Transition to College Mathematics and Statistics

Report
Transition to College
Mathematics and Statistics
for Non-STEM Students
Christian R. Hirsch
Joint Mathematics Meetings
January 9-12, 2013
San Diego, CA
Session Overview
 Why TCMS?
 Design of TCMS
 Units and Key Topics
 Comments and Questions
2
Transition Issues
• Unacceptably high enrollments in non-credit bearing
courses
3
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.
1
2010 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
4
Transition Issues
• Unacceptably high enrollments in non-credit bearing
courses
• Three years of college preparatory mathematics is
insufficient for college readiness
5
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
6
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
7
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.
1
2010 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
8
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.
9
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)
10
Transition to College Mathematics and Statistics
is an alternative fourth-year mathematical
sciences course designed to support collegereadiness of non-STEM students. It is the product
of three years of research, development, and
evaluation funded by the National Science
Foundation.
11
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
12
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
13
14
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?
15
Production Conditions
• Assembly of each SM game system takes 0.6 hours of
technician time and assembly of each DM 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.
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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
23
Transition to College Mathematics and
Statistics
Units 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.
24
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.
25
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.
26
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.
27
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.
28
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, symmetrickey and public-key cryptosystems; error-detecting
codes (including ZIP, UPC, and ISBN) and errorcorrecting codes (including Hamming distance); and
trees and Huffman coding.
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Transition to College Mathematics and
Statistics
Unit 7: Spatial Visualization and Representations
Extends student ability to visualize and represent threedimensional 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.
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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 voteanalysis methods such as majority, plurality, runoff, points-forpreferences (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 nonzero-sum games, Nash’s theorem, and the minimax theorem.
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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.
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• 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.
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Comments and Questions
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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.
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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.
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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.
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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.
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• 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.
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