1 College Algebra A story of redesigning a course to improve conceptual understanding Scott Peterson Oregon State University
2 History and challenges in College Algebra Types of redesign Our redesign of College Algebra at OSU Implementation of the redesign Our re-redesign of College Algebra Looking forward
3 History of College Algebra MAA Reports, 2007 Algebra Gateway to a Technological Future edited by Victor J. Katz College Algebra William Haver, Donald Small, Aimee Ellington, Barbara Edwards, Vernon M. Kays, John Haddock, Rob Kimball
4 MAA Reports, 2007 Approximately 700,000 students annually enroll in College Algebra courses, most of which focus on algebraic manipulation. Students are told to learn the procedures for factoring polynomials, simplifying radicals, solving equations with absolute values, and solving inequalities. Students are expected to learn to follow the same procedure demonstrated to them by the instructor. In these courses, students are not expected to use the solutions in any context outside of mathematics. Furthermore, by the most conservative estimates, fewer than 50% of the students who enroll in the course receive a grade of A, B, or C.
5 History of College Algebra at OSU Math 111 College Algebra here at OSU has been taught in the following fashion since about credit hour course meets 4 times each week MWF Lecture, 50 minutes, capped at 210 students Tue Recitation, 50 minutes, ~ 35 students each, 6 recitations per lecture section Recitation generally involved all/some of the following o answer a few questions from students o group activity o short quiz
6 Grading Two midterm exams and a final exam Exams accounting for approximately 75% of grade Other 25% comes from homework and/or quizzes and/or recitation Results DFW rate for the last nine years averaged 40% Student who do pass often still have difficulties with the algebra in Calculus
7 College Algebra Guidelines MAA Reports, 2007 These guidelines represent the recommendations of the MAA/CUPM subcommittee, Curriculum Renewal Across the First Two Years, concerning the nature of the College Algebra course that can serve as a terminal course as well as a pre-requisite to courses such as precalculus, statistics, business calculus, finite mathematics, and mathematics for elementary education majors. Fundamental Experience College Algebra provides students a college level academic experience that emphasizes the use of algebra and functions in problem solving and modeling, provides a foundation in quantitative literacy, supplies the algebra and other mathematics needed in partner disciplines, and helps meet quantitative needs in, and outside of, academia. Students address problems presented as real world situations by creating and interpreting mathematical models. Solutions to the problems are formulated, validated, and analyzed using mental, paper and pencil, algebraic, and technology-based techniques as appropriate.
8 Course Goals Involve students in a meaningful and positive, intellectually engaging, mathematical experience; Provide students with opportunities to analyze, synthesize, and work collaboratively on explorations and reports; Develop students logical reasoning skills needed by informed and productive citizens; Strengthen students algebraic and quantitative abilities useful in the study of other disciplines; Develop students mastery of those algebraic techniques necessary for problem-solving and mathematical modeling; Improve students ability to communicate mathematical ideas clearly in oral and written form; Develop students competence and confidence in their problem-solving ability; Develop students ability to use technology for understanding and doing mathematics; Enable and encourage students to take additional coursework in the mathematical sciences
9 MATH 111 LEARNING OUTCOMES: A successful student in MTH 111 will be able to: 1. Solve linear, absolute value, quadratic, polynomial, radical, rational, exponential and logarithmic equations; and solve linear, polynomial, rational and absolute value inequalities. 2. When given a symbolic relation between 2 quantities, formulate the correct equation or inequality based on the language of the question, solve the equation or inequality and then decide if the result from that process is an reasonable answer to the initial question. 3. Correctly interpret and use symbolic/numeric/graphic representations of relations. 4. Apply the concepts of domain, range, translations, reflections, and inverses to given functions. 5. Recognize and correctly state symbolically functions whose graphs are given and that are related through translations and/or reflections. 6. Investigate connections between roots, factors, graphs and symbolic representations of polynomial functions, and be able to create polynomial functions when given information about the function s roots and/or factors and/or graph. 7. Develop, recognize and extract correct information from the standard forms for equations of circles, lines, and parabolas. 8. Find and list symbolically the vertical, horizontal, inclined asymptotes of rational functions expressed symbolically, graphically and numerically. 9. Translate the language of direct and inverse relations into algebraic relationships, and then answer questions based on that relationship. 10. Develop and use models from linear, exponential and quadratic data or graphs.
10 Mathematics Baccalaureate Core Learning Outcomes Identify situations that can be modeled mathematically. Calculate and/or estimate the relevant variables and relations in a mathematical setting. Critique the applicability of a mathematical approach or the validity of a mathematical conclusion.
11 Types of Course Redesign National Center for Academic Transformation Supplemental: The supplemental model retains the basic structure of the traditional course and a) supplements lectures and textbooks with technology-based, out-of-class activities, or b) also changes what goes on in the class by creating an active learning environment within a large lecture hall setting. Replacement: The replacement model reduces the number of in-class meetings and a) replaces some in-class time with out-of-class, online, interactive learning activities, or b) also makes significant changes in remaining in-class meetings. Emporium: The emporium model eliminates all class meetings and replaces them with a learning resource center featuring online materials and on-demand personalized assistance, using a) an open attendance model or b) a required attendance model depending on student motivation and experience levels. Fully Online: The fully online model eliminates all in-class meetings and moves all learning experiences online, using Web-based, multi-media resources, commercial software, automatically evaluated assessments with guided feedback and alternative staffing models. Buffet: The buffet model customizes the learning environment for each student based on background, learning preference, and academic/professional goals and offers students an assortment of individualized paths to reach the same learning outcomes.
12 Math 111 Redesign Stage one planning - Structure of redesign Tom Dick, Bill Bogley, Keith Schloeman Math 111 is a 4-credit hour course Meets 3 times each week One Lecture 50 minutes capped at 210 students (current section 110 students) Two Recitations, Mon/Wed or Tue/Thur 80 minutes 6 recitations per lecture section ~ students each (current ~20 each) one Teaching Assistants per recitation Required computer lab time, 2 hours per week
13 Stage two planning Content and pedagogy Scott Peterson, Peter Banwarth, Keith Schloeman Three main parts to the design o Computer lab activities (CLA) o Recitation activities (RA) o Lecture Emphasis on the following o Interactive Engagement (IE) o Applications and Modeling (A/M) o Conceptual Understanding (CU) How can we accomplish this?
14 The Rope Activity We did this activity on the first day of class It worked very well to get the students working together within each group One the surface it seems like a straight forward activity, but there are some interesting things to discuss What are some of the not so obvious concepts that you see?
15 Computer lab activities Peter Banwarth Videos to enhance procedural knowledge Khan Academy Created videos where needed Allows use of face-to-face time for IE, A/M, and CU Applets For student experimentation and discovery of concepts For motivation and CU o Example: vertex form of a quadratic function o Link: quadratic function Worksheet using the videos and applets incorporated CU and A/M and IE encouraged designed to prepare students for recitation activities
16 Recitation activities Keith Schloeman Created worksheets for group work for each of the topics (usually 2 topics per week) application problems stress conceptual understanding incorporate modeling
17 Overall planning Scott Peterson Guided planning group discussion of content and content order discussion of structure of recitations lecture Collected concept tests found many at MathQUEST website created some myself, and Peter Concept Warehouse (Milo Koretsky, Sch of Chem/Bio/Envr Eng) use in recitations and lecture with clickers Grading: very similar to previous traditional course two midterm exams and a final exam, 72% of grade one skills quiz toward end of quarter, 8% other 20% comes from online homework, collected CLA, and RA participation
18 The equation y = x 3 + 2x 2-5x - 6 is represented by which graph? A) B) C) D)
19 The relationship between the latitude L of a city in the Northern Hemisphere and its average annual temperature T is modeled by the function T = -0.68L The slope of this linear function means: A. The temperature at the equator would be B. For every degree increase in latitude the average annual temperature increases by C. For every degree increase in latitude the average annual temperature increases by D. For every degree increase in latitude the average annual temperature decreases by 0.68.
20 Solve 3 x + 2 = x = 14 x = 0 x = 0 and x = 4 x = 2 3
21 Weekly structure of course 1. Student preparation before 1 st rec each week read the text?? work through computer lab activity work the online homework problems 2. Recitations work in groups of 4 or 5 on activities may use concept tests before/after/during activity wrap-up whole group discussion last 10 minutes 3. Lecture look back at materiel covered that week look ahead to new material motivation, link to applications
22 Recitation Instructors Three graduate students all three have had experience with math 111 as a teaching assistant and as an instructor One new Instructor was a graduate student here at OSU and had experience as a teaching assistant for math 111 Weekly meetings with recitation instructors planning group Dr. Beisiegal and Dr. Dick
23 What we learned We are good at over estimating! We over estimated the students ability to connect the material in the CLA and the RA the students ability to connect the same concept in different problems within the same activity The number of problems they could complete in 80 minutes We were able to address these issues along the way
24 Changes we made along the way Shortened the activities Started giving explicit information for each problem on the activities learning objective references to the textbook general comments about the concepts involved Included a list of important terms at the beginning of the activity Included some procedural warm-up problems on the RA
27 What we learned cont. We need to incorporate more modeling projects The CT and wrap-up often took a back seat to the activity. We need to implement a strategy to get students prepared for the first recitation each week We need to be much more explicit about the design of the course and expectations, especially the first week of classes We need to change the order of the topics so that students get off to a better (and easier) start Students did become more engaged in the recitations as the quarter proceeded Students in our section scored right at the average of all the sections on the common exam questions
28 A company purchases a new piece of equipment. The value of this equipment (in US dollars) steadily declines as it ages until it no longer is of any value (has a value of 0). The function models the value of the equipment years after the initial purchase. Use this information to answer questions ) After how many years will the value of the equipment be $2,000? a) 10 years b) 20 years c) -10 years d) 15 years e) The value will never be $2,000. 4) What is the value of the equipment after 40 years? a) $2,000 b) $4,000 c) $1,000 d) -$2,000 e) $0
29 A local school district has four middle schools. The tables below give the enrollment in each school at various times. For which school(s) could the enrollment levels over time be modeled by a linear function? a) School A only b) School B only c) School C only d) All three schools e) None of the schools
30 Student issues Common student comments You are not teaching us anything. If you would have shown us that before, the HW would have been much easier. This is a four credit class so why do we have 2 extra hours in the computer lab? Students did not come to recitation prepared this may have been a result of not making connections between the different activities/problems they were not reading the textbook they were not completing the online HW before the recitation Many students were not attending the required CL time most were completing the CLA may have been missing the IE that was encouraged Students had difficulty adjusting to the new system students need to learn how to learn outside of class this may have contributed to low exam scores
31 Looking forward The Re-redesign We established three guiding principles for further planning. 1 Content and design: Build the course around overlying concepts not the list of topics. Ø Function: domain, range, meaning of symbols in context etc. Ø Function (graphical) behavior Ø Modeling: appropriate models Ø Transformations Ø Connections between multiple representations Ø Average rate of change 2 Content, design, & pedagogy: Move through the content so that we can return to the overlying concepts with each new topic 3 Pedagogy: Reduce new types of problems to problems that we have already learned how to solve!
32 Changes we plan to make Changes in structure The first recitation each week will be activity/group learning based The second recitation will be mostly whole group discussion using concept test as a spring board to that discussion The instructor will lead the second recitation along with one TA 2 recitation sections will be combined for a total of 3 recitation sections each with two TAs
33 Changes in content/order We will again start with linear functions as a review and to introduce the concept of rate of change We will move up transformations to follow linear functions We will move inequalities and absolute value to come after quadratic equations We will incorporate 3 to 4 modeling projects These changes do no effect the material covered on each of the exams
34 Other objectives Continue to fine tune the course over winter and spring quarters Create our own applets that are concept specific Develop more concept tests Start preliminary work toward assessing the program Develop a professional development program for instructors and teaching assistants that are new to the redesign Full implementation Fall 2013
35 Plans for program assessment Continue to collect data on the common exam questions Incorporate completely common exams across all sections of math 111 so that we can compare success rates Algebra Concept Inventory? Pre/post affective survey Usefulness mathematics Mathematics as a cohesive whole Confidence mathematical ability Chance of taking a math class beyond your degree requirements Student interviews Track how many students continue taking math courses Track students success in future math courses
36 The End! Thank you very much for your interest. I hope you enjoyed the show! Please feel free to contact me if you have any questions. Scott L. Peterson