### May

#### 30

# Math Education Software has a Long Way to Go

May 30, 2011 | 1 Comment

I was very busy this weekend working on the semi-annual site update (I am SO getting last place in the Search Engine Optimization contest) and starting on my book – **Beyond SAS Basics: Tips, Statistics and a Naked Mole Rat** and on TOP of all of that, I had to take the world’s most spoiled 13-year-old shopping because there are apparently some items of clothing and footwear existing in Santa Monica that she does not own yet.

I’m also working on a proposal for math education software and I got to thinking that there is SO much out there, how can there possibly be the need for any more. In (very) partial payment for the shopping spree, I had The Spoiled One review math games and websites for me. Since I don’t see the need to call out any particular resource just because she happened to randomly land on that one today, the names have been omitted to protect the guilty.

As background, I should tell you that she was recently accepted for a summer program for high-achieving girls, scores above average on standardized tests for math (not as above as WE would like) and has never made a grade below a B in anything. (Because in our house a C means you are grounded until the next report card.) On the other hand, homework is sometimes accomplished only as a means to effect the return of all of her confiscated electronics. In other words, she is a little better on achievement and motivation than the average student, but hardly a paragon of mathematics virtue. And here were her reviews:

**Video of Rap Song on Mathematics Topics** (Because, you know, you kids these days like that)

… Um, distracting. I learned nothing because I couldn’t understand the lyrics.

**Place Value Video Lecture**

Not really for someone my age (13). Kind of stupid anyway.

**Pre-Algebra Game**

Sucks! (She drew a picture here to indicate how much she hated it.) BORING. Doesn’t really work. (Punctuated by another picture)

**Game with Word Problems**

The game was good I guess … (a few minutes later…. ) Never mind. It didn’t give you the right answer after. I HATE THIS SITE.

**Game on Factors and Multiples**

OK. Not creative or fun. (Another picture, that looked something like this

- . .
- |
- —

**Sites on Math in Every Day Life/ Real Life Math**

Eewww NO!! Doesn’t make me like math!

Mucho Math– The only one that didn’t suck

“That one with the Hispanic math teacher and the kid. That one was okay and kind of funny even though the topic it was on wasn’t really at my level.”

I found this last comment extremely interesting because I knew who she meant. I had sat my daughter down at a computer on a web page with over 1,000 videos, games and other math resources and she came up with the same option that I thought was one of the best ones I’d reviewed when I was doing the same thing a couple of months ago. The teacher is Lawrence Perez. The innovation he has included is really quite simple – he has a student in his video.

Having reviewed numerous other options myself, I have to say I agree with my daughter on much of it. The absolute WORST thing you can do in designing mathematics software is have it get the wrong answer, for example, when it asks :

If Y = 5 + x**2 and Y = 14 what is X

and you put -3 and it says

WRONG! The answer is 3

Of course, -3 is also a valid answer and then you have a student who says,

“I hate this program. It sucks!”

Not as bad, but also frustrating are those programs that don’t tell you the answer, but simply come up with the next question.

If you say that both of these problems are examples of poor design, well, I agree with you, but poor design seems to be rampant.

Having a game or video that is too basic is not the problem of the software, of course, but MAYBE whoever marketed it as being at the middle school level. Or, it may just be that there is wide variation among students and was not appropriate for this particular student.

Yes, I’m generalizing from an N of 1 (well, 3 actually, if you include me and my brother, who is a math teacher and has had generally the same responses), but from what I have seen so far, there is a whole lot of math education software out there that is not effective in interesting students enough to use it. Sometimes the game doesn’t even do the minimal job of providing the right answer, something any parent could accomplish with a $1.29 stack of index cards by writing the question on one side and the answer on the other.

Every time I have done this experiment, whether with me, my daughter or someone else, the outcome has been equally underwhelming. Even more underwhelming is the fact that almost NONE of the designers/ producers of these resources even MENTION the thought that perhaps one would evaluate the software and see if it has any impact at all. The attitude seems to be “Here you go”. Period. Kind of depressing.

I guess the good news is that there are about a bazillion more games, videos and other resources out there to try.

### May

#### 28

I’ve been reviewing a number of options for students to learn mathematics.

A lot of sources kind of sucked. At best, these sites were just the same old thing, flash cards but on a computer screen, for example. There is nothing terribly wrong with that, but it is hard to imagine that they have any greater benefit than just using index cards you picked up at any store and writing 2 x 3 on one side and 6 on the other, which is how I think everyone has learned multiplication since we quit writing on slates with a piece of chalk. Hey, maybe we should go back to that. It probably involves less waste. Green math! But I digress …

At worst, these sites were just plain wrong. This was more often true for those that dealt with less basic mathematics, where they would, for example, give a definition for a chi-square that was really for a t-test or say that the median was the most common score in a distribution (It isn’t. That’s the mode.)

Other sites were better, including videos of short lectures and the explanation of whatever the topic they were teaching was correct. (AnnMaria’s first rule of teaching – have something non-stupid to say).

Two examples are:

- Khan Academy site, which is free, has over 2,000 videos and Bill Gates as its BFF.
- Cool Math Guy website has some free samples, for others, you have to pay. The videos I saw are good explanations of such topics on trigonometry.

There are hoards of math game sites out there, many of which are just a computerized version of asking your child over and over what is 47 + 52 until his brain crawls out his left ear and runs away just to escape the boredom.

Then, there are sites like Gamequarium, which offers a LOT of different math games for every topic, most of which look like they would be fun if you were immature, which I am.

ALL of the resources I found suffer from the same fatal flaw which is that they begin with the presumption that the student has some interest in learning math. This seems a reasonable, some might even say ‘sane’, assumption based on the fact that the person has come to a site that is for teaching mathematics. For those people who seek out these sites, they might work.

The problem is with the vast majority of people who WON’T ever voluntarily go to these sites because they really don’t give a rat’s ass if they ever learn math or not. Sometimes, as this excellent article “The Education of Jose Pedrazza” points out, they are much more concerned about whether they are going to be homeless, how their family is going to eat.

Given those circumstances, it’s really hard to focus on if you learn this math, you’ll be able to do next year’s math and so on for the next 10 years until you graduate from college and get a good-paying job. It’s all well and good to talk about delayed gratification when you are sitting here like me drinking Chardonnay at an expensive oak desk, and quite another when your mom is collecting cans to come up with money for dinner.

Some of it, the odds are great that you will NEVER use. I just came across this statement in a publication on research in teaching and learning mathematics.

*“Across all age levels, the best estimates are made in temperature situations and the most difficult estimates involve acreage situations.”*

ACREAGE? Okay, I’m 52 years old, I use math for a living, I’ve bought and sold four houses in my life, including one that had five acres of land with it and was in North Dakota AND NEVER IN MY LIFE HAVE I NEEDED TO ESTIMATE ACREAGE!!

Yes, I am sure there are farmers and landscape architects and people doing surveillance for homeland security applications who may need to estimate acreage. Every time I write something like this, I get hate mail from people telling me this is why they will never hire me to work for them at Google Maps. (Of course, when I look up these people, they never actually work for Google, or anybody. They are invariably some embittered graduate student teaching Mathematics of Acreage Estimation at Boo-hoo U. )

My point is that most of math is taught completely out of context with no real thought to application other than answering a question on the SAT. For some students, like the most spoiled 13-year-old in America, who happens to live in my house, that is adequate enough incentive. One reason is that for her, and many of her peers, it is NOT gratification delayed ten years. At the end of the school year, many neighborhood parents trek to the Apple Store to buy the iPhone 4 or the gadget du jour for Buffy and Justin who got an A in math. In eighth grade, the kids will all take their high school entrance exams, and when the test scores come and acceptance letters come out, there will be ANOTHER round of iPhone -buying and trips to The Grove. A couple of years after that, many of those same kids will get their first car, with the stern admonition that, “Your grades better stay up or you will be walking to St. Alphonso’s Catholic High School “.

I was a little depressed after I read this article on the Los Gatos Patch, where the mother happily admits that she could not do her 14-year-old son’s Algebra class. It tells me not only that we find it perfectly acceptable not to know math (while it is NOT okay to say that you forgot how to read) but also that the mom obviously has no need for Algebra in her daily life. On the other hand, I was majorly impressed that she got her son to make dinner and to clean up – twice.

Some people just like math – I did and I still do. That’s only incentive, though, to study the parts that interest you. For example, I watched a video on trigonometry for about five minutes. Then I was bored. It was exactly like the movie, Freaky Friday, where the middle-aged mother changes bodies with her teenage daughter, and in algebra class tells the teacher, “No, believe me, I will NEVER use this.”

I use algebra nearly every day of my life. I use matrix algebra, not every day, but certainly weekly, and calculus fairly often, too. On the other hand, I have NEVER and I do mean, NEVER, needed to know a sine, cosine, tangent, arctangent for any reason whatsoever, not even when I was an industrial engineer. This isn’t to say that no one ever uses these. I asked the house rocket scientist when was the last time he used any of these and he said that everyone in the real world uses all of these every day. Well, EXCUSE ME!

Perhaps we have it backwards. Instead of railing about the poor performance of our kids on tests and teaching to the test, maybe we should turn things around. Perhaps we should start with why they need to know how to calculate acreage, t-tests or cosines. Give them some projects where this information as applied. Maybe then not only will they actually give a rat’s ass if they learn it or not, but they’ll also still remember it when they have 14-year-old kids of their own and be able to use that information on the job when people like me hire them.

Wouldn’t that be a nice change of pace?

### May

#### 5

# Is it true that we suck at math?

May 5, 2011 | 8 Comments

*“If everyone knows a thing it’s almost for sure it aint so.”*

*“It’s not so much the things you don’t know that hurt you as the things you know for sure that aint so.”*

I don’t take anyone’s word for anything. Take those quotes, for example, which I’ve both heard attributed to Mark Twain, Will Rogers and several others, ironically by people who were just certain they were correct.

One thing everyone knows is that Americans suck in math. We are so far behind Asia, we are continually told, that we are soon all going to be learning how to say, “Would you like fries with that?” in Chinese.

There was an article in the Los Angeles Times today that profiled a mother who had an Excel spreadsheet with a schedule for her child from 8:00 a.m. to 11 p.m. seven days a week. She said she started in kindergarten, because life is hard and students need to learn to deal with it. Her son, as a tenth grader, scored a perfect 800 on the SAT. Rather than convincing me further that we suck at math, it made me question the goal of propelling a child to perfect scores.

One thing writing a dissertation on intelligence testing taught me is that test scores are very, very far from absolute and objective. Two critical points to keep in mind:

**1. Some group of people decide what is tested, inevitably the group of people that has the most power. **If we insisted that being fluent in more than one language is a factor in achievement scores, Hispanic children would be getting admitted to elite institutions in droves. Before you discard this as a silly notion, think about the arguments made for including high math scores – these are relevant to courses students take, to careers. An argument could be made for functioning in a global market place, for the ability to read texts in the original Spanish (or whatever second language a student reads). I could write a whole dissertation on this – oh wait, I did ! – the point is we make decisions about what goes into the tests and those decisions favor some people and not others.

**2. The scores we use to evaluate both at an individual and larger (school, country) level are almost never how many questions were answered correctly, which you might logically think is your test score.** There you go with the logic again. Cut it out. In fact, scores depart several steps from the number of correct answers. First, there is the issue of partial credit, yes or no and if yes, for what. Second, there is the step of standardizing scores. Usually this means setting the average at some arbitrary value, say 100. If the average student gets 17 questions right, then that is set as a score of 100. The standard deviation, the average amount by which people differ from the average, is also set at an arbitrary value, say 10. (If you’re not familiar with these ideas, think of your family. We’re kind of short in my family, and if you went to a family re-union you’d probably find that the average woman is around 5’3″ give or take two inches. So, you can think of five feet three inches as being the average and two inches as being the standard deviation. If you are reading this and from a country on the metric system, 5’3″ is equal to a furlong plus a bushel, a peck and a hug around the neck.) To return to my long-forgotten point – if 84% of the people score 22 points or lower, than answering 22 questions correctly is given a score of 110. (The mean of 100 + one standard deviation of 10). The scores you see reported aren’t that closely related to the number of questions answered correctly and they tell you almost NOTHING about what precisely people do or do not know.

I think most statisticians know this. I am certain that nearly everyone who does analyses of educational tests knows this. But I am equally certain that the average person reading the newspaper does not. This is important because it has to do with our sucking or not.

My assumption, based on what I read in the papers and hear on TV is that American kids just don’t know basic math. So, I downloaded the TIMSS (Trends in International Mathematics and Science Study ) data and I also downloaded the items that had been released, to see what it is that American kids do and do not know. Here are a few examples:

Students were shown a rectangle divided into twelve squares. Five of those twelve squares were shaded. Then, they were given five choices of circles that were partly shaded and asked:

*“Which circle has approximately the same area shaded as the rectangle above?”*

To solve this problem you need to figure that the rectangle has 5/12 shaded and understand that 5/12 is a little less than one-half. (The figures show a circle that is 7/8 shaded, 3/4, exactly one-half, a little more than one-half and a little less than one-half.)

**This question was answered correctly by 80.2% of American eighth-graders.**

The next question asked :

*A gardener mixes 4.45 kilograms of rye grass with 2.735 kilograms of clover seed to make a mix for sowing a lawn area. How many kilograms of the lawn mix does he now have?*

**This question was answered correctly by 71.9% of American eighth-graders.**

I must admit that I was surprised the figure was that low, although not extremely surprised, since I know many, many adults and some young kids who never do math like this. Every phone, every computer has a calculator on it and they just think this is a useless skill, like cursive. I happen to disagree and the world’s most spoiled thirteen-year-old is not allowed to use a calculator to do or check her math homework.

Another question dealt with inequalities:

*X/3 > 8 is equivalent to….*

To get this answer correct, you need to understand the idea of inequality and how to solve an equation with one unknown. Essentially, you need to reason something like 24/3 = 8 so X > 24 . This, of course, presupposes you also know that 24/3 = 8.

**This question was answered correctly by 42.8% of American eighth-graders. **

A question that was answered by even fewer was:

What is the perimeter of a square whose area is 100 meters?

To answer this you need to know:

- The formula for finding the area of a square
- The concept of a square root
- That the square root of 100 is 10
- The area for finding the perimeter of a square (or rectangle, either would work).

**This question was answered correctly by 26.5% of American eighth-graders. **

One last question,

*A bowl contains 36 colored beads all of the same size, some blue, some green, some red and the rest yellow. A bead is drawn from the bowl without looking. The probability that it is blue is 4/9. How many blue beads are in the bowl?*

**This question was answered correctly by 49.4% of American eighth-graders.**

Are these percentages bad or good? Honestly, I thought the questions were pretty easy and I was surprised by the low percentages on some of them – but I do math for a living and I was in 8th grade almost forty years ago. So, I have known this stuff a very, very long time. I THINK some of the questions were actually what was taught in ninth or tenth grade when I was riding a brontosaurus to school, so the fact that eighth graders today don’t know this information doesn’t convince me we’re all a bunch of drooling idiots.

Here is a blasphemous question for you – Does it matter if you know the answers in eighth grade? I’m serious. Is it worth having your child study from 8 a.m. to 11 p.m. so that he or she knows all of this in the eighth grade instead of the ninth grade?

A few weeks ago, I was looking for data for a proposal I was writing and came across a state Department of Education website that had a note on its pages on test scores that said proficiency meant something different according to the federal government definition and that many people could function perfectly fine will being scored below proficient in math.

At the time I dismissed this as an excuse for poor performance. Today, when I looked at the questions and the results, I was not so sure. My two older daughters are a journalist and a history teacher. Both have degrees from good institutions (NYU and USC). I believe neither of them could answer the question about finding the perimeter of a square with an area of 100. Perhaps they could have answered it when they took their SATs or while they were taking the one mathematics course they took as undergraduates. I’m not sure. I’m fairly certain if they ever knew this information, they’ve totally forgotten it. The truth is, as much as I hate to admit it, that neither of them at any point in their lives will feel the lack of this knowledge.

On the other hand, my daughter who knocks people down for a living (she competes professionally in mixed martial arts) could almost certainly answer these questions off the top of her head, just because she likes math and has always been good at it.

What percentage of Americans (eighth-graders or not) SHOULD be able to answer these questions?

I have no idea what the answer to that is.

Some people would say 100%, because they need to know this information to do well on the tests to get into a good college. I’m not sure that is true. More and more, people are asking WHY you need to do well on the tests. If I want to be a sportswriter or a history teacher or a doctor, what good does it do me to be able to calculate the perimeter of a square given the area?

I think the mother in San Marino may be part of an education bubble that will burst just like the housing bubble has. I am far from the only person to be suggesting this. Not only has the cost of higher education reached astronomical levels where it exceeds the cost of a home in most parts of the country, but it also, for selective institutions, is costing more of your life. Not only are fewer people going to be able to pay it, but, perhaps like the housing bubble, more people are going to say, “This isn’t worth it.”

I did not work from 8 a.m. to 11 p.m. I spent several hours today reviewing grants. Then I went running down to the beach, because it was a beautiful day. I had a Corona while reading the LA Times. I analyzed the TIMSS data and I watched The Daily Show. I also checked my daughter’s math homework and pointed out the one answer she had incorrect. She figured it out and fixed it on her own.

Life is not hard. Life is good.

### Dec

#### 21

# They’re not dumb, they’re REALLY different

December 21, 2010 | 1 Comment

*A couple of nights ago, I had a nightmare. I dreamed that I couldn’t do math. I was having lunch with some colleagues and the bill was $24.82. Everyone handed me money and I had $25.67. I was trying to subtract the bill amount from what was in my hand and divide it by three, but I couldn’t. Every time I thought I started to have the answer, the numbers flew right out of my head. Since it was a dream, I could see them flying, with little wings and everything. As time passed, my colleagues started to get impatient, ask me if I was done yet, make jokes. I remembered that book, Charlie, and started thinking, this is what it must be like to be mentally retarded. I was so upset, I woke up.*

I’ve been slacking on the reverb10 project. I read about it and it sounded interesting. The idea is that every day there is a different prompt and you’re supposed to post on your blog related to that. I have a blog. Three, actually, though that’s another, unrelated story. I thought it would be good for me to write more, since, oddly, I often learn things better as I write about them. Well, it has been really interesting, but in a different way than I thought.

As I read the prompts, and the other bloggers responses to them, I was very strongly reminded of Sheila Tobias’ book, **They’re not dumb, they’re different: Stalking the second tier**. In brief, her book is about her study of why very bright people nonetheless choose not to study science and why they have a hard time with it. She had scientists sit in on literature classes and people with doctoral education in subjects like English sit in on introductory science classes. It was a really fascinating study and reading it, I could totally identify with the science Ph.D.’s frustration with English 102. It was just like Dave Barry said about college, that he chose English as a major because it had no actual facts in it, unlike Chemistry, where they get really snippy if your chemical formula for, say, what happens when you combine two hydrogen atoms and one oxygen comes out to be really different than everyone else. If you say, “Maple syrup!” or “The Queen of England”, they do not give you points for creativity, quite the opposite.

I tried to avoid every single art and humanities course in college. I did take Japanese as a language, since I went to Japan to study for a year. Since mathematics was in the College of Arts and Sciences, that took care of that distribution requirement. They caught me my last semester in my senior year and made me take English Comp, which I managed to do as an independent study with a sympathetic English professor.

So, I looked at the reverb10 prompts and did not do that many of them. I wasn’t quite sure they were talking to me. For example, when the prompt was about what you appreciate, it occurred to me that I appreciate Euclid, logistic regression and my husband, not necessarily in that order. My suspicion that I was playing on a team by myself here occurred when I typed reverb10 and logistic regression into Google and all the hits that came up were me.

So, I’ve been reading these posts by other bloggers and I truly feel like Temple Grandin in Oliver Sacks book, **An Anthropologist on Mars**. I read this blog by a 20-something person who feels guilty about not meeting with people she used to know. The same blog had a link to an awesome article on a man who decorated his basement with $10 worth of Sharpies. Awesome for him, but I’m guaranteeing you that if I tried that my house would just look like Matt Groening or Hugh MacLeod went completely psychotic.

It reminded me of The Perfect Jennifer when she was about nine years old deciding she wanted to teach herself to play The Sting, by Scott Joplin. So, she got a copy of the movie with Robert Redford and Paul Newman and played that part of it over and over until she could play the song by ear. I couldn’t imagine ever even thinking of wanting to do that, much less doing it. Even though her dad had died recently and I did not have a lot of money, I went out and bought her a piano.

There was another reverb10 prompt on what have you made this year. I thought to myself, *“Does dinner count?”*

Lots of people had made lots of things. some of them, like basement-Sharpie-guy, just amazing, and others that you could have bought at the dollar store made by some kid in China and I didn’t want them anyway.

So, I typed in “math” and “reverb10” and came across an interesting blog by a math teacher who quit her doctoral program to go back to teaching. Even though I did finish my doctorate, and, in fact, enjoyed it, I could totally related. Jane Mercer, one of the people on my doctoral committee, and a profound influence on my life, had a sign in her office that simply said,

*“No matter how far you’ve gone down the wrong road, turn back.”*

Then, her next post was about making wreaths out of buttons and I thought,

*“Why would you even do that? No, seriously, why?”*

And it occurred to me, because I am not really all that slow on the uptake, despite my nightmares, that there are some people who would think the same about me.

Tomorrow, when I am sitting in the airport, I am going to write a blog about quasi-separation and other problems with logistic models. I’m really looking forward to it. Usually when you read papers on some statistical procedure they have these stupid, perfect little datasets that are set up not to offend anybody so they are something like the auto.dta dataset from Stata, and everything works out perfectly to be highly correlated with no problems of multi-collinearity and the chi-square is always significant and the R-square is always really awesome and something like .80. So, you get graduate students who have an R-square of .42 for their dissertation data and they are disappointed instead of simultaneously having orgasms and doing the little happy dance like the situation warrants.

My paper is going to start out early on with real life, like getting a chi-square with the probability > .97 and the “NOTE: This model may not be valid” on your output, which causes you to comment to yourself,

*“Yeah, no shit.”*

In writing this paper, though, I am really, really trying to keep in mind button-wreath-woman and basement-Sharpie-writing-guy and person-who-feels-guilty-over-coffee and think what would make it interesting and relevant to them. I think I will write better papers in the end.

So, that’s what I learned from reverb10.

### Dec

#### 15

# Advice on marriage, Euclid & logistic regression

December 15, 2010 | 2 Comments

My friend, Gokor Chivichyan, a mixed martial arts instructor, once gave this questionable advice to his students,

“Women usually just say they’re fat to get attention. So, me, I agree with them. If she says she’s fat, I say, yes, you fat but we like you anyway. If she’s really fat, though, you just have to dump her. Not if she’s your wife, though. Then, it’s too bad but you have to keep her anyway and take care of her because she’s the mother of your kids.”

(For the record, I have met Gokor’s wife and she is both lovely and charming.)

We tend to keep our private life extremely private. Dennis has been referred to as “your alleged husband” by my friends, who have never met him. Recently, another friend of mine, my former business partner from Spirit Lake Consulting was visiting and, after knowing me for 20 years, met my husband for the first time. My friend commented,

*“Your husband really loves you.”*

I was a bit miffed that he found this so surprising. I think one reason this is a surprise is that we so often categorize people into single boxes. I was a world-class athlete while my husband hates all exercise. Being a sixth-degree black belt, I once suggested to him that perhaps he could learn judo for exercise. His response was,

I’m not doing anything where people touch me unless I get to have sex with them at the end of it.

As I was laying in bed this morning with my eyes closed, trying to avoid morning, Dennis was carrying on about the Complete Works of Euclid, which proofs were not really proofs, but axioms, the incompatibility of irrational numbers with early Greek geometry, the inefficiency of using geometry for certain proofs rather than algebra or calculus. This is why my husband loves me. Not only did I not find this boring and throw a pillow at him, which most of the women of my acquaintance might have done, but actually opened my eyes, made a comment or two about how it fit exactly with what I was thinking about, which happened to be …

The geometric concept of a line is that it extends infinitely in both directions. What most people think of as a line, with two end points, is really a line segment. Most of us learned this in high school or middle school and don’t really think about it much. However, sometimes it becomes relevant.

Let’s say you were trying to predict a dichotomous dependent variable. Since it is around Christmas time, let’s pick whether a person is traveling home for the holidays or not, which we have coded 1= no, 2 = yes. That might be a very useful fact for people to know who were in either the travel or family therapy industries to target their marketing/ determine outpatient clinic hours.

This is a dichotomous variable and you can see that it plots pretty terribly against a continuous predictor variable – say, income.

Y = a + bX

is just plain wrong here. It doesn’t fit. Very, very far from the assumption that a line extends infinitely we are stuck with a stupid line that goes from 1 to 2.

How about probability then? We could use the probability of going home for Christmas by income. That will extend from 0 to 100, which is certainly closer to infinite.

Well, this is better. It sort of approximates a line.In the graph above, you can see the obtained regression equation

Y = -.1236 + .0313X

(I know you were dying to know.)

You can also see the predicted values it gave me for incomes below $5,000 are negative. I guess those are the people who are not coming home even IF hell freezes over. The probabilities for people with incomes over $40,000 are above 1.0. I guess that means they are going home twice, once to mom’s house and once to dad’s place in the Hamptons with his trophy wife.

So, we have one case, with just the binary outcome, which is clearly not linear. We have another case, predicting the probability of the outcome, which may be linear, but is actually a line segment and not a line. That may be true in theory for lots of things. I doubt income extends from negative infinity to positive infinity, although Bill Gates and Warren Buffet are doing their bit to extend the right side of the distribution for themselves while the Republicans and certain banking and investment firms are making a best effort to extend it on the left for all the rest of us.

There are a whole bunch of reasons that using linear regression is wrong when you have a binary dependent variable, and the fact that it is flat not a linear relationship is just one of those.

Now, if I were an ancient Greek, I would include a lot of geometric examples, not really proofs. If I were an ancient Greek that had access to JMP 8 software I might include another variable graphed against probability and say, “Looky here”, or however you say that in Greek.

This is a very important point – Greek or not – even though the relationship charted above is very high – R-squared = .78 to be precise, it is clearly not a linear relationship. It is an S-curve and it looks very much like a logistic relationship.

Three very important points emerged from this:

- The potential to teach kids the basic understanding of some of the more abstract concepts of mathematics by pictures. I can see how you could start with these graphs and do a linear relationship, then log one variable, log both and start to see the different types of pictures. Those Greeks were on to something. Too bad they didn’t have JMP. Never know what they could have achieved. (Click here for link to random JMP page.)
- The idea of using graphs to teach students is intriguing, and yet I am puzzling how I could drag the world’s most spoiled twelve-year-old away from the Disney channel downstairs and get her to see it that way. The use of graphing calculators in mathematics is not new, but neither does it seem to be particularly effective. This is all fascinating TO ME because I see the end point of making predictions. Perhaps we should spend the first few weeks of mathematics on why what we are about to do is important?
- I was thinking that I had failed miserably on most of the #reverb10 prompts because, well, frankly, I’m more interested in examining logistic and linear relationships than ruminating on my life. Then, it came to me – what’s the one thing I have come to appreciate in the past year? That I’m married to someone who would wake me up by bringing me coffee in bed and talking about the complete works of Euclid!

### Dec

#### 10

# Math and Computer Programming through Black Belt Eyes

December 10, 2010 | 1 Comment

In my misspent youth, I was the first American to win the world judo championships. This came about since I had a propensity to run my mouth off, which often led to fights. Those people who said I better be able to “walk the walk if I was going to talk the talk”. Well, I took them seriously. In retrospect, they probably wish they had instead advised me to just shut the hell up. Too late now.

In a presentation to our judo board of directors, a marketing specialist commented that a major problem was that materials were developed “with black belt eyes”. In other words, we had pictures of flashy throws by top athletes which made us, the black belts, think the person was extremely skilled and reminded us of when we were young athletes. The market analyst said to us,

The reaction of the average person seeing these is either, “ouch! That looks like it hurts!” or “An older person (or younger person or overweight person or out-of-shape person) like me could never do that.”

For computer science, I think we have the exact same problem. For example, being a decent programmer requires, as a minimum, some basic level of algebra and statistics. You need to understand scientific notation, subscripts, superscripts, a few symbols like ∑ . I’m not talking even Calculus here, but stuff like what the mean of X is and how computation is affected by the distribution of parentheses. This is stuff that a lot of us learned in seventh through tenth grade. What if you didn’t? What if you went to a school where you were taught by seven substitutes in nine months and you never got to it? What if your school didn’t have textbooks? I’m talking about American schools. Probably not the ones your kids go to, if you are reading this, but American schools nonetheless.

Sometimes you have a very good, knowledgeable hard-working teacher and you still didn’t learn it. My older brother is a math teacher and you couldn’t ask for a better one. He often mentions how unmotivated his students are. Let’s think about those black belt eyes for a moment. Most of Algebra is taught completely separate from anything remotely important to the child’s life. I have a daughter in seventh grade at a good school and her textbook has the “appropriate” distribution of different names that is supposed to make it relevant for kids,

Suppose Keisha is making $6 per hour more than Diego. Diego was paid $18 for three hours of working after school at his family’s market. How much money does Keisha make per hour?

We tell kids that they will use algebra in their life and they are skeptical because they don’t really see their parents or clerks at the grocery store working problems out on pieces of paper. When we get to students in community college, they are already calling bullshit on us. That twenty-something student in your Developmental Mathematics class knows that he does not use Algebra in his job at 7-11 and he is pretty sure that his fascist boss doesn’t either.

Don’t even get me started on statistics. Oops, too late!

Sometimes I have to wonder if anyone who writes those statistics textbooks ever met an actual student. Yes, I find puzzling out pages of equations challenging and interesting. Many equations I can glance at and say to myself, “Sum of squares error” and move on. Most people are not me. It goes back to that issue of “black belt eyes” again. Much of what is in an average statistics book – Poisson distributions, the central limit theorem, matrix algebra approaches to multiple regression – students won’t need to know for years, if ever. Several bad things have happened over the years, and I don’t even know where to start.

We have tried to cram so much into textbooks, and into introductory courses, I guess to please whoever thinks we water down the curriculum, that it is virtually impossible to teach it all well in the time allowed.

It is possible, and even rewarded, for students to get by on memorizing formula, facts and sample problems without really understanding what is going on.

Students who come in without the prerequisites are just plain screwed unless the teacher or a TA has the time and kindness to spend hours getting the student up to speed. With cuts in budget and the epidemic of part-time faculty at the college level, let’s just abbreviate it to students who come in without the prerequisites are just plain screwed.

How about this for a completely different idea…. Let’s teach a year of Integrated Computer Science for ten credits each semester. Let’s include programming techniques, algebra and statistics. Let’s have real word problems like deciding how many representatives each state gets and, unlike this Census video, let’s NOT skip over the math. And then let’s write a program to do it. And see if we get the same answers as the U.S. census.

Quit assuming that the only things students are interested in programming are computer games. Quit assuming that students will spend two years struggling in Developmental Mathematics, College Algebra, Statistics and Computer Science 101 courses so that MAYBE in a year or two they will get to write a database program. Read Shelia Tobias’ book “They’re not dumb, they’re different: Stalking the second tier “.

And if you have the attitude that anyone who isn’t willing to learn several semesters’ worth of apparently (to them) useless material, didn’t come into college with all the prerequisites already learned or doesn’t immediately grasp new equations, proofs and concepts has no place in computer science – do us all a favor and don’t go into teaching.

### Dec

#### 6

# I Wonder What Would Have Happened If I Sucked at Math

December 6, 2010 | 2 Comments

On the front page of the Los Angeles Times today was a story about three of the middle schools in Los Angeles serving the highest proportion of students in poverty. My daughter, “The Perfect Jennifer”, did her student teaching at one of the three and teaches at a second.

She said to me today,

Mom, the beginning of your story is very common among the students I teach. The families don’t have a lot of money, they have problems at home, the dad isn’t always around, they end up in foster care, have problems with the police, go to juvenile hall. Usually, these stories don’t end with – And she got a Ph.D., became a statistical consultant, runs her own company and lives by the beach in Santa Monica. You do know that’s not the way these stories usually play out, don’t you?”

The reverb10 prompt for yesterday was what you wondered in 2010. I know that was yesterday but I tend to live by my own rules and time lines as much as the law and the necessity to make a living will allow. It’s also Computer Science Education Week where we are treated to videos of real live computer scientists telling us how great it is to be in computer science. After watching one, the house’s resident rocket scientist commented,

“The first mistake they made in producing this video was allowing those people to dress themselves.”

What I wonder about is what would have happened to me if I had sucked at math. I think back to when I was young and, in most ways, less promising than the students my daughter has today. My family didn’t have money or connections in this country. I was female, short, chubby, near-sighted, Latina – in a time when it was still legal to advertise jobs for men only and people thought it was okay to say things like,

“You shouldn’t be offended by comments about Hispanics. No one thinks of you as Hispanic because you are so intelligent.”

I was somewhat less sweetness and light back then than I am now and my most likely reaction to comments like that was either to say, “What the fuck?” or punch the speaker in the face (hence the acquaintance between myself, the foster care system and the juvenile authorities).

So, what happened?

Well, I had a sixth grade math teacher named Sister Marion who thought I should make A’s, and I knew better than to argue with a nun. In middle school, I had an Algebra teacher named Mr. Cartwright who just assumed I should excel in Algebra and demanded to know what my problem was any time I got less than an A on a test. I went to an alternative school, Logos High School, back when it was in the inner city, before they decided to move the school to the rich suburbs and do well instead of doing good. There, I had a math teacher named Chris, who was a conscientious objector to the Vietnam War and another math teacher named Phyllis who taught matrix algebra. We were just getting the chance to program computers when I was in high school, through an arrangement with St. Louis University, down the street.

I took the SATs, did well, got admitted to Washington University in St. Louis and took some classes on programming – BASIC and FORTRAN – just because. I took Calculus and Statistics because I thought these might be useful some day, but, if not, they were kind of interesting courses. I took regional economics and urban economics and learned about actual applications of matrices where you had the sales from region A to other people in region A, then their sales to region B in the next cell, their sales to region C — and it all started to make sense. I did not ace all of my courses in college. In fact, I pretty much majored in parties (don’t tell my mom) and I worked full-time.

BUT … and I think it goes back to Sister Marion … I always assumed there wasn’t any subject I couldn’t learn if I put my mind to it. When I was at General Dynamics and nine months pregnant, the managers were really freaked out about having a very, very pregnant woman engineer climbing around on the machines. One manager said to me that it was a liability because I could fall down. I told him that I had been walking since I was a year old and that I hadn’t fallen down since. I know the reason they sent me to that SAS programming class was to get me out of the factory.

I didn’t start out with looks, money, connections or even good behavior. By the time I was an engineer, I still didn’t have the sense not to be a smart ass to upper management. What I did have going for me was that I was good at math and learned to program a computer very well. There were not enough people that could say that, so I was tolerated and helped until I learned to dress myself and shut the hell up on occasion.

One of the few poems I remember ever learning was from Robert Frost and it ended

A path forked in the woods and I

I took the one less traveled by

And that has made all of the difference.

If I had studied poetry instead of math and computer programming, I don’t know where I’d be but I don’t think it would be here.

I wonder, if I had sucked at math, would I still be able to take trips to Tunisia, Costa Rica, Beijing and Athens just because I felt like it.

I wonder if I would have been able to go to the Bahamas and seen the marching flamingos at the Bahama Zoo. I really wonder who the hell sits around a zoo and suddenly says,

“You know what we should do, today? We should try to teach the flamingos to march.”

Seriously, how does that ever enter your brain? I REALLY wonder about that.

Trivial pursuit answer of the day: The flamingo is the national bird of the Bahamas.

### May

#### 25

For the first time in two years, an application came in my email for a technical position from a person under 30 who was an American citizen. This isn’t because I don’t look for people. I have talked to lots of young people I know who are pretty good with computers and asked if they would be interested in learning about statistical software. We would train them. Nope. They want to go to law school (lots of them), get an MBA (lots of them) with the odd few who want to be teachers, journalists or artists.

Last night, I was reading a data mining book that had NO equations and I had one of those mental stumbling blocks, you know, like when you can’t remember the name of your youngest child? Well, that happens to ME all the time, anyway. I doubt it is due to all the drugs in college because I’ve always had that problem. [Not that I ever personally did any drugs, of course. I am referring to second-hand smoke.]

Just out of the blue for no reason I was not 100% sure of the definition of an inverse of a matrix. So I asked my husband,

“*Hey, the inverse of a matrix is the matrix you multiply it by to get the identity matrix, right?”
*

He answered,

“Yes, but sometimes there is no matrix you can multiply by to get the identity matrix. Then the inverse is undefined. That usually doesn’t happen unless your variables are correlated.”

“Yes, but sometimes there is no matrix you can multiply by to get the identity matrix. Then the inverse is undefined. That usually doesn’t happen unless your variables are correlated.”

I guess he added the part after “Yes”, just in case a whole section of my memory had been wiped out. Of course the whole problem with multicollinearity in regression is obvious if you know this because you cannot invert a matrix so you cannot solve the normal equations to get your coefficients.

I sat in a graduate course today taught by a very knowledgeable professor, surrounded by graduate students at a selective university in a course they paid a lot of money to take. Several times, he said something like this:

*“What is regression? You have some X’s and there is a black box and then you get a predicted Y.”
*

I am looking at his drawing on the board and thinking to myself, no, it is not a black box. When I looked at his black box, this is what I saw:

And I thought

A. You take the X matrix and transpose it. You know you need to transpose it because you can only multiply a matrix if the number of rows in one matrix equals the number of columns in the other. You multiply that (the transposed matrix) by X (the original matrix).

B. You then take the inverse of the result from step A.

C. Then you multiply the inverse of the product of the transposed X matrix and the original X matrix by the transpose of X.

D. You multiply that by the Y vector

and that gives you the vector of regression coefficients.

Here is a really good explanation of least squares estimates in matrix notation, by the way. Thanks to Pennsylvania State University.

I do not blame the professor at all for not saying any of that because he has two problems with this course, neither of which have ANYTHING to do with his competence as a professor or of the ability of the students. I know because I have experienced this problem growing and growing over the past 25 years.

1. We are cramming a ludicrous amount into courses with names like “research methods” or “data mining” or “statistics”. The poor soul teaching this course must cover data mining, data warehousing and business analytics in one course. That is impossible. Because students are often working full-time while going to graduate school and because schools have gotten more and more expensive, there is a lot of pressure to cut the number of courses. So, what used to be three courses is now one. When I learned multiple regression, it was a course all by itself. The normal equations, above, are not basic but not incredibly difficult, either. Certainly the vast majority of graduate students could learn to transpose a matrix and multiply the result. When I was in graduate school we had the luxury of spending an entire three-hour class just going over these equations and even some of the next week’s class for students who had questions. When we put too much into a course it is impossible to cover ANY of it in-depth. I have seen the same problem in my children’s math textbooks from fifth-grade on up. We wised up with the youngest one and now spend time at home making sure she understands not just the definitions and rules of, say plane geometry, but also how she can apply those. We fool ourselves by saying we are rigorous by cramming 42 topics into one textbook but all that happens is that people learn a little bit about a lot of things and a lot about nothing. I’m not joking here, I think this is why so many people want to go into management and “See the big picture” and will tell you, “I’m not a detail person”. Writing code that runs – that takes details, something as simple as ending a statement with a semi-colon, with knowing the difference in SPSS between rules for batch processing versus interactive. Details matter.

2. Again, because people want to “get out and get it over with” we are requiring fewer and fewer in terms of prerequisites. Many colleges no longer require any mathematics beyond algebra – if that! As I said before, I think College Algebra is an oxymoron. You should have learned algebra in high school. Certainly, many students never learned matrix algebra. When I was in graduate school, the professor could write equations in matrix notation because we were supposed to have learned it as undergraduates and the majority of us did. There was an entire course in descriptive statistics and if you didn’t have it as an undergraduate, guess what, you had to take it. And if it meant that you didn’t finish your graduate degree as soon as you would have liked, oh well. If you hadn’t learned it somehow, there was a teaching assistant and you went to him or her to help you understand the class.

So …. we don’t give our students the prerequisites at the lower level, at the upper level we cram three times as much in a course as they could really hope to comprehend in that short of a time. In the end, they don’t know very much about math and they are convinced that they aren’t any good at it because they don’t have the talent and math is hard. The truth is that math isn’t all that hard, it just takes time, like anything else, and we have no idea if they could be good at if we gave them the time and really tried to teach it to them, starting with,

“The identity matrix has all ones in the diagonal and zeroes in the off-diagonal elements.”

Here is my modest proposal to fix all of this:

1. Have LESS material taught in each math class, that is, fewer topics.

2. Require MORE classes of students

3. Do NOT let students waive or skip prerequisites unless they test out of them. (Do let students test out of classes, by the way. I always encourage that.)

4. Don’t write the mathematics out of courses. Leave it in. If you do #1 -3, students WILL understand it.

### Apr

#### 30

# Mom! Dad! I need help!

April 30, 2010 | 3 Comments

This is why:

A. I support affirmative action

B. I think some kids succeed in math and science while most don’t.

For the past several days, this call has been heard in our house at least once every five minutes,

*“Mom! Dad! I need help!” *

It is science project time for the sixth grade at St. Anne’s School. This year, the world’s most spoiled twelve-year-old has gotten on sciencebuddies.org and decided to do her project on how the density of a solution can be determined by the index of refraction. Plus, it involves lasers, so it is hard to beat that. So, every five minutes we hear,

*“Mom! Dad! I need help!”*

“Yes, what is it?”

*“I found on the internet that Snell’s Law is sine(theta1) divided by sine(theta2). What’s a sine function?”*

So, I sat down with the white board in the living room floor. (WHY do we have a white board on our floor? Who put it there? I don’t know.) and wrote:

y = f(x)

She asked,

*“So you multiply f and x, right?”*

We realized then that she had only gotten so far in school and to her the notation f(x) meant you multiply f by whatever is inside the parentheses. So, I explained the idea of a function, drew out a linear function, a curvilinear function and a sine function.

A few minutes later,

*“Mom! Dad! I need help!”*

*“Yes?”*

*“What is theta1 and theta2?”
*

So, we find an article on wikipedia that explains Snell’s law and has a diagram showing theta1 and theta2. We explain that theta is a Greek letter, that in math people use Greek letters a lot to stand for things. We point out which one is theta1 and which is theta2 on the diagram in the article. Satisfied, she writes up the first part of her science project – her question and hypothesis.

While she’s at work, Dennis gets on line and orders prisms and lasers from MiniScience.com . In an amazing burst of restraint he only orders two prisms and four lasers and nothing else. Realizing we need something to measure density, I walk down to Sur La Table and pick up a measuring cup that has a digital readout in the handle that tells the weight in grams of what you put into the cup.

She spends a good bit of Sunday afternoon setting up her apparatus and taking measurements, the first part of that is messing with the cup, putting in sugar, checking the weight in grams, dumping in water, calculating the percentage. It involves making a mess and not getting yelled at, a combination hard to beat. Calculating ratios and density is secondary. Being only slightly higher on the maturity scale, Dad helps.

Eventually, he holds the laser while she marks the spot it hits on the ubiquitous white board. They repeat this with solutions varying in density.

*“Mom! Dad! I need help!”*

“What is it?”

“On this paper it says I am supposed to write down which are my dependent and independent variables and which are my controls. Which is it?”

“Well, the thing that you changed would be your independent variable -”

“The density, how much sugar was in the solution. So the independent variable is the thing that changed and the dependent variable is the thing that stayed the same?”

*“No, variables change. That’s what vary means, to change…”*

A discussion of variables versus constants ensues.

Over the days that Julia works on her science project she learns about math, including trigonometry, measurement in grams and centimeters, refraction and more. She uses the Internet and finds some sites that interest her. She spends a lot of time on sciencebuddies.com , looking at other projects she doesn’t choose. She reads about Snell’s Law and refraction on wikipedia. Does she understand it all, even with explanation? Nope, but she understands a lot more than she did last month. In making her project board, she uses OpenOffice, decides she doesn’t like that and switches to Microsoft Office. She tries the chart feature within PowerPoint, decides that sucks and does her chart in Excel. She learns how to edit a chart in Excel….

*“Mom! Dad! I need help!”*

“Yes?”

“How do I fix this chart?”

*“Right-click on it. Pick select data. Click where it says X axis category labels.”*

We have another discussion about X axis and Y axis, categorical data versus numeric data. And so it goes until the project is done. Every year, Julia is required to do a science project because **every** child at her school is required to do a science project. Similarly, every child in her school takes Algebra in the eighth grade because that is the only math class that is offered. A teacher at a public school district bragged to me recently that her district did the same, every child took Algebra in the eighth grade. I found that fascinating because a few years ago I had done an evaluation for a program at the high school that was addressing the problem that 65% of the NINTH-GRADERS were failing Algebra. So, the solution, apparently, was to teach Algebra in the eighth grade.

This is like back in the 1970s when the solution to children from lower-income families entering kindergarten behind those from middle-class families was to have mobiles over the crib and other accoutrements of the typical suburban nursery.

What Julia has that those children don’t have is both a school that requires more of her and a home environment that provides the support to meet those requirements. There are three computers within reach of where I am sitting, with Unix, Macintosh and Windows operating systems all with either Open Office or Microsoft Office. There is a wireless network in the house. While the stuff makes it easier to do her project, it is not just the stuff and it is not just the requirement to do a science project.

She also has two parents sitting around who are willing (albeit grudgingly at times) to drop what they are doing and explain anything from the concept of f(x) to how to label the X categories on a graph in Excel. While I am writing this, because a documentary on the financial market is on TV, Julia and her father are arguing about economic theories based on rational behavior versus Schiller’s theory of irrational economic behavior. It involves some rather immature discussions of what he might do to the stuffed monkey he is offering to buy from her and tossing of the monkey back and forth.

In the past couple of weeks alone, Julia has probably received 20 hours of tutoring in math and science. Vygotsky would be pleased. Two years from now, she’ll be taking exams to get into high school and I am > 99% sure that she will get into the high school that we have already picked out. Is that arrogant? Nope. With nine years of the advantage of a good private school and day after day of patient (usually!) explanation of functions, sine(theta), X axis, angles of refraction and more I expect she will do well. Currently, she also has an older sister in the house who was a history teacher and is living at home while she finishes her masters. She makes sure to check Julia’s social studies homework and quiz her on that.

Why am I in favor of affirmative action? Because I am not stupid. The world’s most spoiled twelve-year-old has had years of individual tutoring, just about every resource money can buy and excellent, caring teachers every single day. I realize that any child that comes from a low-income home with parents who have never graduated from college and does just as well as Julia on the high school or college entrance exams is probably more motivated, smarter or in some way exceptional.

That’s the way the world is, right? I’ve never been too happy with that answer. So, I am sending an email to the urban schools program at the university offering to teach their teachers how to use SAS On-demand for Academics (hey, it will be free beginning in August). Yes, it is a small thing. I am pretty sure, though, that big changes come from a combination of small things added together.

Maybe you could do something to help. Probably you have your own spoiled twelve (or ten or eight) year old that hollers every five minutes that she needs help, but maybe there’s some little bit you could do to help someone else’s, too.

### Nov

#### 27

# Math websites , and other things, for which I am grateful (Hint: the new STEM initiative isn’t one of them)

November 27, 2009 | 2 Comments

Of course I am most grateful for my family. As daughter number two, a.k. a. , “The Perfect Jennifer”, commented yesterday,

*“This is the only family I know where everyone in the family actually talks to one another.”*

It’s true we don’t have any made-for-TV movie problems. No one is in rehab, no divorces, incarcerations, homelessness, domestic violence. The depth in our household is the occasional excess whining.

On the other hand, after reviving from a food-induced coma the various daughters had other plans. Jennifer was heading out to a club to catch up with some friends. I was surprised that any place would be open on Thanksgiving Day, but Jenn pointed out that plenty of people don’t have families, and other people have families that drive them to drink.

So, I went back to some notes I was writing on matrices and realized that I am also extremely thankful for people who take the time and effort to make their knowledge freely available on the web. I was extremely skeptical of the announcement this week that President Obama is supporting STEM (science, technology, engineering & mathematics) education. I picked this link out of the 500+ on the web because it included the interesting comment that most parents would rather talk to their children about drugs than mathematics and science.

While I wish the best of luck to President Obama and all of his corporate cheerleaders, I think the preceding statement is one half of the reason I suspect nothing will come of this. The other half is that the vast majority of teachers I have met don’t want to teach STEM and don’t want to learn it.

This makes me doubly thankful for those who are good teachers and generous enough to share themselves. Let’s take a simple tour of some nice websites with a topic, say, matrices.

Start with onlinemathlearning.com – the videos are excellent for a student who already has some interest in math and perhaps a basic understanding. There are no animated leaping leopards from rain forests here. You know, I am not sure those help. At worst, they give students the message that math in itself is not inherently interesting enough to learn. The onlinemathlearning site gives this explanation of a singular matrix:

*“If the determinant of a matrix is 0 then the matrix has no inverse. It is called a singular matrix.”*

This is followed by a very understandable video which shows that to invert a matrix one needs to multiply by 1 divided by the determinant. If the determinant is zero, it can’t be done. For those who did not know what a determinant is, that is explained in an earlier page. (Really, you should take a look at the video. It is quite a nice explanation.)

Somewhat surprisingly, given all the dissing it gets in academic quarters, wikipedia has some great math and statistics articles. For example, this one on positive definite matrices gives the following definition,

“An *n* × *n* real symmetric matrix *M* is *positive definite* if *z*^{T}*Mz* > 0 for all non-zero vectors *z* with real entries (), where *z*^{T} denotes the transpose of *z*.”

followed by some equally understandable examples. [Note: For those of you who are shaking your heads and saying, ‘THAT’S understandable?’ , trust me that I left out a lot of websites that seemed to be written with the attitude that if you didn’t already understand everything about mathematics it was your own damn fault and too bad.

A symmetric matrix by the way, is not, like you might suppose, simply one where it has the same number of rows and columns. No, rather it is a particular KIND of square matrix where the matrix equals its transpose. ]

If you would like to know a little bit more about positive definite matrices than you get from wikipedia, you can check out this page on “Not positive definite matrices, causes and cures ” . This is a link from Ed Rigdon’s SEM FAQ page. (It might be said that frequently asked questions about structural equation modeling is an oxymoron, unless the question is ‘What the hell is structural equation modeling?’ )

Now, it no doubt represents a failure in my education that I do not know who Ed Rigdon is. On the other hand, I’m pretty sure he doesn’t know who I am either. Regardless of our mutual non-acquaintance, he gets major kudos from me for the following statement (emphasis added) wherein he touches on one of the major flaws in much use of statistical software today:

*“Now, some programs include the option of proceeding with analysis even if the input matrix is not positive definite–with Amos, for example, this is done by invoking the $nonpositive command– but it is unwise to proceed without an understanding of the reason why the matrix is not positive definite. … Sample covariance matrices are supposed to be positive definite. For that matter, so should Pearson and polychoric correlation matrices. That is because the population matrices they are supposedly approximating *are* positive definite, except under certain conditions. So the failure of a matrix to be positive definite may indicate a problem with the input matrix.”*

Just because you can do something doesn’t mean you should. LISREL quite sensibly quits under the circumstance when the covariance matrix is not positive definite, and issues you a message to that effect, at which point you should feel shame.

My point, which I do have buried in here, is that STEM education is not about “making science cool”, it is about understanding stuff.

Here is what I am going to do right now for STEM education. I had a talk with my 11-year-old last week about possible questions that could be answered because we can tap into the high performance computing cluster from home and there are all sorts of enormous datasets, including census data. I suggested perhaps her class would like to come up with some questions. Julia made an A in math and did okay on her standardized tests (defined as not nearly as above average as I consider acceptable) and her lowest score was in ‘data interpretation’. Since I haven’t heard back from her teacher, Julia and I are going to hypothesize about such things as the number of 11-year-olds in the country and how many of them live in different regions, the average income, standard deviation of income, where she stands relative to that. Then, I am going to write a program to find all of the answers and run it on SAS 9.2 which we are still testing (no bugs so far) . Since I am still testing it and haven’t used the map library, this will be a nice thing for the university that I am working for free on Thanksgiving weekend and Julia’s knowledge of data interpretation and hypothesis testing will improve.

Whether she thinks it is cool or not.

« go back — keep looking »## Blogroll

- Andrew Gelman's statistics blog - is far more interesting than the name
- Biological research made interesting
- Interesting economics blog
- Love Stats Blog - How can you not love a market research blog with a name like that?
- Me, twitter - Thoughts on stats
- SAS Blog for the rest of us - Not as funny as some, but twice as smart. If this is for the rest of us, who are those other people?
- Simply Statistics, simply interesting
- Tech News that Doesn’t Suck
- The Endeavor -John D Cook - Another statistics blog