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Online Calculators and tools

I have found that while student graphing calculators have become powerful tools not only for advanced math, but for determining derivatives (and integrals!), having those tools available where the results can be put up on a larger screen, or where multiple lines can be displayed in color. I recently came across http://www.numberempire.com/linktous.php where there is a collection of such calculators. While neither WyzAnt or I can guarantee all the calculators will (or will continue to) work as the site describes, I have recently been using the definite integral and graphing calculator, which produced nice, clear & colorful results, which helped clarify a problem we were having with an otherwise very functional black-and-white graphing calculator. A word to the wise, the online calculators typically require very specific methods of entering functions. For example these calculators often require extra parentheses () or symbols. For example, if I wrote 5 cos 4x, you would... read more

Using current events to explain statistical ideas

It is often examples that make ideas understandable to students and current events can be a good source of examples. Case in point. Today in Wisconsin, the issue of the day is the outcome of the recall elections and problems with the exit polling. As a tutor, the outcome isn’t interesting, but exit polling like all surveys is key to the usefulness of statistics! In fact, it gives a great opportunity to illustrate some of the basic (and non-mathematical) ideas and concepts of statistics — usually the ideas presented at the beginning of most introduction-to-statistics courses. Statistical inferences are grounded in some basic definitions and assumptions (in bold). A population is a defined collection of individuals that we want to know some data about and a sample is a group taken from the population that we are going to actually collect data from (Sullivan, 2010, p. 5; Triola, 2010, p. 4). If we wanted to know the actual data about a population, which is called a parameter,... read more

Science problems as Math problems

I've been working recently with a student who "presented" as a student struggling with physics. But in many ways, the physics is less of an issue than applying mathematics to the physics concepts. Their text is Glencoe's Physics Principles and Problems, which some reviewers describe as much as a math text as a science text. After helping with several chapters of homework, I would say that the problems at the end of each section or chapter tend to focus on those where mathematics can be applied to the physics, and that as a result if the tests are based upon those questions, the test will be as much a test of the student's understanding of the mathematics as their knowledge of the science. In many ways, it was the time consuming nature of the mathematics that was creating problems for the student. This is not an uncommon phenomenon in that the development of math allows for new models of science, or that scientific theories require mathematical descriptions of... read more

Lies, BLEEP lies, and Statistics

Mark Twain is often credited as saying that there are "lies, damn lies, and statistics" but as someone who tutors in statistics, I see it more as there are people who tell lies and lies with statistics. Statistics themselves are only numbers, and while calculations can be mistaken, the wrong formulas can be used and yes numbers can be used to mislead people; the numbers themselves do not lie. The problem for most people with statistics is that it is an unusual way to think about and manipulate numbers. This week, I have been helping a student better understand the implications of an average of a sample, also referred to as the mean in order to prepare for an upcoming standardized test. Generally, a sample consists of individuals 1, 2, 3, …, n, who each have some numerical characteristic x1, x2, x3, …, xn. For example, a sample of individual's resting heart rate (measured in beats per minute, bpm) could be as follows: Individual 1 has a rate of 43 bpm,... read more

Tricky test makers and their tricks

Particularly with regards to standardized tests, the test makers are not only testing for competency, which could be tested with straightforward questions, but are also testing for excellence, to separate out each test taker by their percentile rank, and to sort among percentiles the difference between Top 20%, Top 10%, Top 5%, Top 1%, and even to smaller units. Thus, test makers often use not only the problem, but they often use a non-standard or non-intuitive presentation of the information and often deliberately pick potential answers that either mislead the test taker or that confirm a common mistake that test takers might make. But this effort is also a test taker's biggest advantage and opportunity. An ambush is not an ambush if the person being ambushed knows about it ahead of time. Thus, when preparing for and taking standardized tests, the test taker should be on the lookout for non-standard presentations of questions, and be weary of selecting an obvious or... read more

Training your intuition

For most people, solving a problem or a question is not difficult if they have a model to follow and the correct data to plug into the model. Take one of the most basic functions, paying for something at a cash register. If the cashier tells you the Happy Meal costs (with tax) $4.23, and you hand the cashier a $10.00 bill, I suspect that most cashiers will give and most people will expect their $5.77 in change. Oh, you can confuse people and make the problem more difficult (7 dimes, a nickel and two pennies, rather than 3 quarters and two pennies), but these are just "tricks." This works, because for the vast majority of people, this is an "ordinary" occurrence something we've either done or witnessed hundreds of times, and we can intuitively extend our addition and subtraction rules to a new problem. Unfortunately, most classroom topics are taught like the math example above using clear, intuitive, and easily understood examples, but tested using... read more

Tutoring boys, tutoring girls

A growing area of educational thought is reconsidering the pros and cons of single-sex education, i.e. all-male or all-female schools. In Madison this week, the president and CEO of the Madison Urban League proposed opening an all-male school for grades 6-9 aimed at African-American students (Troller, 2010). The hope is that such a school can take advantage of the ways that young men learn differently from young women and provide dedicated adult-male support that young men who often lack such support need (Troller, 2010). Obviously, this idea raises issues about the history of racial segregation, the fight for integrated schools, and the challenge of civil rights for people of all races in the United States. But single-sex education is not solely a desire for more all-male schools, rather there is a movement towards creating more all-female schools as well (Meehan, 2007). As Meehan (2007) observes, girls behave and learn differently in the classroom, and as a result... read more

Defining Variables

Often students are confused with variables: those x's, y's, z's and other letters that begin to replace numbers beginning in algebra and continuing on into geometry, algebra II, Trigonometry, Precalculus, and Calculus. While there are several aspects to variables, one of the best ways to start is to understand a variable as a placeholder for a number. Take the equation: 7 + 3 = 10. Inserting x into this equation for the number 3 produces the "same" equation 7 + x = 10. Yet this equation also shows how a variable is "defined" in that x must equal 3 for the equation to work. More often variables are defined, however, not by a specific number, but with an idea that covers either a to-be-determined range of numbers or an as yet unknown number. Try this example. Suppose I go to the store planning to put a cheese and cracker tray together for an upcoming football game. I have 24oz. of cheese and 28oz. of crackers and I expect each person to eat 4... read more

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