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

Filed Under Algebra, Dr. De Mars General Life Ramblings, statistics

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.

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i love mathematics.

[...] deletion is that you may end up with a matrix that is not positive definite. This is a bad thing. I wrote a blog a while back on the sadness of non-positive definite matrices. This page from Ed Rigdon’s structural equation modeling site explains a little more about [...]