| Credit- Degree applicable | | Effective Quarter: Fall 2020 | I. Catalog Information
| MATH 44 | Mathematics in Art, Culture, and Society: A Liberal Arts Math Class | 5 Unit(s) |
| | (See general education pages for the requirement this course meets.) Prerequisite: Intermediate Algebra (MATH 109, MATH 114 or MATH 130) or equivalent.
Advisory: EWRT 211 and READ 211, or ESL 272 and 273. Lec Hrs: 60.00
Out of Class Hrs: 120.00
Total Student Learning Hrs: 180.00 This course is a survey of selected topics from contemporary mathematics, including problem-solving techniques and connections between mathematics and culture. It includes a selection of introductory topics from symmetry; graph theory; chaos and fractals; topology; number theory; geometry; combinatorics and counting; the mathematics of social choice; data analysis, probability, and statistics; consumer mathematics and personal financial management. |
| Student Learning Outcome Statements (SLO)
| | | Analyze contemporary mathematical problems, apply problem solving techniques using a variety of methods, and communicate the results mathematically through a variety of forms. |
| | | Demonstrate and correctly apply basic mathematical techniques in at least five of the following ten areas: symmetry, graph theory, fractals and chaos theory, topology, number theory, geometry, combinatorics, methods of social choice, probability and statistics, economics and personal finance. |
| | | Examine and evaluate myths and realities about the contemporary discipline of mathematics and its practitioners. |
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II. Course Objectives | A. | Examine problem solving techniques using a variety of methods and communicate mathematically through a variety of forms. |
| B. | Investigate and solve problems in at least five of 10 areas of contemporary mathematics and its applications. |
| C. | Examine at least two of the topics listed below related to myths and realities concerning mathematics. |
III. Essential Student Materials | | Scientific or graphing calculator |
IV. Essential College Facilities V. Expanded Description: Content and Form | A. | Examine problem solving techniques using a variety of methods and communicate mathematically through a variety of forms. |
| a. | Mathematical discovery and invention |
| b. | Logical, axiomatic thinking |
| c. | Inductive thinking and searching for patterns in data |
| d. | Mathematical experimentation |
| 2. | Mathematical communication |
| a. | Reading, writing, creating visual images, and oral presentation |
| b. | Collaborative activities |
| 3. | Historical/cultural topics, such as |
| a. | Logic, mathematics and culture |
| b. | Visual or geometric "proofs" |
| B. | Investigate and solve problems in at least five of 10 areas of contemporary mathematics and its applications. |
| 1. | Distinguish between types of symmetry, and use them to analyze patterns, art, and cultural artifacts. |
| a. | The analysis of geometric patterns |
| b. | The rosette groups: rotational symmetry |
| c. | Translation, reflection, and glide symmetries |
| d. | Linear or frieze patterns |
| e. | Tessellations of the plane: regular, semi-regular, irregular, and Penrose tilings |
| f. | Combining symmetries: introduction to symmetry groups |
| 2. | The graphic designs of M.C. Escher's and other artists |
| 3. | Analysis of designs from around the world |
| 4. | Investigation and report of use of symmetry in each student's family or cultural group |
| 2. | Recognize and utilize graphs, digraphs, and trees in problem solving, and use graph theory to analyze cultural designs and social relations. |
| b. | Counting degree, vertices, edges, and faces: Euler's Theorem |
| c. | Eulerian and Hamiltonian circuits and their applications |
| d. | Scheduling problems, coloring graphs, and related applications |
| e. | Historical/cultural topics, such as |
| 1. | Kinship systems from several cultures |
| 2. | African sand designs and Euler circuits |
| 3. | Social relationships and the small world phenomenon |
| 3. | Recognize, analyze, and create fractal patterns, and relate fractals and chaos theory to the iteration of simple processes. |
| a. | Iteration of simple processes |
| 3. | Mandlebrot and Julia sets |
| c. | Mathematical chaos: periodicity and disorder |
| d. | Historical/cultural topics, such as |
| 1. | Fractals in African design |
| 3. | Applications of fractals |
| 4. | Identify the topological properties and parameters of surfaces, networks, knots, links, and mappings, and use them to analyze and create culturally significant designs. |
| a. | Topology: equivalence under distortion |
| c. | Planar and non-planar networks |
| e. | Fixed points of mappings |
| f. | Historical and cultural topics, such as |
| 1. | String figures from around the world |
| 5. | Use the theory of numbers and modular arithmetic to analyze patterns, and to encode and decode information. |
| a. | Numerical patterns in nature: the Fibonacci numbers |
| c. | Modular arithmetic and applications to error-detecting codes |
| d. | Data encryption: the RSA code |
| e. | Historical and cultural topics, such as |
| 2. | Chinese remainder theorem and the historical roots of modular arithmetic. |
| 3. | Pattern analysis of artwork from many cultures using modular arithmetic. |
| 4. | The use of binary multiplication and division schemes in ancient Egypt. |
| 5. | Calendar calculations in many cultures |
| 6. | Investigate geometric properties and patterns involved in right triangles, the Fibonacci numbers, spirals and helices, polyhedra, transformations of scale, and non-Euclidean geometry. If this topic is covered, at least one of the historical/cultural subtopics and five other subtopics chosen from a-i will be covered. |
| a. | Pythagorean theorem and applications |
| b. | The Fibonacci numbers, the golden mean and phyllotaxis |
| c. | Spirals and helices in nature and art |
| d. | Polyhedra and the Platonic solids |
| e. | Growth, size, and shape: dimensional analysis |
| g. | Flatland and the fourth dimension |
| h. | Computational geometry: the art gallery theorem and other applications |
| i. | Historical/cultural topics, such as |
| 1. | Discovery and use of "Pythagorean theorem" prior to Pythagoras in China and Babylonia |
| 2. | The development of spherical trigonometry in the Islamic world. |
| 3. | Traditional and theoretical Origami |
| 7. | Solve problems using counting principles, permutations and combinations. |
| a. | Multiplication and addition principles |
| b. | Pigeonhole principle and applications |
| c. | Permutations and combinations |
| d. | Binomial coefficients and Pascal's Triangle |
| e. | Calculating probabilities and other applications |
| f. | Historical and cultural topics, such as |
| 1. | Development of properties of "Pascal's Triangle" in China, Middle East, and India prior to Pascal |
| 2. | Application of Fibonacci numbers to prosody in ancient India |
| 8. | Identify mathematical techniques used in social choice, and critique methods of voting, sharing, and apportionment. |
| a. | Voting methods and paradoxes |
| 4. | Plurality with elimination |
| b. | Voting paradoxes and Arrow's impossibility theorem |
| c. | Weighted voting systems |
| 2. | Shapley-Shubik power index |
| 2. | Divisor methods, such as the Jefferson Method, the Webster Method, the Hill-Huntington Method. |
| f. | Historical and cultural topics, such as |
| 1. | Voting methods around the world |
| 2. | Voting methods in sports, entertainment, and culture |
| 3. | US election controversies and the Electoral College |
| 4. | Apportionment controversies in the United States |
| 9. | Determine measures of central tendency and dispersion of data, evaluate survey and sampling methods, understand the meaning and application of probability, find and analyze examples of the use of statistics in the media. |
| a. | The population and collecting data |
| b. | Descriptive statistics: graphing and summarizing data |
| c. | Measuring uncertainty: probabilities and odds |
| d. | The normal distribution |
| e. | Historical and cultural topics, such as |
| 1. | History of opinion polls |
| 2. | Cultural forms of risk-taking with money, which may include topics like Gambling and Casinos, Lotteries, Chain Letters, Pyramid schemes. |
| 3. | Display of data and the development of number systems |
| 4. | Games of chance of indigenous America and other areas |
| 5. | History of statistics and biographies of statisticians |
| 10. | Apply Mathematical Models to Economics and Personal Finance |
| a. | Interest rates: Compound interest and exponential functions |
| 1. | Compound interest and exponential functions |
| 2. | The constant e, natural logarithms and continuous compounding (optional) |
| 3. | Rules of 70 and 72, and doubling times (optional) |
| 4. | The effect (short and long term) of compounding |
| 5. | Variable vs. fixed interest rates |
| 1. | Annuities and geometric series |
| 2. | Amortizations and Installment Plans, which may include the topics of Mortgages, Student Loans, Consumer Loans. |
| 3. | Points and fees (optional) |
| 4. | Annual Percentage Rate (APR) |
| 5. | Comparing different options |
| c. | Savings and Investments |
| 2. | IRAs and other savings plans |
| 3. | Analyzing investment choices, which may include the topics of Performance, Risk and volatility, Diversification |
| d. | Historical and cultural topics, such as |
| 1. | Evolution of National and Global Debt Models, which may include topics of the National Debt, International Trade Deficit, Use of Lotteries to Fund Social Programs. |
| 2. | Evolution of Investment Models, which may include the topics of IRA's, 401(k) and other savings plans, Capital Asset Pricing Model and Modern Portfolio Theory, The Black-Scholes formula and options trading, The Markowitz Model for Efficient Portfolios, Global Investing. |
| 3. | Innumeracy in financial matters, which may include the topics of Ponzi and pyramid schemes, Internet scams. |
| 4. | The growth of econometrics, which may include the topics of "Quants" on Wall Street, The Nobel Prizes in economics. |
| C. | Examine at least two of the topics listed below related to myths and realities concerning mathematics. |
| 1. | Mathematical autobiography: an examination of the student's mathematical background |
| 2. | Contemporary mathematicians: reports on living or recent mathematicians, their work and background |
| 3. | Contemporary mathematical topics: report on the student's choice of topic concerning a recent mathematical development. |
| a. | Current knowledge on links between the brain and mathematics |
| b. | Math anxiety and the psychology of mathematical achievement |
| c. | Ethnic background and mathematics achievement |
| d. | Gender differences and mathematics achievement |
| 5. | Mathematics and subculture |
| a. | Street math versus school math |
| b. | Mathematical thinking within cultural groups such as |
| 3. | Disabled community (mathematical structure of Braille or sign language) |
| 4. | Specific national, ethnic, or other groups in world culture |
| 6. | Mathematics within occupations (may involve field trip or guest speaker), for example |
| 7. | Electronic mathematical resources |
| 8. | Mathematics and the arts |
| a. | Contemporary plays (e.g. Proof, Arcadia) |
| b. | Literature (e.g. Fantasia Mathematica, The Mathematical Magpie) |
| c. | Films (e.g. A Beautiful Mind, Pi, Good-Will Hunting, Enigma) |
| d. | Television and radio shows |
| f. | Performances that deal with mathematical themes |
| 9. | Mathematics of Inequity |
| a. | In law (e.g. Examining court cases of age bias and other discrimination cases argued statistically) |
| b. | In Education (e.g. Looking at "curving" and standardized testing) |
| c. | In Ecology (e.g. Eco-racism and the Bayview-Hunter's Point case) |
| d. | In Social Science (e.g. Mathematical analysis of hierarchies, such as looking at statistical comparisons of data on the "developing world" and, "developed world" and other examples) |
VI. Assignments | A. | Homework and critical thinking problem-solving exercises from the text that include written explanations of concepts and justification of conclusions.
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| B. | Periodic quizzes and/or in-class assignments |
| C. | Required readings from text and other sources. |
| D. | Review questions from the text and/or other sources based upon lecture, reading and/or other materials designed to help students integrate the methods, ideas and techniques learned in class to solve problems.
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| E. | Written reports or essays on a contemporary or historical mathematical source based on library and/or web site research which may also require the student to prepare and present the report orally. Such presentations may require visual aids, demonstrations, etc.
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| F. | Group projects, laboratory projects, and extensive oral presentations that include written descriptions of methods and results, and justification of conclusions. |
VII. Methods of Instruction | | Lecture and visual aids
Discussion of assigned reading
Discussion and problem-solving performed in class
In-class exploration of internet sites
Quiz and examination review performed in class
Homework and extended projects
Fieldwork and field trips
Guest speakers
Collaborative learning and small group exercises
Collaborative projects
Problem solving and exploration activities using applications software
Problem solving and exploration activities using courseware
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VIII. Methods of Evaluating Objectives | A. | Homework and critical thinking problem-solving exercises will be evaluated for accuracy, completion, and justification of conclusions in order to obtain regular assessment of the student’s comprehension of material covered in lecture. |
| B. | Quizzes and in-class assignments will be evaluated for accuracy and completion in order to assess student’s comprehension of material covered in lecture and to provide feedback to students on their progress. |
| C. | Participation in and contribution toward classroom discussions and collaborative group written analytical work involving comparative source materials such as the text or recent news articles. |
| D. | A minimum of one in-class one hour exam composed of both computational and concept based questions which will require the student to demonstrate ability in integrating the methods, ideas and techniques learned in class. Questions may also require the student to communicate ideas and conclusions in short essay format.
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| E. | Reports and essays will be evaluated for accuracy and completion. Oral presentations will further be evaluated for clarity and effectiveness of visual aids and demonstrations. |
| F. | Group projects and laboratory projects will be evaluated for accuracy, completeness, and proper use of techniques and methods discussed in class. |
| G. | A minimum of two of the following:
1. Research project or essay to be presented orally to the class which will be evaluated for accuracy and completion. Oral presentations will further be evaluated for clarity and effectiveness of visual aids and demonstrations.
2. Extended group project or laboratory project which will be evaluated for accuracy, completeness, and proper use of techniques and methods discussed in class.
3. Additional one-hour in-class exam and/or take-home exam composed of both computational and concept based questions which will require the student to demonstrate ability in integrating the methods, ideas and techniques learned in class. Questions may also require the student to communicate ideas and conclusions in short essay format. |
| H. | Two-hour comprehensive final exam composed of both computational and concept based questions which will require the student to demonstrate ability in integrating the methods, ideas and techniques learned in class. Questions may also require the student to communicate ideas and conclusions in short essay format.
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IX. Texts and Supporting References | A. | Examples of Primary Texts and References |
| 1. | *"The Heart of Mathematics", 4th edition, Burger and Starbird, publ. by Wiley, 2013. |
| B. | Examples of Supporting Texts and References |
| 1. | "Excursions in Modern Mathematics", Tannenbaum and Arnold, publ. By Pearson, 2010.
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| 2. | "For All Practical Purposes", 8th edition, COMAP, publ. By Freeman, 2010. |
| 3. | "Fractals for the Classroom", Peitgen, Jurgens, Saupe |
| 4. | Multicultural Mathematics Bibliography compiled by Karl Schaffer, at (http://nebula2.deanza.edu/~karl/) |
| 5. | Robert Devaney's web sites on chaos and fractals: (http://math.bu.edu/people/bob/) |
| 6. | "Problem Solving Strategies: Crossing the River with Dogs and Other Mathematical Adventures", by Ted Herr and Ken Johnson, publ. by Key Curriculum Press. |
| 7. | "Mathematical People, More Mathematical People", edited by Alexanderson. |
| 8. | "What's Happening in the Mathematical Sciences", four volumes, ed. by Barry Cipra, pub. by the American Mathematical Society. |
| 9. | "Geometry Labs", Picciotto, publ. by Key Curriculum Press. |
| 10. | "Symmetry, Shape and Space", Kinsey and Moore, publ. by Key Curriculum Press. |
| 11. | Recreational Mathematics sites: (http://www.mathpuzzle.com/) |
| 12. | String figures from around the world: (http://www.isfa.org/) |
| 13. | Polyhedra: (http://www.georgehart.com/virtual-polyhedra/vp.html) |
| 14. | Online mathematics columns: (http://www.maa.org/news/columns.html) |
| 15. | Topics in mathematics: (http://www.mathacademy.com/pr/) |
| 16. | Origami and mathematics: (http://mars.wne.edu/~thull/origamimath.html) |
| 17. | African Fractals: (http://www.rpi.edu/~eglash/eglash.dir/afractal/afractal.htm) |
| 18. | Egyptian and Babylonian mathematics: (http://www-gap.dcs.st-and.ac.uk/~history/HistTopics/Babylonian_and_Egyptian.html) |
| 19. | Chinese mathematics:(http://www.roma.unisa.edu.au/07305/chinese.htm)
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| 20. | Ethnomathematics on the web: (http://www.rpi.edu/~eglash/isgem.dir/links.htm) |
| 21. | History of statistics and stories of statisticians compiled by Diane Mathios: (http://faculty.deanza.fhda.edu/mathiosdiane/stories/) |
| 22. | "Proof", the 2001 Pulitzer and Tony award winning play by David Auburn. |
| 23. | "Arcadia", play about chaos theory by Tom Stoppard. |
| 24. | A large bibliography keyed to specific topic headings and a packet of suggested classroom activities, guides, and overheads will be kept in the division office for use by interested faculty. |
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