Transition to College Mathematics and Statistics for Non-STEM Students

1. Transition to College Mathematics and Statistics for Non-STEM Students Christian R. Hirsch Joint Mathematics Meetings January 9-12, 2013 San Diego, CA

2. Session Overview • Why TCMS? • Design of TCMS • Units and Key Topics • Comments and Questions

3. Transition Issues • Unacceptably high enrollments in non-credit bearing courses

4. AMS Survey of Enrollment Patterns1 • In Fall 2010, approximately 11% of four-year college mathematics and statistics enrollments were in pre-college courses. • In Fall 2010, approximately 61% of two-year college mathematics and statistics enrollments were in pre-college courses. 12010 Preliminary Results from the Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States. http://www.ams.org/profession/data/cbms-survey/cbms2010-work

5. Transition Issues • Unacceptably high enrollments in non-credit bearing courses • Three years of college preparatory mathematics is insufficient for college readiness

6. In the 2009-2010 administration of the Ohio Early Mathematics Placement Testing, based on the expressed intended majors of test-takers, 63% of the test-takers would place in a remedial (non-credit bearing) course in college if they took no further mathematics in high school. E. Laughbaum, personal communication, January 5, 2013

7. Transition Issues • Unacceptably high enrollments in non-credit bearing courses • Three years of college preparatory mathematics is insufficient for college readiness • Preparation for college mathematics is not necessarily preparation for calculus and vice versa

8. Fall 2010 Four-Year College Mathematics and Statistics Enrollments1 • Approximately 25% of students took mainstream or non-mainstream Calculus I or II. • Approximately 10% of students took a noncalculus-based course in statistics. • Approximately 7% of students took a Liberal Arts Mathematics course. • Approximately 6% of students took a Finite or Business Mathematics course. • Approximately 4% of students took an Elementary Education Mathematics course. These data suggest that for a large number of students, success in college is dependent on mathematics coursework that is independent of calculus instruction. 12010 Preliminary Results from the Statistical Abstract of Undergraduate Programs in the Mathematical Sciences in the United States. http://www.ams.org/profession/data/cbms-survey/cbms2010-work

9. Transition Issues • Unacceptably high enrollments in non-credit bearing courses • Three years of college preparatory mathematics is insufficient for college readiness • Preparation for college mathematics is not necessarily preparation for calculus • Weak alignment between university mathematics placement tests (and sometimes courses) and both high school mathematics priorities and professional recommendations for undergraduate mathematics.

10. MAA Curriculum Foundations Project Summary Recommendations • Emphasize conceptual understanding • Emphasize problem-solving skills • Emphasize mathematical modeling • Emphasize communication skills • Emphasize balance between mathematical perspectives (e.g., continuous and discrete, deterministic and stochastic) Ganter & Barker (2004)

11. Transition to College Mathematics and Statistics is an alternative fourth-year mathematical sciences course designed to support college-readiness of non-STEM students. It is the product of three years of research, development, and evaluation funded by the National Science Foundation.

12. Development Process and Consultants Algebra and Functions Bernie Madison – University of Arkansas Discrete Mathematics Steve Maurer – Swarthmore College Geometry Doris Schattschneider – Moravian College Statistics and Probability Christine Franklin – University of Georgia

13. TCMS Design Features • Development of mathematics as an active science of patterns • Course organized around interwoven strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics • Mathematical strands developed in coherent, focused units that exploit connections to the other strands • New mathematical ideas introduced in the context of problem situations • Focus on applications and mathematical modeling • Emphasis on small-group collaborative learning and sense-making • Full and strategic use of technology tools

15. VIDEO GAME SYSTEM PRODUCTION PROFIT The manager of TK Electronics must plan for production of two video game systems, a standard model (SM) and a deluxe model (DM). Given the following production limits for assembly time, testing time, and packaging time, how should the manager plan production to maximize profit for his company?

16. Production Conditions • Assembly of each SM game system takes 0.6 hours of technician time and assembly of eachDM game system takes 0.3 hours of technician time. The plant limits technician time to at most 240 hours per day. • Testing for each SM system takes 0.2 hours and testing of each DM system takes 0.4 hours. The plant can apply at most 160 hours of technician time each day for testing. • Packaging time is the same for each model. The packaging department of the plant can handle at most 500 game systems per day. • The company makes a profit of $50 on each SM model and $75 on each DM model.

23. TCMS Design Features, cont’d. • Development of mathematics as an active science of patterns • Course organized around interwoven strands of algebra and functions, geometry and trigonometry, statistics and probability, and discrete mathematics • Mathematical strands developed in coherent, focused units that exploit connections to the other strands • New mathematical ideas introduced in the context of problem situations • Focus on applications and mathematical modeling • Emphasis on small-group collaborative learning and sense-making • Full and strategic use of technology tools • Designed to be used following the third course of an integrated mathematics program or with more conventional programs following an advanced algebra course

24. Transition to College Mathematics and StatisticsUnits and Key Topics Unit 1: Interpreting Categorical Data Develops student understanding of two-way frequency tables, conditional probability and independence, and using data from a randomized experiment to compare two treatments . Topics include two-way tables, graphical representations, comparison of proportions including absolute risk reduction and relative risk, characteristics and terminology of well-designed experiments, expected frequency, chi-square test of homogeneity, statistical significance.

25. Transition to College Mathematics and Statistics Unit 2: Functions Modeling Change Extends student understanding of linear, exponential, quadratic, power, trigonometric, and logarithmic functions to model quantitative relationships and data patterns whose graphs are transformations of basic patterns. Topics include linear, exponential, quadratic, power, circular, and base-10 logarithmic functions; mathematical modeling; translation, reflection, stretching, and compressing of graphs with connections to symbolic forms of corresponding function rules.

26. Transition to College Mathematics and Statistics Unit 3: Counting Methods Extends student ability to count systematically and solve enumeration problems using permutations and combinations. Topics include systematic listing and counting, counting trees, the Multiplication Principle of Counting, Addition Principle of Counting, combinations, permutations, selections with repetition; the binomial theorem, Pascal’s triangle, combinatorial reasoning; and the general multiplication rule for probability.

27. Transition to College Mathematics and Statistics Unit 4: Quantitative and Algebraic Reasoning Extends student facility with the use of functions, expressions, and equations in representing and reasoning about quantitative relationships, especially those involving financial mathematical models and bivariate data. Topics include investments and compound interest, continuous compounding and natural logarithms, and amortization of loans; linearization of bivariate data using log and log-log transformations; and solution of equations involving logarithms, absolute value, and radical expressions.

28. Transition to College Mathematics and Statistics Unit 5: Binomial Distributions and Statistical Inference Develops student understanding of the rules of probability; binomial distributions; expected value; testing a model; simulation; making inferences about the population based on a random sample; margin of error; and comparison of sample surveys, experiments, and observational studies and how randomization relates to each. Topics include review of basic rules and vocabulary of probability (addition and multiplication rules, independent events, mutually exclusive events); binomial probability formula; expected value; statistical significance and P‑value; design of sample surveys including random sampling and stratified random sampling; response bias; sample selection bias; sampling distribution; variability in sampling and sampling error; margin of error; and confidence interval.

29. Transition to College Mathematics and Statistics Unit 6: Informatics Develops student understanding of the mathematical concepts and methods related to information processing, particularly on the Internet, focusing on the key issues of access, security, accuracy, and efficiency. Topics include elementary set theory and logic; modular arithmetic and number theory; secret codes, symmetric-key and public-key cryptosystems; error-detecting codes (including ZIP, UPC, and ISBN) and error-correcting codes (including Hamming distance); and trees and Huffman coding.

30. Transition to College Mathematics and Statistics Unit 7: Spatial Visualization and Representations Extends student ability to visualize and represent three-dimensional shapes using contour diagrams, cross sections, and relief maps; to use coordinate methods for representing and analyzing three-dimensional shapes and their properties; and to use geometric and algebraic reasoning to solve systems of linear equations and inequalities in three variables and linear programming problems. Topics include using contours to represent three-dimensional surfaces and developing contour maps from data; sketching surfaces from sets of cross sections; three-dimensional rectangular coordinate system; sketching planes using traces, intercepts, and cross sections derived from algebraic representations; systems of linear equations and inequalities in three variables; and linear programming.

31. Transition to College Mathematics and Statistics Unit 8: Mathematics of Democratic Decision-Making Develops student understanding of the mathematical concepts and methods useful in making decisions in a democratic society, as related to voting, fair division, and game theory. Topics include preferential voting and associated vote-analysis methods such as majority, plurality, runoff, points-for-preferences (Borda method), pairwise-comparison (Condorcet method), and Arrow’s theorem; weighted voting, including weight and power of a vote and the Banzhaf power index; fair division techniques, including apportionment methods; and game theory, including zero-sum and non-zero-sum games, Nash’s theorem, and the minimax theorem.

32. Voices of TCMS Teachers What do you see as the major strengths of TCMS? • It reaches out to a select population of students that we previously had nothing to offer them. • TCMS is the perfect class for collaborative learning. Students learn to actually read in a math class, they learn how to make mistakes and learn from them as opposed to being discouraged by them, and they also get a deeper understanding of the mathematical material since the topics are all in a real-world context.

33. This course really made the teacher and students think about the mathematics being taught and learned. It gave a lot of students who were unsuccessful in Algebra 2 an opportunity to be successful. They enjoyed most topics and the contexts were very engaging. Many of my students left at the end with a view of mathematics as being useful. • One of the strengths is the connection that TCMS makes to the professional careers that exist today. Students can see the relevance in learning the mathematics, even if it’s not [always] the field of study they are interested in.

34. Comments and Questions

35. Preliminary Findings • 1. Based on the TCMS Post Belief Survey, students across all 6 field-test schools generally found • the Statistics and Coding /Cryptography units to be most interesting, and noted the mathematics reasonable to understand, but particularly noted that the contexts were very very interesting and something they could relate to. • students from traditional schools couched many of their comments (e.g., those related to the real-world problems) as a contrast to the non-context experiences they encountered in their previous Algebra and Geometry courses.

36. 2. Based on the ITED Pre/Post highest level Quantitative Thinking Test (a 40-item test focusing on thinking and reasoning skills), students at • all schools showed pre-post gains at least slightly greater than expected (national) norms (an identical mean in spring to fall would be normal growth). • one CPMP background school made significant gains at the p=.05 level; one traditional school at the p.05 level, and one at the p=.10 level. • Across all schools, gains were particularly notable for students in the 3rd and 4th quartiles.

37. 3. Based on the Conceptions of Mathematics Pre/Post Inventory, students • Most changes within seven grouped (item) categories involved changes toward a conception of Mathematics as Concepts more than Procedures and Sense-Making more than Memorizing. • 4. We are in the process of contacting and collecting 1st semester college data from students.

38. Voices of TCMS Students Would you recommend this course to students (juniors) considering a math course to take for next year? What topics did you find most interesting? Least interesting or most difficult? • Yes, because it seems like we learn things that are much more applicable to daily life than other math classes. My favorite was creating codes. • This course takes skills and ideas learned from previous courses, reviews them, and expands upon the concepts that may have been misunderstood previously. Most interesting: Different voting methods gave different outcomes. Most Difficult: Some of the graphing we were required to do, involving transformations especially.

39. It was a good class to take because the concepts are fairly easy, but they are different from every other math class. • I think this class will prepare you for college with the layout of the class you get to learn many different types of math. Interesting—statistics were most interesting it related a lot to real life. Most difficult were some of the graphs and knowing how to read them.