Math and Science In Your Life

Science, Math and Technology Resources at Empire State College

Archive for the ‘Math’ Category

Statistics and Data Analysis

On-line Resources: Statistics and Data Analysis

Data, data, everywhere…

As students, and as citizens, we are bombarded with statistical data; polls, reports, economic data, and on and on. You need a solid understanding of how information and knowledge are extracted from data to be successful as a student, and to participate fully in the modern world.

Of Special Note: Empire State College offers a newly redesigned and updated statistics course called Statistics: An Activity Based Approach. You can learn more about this course by visiting www.esc.edu.

Spotlight
StatCrunch screen shot StatCrunch

StatCrunch is a fully on-line data analysis toolkit with features comparable to expensive desktop statistics and data analysis software. StatCrunch allows you to upload data files, apply statistical tests, and create plots of various types.

Probably the best way to get a feel for what you can do with StatCrunch is to watch this demonstration/tutorial video from the folks who created the software.


Quantitative Literacy

On-line Resources: Quantitative Literacy

You may or may not like math, and you may or may not believe that you are good at math. But every educated person, and every fully-functioning citizen, must be able to solve problems and answer questions that boil down to data and numbers. The basic skills that support this are called quantitative literacy and those skills include; the ability to find, organize, and analyze data, the ability to understand and work with basic statistical techniques, and the ability to work with large numbers using estimation. We’re not talking about rocket science (or mathematics) here, it’s all about practical math skills for everyday use.

For example, what does it mean when a political poll is said to have a margin of error of three points. Does this guarantee that the forecast made by the poll is within three percentage points of what the entire population believes? No, it does guarantee that. Political polling makes use statistical methods that produce results with a known accuracy -most of the time. But polling techniques only work if the sample on which the poll is based is truly representative of the entire population. And it is surprisingly easy for bias to slip into the sample selection process. When this occurs the “margin of error” is meaningless and several recent high-profile polling failures have been attributed to poor sampling techniques.

Furthermore, the “margin of error” number that is commonly cited along with polling results is really only half the story. Statistical techniques produce a margin of error at some “level of certainty”, and that level of certainty is not 100%. If the level of certainty for a poll is not stated you can assume that it is 95% (standard practice for most political poles, but beware!). So a full statement of poll results should be something like, “at a 95% level of certainty the attribute reported in the poll has a margin of error of 3 points”. Even when the sample is unbiased, standard polling techniques will produce invalid results every once in a while based simply on chance (at the 95% level that once in a while is one time in 20). When chance conspires to produce a non-representative sample the result obtained through the poll can be misaligned with reality by any amount. There is no limit on how “far off” the result might be.

Spotlight
quantitative literacy thumbnail   Quantitative Reasoning and the Environment
This book, spanning under 200 pages, covers the topics that an environmental science student might use in a range of contexts. That content also happens to coincide closely with the body of knowledge that might call quantitative literacy.

Mathematical Models for Mere Mortals

Mathematical Models for Mere Mortals

These days, the use of mathematical models to simulate complex systems, and to forecast future events, is an everyday idea. Perhaps you’ve heard your TV weather reporter say something like, “the models show the storm passing to the north of us, but we’ll have to wait and see.” Or maybe you’ve heard of the models that economists use to forecast future trends in the economy (sometimes incorrectly). If you’ve read anything at all about climate change, you’ve no doubt heard about the climate models that are used to simulate the effect of increased levels of carbon dioxide in the atmosphere.
Mathematical models are idealizations of reality, and modeling lies at the very heart of modern science, engineering, and economics. When we talk about using math to better understand the world around us, at some level we are talking about building and using mathematical models.
Of Special Note: For students with a background in college math, The Empire State College Course, Math Modeling, is for students ready to go beyond the basics and learn about the models used in science and economics.

Spotlight
EdGCM Logo   EdGCM -
Climate Modeling Made Easy(ier)
The EdGCM Climate Model is a remarkable tool. You can download and run this research quality climate model developed jointly by NASA and Columbia University. But even with the extensive support provided on the EdGCM web site, getting started and understanding how the model works can be a challenge. We’ve created several tutorials designed specifically to help interested adults and adult students get started. Access our introductory information
here
.

Maple Resources for Empire State College Students

Maple Resources for Empire State College
Students *

Students at Empire State College use Maple (math software) in on-line math courses and we are able to provide Maple to students at reduced cost. In addition, we are using a Maple technology called MapleNet to provide access to interactive, math-rich, content through a web browser interface.

Students taking courses that require Maple will receive instructions for obtaining Maple as part of their course materials and should order Maple from the bookstore to obtain the software at very low cost. Students not taking math courses can also contact the Empire State College bookstore for information on obtaining Maple from the College. The links below provide access to resources designed specifically to assist students using Maple in on-line courses.

https://commons.esc.edu/cdlmaplecenter/

http://www.maplesoft.com/


Discrete Math

On-line Resources: Discrete Math

Think about our world and name some things that can be counted; the number of pixels in an image, the votes cast in an election, the number of fingers on your hand, the number of different ways in which a network can be traversed. From a mathematical perspective, in each of these examples we are talking about sets of discrete objects. The study of discrete objects and how they can be counted is fundamental to the study of discrete mathematics.

Now consider things that we measure; the speed of a cannon ball when it leaves the cannon, the distance from the earth to moon, the volume of blood in your body. In these cases the results of the measurement -the number that you get- depends on the accuracy and the precision of the measurement. The speed of a car could be 35 mph, 35.1 mph, 35.111 mph,and so on. Given a better measuring device you could always add another digit of precision to your measurement. For this reason we say that measurement usually gives us a “continuous” result. In terms of our study of mathematics, algebra and calculus are among the mathematical tools that we use to analyze and understand questions that involve continuous quantities.

Discrete math is also commonly associated with computers. That’s because, believe it or not, computers cannot handle those messy imprecise numbers that go along with continuous math. Sure you probably use your computer to work with numbers containing decimal points all the time, but at the lowest level your computer sees 31.111 as five integers with a decimal point after the first two. That’s a discrete quantity. The messy part comes in when we use our computers to do calculations with irrational numbers like Pi. As you probably know, Pi cannot be defined as a exact value; the decimal places go on forever. When a computer does math with numbers like that it has to decide how many digits of precision will be used for each calculation. And when lots of these “rounded off”numbers come together in some sort of calculation problems can occur.

Another area where discrete math is used is in the definition of the recipes that are used to write computer programs; called algorithms.Students who pursue degrees in information technology or computer science are usually required to take one or more courses in discrete mathematics.

Spotlight
image processing example   Image Analysis Using Maple may sound like an arcane topic, but if you own a digital camera you already have an advanced digital image analysis system. Image analysis is central to how digital imaging works. Some cameras even give you access to some of the image processing features allowing you to shift the contrast, the color balance, or the luminosity of an already captured image.

For Empire State College students studying discrete math, and using Maple math software, you can apply these sorts of image transforms using Maple. Maple provides the basic image processing functions found in your camera and more advanced ones not typically found in graphics software.

You can download a worksheet from the Maple Applications Center (free registration required) that demonstrates these features. On the Application Center home page search for “image” and select “image tools” from the list. You can download the worksheet for a demonstration of the different ways that Maple can be used to manipulate images.


Geometry

On-line Resources: Geometry

As a field of mathematical study, geometry sometimes seems a bit dusty. After all, the basic rules of geometry, at least the Euclidean kind you learn about in school, were written down over 2000 years ago by, you guessed it, the Greek mathematician Euclid .

But geometry is alive and well. It is through the study of geometry that most students are introduced to the idea of a mathematical proof, one of the doorways to higher math and indeed to western thinking as a whole.

Furthermore, if we gaze beyond the Euclidean plane with its points and lines – and use different sets of axioms (basic laws assumed to be true)- entirely new vistas appear with applications across the full range of science. Turns out, Euclid was really on to something.

Spotlight
Geogebra screen shot   GeoGebraIt’s hard to convey how useful this program is for learning math. Geogebra maintains a dynamic connection between plots and geometric constructions on the one hand and the coordinates and functions that define those constructions on the other. You can manipulate a drawing and the algebra changes and you can also go the other way.

Algebra/Pre-Calc/Calculus

On-line Resources: Algebra/Pre-Calc/Calculus

In the minds of many, calculus is college mathematics. Using calculus we can calculate exactly how much force to apply to send a spacecraft to mars, or the rate at which a medication will be absorbed into the bloodstream. Calculus connects mathematics to our natural world.

You might be surprised to learn that all of calculus is built on three not so difficult concepts: limits, rates of change, and areas under a curve. As one well-known mathematics professor has put it, “A typical calculus textbook has 5 pages to explain the concepts and 1200 pages of examples.”

Of Special Note: For students with a background in college math, The Empire State College Course, Math Modeling, is for students ready to go beyond the basics and learn about the math models used in science and economics.

Spotlight
golden ratio thumb   Numb3rs
The popular TV series Numbers has spawned several web sites that provide background and classroom activities tied the mathematics featured in each episode. A group at Cornell University has taken a deeper look at some of the math. Be sure to scroll down to see the resources for season four. These pages include the use of math demonstrations created specifically for the show at the Wolfram Demonstrations web site.

Math

On-line Resources: Math

The Internet is remaking the way that mathematics is shared, taught, and learned. Need to differentiate an equation? A half dozen web sites provide tools that can do that for you. Need information on an obscure mathematical term? In many cases a search will find more information than you could read in a week. Using what is available wisely is the key.

Spotlight
Google Docs Spreadsheet   Spreadsheet TutorialsAs a college student you will find many uses for a spreadsheet. Spending a little time improving your understanding and skills of how spreadsheets work can provide big dividends. To help you make effective use of your spreadsheet program we’ve provided tutorials that demonstrate a wide range of basic techniques. From working with data copied from web pages, to creating histograms, our tutorials show you just what you need to know without a lot of extra information.

Spreadsheet Tutorials – Techniques for Students

The Empire State College technology helpdesk provides some support for Excel spreadsheet software, and students can find training and assistance directly from Microsoft at http://office.microsoft.com/en-us/excel-help/ . Unless the course requires Excel, students may generally use other spreadsheet software to solve problems, such as the one from OpenOffice http://www.openoffice.org/  or other software products.