This course is designed to provide a hands-on experience to the student for gaining insight into the world of data for building models; and for providing a glimpse into the practical power of the world of numbers. These models are relevant for some of the needs and challenges of India. A significant portion of the learning process is based on open e-books that are highly interactive and that transfer knowledge into practice. This course aims to seamlessly blend theory and practice where often practical knowledge precedes theoretical insights. Additionally, the project and application oriented hands on learning process would introduce the joys and advantages of working cohesively and productively in groups to relate mathematics to real world problems.
2. Objective and Expected outcome
- Learning mathematics by connecting it to real life problems linked to society
- Extending blackboard teaching to solve practical problems using mathematics
- Triggering analytical thinking
- Encourage experimentation and hands-on projects mode of learning
- Appreciation of role of data, learn how to collect data and analyze it
- Correlate real world observations with the theoretical knowledge
- Mathematical analysis of data for simple quantitative inference
Every Semester, teaching will be spread over 16 weeks including two weeks for review.
3. Themes & Sub-themes
I. Numbers 6 lectures
Interesting properties of prime numbers without proofs
Ramanujan’s work on the Prime Number Theorem
Euclid’s division algorithm
Mathematical illustration through intuitive examples
Visualisation through tiling analogy
Encryption and Prime numbers
Gentle introduction of 2 x 2 matrices
Constructing the RSA Algorithm
Implementations of RSA algorithms
Other methods of encryption
II. Data and patterns 6 lectures
Historical perspective and importance of data
Kepler’s law for planetary orbits from Tycho Brahe’s astronomical observation Ramanujan’s work on Prime numbers through data
Data collection techniques
Formulation of problem – goals, targets
Methods to collect data – questionnaire, observations, recording, etc.
How much of data is enough for the given problem
Population and sample
Collecting and organising data in various situations through practical methods, from the internet and from other sources.
Use of spreadsheets for practical work related to the above concepts.
III. Statistics 8 lectures
Organisation of data
Frequency table Grouping
Visualisation of data
Pictorially displaying data: dot plots, bar graphs, line graphs, pie charts
Misinterpretation of data by distorting the figures: Scaling and axis manipulation,
Line graphs and cropping
Analysis of data
Mean, median, mode, variance, standard deviation.
Histogram, skewed distribution
Comparing two distributions
Statistical analysis of daily life data
Statistical analysis of stock market data
Statistical analysis of weather data
Statistical analysis of data for better governance
IV. Probability 4 lectures
Interpreting probability, Sample Space, Events
Understanding the tossing of a coin and throwing of dice for large number of trials
Probability in a situation where there are equally-likely outcomes
Probability of two independent events
Compute probabilities from insufficient information
Validity of computed probability
4. Suggestive Projects
- Algorithmic approach of Sieve of Erastosthene’s.
- Ramanujan’s theory of prime numbers: Use of prime numbers in coding and decoding of messages.
- Bertrnad’s postulate
- Download http://pib.nic.in/prs/2011/latest31mar.pdf. Analyse various information that have been extracted from the Census, 2011. Understand as to how these information have been presented.
- Visit the census site of India http://www.censusindia.gov.in/Census_Data_2001/Census_Data_Online/Language/State ment3.htm Depict the information given there in a pictorial form.
- Prepare a questionnaire to collect information about money spent by your friends in a month on activities like traveling, movies, recharging of the mobiles, etc. and draw interesting conclusions.
- Check out the local newspaper and cut out examples of information depicted by graphs. Draw your own conclusions from the graph and compare it with the analysis given in the report.
- Analysis of population migration data – positive and negative influence on urbanization
- Each day newspaper tells us about the maximum temperature, minimum temperature, humidity. Collect the data for a period of 30 days and represent it graphically. Compare it with the data available for the same time period for the previous year.
- Draw a career graph of a cricketer (batting average for a batsman and bowling average for a bowler). Conclude the best year of his career. It may be extended for other players also – tennis, badminton, athlete.
- Share market data analysis – correlation and extreme fluctuation
- Vehicle registration data – correlating with pollution and number of accidents
- Visit a village near Delhi and collect data of various crops over past few years from the farmers. Also collect data about temperature variation and rain over the period for a particular crop. Try to find the effect of temperature and rain variations on various crops.
- How safe are privately owned public transport versus government owned public transport? Collect the data from archives about accidents of Blue Line buses and compare with those of DTC buses. Verify whether DTC buses are significantly safer.
- Visit Kirana shops near your home and collect the data of sale of certain commodaties over a month. Try to figure out the stock of a particular commodity which should be in the store in order to maximize the profit.
- Mendelian Genetics: Genes are molecular units of heredity and carry certain information. They occur in pairs. Gregor Mendel studied about the inheritance in pea plants. One of the characteristics about their inheritance is smooth (S) and wrinkled (W). Suppose a plant has a heterogeneous gene that contains both the characteristic S and W. Find the probability of having a heterogeneous offspring (SW) or a homogeneous offspring (SS or WW) in the first generation, second genration, third generation, ….
- Choose any week of your ongoing semester. Collect data for the past 10 – 15 years for the amount of rainfall received in Delhi during that week. Predict amount of rainfall for the current year.
- Gambler’s fallacy
- Stock price movement using the Binomial Distribution
- Weather prediction (prediction of monsoon from past data)
- Risk assessments by insurance firms from data
- Predicting stock market crash
- Predicting outcome of election – exit polls
- Predicting mortality of infants
- Studying various games that use the concept of Probability – Lotto, Throwing dice, Khul Ja Sim Sim, etc
- Data from iterative map for population growth – dynamics from plots
- Statistical analysis of alphabets/ words appearing in a given text
- Statistical methods for drug testing
- Modelling population growth through data
- Modelling spread of disease through data
- Validation of existing models through data
5. Reading List
Berlinghoff, W. P., Grant, K. E., & Skrien, D. (2001). A Mathematics Sampler: Topics for the Liberal Arts (4 ed.). Ardsley House.
Maki, D. P. (2006). Mathematical Modeling and Computer Simulation (1 ed.). Thomsons Brooke Cole.
Parks, H. M. (2007). A Mathematical View of Our World (1 ed.). Thomsons Brooks Cole.
Staszkow, R., & Bradshaw, R. (2004). The Mathematical Palette (3 ed.). Thomsons Brooks Cole.
Tannenbaum, P. (2010). Excursions in Modern Mathematics. Pearson.