This is the most depressing chart I have seen in a long time. Below are the results of our pretest on knowledge of fraction operations of 322 students in grades 3 through 7, attending schools on and adjacent to American Indian reservations.
These are questions like,
“Drag 6/1 to the correct spot on the number line.”
Which was one of only two questions that at least 50% of the children answered correctly.
Identify the letter that marks 7/8 on a number line
14% of the children answered that right.
Then there are the word problems,
“Bob and Ted painted a wall. Bob painted 1/5 of the wall and Ted painted 2/5 of the wall. How much of the wall is left to paint?”
38% of the children answered that correctly.
Looks like they did better on item 7, which asks which of these statements is true
5/ < 3/4
2/8 < 1/4
3/6 = 6/
2/ = 4/5
26% of them got that correct. Guess what? That was one of the few multiple choice items on the test, so random guessing would have gotten it correct 25% of the time.
This is a test of what is ostensibly third- through fifth-grade math. Two-thirds of the test is at the fourth-grade level or below. As our results indicate, the majority of the students who took the test would not understand what that statement means.
For the 163 fifth-graders who took our pretest, the mean score was 28%.
For the 114 fourth-graders, the mean was a dismal 14.7%.
It wasn’t that the students didn’t try. I looked and there were very few places they left the items blank. They simply did not know.
These students came from several different schools, and while there may be differences between schools, there is nothing to suggest one school with abysmal results pulled down all of the others.
I called our lead cultural consultant, Dr. Erich Longie, out at Spirit Lake, and told him that I was concerned about presenting these results to the schools that they might want to shoot the messenger. After all, it is important to us that these schools continue to provide us their input and guidance. He told me not to worry about it too much.
“They know,” he told me, “As someone who has been a teacher and administrator in schools on the reservations, I’m not surprised by the results and I can’t imagine these schools will be, either. What we all ought to be worried about is making sure that the post-test scores don’t look like this.”
So … students will start playing Fish Lake in the schools next month. No pressure here.
Excuse me while I get back to work.
I was going to call this new category for my blog
“Mama AnnMaria’s advice on not getting your ass fired” but it turned out to be too long to fit in the box.
It may surprise young people in the work place to find out that people who admit to having screwed up are often valued more as employees than those who are blameless.
Who cares whose fault it is?
One of the things that drives me crazy is when the first thing (and sometimes the second and third thing) an employee does in response to a problem is to find proof that it was not his or her fault. There are a whole lot of reasons why this is stupid, bad and will eventually get your ass fired.
Are you exclaiming.
What? Why would you fire the one person who never makes a mistake?
Well, for starters, you are clearly delusional. Everybody makes mistakes so if you are convinced you NEVER make mistakes, it is never your fault, then you have a tenuous grasp on reality that you may suddenly lose one day and begin mowing down your co-workers with an Uzi, convinced that they are evil demon zombies out to eat your non-mistake-making perfect brain. As a responsible employer, I cannot take that chance.
Next is the fact that you are wasting time and energy. You could have found the missing data and gotten it to Dr. Cflange. Instead, you put your effort into finding that email from seven months ago where Bob said we didn’t need to worry about sending the data to Dr. Cflange to prove that it wasn’t your fault that the data was not sent to our collaborator, after all, Bob told you not to bother. So, here we are, three hours later and Dr. C still hasn’t gotten the data. Besides, the fact that Bob told you that seven months ago when Dr. Cflange was in Uzbekistan does not absolve you of responsibility of sending out that data any time until the end of the world. Plus, Bob hates you now.
Which brings me to my next point – if you are always claiming you are blameless, then by implication, you are blaming someone else. Your boss is not stupid.
It’s like that time when my mom came home and the front window was broken. She asked what happened and we all swore up and down that we had nothing to do with it. She asked,
“So, you were all just standing around and the glass just fell out of the window?”
We all swore that yes, it had happened exactly like that.
(Mom, if you are reading this, it wasn’t me that pushed one of the Slattery boys into the window. Just so you know.)
Unlike me, who did not throw said sibling under the bus, if you are pointing at Bob and saying,
“It was him, it’s his fault, not me!”
Then, guess how likely Bob is to be inclined to help you out in the future. So … people who are always blaming everyone around them are not going to have as good teamwork with their co-workers.
Listen carefully here, because this next part is really important. Let’s assume the people you work with are not idiots, that there is a reason you are working for them instead of them working for you. Let’s call that reason -“experience”. Not being idiots, your bosses realize that everyone makes mistakes.
Employers are not looking for people who never make mistakes. Those people don’t exist. They are looking for people who can fix problems.
Final two reasons never taking responsibility for any mistake is going to eventually get your ass fired –
If every time an issue comes up it’s like an argument before the Supreme Court to get you to address it because you are so involved in gathering your evidence why it was not your fault, eventually people will quit pointing out problems to you because it’s just not worth the hassle.
If you never believe that any problem is your fault, then you will never get any better at preventing them, because none of the problems that occur have anything to do with you.
The most impressive interactions I have with employees often begin like this:
“That was my mistake that X happened. I would like to take the responsibility of fixing it by doing Y.”
Those people are probably never going to get their asses fired.
Now you know. Act accordingly.
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).
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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.
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.
Since it is the weekend, I decided to blog about weekend stuff. Look for more statistics tomorrow. For most of the past quarter-century, I have been roped into being a volunteer for one organization or another. Here is a very, very partial list:
- American Association on Mental Retardation
- National Council on Family Relations
- United States Judo Federation
- Community Outreach Medical Services
- American Youth Soccer Organization
I’ve been everything from Chair of the Board to chaperone. I’ve spoken at more conferences than I can count, certainly giving a few hundred presentations. I’ve raised hundreds of thousands of dollars.
Given that experience, I’ve concluded that volunteers fall into three broad categories. Recognizing that fact is probably key to having a successful non-profit organization, because for most non-profits, volunteers are essential.
Category 1: People who are very excited to be a volunteer. These individuals derive a lot of their self-esteem from their position in the organization. Their enthusiasm may stem from a genuine passion for the mission of the organization, be it youth sports, individuals with disabilities or health care. Alternatively, the volunteer position may be an exciting departure from a boring day job, an opportunity to use more of their talents. Generally, it is both reasons. They are willing to do a lot of work. They are also willing to put up with authoritarian and unprofessional interactions with the organizations, because they are so enthusiastic and often, they are accustomed to being bossed around and devalued on their “day jobs”. There is a limit to their tolerance, though.
Category 2: People who are not at all excited to volunteer but have skills and talents your organization needs. These individuals are there out of obligation – they have a child on the team, a friend on the staff, or they really care deeply about the mission of the organization. These people do valuable work for the organization like raising money, providing free legal or accounting services. They have very little tolerance for authoritarian and unprofessional interactions with the organizations, because they would rather be somewhere else in the first place and they are accustomed to being the boss or highly valued on their “day jobs”.
Category 3: People who show up and don’t do any real work.
It seems pretty clear to me that organizations need both of the first two categories, and the more the better.
Not everyone sees it that way, obviously. Let me give you just a few examples, and again, this is a very partial list. I have witnessed
- Volunteers told how to dress for an event,
- Week-long required continuing education classes costing several hundred dollars,
- Required training held hundreds of miles away,
- Required drug testing (this was after I had been asked to coach for free at an event and pay my own way. My response was “Are you fucking kidding me?”),
- “Two-hour” meetings running six hours late,
- Volunteers told to “Show up at 7 a.m. , don’t be late and be sure you don’t leave early”,
- Volunteers chastised, yelled at, berated by a board member or staff member
Do’s and don’ts
Well, first of all, don’t do any of those things above.
Second, say “Thank you.” A lot.
Think of these individuals just like donors who are giving you thousands of dollars, because they are. It would costs you a lot of money to replace their services. Treat them as you would professionals providing services for you. Would you ask your accountant to take a drug test? Would you tell your attorney to be sure he dressed professionally when he represents you in court? Don’t assume just because someone is working for free that he is a degenerate or an idiot.
It’s funny that most organizations seem to think what volunteers want is an engraved plaque or a certificate printed out from PowerPoint. Really, a little common courtesy goes a long way.
I find it weird when I make people nervous. I’ve had people shake and stutter so much that I thought they had some sort of disability, only to find out later that it was a reaction to meeting me!
My family and friends say I’m intimidating, which I also find bizarre. I am, literally, a little old grandma.
Are you kidding? You’re just amazing! Do you think we forget that you were the FIRST American to win a world judo championship, have a Ph.D. , published a book last year, started a company that made a million dollars in less than two years, then started another company to make games, came out with your first game this year, published scientific articles. Oh, and you raised four successful kids, one of whom is also a world champion and making movies.
He went on to an embarrassing degree about a lot more stuff. I’m not one of those fake humble-brag people, like the super-models who claim to be “so fat”.
It’s just that …. it’s not like that when it’s happening. Even to me, if I stopped and piled it all up like that, it sounds impressive, but day to day, it’s not really like that at all.
Whether it’s winning a world championship, earning a Ph.D., building a company or making computer games, it’s not amazing when you’re in the middle of it.
For example, I spent the last week fixing up our next game, Fish Lake. I improved the graphics, added gravity so that when a character walks off a hill, it falls down instead of walking around on the air. I added artificial intelligence to make the animals run around at random instead of just stand there. I modified the css so that the input boxes for the math problems stand out better. All of those are minor fixes in the grand scheme of things. The purpose of the game I was working on is to teach fractions, which are a super important part of understanding math, but if it’s not a fun game, kids aren’t going to play it.
Tomorrow, my day starts with reviewing the quizzes one person wrote, followed by reviewing a PowerPoint and video clip someone else wrote to teach about reading graphs and then testing some software for podcasts.
Hopefully, enough days like this piled on top of one another and we’ll have an amazing game.
It’s just like in my judo competition time, when I trained three times a day, every day. Looking back, winning a gold medal and being best in the world was amazing.
In the middle of it, though, it’s just getting up and working hard all day. Repeat a few thousand times.
Every day, every week, I face the same question that all entrepreneurs ask themselves –
“How do you know when you are done?”
Most days, I start work around 10 am and finish about 14 hours later. Usually, I take off an hour for lunch and an hour for dinner, or take a few hours in the middle of the day to get away from the office. Sunday, it was taking my grandchildren to the Natural History Museum and the park. I average 10-11 hours a day, seven days a week. Even then, there is no end in sight to the tasks I want to accomplish, goals I want to achieve. When there’s no time clock to punch, no boss looking over your shoulder, how do you decide when it’s time to hang it up for the day?
One answer is when you are just exhausted and making more mistakes than you are progress. Frankly, the prospect of just working every night until I fall asleep from exhaustion isn’t very appealing. I did that in the year after my husband died, and even though it was probably a preferable (and more profitable) way of coping than drinking myself into a stupor every night, I can tell you that it’s not a lifestyle I would recommend. The reality is that there is never, ever going to be a day at the office when I say,
Okay, that’s it. No more work to do here. Time to head to the beach.
Some people (who are not me), would say that you should take off to celebrate achievements. For example, last week, I
- found out that a project we had worked on for a client had been wildly successful,
- submitted a grant proposal to create a game for English language learners, including receiving written agreement from teachers in three school districts in three different states to assist with development,
- finished 1/4 of the lectures for a course I will be teaching soon,
- made major improvements in one level of the Fish Lake game, which we will be able to use for Spirit Lake as well,
- found out that a huge school district is now using Spirit Lake,
- renewed a consulting contract,
- created css to improve our web pages in the Fish Lake game,
- did the usual stuff of meetings, approving payroll, answering email, reviewing staff tasks on basecamp, updating a few things in the company wiki, approved a couple of employment contracts.
And all of this was accomplished with having spent all of Monday in airports and on planes flying back from Kansas City where I had been as coach for a judo team of seven students from Gompers Middle School. So … did I take off early? No, because I still needed to
- submit a revised budget for a contract,
- submit another revised budget for a grant,
- rewrite the PHP for a client database,
- get ready for an investors’ meeting,
- figure out what is wrong with the gravity in one level where the player is literally walking on air.
My unhelpful point here is that I DON’T necessarily take off to celebrate and I definitely don’t take off when I have something that could be very important to our company, like a meeting with a potential investors during which I want to get as much information (and not look like an idiot) for that time down the road when we do need to bring in outside investors.
What I DO try to to do is stop working by midnight every night. There just seems to be something dysfunctional about not leaving the office the same day you came in, even if you come in at 10 a.m. I don’t take off to celebrate so I can take off when I feel that I need a break.
One thing I can guarantee you for an absolute fact is that you will be less effective if you don’t get enough sleep. You’ll make mistakes you never would have made if you were not so tired. Knowing this, another reason that I try to quit working at midnight so I can be asleep by 2 a.m. That gives me 8 hours to sleep before I get up and hit it again at 10 a.m.
Staying up until 5:30 a.m. as The Invisible Developer sometimes does strikes me as counter-productive. You’re just going to sleep later the next day, so why not just go to sleep now and start up again when you are rested enough to be more effective. Even if I do say so myself, this post I wrote about doing one more thing before you go to bed is worth reading. Often, that one more thing will be to make the list of the things that are a priority for tomorrow. I then can knock off with confidence that I’ll get on those things first thing the next day.
I work hard, I work a lot, but I have learned not to make myself crazy trying to get everything done, because … at the end of the day, there’s another day. That’s how time works.
Following a discussion using matrix algebra to show computation in a Multivariate Analysis of Variance, a doctoral student asked me,
“Professor, when will I ever use this? Why do I need to know this?”
He had a valid point. I’m always asking myself why I’m teaching something. Is it because it interests me personally, because it is in the textbook or because students really need to know it.
Let’s take some things about matrix algebra we always teach students in statistics.
What conformable means and why it might matter
Two matrices are conformable if they can be multiplied together. When you multiply two matrices, the row of the first matrix will be multiplied by the column of the second matrix. You sum the products and that is the first element in the matrix. You repeat this until you have multiplied all of the rows in the first matrix by all of the columns in the second.
So — you can multiply a 2 x 3 matrix by a 3 x 2 matrix but not vice versa.
Multiplying a matrix of dimension a x b and a matrix of dimension c x d will give you a resulting matrix with a rows and d columns, that is, of dimensions a x d .
This can give you results that sometimes seem counter-intuitive, like that the product of a 1 x 3 matrix and a 3 x 1 matrix is a 3 x 3 matrix.
It may seem weird that the result of matrix multiplication can either be a larger matrix than both of the matrices you multiplied, or smaller than both of them, but there it is.
If both matrices are square, that is, of dimension n x n, then the resulting product will also be an n x n matrix.
And, of course, any matrix can be multiplied by its transpose because the transpose of an m x n matrix will always be n x m .
If a square matrix is of full rank, it means that none of the rows are linearly dependent. If you DO have linear dependence, it means you have redundant measures. Now, I could go on to prove this mathematically and all of it is very interesting to me.
I question, though, whether you really need to know anything about matrix algebra to understand that redundant measures are a bad thing.
Do you need matrix algebra to explain that we are going to apply coefficients (do you even need to refer to it as a vector?) to the values of each variable for each record and get a predicted score such that
predicted score = b0 + b1X1 + b2X2 …. b.Xn
When I was in graduate school, calculators that did statistical analyses, even as simple as regression, cost a few hundred dollars which was the equivalent of three months of my car payment. Computer time was charged to your department by the hour. So … my first few courses, I did all of my homework problems using a pencil and paper, transposing and inverting matrices – and it was a huge pain in the ass.
Then, I got a job as a research assistant and one of the perks was hours of computer time. I thought I’d died and gone to heaven. It took me less than half an hour to get all of my homework done using SAS (which ran on a mini-computer and spit out printouts that I had to walk across campus to pick up).
My students are learning in a completely different environment. So … do they need to learn the same things in the same way I did? This is a question I ponder a lot.