John M.'s Blog at WyzAnt.com

Online Calculators and tools

posted by WyzAnt tutor: John M.

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 assume that the 4 multiples the x, the cosine of which is then calculated and then multiplied by 5. But for these calculators, this order of operations requires all of the relationships spelled out. So it would be 5*cos(4*x) instead.

Good calculating!
John

Using current events to explain statistical ideas

posted by WyzAnt tutor: John M.

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, we would need to undertake a census “the collection of data from every member of the population.” (Triola, 2010, p. 4). If the population is large, this would be a daunting and expensive task. Instead, statistics can be used (1) to select a sample of the population that is representative of the population, (2) to collect data about the sample which is summarized in a statistic, and (3) to determine how to extrapolate the population’s parameter from the sample’s statistic. (Sullivan, 2010; Triola, 2010).

But often people (including individuals who should know better) make a critical error by focusing only on the second step because it seems to be what we want to find out, usually as quickly as possible. So for the example of the day, we want to know how people voted so we know the result right away. It seems to make sense that the easiest way to find the result would be use an exit poll, i.e. ask a subset of the voters (a sample) how they voted and calculate a summary of how they voted (a statistic) and use that statistic to estimate the parameter how all the voters (the population) voted. But over-focusing on the second step, creates real problems, because sample selection is crucial either to ensure the sample is representative of the population (step 1) or to ensure the mathematical reliability of extrapolating the parameter from the statistic (step 3).

For exit polls in particular, steps 1 and 3 are difficult. First, representativeness can involve a catch-22: we need to know what the population is like to pick a good sample, but we want to pick a good sample to find out what the population is like. This issue often appears in the press as individuals complaining about the make-up of the survey, e.g. complaining that the sample included too many individuals reporting a particular political affiliation. But it is possible to substitute a random sample for a truly representative sample, and utilize mathematical techniques to show that a statistic from a random sample would be sufficiently reliable (using step 3). Second, unfortunately human beings are really bad at true “randomness” and thus our samples are usually much less random than we think. For something like an exit poll, people volunteer to answer, and so even if a surveyor were to “randomly” select voters as they leave the polls, the only data collected would be about people who “conveniently” choose to answer the questions. Additionally, other factors can interfere with randomness such as when people vote or whether they vote in person or absentee. So, the extent our sample deviates from randomness or representativeness diminishes the reliability of the statistic to estimate the parameter.

To be fair, most statisticians understand these problems, but when the statistics are presented for wider circulation, these problems are ignored, de-emphasized, or misunderstood. Clarifying these issues is often what the first chapter of a good statistics class is all about, so that future users of statistics can be aware of these potential problems. Exit polls in particular are particularly vulnerable to have their problems exposed because they are always followed by an actual census. After all, the vote counting is what we are trying to predict using the exit poll, so that an estimate of that count would be less accurate isn’t surprising, and all the more so given the difficulties of randomness and representativeness.

Works Cited

Sullivan, Michael (2010). Statistics: Informed Decisions Using Data (3rd ed.). Prentice Hall: Upper Saddle River, NJ.

Triola, Mario (2010). Elementary Statistics (11th ed.). Addison-Wesley: Boston, MA

Science problems as Math problems

posted by WyzAnt tutor: John M.

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 the phenomenon in order to produce testable hypotheses. The further you go in math or science, this really does not change. I have another student whose assigned calculus problems are as much applied engineering problems where the demonstrated calculus would be used to solve them.

While the theoretical relationship between physics and math is interesting, the key point to remember is that a student should present the work in a way that demonstrates their knowledge of what the course is trying to teach. So for a physics class, the first goal is to show that you know the fundamental physics concept (e.g. F=ma, or conservation of momentum or energy), and then show the details of your math work, so that if you make a mistake, the teacher knows that you understood the physics, but you made a calculation error, or couldn't complete the answer because of time. Conversely in calculus, the focus needs to be that you show you know how the math works, and can go through the steps to solve a problem, even if you've misinterpreted or misapplied the underlying physical/engineering equation.

Lies, BLEEP lies, and Statistics

posted by WyzAnt tutor: John M.

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, 2 has 47, 3 has 52, 4 has 33, …, and n has z. An average then is the sum of all the individual rates (Sum) divided by the count of individuals (Count). So, given the numbers above and n=5 and the average is 45, I can calculate z. Sum is equal 43+47+52+33+n, the count is equal to 5, and the average is equal to 45, so the Sum is equal to 5x45=225 and 225-43-47-52-33=50=z.

Notice, however, some properties of the average. First of all, unless all the sample values are equal, there is always a smallest and largest number, and the average is always between them. I can always rearrange the samples in order from smallest to largest (e.g. 33, 43, 47, 50, 52). And no matter how many 33's I add to the sample, as long as there is one number greater than 33, the average will always be between 33 and the largest number, and similarly, no matter how many 52's I add to the sample, the average will always be between 52 and the smallest number. Thus, by definition, it is impossible for everyone to be above or below average. Thus, qualitatively, being above average, at average, or below average, is not of itself good or bad, it is merely a mathematical way to compare an individual sample value against a value that mathematically summarizes a central point of all the sample values.

No sooner do I work on this, than the local paper editorializes indicating a complete lack of understanding of this very principle. The complaint was that Wisconsin's reported drunk driving rate was over 20% (i.e. 1 out of 5 people surveyed admitted to driving while intoxicated in the last year) and that this was the highest such rate among all 50 states in the U.S. Now, to be clear, a 20% rate is qualitatively scary, but out of 50 states, some state has to have the highest rate, so by definition it can say nothing additionally worse to say that the 20% rate was the worst among the 50 states, because some rate was going to be worst. The 20% rate only means something in comparison to the other values, or their summary, the mean. After all, how you would evaluate that result if the sample of the 50 states reported values between 18 and 20 versus if the reported values were between 5 and 20 percent? In the first group, it would mean very little to say that 20% was the worst score because at worst it is only slightly higher than its mean (at least 18%), whereas in the second group, the difference could be much larger, and by implication more improvement was possible. Thus, the 20% value only provides additional information in the context of the average, and other measures of how the data is spread out.

Tricky test makers and their tricks

posted by WyzAnt tutor: John M.

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 easy answer, as signs that the test maker may have set a trap for you. Even under time pressure spending 15-20 seconds analyzing the problem to ensure that you know (1) what the exact question is, (2) how to order the information correctly in order to solve the problem, and (3) to determine if there is a trick that makes the problem easier can be a good way to improve your overall score and separate yourself into a higher percentile result with little additional effort on your part, except to stop and think.

Training your intuition

posted by WyzAnt tutor: John M.

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 confusing tricks to render our intuition useless. Thus, one of the most important skills a tutor can provide is to help "train" a student's intuition to help identify when an answer might be incorrect and why. Take a statistics/algebra problem. Suppose you are asked about the frequencies, mean and standard deviation a 250 individual results on a test (graded from 0-100) and what those numbers suggest about how the individuals performed from a population of 10,000 students who took the exam. Your student does their math but makes a mistake and determining that the mean score was 75 and the standard deviation was about 5 points, and reports frequency data shows that there were about 25 scores less than 60 and 25 scores above 90. At first blush, this might makes sense as the low scores "cancel out" the high scores and so a mean of 75 might be reasonable.

But these results conflict with some basic rules about how samples should behave that need to become intuitive. First, the central limit theorem implies that any sampling distribution to find its mean, no matter how unusual, will become approximately more normal (i.e. bell curve shaped) as the sample size increases (Sullivan, 2010). Second, a normal distribution is bell shaped, with the mean at its highest peak, and empirically with 68% of the data within +/- 1 standard deviation of the mean, 95% of the data within +/- 2 standard deviations, and 99.7% of the data within +/- 3 standard deviations (Sullivan, 2010). Third, beyond empirical data, Chebyshev's Inequality requires --- as a matter of the mathematics of the mean and standard deviation --- that any distribution no matter how skewed, requires that at least 75% of all data lie within 2 standard deviations of the mean, and 88.9% of all the data lie within 3 standard deviations of the mean (Sullivan, 2010).

Using these ideas, we would expect the sample above to be normally shaped, and thus that almost all the data (99.7%) should be between 75 +/- 15 (3x5) or between 60 and 90. If 20% of the data is outside that range (10% less than 60 and 10% above 90) we should be suspicious. Furthermore, mathematically if the mean is 75 and the standard deviation is 5, almost at least 88.9% of the data must be between 60 & 90. But here we have frequencies that indicate that 20% of the data is outside that range, i.e. that only 80% of the data is within that range. So an empirical suspicion that something is wrong, must be certainly wrong.

Thus, once our intuition is trained to look for such anomalies, we can look back over our work and ask where the mistake was made. Clearly, the standard deviation is too small to be correct, thus, we should focus our error checking on how we might have made it smaller: did we divide instead of multiply somewhere? did we subtract when we should have added? or Did we transpose digits making numbers smaller? etc. Furthermore, because the standard deviation is based upon the mean, an error in calculating the mean might have influenced this result as well.

In helping students, it is just as important to help them know when they are making a mistake on their own as when they have calculated the right answer. Today with the rise of computer calculations often people only see the end results and treat them as "Deus et Machina," God by machine, and assume the numbers must be right. An important part of reviewing numbers is to humanly understand, intuitively, why those numbers make sense beyond the fact that the computer produced the result, and that is what students need to succeed on their own during exams, particularly standardized tests like the ACT, SAT, MCAT, etc.

Tutoring boys, tutoring girls

posted by WyzAnt tutor: John M.

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 can be left behind especially in math and science. Often girls are focused on the relational aspects of problems: who will be harmed, who will benefit, etc. (Meehan, 2007). As a result, they often pause before asking questions, and when they ask questions they are often more interested in the relationships rather than the mathematical or scientific issue raised by the question (Meehan, 2007). The advantage of an all-female-student environment is that it can take advantage of the different female learning style and teachers can adapt their methods in math and science courses towards this relational approach (Meehan, 2007). In fact, Meehan describes how, when an all-female school is unavailable, how to encourage coeducational schools to consider how to adapt their teaching to the needs of their female students.

Thus, as a tutor who emphasizes in teaching math, science and test-taking for both boys and girls, it is important to consider cultural and sexual differences in the way my individual students learn effectively. In my experience, professional educators, often because of having too many students per class and with limited instruction time, cannot accommodate all their students' individual differences effectively. So parents come to me to help their children in a particular field, but the main need for their children is tutoring designed more towards their individual, cultural, or sexual learning styles. Not only does this help my students with their current academic struggles, but in the long run it can help them identify how to learn more effectively on their own.

References

Troller, S. (2010, November 12). "Why charter school for African-American boys needed." The Capital Times. Retrieved from www.madison.com & currently available at http://host.madison.com/ct/news/local/education/local_schools/article_740aed70-edd1-11df-9e8d-001cc4c03286.html

Meehan, D. (2007). Learning like a girl: educating our daughters in schools of their own. New York: Perseus.

Defining Variables

posted by WyzAnt tutor: John M.

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 oz. of cheese and they need 2oz of crackers for every 1oz. of cheese, so how many people can I serve cheese to, and how many more crackers do I need for all my cheese? Before scrolling down, think about how you would solve this in real life and then think about how you would work this problem for class.

I would begin to define y as the number of people and z as the additional amount of crackers in ounces that I need to buy, i.e. the two questions being asked that I do not know. Then I would link those variables to what I do know. I know that for every person y, they will eat 4oz of cheese, and that should equal the total amount of cheese (24oz), i.e. 4y = 24. I also know that for each of the 24oz's of cheese, I need 2oz of crackers. I also know that the total amount of crackers for that cheese is the 28oz I have plus the z crackers I need to buy. Thus, 24 x 2 = 28 + z. Solving these equations, I find I have enough cheese for y = 6 people, and that I need z = 20 oz. of crackers.

While your teachers may try to complicate things with unneeded information, additional variables or other mathematical operators, most problems involve identifying what you know (the amounts of cheese and crackers you have and the rates at which those items are needed), defining as variables what you need to find out, and preparing equations that relate what you know to the variables you don't know.