I have been teaching at the post-secondary level since 1987, at schools ranging from a small liberal arts college in North Dakota to the second-largest non-profit university in the country. I’ve taught at private schools and public ones, and courses ranging from first year undergraduate to doctoral students. In all of those situations, some students aced the courses and some students failed. The difference between those students was NOT as some might believe, that the students with A’s had some sort of magical math gene the others didn’t. Nope. Here are seven tips how not to fail a college math class.
- Have the textbook when the class starts. Textbooks are required for a reason. That reason is primarily that the instructor does not have the time to tell you in the lecture everything that might be useful. Every course I have taught, at least one student tells me that he or she does not have the textbook yet. This makes me wonder, “Did you not know you were going to take this course?” , because I am pretty certain that I told the university the book that would be required two months ago. Even if you have an excellent reason for not having the textbook, falling a week behind in the reading makes the class more difficult.
- Read the assigned readings. You are supposed to read them. That is what “assigned” means. See #1. Also, some of the stuff you learn might not be so easy. This is why it is good to go over it twice, once in the lecture and once by reading it.
- Attend all of the lectures. It can’t hurt. See #2. Very few professors are so terrible that you cannot learn anything from them. If you think the professor is difficult to understand, perhaps it is because you did not read the assigned readings before the class so this is the first time you have been exposed to this material. Maybe you missed the last lecture where he or she explained the information that is PREREQUISITE to understanding the information covered in this lecture.
- If you still don’t understand, read the textbook again. I was an excellent student in statistics. It is what I specialized in for my Ph.D. (along with Tests & Measurement). The only statistics courses I did not get an A in, I got an A+. And still … there were many times when I read the textbook, thought I understood it, tried the problems at the end of the chapter and realized I didn’t understand it so well after all. So, I read the chapter again. Sometimes for a third time.
- Don’t try to cram at the last minute. Math builds on itself. If you did not understand chapter two, you are going to have a hard time with chapter three. If you just read it for the first time at 3 a.m. the night before the final exam, I’m guessing you didn’t understand chapter two very well.
- Ask for help as soon as you don’t understand something. How to ask for help is a whole post in itself.
- Don’t study drunk or high. This may sound like really unnecessary advice but I see people doing it. Most often it is because they are young and stupid, so drinking and getting high is part of what they do in college. Sometimes, they have fallen behind, are stressed out about not doing well in their math classes (often due to numbers 1 through 6 above), so they have a drink or smoke a joint so they can relax a little before tackling the books. “Hey, you know what would improve my ability to estimate variance? The same substance that so impairs my ability to estimate distance that they make it illegal to use while driving!”
A common factor in the first six of these is that math is cumulative. You can have messed up on the section in a literature course on whatever it is you were supposed to learn about Jane Eyre , pick up the next assigned book, Great Expectations, and still get an A on the test on that book. (I don’t say this from personal experience, having avoided English courses like the plague, but I have witnessed it done by other people. )
So … the next time you take a math class, try the tips above and see what happens. Maybe it is hard. Maybe it takes you a lot more work than you had anticipated. That is good, because when you graduate from college you will learn that the hard stuff is what people pay you to do. You can read Jane Eyre on your own time. (Sorry, English teachers).
Shameless plug – It’s Small Business Saturday
Learn about math AND support small business
Any time you learn anything new it can be intimidating. That is true of programming as well as anything else. It may be even more true of using statistical software because you combine the uneasiness many people have about learning statistics with learning a new language.
To a statistician, this error message makes perfect sense:
ERROR: Variable BP_Status in list does not match type prescribed for this list.
but to someone new to both statistics and SAS it may be clear as mud.
Here is your problem.
The procedure you are using, PROC UNIVARIATE , PROC MEANS is designed ONLY for numeric variables. You have tried to use it for a categorical variable.
This error means you’ve used a categorical variable in a list where only numeric variables are expected. For example, bp_status is “High”, “Normal” and “Optimal”
You cannot find the mean or standard deviation of words, so your procedure has an error.
So … what do you do if you need descriptive statistics?
Go back to your PROC UNIVARIATE or PROC MEANS and delete the offending variables. Re-run it with only numeric variables.
For your categorical variables, use a PROC FREQ for a frequency distribution and/ or PROC GCHART.
I’ve been busy my whole life. Right now, I’m finishing the last week of a course I’m teaching on biostatistics, writing a lecture for a course on multivariate statistics that starts next week, fixing bugs in our next game, Fish Lake, working on a new project for free resources for teachers, and working on a final grant report. Writing this, I just remembered a couple of things I needed to do.
Driving 90 miles to take The Spoiled One back to school and then turning right around and driving 90 miles home seemed like a waste of time that I did not have. The Invisible Developer pointed out that he had work to do also on the spear fishing part of the game and that he had picked her up on Friday.
So … away we went, and since she recently got her learner’s permit, The Spoiled One drove on the freeway for the first time. This was interesting in itself, since the 101 regularly makes the list of 10 most congested freeways in America.
Not only did she get nearly two hours of practice in driving, but I also got filled in on all of the latest news on her soccer team, college fairs, the campuses she was interested in visiting and life in general. If your child is 16 and still talks to you in a civil tone for two hours straight, count yourself among a lucky minority of parents.
Having raised four daughters, I know whereof I speak.
When we got to the school, she immediately began complaining (she’s not called The Spoiled One for nothing). According to her, she is living in “hell”. (See picture below for what hell looks like. It is surprisingly more scenic than I had imagined.)
What is so infernal about her school, I asked. They make her study. Even on Sundays. There is a study hall from 7 to 9 pm and she has to walk across the yard to get to the building. Yes, like prison.
Just as she was telling me this, I saw something in front of her dorm. It was a deer! I said we should go take pictures of it and she said we’d never be able to get close enough, and besides we were wasting time. She had to get to study hall and put away her clothes and books in her dorm room. Besides, her religion teacher had told the students to stay away from the deer because coyotes track them and students who got too close could get attacked by coyotes. (You would think a nun wouldn’t just go around making shit up, now wouldn’t you? Having spent a good bit of the last twenty-five years in North Dakota, I’m justifiably skeptical of the deer-coyote-mauled prep school student triumvirate.)
Just then, the deer walked through the gate on to the baseball field and I spotted a second one in there. So, we sneaked up on them and took pictures.
That’s when it occurred to me that sometimes the best use of my time is to “waste it”. Really, what better way to spend my time than talking to my daughter and watching deer grazing as the sun sets in the mountains.
But now, I really do need to finish that lecture.
In August, I attended a class at Unite 2014 (on Unity game development) and the presenter said,
“And some of you, your code won’t run and you’ll swear you did exactly what was shown in the examples. But, of course, all of the rest of us will know that is not true.”
This perfectly describes my experience teaching. For example, the problem with the LIBNAME.
I tell students,
Do not just copy and paste the LIBNAME from a Word document into your program. Often, this will cause problems because of extra formatting codes in the word processor. You may not see the code as any different from what you typed in, but it may not work. Type your LIBNAME statement into the program.
Apparently, students believe that when I say,
Do not just copy and paste the LIBNAME statement.
either, that what I really mean is,
Sure, go ahead and copy and paste the LIBNAME statement
or, that I did mean it but that is only because I want to force them to do extra typing, or because I am so old that I am against copying and pasting as a new-fangled invention and how the hell would I know if they copied and pasted it anyway.
Then their program does not work.
Very likely, their log looks something like this:
58 LIBNAME mydata “/courses/d1234455550/c_2222/” access=readonly;
59 run ;
NOTE: Library MYDATA does not exist.
All quotation marks are not created equal.
What you see above if you look very closely is that the end quote at the end of the path for the LIBNAME statement does not exactly match the beginning quote. Therefore, your reference for your library was not
but rather, something like
/courses/d1234455550/c_2222/ access=readonly run ;
Which is not what you had in mind, and, as SAS very reasonably told you, that directory does not exist.
The simplest fix: delete the quotation marks and TYPE in quotes.
LIBNAME mydata ‘/courses/d1234455550/c_2222/’ access=readonly;
If that doesn’t work, do what I said to begin with. Erase your whole LIBNAME statement and TYPE it into the program without copying and pasting.
Contrary to appearances, I don’t just make this shit up.
Computing confidence intervals is one of the areas where beginning statistics students have the most trouble. It is not as difficult if you break it down into steps, and if you use SAS or other statistical software.
Here are the steps:
1. Compute the statistic of interest– that is mean, proportion, difference between means
2. Compute the standard error of the statistic
3. Obtain critical value. Do you have 30 or more in your sample and are you interested in the 95% confidence interval?
- If yes, multiply standard error by 1.96
- If no (fewer people), look up t-value for your sample size for .95
- If no (different alpha level) look up z-value for your alpha level
- If no (different alpha level AND less than 30) look up the t-value for your alpha level.
4. Multiply the critical value you obtained in step #3 by the standard error you obtained in #2
5. Subtract the result you obtained in step #4 from the statistic you obtained in #1 . That is your lower confidence limit.
6. Add the result you obtained in step #4 to the statistic you obtained in #1. That is your upper confidence limit.
Simplifying it with SAS
Here is a homework problem:
The following data are collected as part of a study of coffee consumption among undergraduate students. The following reflect cups per day consumed:
3 4 6 8 2 1 0 2
A. Compute the sample mean.
B. Compute the sample standard deviation.
C. Construct a 95% confidence interval
I did this in SAS as so
data coffee ;
input cups ;
proc means mean std stderr;
var cups ;
I get the follow results.
|Analysis Variable : cups|
|Mean||Std Dev||Std Error|
These results give me A and B. Now, all I need to do to compute C is find the correct critical value. I look it up and find that it is 2.365
3.25 – 2.365 * .94 = 1.03
3.25 + 2.365 * .94 = 5.47
That is my confidence interval (1.03, 5.47)
If you want to verify it, or just don’t want to do any computations at all, you can do this
Proc means clm mean stddev ;
var cups ;
You will end up with the same confidence intervals.
Prediction: At least one person who reads this won’t believe me, will run the analysis and be surprised when I am right.
A recent tweet about mixed martial arts decisions set me to thinking about probability. @Fight_ghost tweeted that a TV commentator made no sense when she said that she thought a fighter should have won by split, not unanimous decision. Others on twitter agreed with him that was a stupid comment, and asked did she think judges should say the other fighter only 2/3 won or what.
I thought it did make sense in statistical terms. Think of it this way:
The “true score” of the population in this case is the mean of what an infinite number of judges would rate a fighter’s performance. Of course, there is going to be variation around that mean. Some judges may tend to weight take downs a tiny bit more. Judges vary in their definition of a significant strike. Some judges are just going to be clueless or inattentive and give a score that is far from accurate. On the average, though, these balance out and the mean of all of those infinite judges’ scores should be the true score. Let’s say our fighter, Bob, had a true score of 27. The most common score we should see a judge give him is 27, but a 26 or 28 would not be totally unexpected. Given that the standard deviation of fight scores is low, we would be surprised to see him given a score of 25 or 29 and completely floored if he received a 24 or a 30.
Let’s say we have a second fighter, Fred. His true score is 29. The most common score we should see for him is a 29, but again, a 28 or a 30 would not be unexpected because there is variation in our sample of judges.
Here is the point … when fighters are far apart in the true score of their performance, judges should very seldom have a difference of opinion in who won. Even when Bob is scored high, for him, at 28 and Fred is scored his average of 29, Fred still wins. Let’s say the standard deviation of judge’s scores is 1. I believe it is really lower than that and I do know that the winner of a round has to get 10 points, but for ease of computation, just go with me.
For Bob to win, he must be rated at least two standard deviations above his true score (which occurs 2.5% of the time) and Fred must be rated below his true score, which occurs half the time. Since the scores for Bob and Fred are independent probabilities the probability of BOTH of these events happening is .025 x .5 = .0125
The other way for Bob to win is if Fred scores two standard deviations below his true score, which will occur 2.5% of the time AND for Bob to score above his true score. Again, the combined probability is .0125. SO …. only 2.5% of the time (.0125 + .0125) would Bob win. Since judges’ scores are independent, the probability of any one scoring it for him, causing a split decision is .025 + .025 + .025 = 7.5%
(If all three judges scored it for Bob, that would be a very, very low probability of .o25 * .025 * .025 because, again, the judges scores are assumed independent of one another. In only 0.063% of the cases would this occur. We should probably subtract that and the probability of two of them scoring it for Bob to be exact, but I have to finish grading papers tonight so we’ll just acknowledge that it is not exactly 7.5% and move on.)
Let’s go back to the fight that actually happened. I didn’t see it so I am going to take some people’s word that it was a close fight. They might be lying but let’s assume not.
In this case, Bob, who has a true score of 27, is not fighting Fred, but rather, Ignatz, who has a true score of 27.3 (with three judges, he’d get a 27, 27, 28 score). There is great overlap in Bob and Ignatz’s scores. To outscore Ignatz’s average score, Bob would need a score of 27.4 – well, a z-score of .4 occurs about 35% of the time. Half of the time Ignatz is going to score 27.3 or lower so the probability of him both having an average or below score AND Bob having a 27.4 or high score is .5 *.35 or .175. So 17.5% of the time, a judge would give Bob a higher score. Since there are three judges, the probability of ONE of them giving him a higher score would be .175 + .175 + .175 = 52.5%
There is also the small probability that it could go unanimous the other way, but that’s not really pertinent to our argument.
The point is simply this … if two fighters’ true scores are close, it is much less likely that you will see a unanimous decision than if their true scores are really far apart. The closer they are, the more that statement holds. So, no, it is not a stupid comment to say that you believe someone warranted a split decision rather than a unanimous decision. It may simply mean that you think the fighters’ were so close that you were surprised there was not any variance in favor of the only slightly better fighter.
Really, I think most people would find that a reasonable statement.
Extra credit points:
Give one reason why the Central Limit Theorem does not apply in the above scenario.
Answer this question:
Does the fact that the distribution of errors is necessarily non-symmetric in Fred’s case (cannot score above 30) negate the application of the Central Limit Theorem?
Lately, I’ve seen a lot of examples of this …
Malicious obedience is discussed on the englishstackexchange page (who even knew this existed) as
“….when people set their boss up to fail by doing exactly as he or she says even though they know in their hearts that their actions are incorrect or not optimal.”
I would add that it also includes taking zero personal responsibility. For example, let’s say you are the administrative assistant in an organization and you have been running lots of personal errands during work hours. The boss tells you that you need to stay at your desk. However, part of your job is to take the mail to the post office and in today’s mail is a major grant proposal that needs to be postmarked today. You don’t mail it and when the company loses out on a huge amount of money you protest self-righteously that you were told to stay at your desk.
In this case, as very often happens in the work place, you had two conflicting directives – one to stay at your desk and a second to take the mail to the post office.
Of two conflicting orders, you CHOSE to do the one that caused the company harm.
I have seen this sort of thing played out over and over. Never once have I seen the individual involved accept any responsibility.
An article in Infoworld gives a great way to discuss this with an employee , I quoted them here because I could not have said it better myself
“I don’t know what you think you’re going to accomplish, but what you are going to accomplish is finding yourself another position – this isn’t acceptable, and I really don’t care how good you are at loopholing policies and guidelines to prove you didn’t violate any of them. What I care about is getting the job done well, and that isn’t what you’re doing. …You’ll need the documentation because employees who act this way are brilliant at denial – both to you and to themselves. And know in advance that the odds aren’t all that good – mostly, you’re putting yourself through this to satisfy yourself that you did the right thing. “
I really don’t know what other people who are maliciously obedient are trying to accomplish. As others have written, I think they are trying to sabotage their bosses because they are unhappy in their positions. As I have said before, if you are that unhappy in a job – quit.
In my youth, I have been that pain in the ass employee who did not work up to their potential due to being unhappy for a variety of reasons – not being paid enough, not having my own office, not having an expense account, working for a boss who was technologically illiterate – you get the idea. The point is, I was at fault – yes, even in the one position where my boss was an idiot (I’ve usually been amazingly lucky when it comes to bosses, but there will always be that one).
I had taken the job at that salary, with those benefits, with that boss (okay, in that case I might say the truth in advertising rule was violated because the boss did not announce during the interview, “I AM AN IDIOT,” but it was also my fault for not asking more questions.)
I can tell you what I was trying to accomplish and it is embarrassing to admit – I was trying to prove I was smarter than my boss. (Even the smart bosses I had – and that was all but one of them – I thought would have been smarter to have paid me more money, given me an expense account, etc. ) I was acting stupid. The time I spent hanging around trying to prove I was smarter than my boss was wasted.
My point, which you may despaired of me having by now, is that the right thing for me to have done was either do the job to the best of my ability or quit.
Since I have written today about being a dumbass as an employee, in the interest of fair time, I guess I will have to write next about being a dumbass as a boss.
Speaking of bosses and business – check out Spirit Lake: The Game version 4.0
Last year, I went from teaching in classrooms in a pretty building with a library on the ground floor to teaching on-line. I also went from the semester system to teaching the same content in four weeks. This has led curious friends of mine, used to teaching in the traditional format, to ask ,
How does that work? Does it work?
Initially, I was skeptical myself. I thought if students were really serious they would make the sacrifices to take the class in a “regular” setting. Interestingly, I had to take a class on a new system and had the option to sign up for a session held on a local campus or on-line. After looking at my schedule, I chose the on-line option. No one has ever accused me of being a slacker – in fact, it may be the only negative thing I’ve not been called. Still, I thought it was possible I might have conflicts those days, whether meeting with clients, employees or investors. The option of taking the course in smaller bits – an hour here or there – was a lot more convenient for me than several hours at a time. To be truthful, too, I didn’t really want to spend hours hanging out with people with whom I didn’t expect I would have that much in common. It wasn’t like a class on statistics that I was really interested in.
So … if we are willing to accept that students who sign up for on-line, limited-term classes might be just as motivated and hard-working as anyone else, do they work? I think the better question is how they work or for what type of students they work.
National University, where I teach, offers courses in a one course one month format. Students are not supposed to take more than one course at a time and , although exceptions can be made, I advise against it. The courses work for those students (and faculty) who can block off a month, and then, during that month DEVOTE A LOT OF TIME TO THE COURSE. Personally, I give two-hour lectures twice a week. If a student cannot attend – and some are in time zones where it is 2 a.m. when I’m teaching – the lectures are recorded and they can listen to them at their leisure. Time so far – 16 hours in the month. Normally, a graduate course I teach will require 50-100 pages of reading per week. Depending on your reading speed that could take you from one to four hours.
I just asked our Project Manager, Jessica, how long she thought it took the average person to read 75 pages of technical material she said,
“Whatever it is, I’m sure it’s a lot more than you are thinking!”
Talking it over, we agreed it probably took around 3-5 minutes per page, because even if some pages you get right away, others you have to read two or three times to figure out wait, that -1 next to a capital letter in bold means to take the inverse of a matrix while the single quote next to it means to transpose the matrix. These are things that are not second nature to you when you are just learning a field. Discussing this made me think I want to reduce the required reading in my multivariate statistics course. Let’s say on the low end, then it takes five hours to read the assigned material and review it for a test or just for your own clarity. Now we are up to 20 hours a month + 16 = 36 hours.
I give homework assignments because I am a big believer in distributed practice. We have all had classes we crammed for in college that we can’t remember a damn thing about. Okay, well, I have, any way. So, I give homework assignments every week, usually several problems like, “What is the cumulative incidence rate given the data in Table 2?” as well s assignments that require you to write a program, run it and interpret the results. I estimate these take students 4-5 hours per week. Let’s go on the low end and say 16 + 20 + 16 = 52 hours
There is also a final paper, a final exam and two quizzes. The final and quizzes are given 5 hours total and it is timed so students can’t go over. I think, based on simply page length, programs required and how often they call me, the average student spends 14 hours on the paper. Total hours for the course 52 hours plus another 19 = 71 hours in four weeks.
IF students put in that amount of time, they definitely pass the course with a respectable grade and probably learn enough that they will retain a useful amount of it. The kiss of death in a course like this is to put off the work. It is impossible to finish in a week.
My personal bias is that I require students actually DO things with the information they learn. It is not just memorizing formula and a lot of calculations because I really do think students will forget that after a few weeks. However, if they have to post a question that is a serious personal interest and then conduct a study to answer that question, the whole time posting progress and discussions on line with their classmates , then I think they WILL retain more of the material.
So, yes, students can learn online and they can learn in a compressed term. It IS harder, though, I think, both for the students and the instructor, and takes a lot of commitment on the part of both, which is why I don’t teach very many courses a year.