches) by a wind of 39 mph? Would you be able to hold the door against the wind ?
Preliminary Thoughts and Opinions Ask participants to share their experiences with hurricanes Introduce the concepts of wind speed and air pressure (click here)click here Elicit student opinion (individually and then as a group) on what happens to the force exerted by the wind as the wind speed increases. Confront students with the graphic from the USA Today on wind speed and air pressure (click here).click here
Activity Enter the data from the USA Today graphic into Excel or a graphing calculator Construct a reasonable model to predict the force from the wind (psf) from wind speed (mph).
Questions Read the graphic more carefully and find the formula used by Ahren. How does that model compare with your’s ? You may ask students to do more research on Ahren’s model.
Applying the Model Based on your model, ►What would be the force of the wind in psf if the wind speed were 100 mph ? 200 mph ? ► If the force of the wind were 50 psf, what would the wind speed be ? 75 psf ? ► Make a table and calculate the change in the force of the wind for each increase of 5 mph in wind speed. Start the table at 40 mph. Describe the change in the rate of change of force.
Wind SpeedForce Rate Of Change mphpsfpsf/mph
More Difficult Questions Examine the information from the Atlantic Oceanographic and Meteorological Laboratory (click here).click here Is it feasible to find a model to predict the median damage of a hurricane based on its wind speed ? Explain. Is it feasible to find a model to predict the number of “extreme hurricane impacts” for any decade? Explain.
A team of AMATYC members have created resources for faculty who wish to refocus their college algebra course. These resources are shared with faculty as part of an NSF-sponsored workshop. This presentation will give faculty in NC a preview of these resources and consider how a refocused course should be defined in terms of content, pedagogy, assessment, and technology.
How do you know when learning has taken place ? When students can get the correct answer to a problem ? When students can get a satisfactory grade on a test ? When students feel good about the course ? When changes in the student’s way of thinking and/or habits of mind are altered ?
How do you know when learning has taken place ? When changes in the student’s way of thinking and/or habits of mind are altered. What changes do we want to cause/observe in our students?
What are the big ideas in your College Algebra Course ? Solving quadratic equations Solving exponential equations Simplifying rational expressions Simplifying logarithmic expressions Finding solutions to systems of equations Interpreting and using data shown in tables, graphs, or formulas Understanding variables and relationships between variables Understanding rate of change Strengthening their problem solving skills Using mathematics to support conclusions
A business runs an advertisement in a local paper every Thursday. The ad is 1.75 inches (height) by 2.75 inches (width) and costs $300. However, next month, the business is celebrating 10 years of service and is prepared to spend at least twice as much for a larger advertisement. The advertising manager has the following restrictions placed on ads in the paper: all ads are measured in 1/32 of an inch the width of an ad is limited to 8/32 of an inch intervals the paper gives a 10% discount on ads that are priced over $500. In addition, the advertising manager wants the advertisement to stay as close as possible to the golden ratio - so that it looks good to the eye. Find the size of the ad that the manager should use and be prepared to support your conclusion. Teaching Hint: Have students write a two-Minute paper at the end of this class: Describe how your group solved this problem and any disagreements you had
Twenty-five people are involved in a meeting. The meeting is planned from 10 a.m. until 3 p.m. so a decision must be made about what to do about lunch. A conversation clearly showed that 22 of the 25 wanted a light lunch at a nearby restaurant that serves great salads. Three of the twenty-five felt they needed more than that and wanted to go to a buffet where they could eat a little of everything. As the organizer of the meeting, what do you decide to do about lunch?
Your Thoughts Is there a need to revitalize college algebra? Is there a difference in the mathematical needs of students who are in programs that require calculus and those who are not? What are those differences?
Beg Alg Int Alg Pre Calc Calc Beg Alg Int Alg Coll Alg Pre Calc Calc Beg Alg Coll Alg Pre Calc Calc Coll Alg Coll Trig Calc Pre CaIc Calc. Coll Alg Bus Calc Int Alg What is College Algebra?
Augsburg College, 2005 (College Algebra as a pre req to Pre-calculus) Doree, S Who takes College Algebra?
Who takes College Algebra at Wake Tech?
Why do students take College Algebra?
Facts about C A at Wake Tech What is your intended major at the school to which you plan to transfer?
Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 27) CUPM Curricular Guide Unfortunately, there is often a serious mismatch between the original rationale for a college algebra requirement and the actual needs of the students who take the course.
Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 27) CUPM Curricular Guide A critically important task... is to clarify the rationale for the requirements, determine the needs of the students who take college algebra, and ensure that the department’s courses are aligned with these findings.
Undergraduate programs and Courses in the Mathematical Sciences: CUPM Curriculum Guide, 2004 (p. 28) CUPM Curricular Guide Offer suitable courses... designed to Engage students in a meaningful and positive intellectual experience; Increase quantitative and logical reasoning abilities needed for informed citizenship / workplace; Strengthen quantitative and mathematical abilities that will be useful to students in other disciplines; Improve every student’s ability to communicate quantitative ideas orally and in writing; Encourage students to take at least one additional course in the mathematical sciences.
“for all students” The ideas behind Calculus reform permeated the discipline, over time. Writing, group work, projects, the use of technology to allow some topics to be minimized and/or deleted, formative assessment, … The effort to refocus college algebra will also have an affect on prerequisite courses as well as subsequent courses.
Arnold Packer, from a conference on college algebra at West Point, Look at a typical description: This course is a modern introduction to the nature of mathematics as a logical system. The structure of the number system is developed axiomatically and extended by logical reasoning to cover essential algebraic topics: algebraic expressions, functions, and theory of equations. Who decided that "algebraic expressions, functions, and theory of equations" is essential, and if so, essential to whom or for what?
Arnold Packer, from a conference on college algebra at West Point, 2002 (http://www.maa.org/t_and_l/college_algebra.html ) The course covers the following topics: Radicals, Complex Numbers, Quadratic Equations, Absolute Value and Polynomial Functions, Equations, Synthetic Division, the Remainder, Factor, and Rational and Conjugate Root Theorems, Linear-Quadratic and Quadratic-Quadratic Systems, Determinants and Cramer's Rule, and Systems of Linear Inequalities. That is a long list of topics; yet, it is only half the topics listed in a typical college algebra syllabus. How much can students learn in one or two days on a topic and what will they remember? How much can students learn in one or two days on a topic and what will they remember? If a student passes the course, what, from the course, will they remember and be able to use at a later date? If a student passes the course, what, from the course, will they remember and be able to use at a later date? For too many students it looks like-and is-a painful experience that they would prefer to skip.
What should students be able to do? Create, analyze, and interpret basic mathematical models from informal problem statements; Argue that the models constructed are reasonable; and Use the models to provide insight into the original problem. Algorithmic skills are much less important than understanding the underlying concepts. Curriculum Reform and the First Two Years (Curriculum Foundations Project)
“REFOCUSED” what does that mean?
R e f o c u s e d Let y = 220 e.337x Find x if y = 1200 The population of Wake County has grown rapidly over the last 20 years. The population in 1980 was 301,400; in 1990, 426,300; and in 2000, 627,800. Find an exponential model for this data and predict when the population of Wake County will reach 1,200,000. What factors may influence this prediction?
R e f o c u s e d Solve: Y = 3.5 X + 25 Y = 4.75 X A trivia shop figures the cost to sell replicas of a famous picture is $1.50 per picture plus $250 for the initial investment. The revenue is expected to be $5.00 per picture. Graph the cost and revenue functions and find the break-even point. Next, graph C(n) – R(n) and explain the significance of the horizontal intercept.
R e f o c u s e d Find the slope of the secant line through the curve Y = x – 3x 2 at x = 11 and x = 12. A company has determined that an appropriate cost function for a product is C(n) = n – 3n 2 where n is the number produced. Construct a spreadsheet with three columns to show n, C(n), and the marginal cost. Over what region is the cost increasing at a decreasing rate? Over what region is the marginal cost decreasing?
R e f o c u s e d Find the slope of the line that passes through the two points (74, 11.85) and (75, 11.88) The time of sunrise and sunset in Raleigh is shown below. Sunrise Sunset March 12 7:29 am 7:20 pm March 13 7:28 am 7:21 pm Construct two ordered pairs (# of days since 1/1, time between sunrise and sunset in hours) and find the slope of the line joining those two ordered pairs. Explain the meaning of the slope of this line in the context of the problem.
R e f o c u s e d Graph: Y = 2x 2 + 3x - 5 Use the Excel template provided, with the interactive scroll bars, to control the values of a, b, and c in the polynomial function y = ax 2 + bx + c. After experimenting, write a brief explanation of what each does to the graph.
By Promoting Best Practices
Understanding over Memorizing
Best Practices Integration over Isolation
Best Practices Depth over Breadth
Best Practices Application over Recognition
The Vision - students 1. Students empowered with the necessary mathematical knowledge, confidence, and skills, enabling them to continue in more advanced mathematics or quantitatively based courses, to get degrees in the areas, and to be successful in mathematics – dependent careers.
The Vision - curriculum 2. A curriculum that changes from one of symbolic manipulation, skill building and emphasis on mechanics, and memorization of algebraic techniques to one that emphasizes variables and functions, mathematical models and representations, data-based interdisciplinary applications that are relevant and meaningful, and more and better use of technology.
The Vision - teachers 3. Energized and enthusiastic teachers with high expectations who are using multiple approaches, teaching and learning with technology, accomplishing learning objectives using exercises, small group activities, and projects connecting mathematical ideas within the discipline and across disciplines, using a variety of assessment methods, and discussing issues, questions, and ideas with colleagues.
The Vision - methods 4. Students who are actively involved in learning algebra through individual and group activities which involve mathematical modeling, who are solving problems arising from a variety of disciplines, and using computers and calculators to generate numerical examples, graph data points, and conjecture and reason about mathematics. HBCU College Algebra Reform Project Newsletter, 1996
What should be re-focused? CONTENT PEDAGOGY TECHNOLOGY ASSESSMENT
CONTENT Mathematics courses and programs in the first two years of college need to develop students’ quantitative and workplace skills and actively engage them in the mathematics they will encounter outside the class- room. Faculty may need to teach content that is different from what they were taught, teach more than they were taught, and teach differently than the way they were taught. Students should understand some of the big ideas of mathematics through a curriculum, a variety of problem-solving strategies, and significant projects that examine selected topics in depth. Students should have opportunities to demonstrate their mathematical knowledge, as well as their creativity. BEYOND CROSSROADS, CH 6
CONTENT Besides algebra, what other mathematics will students see in and use in their daily lives and in their profession? Statistics Probability Networks – Graph Theory Finance Geometry – of solids and shapes
Integrating Content Applications can often blend content from several areas. This helps to dispel the notion that mathematics is a collection of isolated topics.
Integrating Content Application done on N-spire: Area of a Ring See: area_of_the_ring.tns or: area_of_the_ring.xls The Excel file was written by Scott Sinex after the Maryland meeting
CLASSROOM ASSESSMENT Collecting information from students to measure their learning aids instruction and guides the instructor. feedback on lessons assessing important concepts improving the course
Modules Created by The Right Stuff Task Force 17 modules exemplify the material that might be found in a refocused course Module 0 – Implementing the Rule of Four Module 7 – Hurricanes Module 16 – Rate of Change
Module 0 The Rule of Four
Problem: You have 250 students who need math courses. About 220 of them need a course that strengthens their problem-solving skills, their ability to think critically about solutions and processes, one that emphasizes communicating mathematically, and one that utilizes technology. The other 30 or so, need the skills (algebraic and otherwise) necessary to go on to calculus (or pre-calculus). What do you do?