Nature, Motivation and Teaching Methodology for Engineering Problem Solving and Computer Programming

Transcription

1 Nature, Motivation and Teaching Methodology for Engineering Problem Solving and Computer Programming K. Ming Leung Department of Computer Science and Engineering Polytechnic School of Engineering, NYU K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

2 Why Use Matlab? Five years ago, CS1133 replaced CS1114 as the first computer course for engineering students. It assumes the students have no prior programming experience. It uses Matlab, a computer programming language that is easier to use to write and debug programs. It is especially good for testing ideas and for rapid prototyping. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

3 Testing New Ideas Consider this problem: for the floating-point numbers between 0 and 1, what is the proportion of them that do not give 1 when they are multiplied by their own reciprocal? I derived a result for a tight upper bound: (1 ln(2))/2. This result can be tested easily in Matlab within seconds. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

4 Why Use Matlab? (cont.) Producing graphs and other visualizations are handy for interpretation and analysis of results in engineering applications. From quick and simple, to sophisticated publication-quality graphics can be produced. For example, how does the function f (x) = cos(x)exp( 0.3x)) look like from 0 to 15? Matlab has thousands of built-in rather general functions, as well as toolboxes for numerous specialized scientific and engineering applications. Thus Matlab is useful also for upper-level courses in many disciplines (such as solving di erential equations in Mechanical Engineering, processing signal and images in Electrical Engineering). K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

5 The Sunflower Head The arrangement of seeds (florets) on the head of a sunflower can be described mathematically in polar coordinates: 1 r k = p k, k = k, for k =1, 2,...,n, p with the divergent angle = (3 5) ( ), the golden angle. The resulting pattern produces the most e the flower head. How well does this construction work? cient packing of seeds within How sensitive is the dependence on the value of the divergent angle? It turns out the divergent angle cannot deviate by more than 0.04%. 1 Helmut Vogel, A Better Way to Construct the Sunflower Head, Mathematical Bioscience, 44, 179(1979). K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

6 Increasing the divergent angle by 0.04% K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

7 Emphasis on Real Problem-Solving At least 45 percent of undergraduates demonstrated no improvement in critical thinking, complex reasoning, and writing skills in the first two years of college percent showed no progress in four years. 2 Richard Arum and Josipa Roksa, Academically Adrift: Limited Learning on College Campuses, UniversityofChicagoPress,2011 K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

8 Weak Mathematical and Problem-Solving Skills The American work force has some of weakest mathematical and problem-solving skills in the developed world. In a recent survey by a global policy organization, 3 adults in the United States scored far below average and better than only two of 12 other developed comparison countries, Italy and Spain. Worse still, the United States is losing ground in worker training to countries in Europe and Asia whose schools are not just superior to ours but getting steadily better. 3 Organization for Economic Cooperation and Development from K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

9 Problem with Real Problem-Solving The 2003 National Assessment of Adult Literacy is a nationally representative assessment of English literacy among American adults age 16 and older. The study, sponsored by the National Center for Education Statistics, 4 is the nation s most comprehensive measure of adult literacy since the 1992 National Adult Literacy Survey. Over 19,000 adults were asked to answer the following two problems: 4 National Center for Education Statistics website: K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

10 Question 1: Refer to the advertisement for the Carpet Store on page three of the newspaper to answer the following question. Suppose that you purchase DuPont Stainmaster carpet at the sale price. Using the calculator, compute how much in dollars and cents you would save per square yard over the regular price, excluding tax and labor. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

11 Question 2: Suppose that you want to carpet your living room which is 9 feet by 12 feet, and you purchase DuPont Stainmaster carpet at the sale price. Using the calculator, compute the total cost, excluding tax and labor, of exactly enough carpet to cover your living room floor. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

12 Problem with Real Problem-Solving (cont.) It was found that only 62.9% of the adults answered the first question correctly, and only 17.6% correctly answered the second question. The same questions were asked in a study conducted in 1992, with basically the same outcome. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

13 Types of Problems in CS1133 Only word problems are used in CS1133. The following types of problems are not used matching problems true-or-false problems fill-in-the-blank problems multiple-choice problems Most the CS1133 problems are real-world problems. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

14 General Guiding Principles Whenever there is a freedom of choice, exercise ones choice wisely. Computer programs need to be modified and updated throughout its life history as newer versions. They must be written as flexible as possible (no hardwiring). A programmer should not do anything that can be done by a computer. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

15 General Problem-Solving Skills Emphasized General problem-solving skills that have wide applications beyond computer programming are emphasized: Read problem statement very carefully. Know what the given quantities are, the assumptions, and the form and type of output results. If feasible, examine all possible solutions. Try working forward from the given quantities and backward from the required output results. Hand-check each step in the solution of the problem. Solve a simplified version of the given problem. Make up a simplified version that one knows how to solve. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

16 Example: while the sum is less than 1 Conduct the following computer experiment numerous times: Each time repeatedly pick a random number from 0 to 1 until the sum is no longer less than 1. Find the average number of random numbers needed. Find the average sum. 2 n = 3 n = 2 n = 2 n = 4 n = Stone slabs of random thicknesses are used to build a fence whose height must always be greater than 1 meter. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

17 Example: Closer to the center Find the fraction of points inside a square that lie closer to its center than to any one of its edges y x A square castle surrounded by water in the middle of a lake. To help the people who lives near the center of the castle, a well of a certain depth is to be dug. To know how deep to dig, we need an estimation of the percentage of people who live near the well. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

18 Example: Evaluate (1 x) 6 for a given x Use various di erent values of x and evaluate the following equivalent expressions: 1 x 6 6x x 4 20x x 2 6x +1 2 (((((x 6)x + 15)x 20)x + 15)x 6)x +1 3 (1 x) 6 Do they all give the same result? Check the case where x is very close to 1. If the three answers do not agree, which one do you trust? Any polynomial can be written in the two equivalent forms as shown in expressions 1 and 2. In general a polynomial is not expected to have a compact factorized form as shown in expression 3. K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

19 Example: Fraction of points within a hypercube that lies near the edge? Given a unit hypercube (a unit cube in an n-dimensional space), find the fraction of the points that lies near its edge (within a distance of 0.01 from a boundary). What happens as the dimension of space increases? What does the result tell you about high-dimensional space? K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

20 Example: Average distance between 2 arbitrary points in a hypercube Find the average distance between two arbitrarily chosen points inside a hypercube. What happens as the dimension of space increases? What does the result tell you about high-dimensional space? K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23

21 Example: Angle between 2 randomly chosen vectors in a hypercube Find the angles between pairs of arbitrarily chosen vectors lying inside a hypercube. Try 20 pairs. What happens as the dimension of space increases? What does the result tell you about high-dimensional space? K. Ming Leung (Department of Computer Science and EngineeringPolytechnic CS1133 School of Engineering, NYU) / 23