Forty Years of Teaching Math, Computer Science and Engineering Science at a Community College- Part XIII- Linear Algebra

From 2001 until the time I retired I taught Linear Algebra every spring. In the 1970's Linear Algebra was a gateway to higher math for math majors and was taught in a very abstract fashion.  Through the years Linear Algebra became more applied, catering more to engineers and computer scientists.  A second course in Linear Algebra includes all the theory that was previously taught in the first course, and then some.  My course changed with the times.

I was pleasantly surprised in my last few years to have a fine group of computer science students. Some of them came into the class with a list of topics they wanted to learn. These bright students were well read in their field.  There was this barrier that they could not cross, and it was mostly Linear Algebra. Linear Algebra had emerged as a central course for AI in general, and specifically for machine learning.  Linear Algebra is also central for computer vision, computer graphics, gaming and robotics. Suffice it to say, Linear Algebra is a very important course, and its importance is growing as we speak. 

A simple problem in Linear Algebra is to take a bunch of points that represent a surface in 3D, multiply each point by an appropriate matrix, and surprise, the new set of points is the original surface rotated about a given axis.  Encoded in the rotational matrix is the axis of rotation and the number of degrees of the rotation. Humans see by viewing two 2D images - one rotated a fixed number of degrees from the other - and constructing a 3D image in our brain.  In Linear Algebra we can replicate this for a machine. 

Linear Algebra is also useful in just about every engineering course.  It is not required in engineering because there is already 150+ hours required by ABET.  Engineering students are faced with taking 153+ hours or learning the subject on the fly while they are taking 4 or 5 engineering courses in their junior year.  A handful of engineering students ended up in Linear Algebra, although they were not as curious about the subject matter, mostly because they were curious about how they would survive Dynamics, Mechanics of Materials, Circuits and Differential Equations, along with Linear Algebra, all in the same semester!

In my next-to-last semester I had one particular student who, on the first day, asked me if we were going to learn how to factor matrices using QR Decomposition, and I think one other method.  There are several different types of factoring, all with specific uses, and multiple algorithms for each type of factorization.   

Factoring matrices is particularly useful with machine learning. Specifically, it is important in determining what variables are not important in a set of data.  Humans are particularly good at this as we go through life. When we round a corner, for instance, we decide the color of a nearby house has nothing to do with how fast we should go.  Matrix factorization and Linear Algebra in general helps us replicate that process with machines while they learn. 

Anyway,  a student asked me about a specific kind of matrix factoring, so I told him I would oblige.  Weeks later, he asked again, and I said not yet. I can't remember how many times it was brought up, but I stalled until the end of the semester.  

When I did finally show the class how to do QR factorization, the algorithm I used was easily and fully understood by the whole class, because various elements were based on all they had learned throughout the whole semester. During Week 1 of the class, I would have lost them 5 minutes in. By week 15 the class not only understood how to do the procedure, but they fully understood why the procedure worked, because of all the theory that was covered previously in the class.  A topic that was once an impenetrable barrier was now a joy to learn!

I explained how pretty much anything else these students needed to know in Linear Algebra, at least applied to machine learning, AI and any other Comp. Sci. subject, could likely be self taught and fully understood. The course I put together made it so.  It was a great meta-lesson, brought about by the students' curiosity.  I had heard numerous times that our computer science students were ill-prepared and slow.  I certainly did not see this with the dozen or so students I worked with each spring in Linear Algebra. 

The above picture is from one of many short courses available for students entering various fields that require Linear Algebra.  I used to pass one or two of these out, allowing students to follow along as we developed the course. My course always covered about 95% of each of these short courses. In robotics, one of these constituted a beginning graduate-level course. I told students they were welcome to ask any question about Linear Algebra that pertained to any other field, if they chose.  Sometimes they would take me up on that offer, successfully.  Through the years I got all sorts of crazy questions.  It was a very simple and efficient way to use my students' curiosity to keep myself up to date in such matters.  Looking back, it is ironic how I let my students teach me, and then I in turn was able to teach them in a more up-to-date fashion. 

On a humorous note, I would always reserve the last day of class to "attack Canada." There was always a build up to this point, where I would promise to give them enough tools to launch an attack.  The students, of course, knew I was not serious.  It was very funny because it was such a preposterous idea. 

It is easy to formulate such a problem where we are adding soldiers by breeding, and subtracting them due to combat loss and natural deaths. Since it is an easy problem to visualize, it is a good problem. The usual solution requires solving a linear, first order system of differential equations. Using eigenvectors (part of Linear Algebra,) the whole solution can be solved very simply, without using any calculus.  

Since our population is about 10 times that of Canada, we needed to make Canadians great lovers and even better fighters to win a war.  Canadian beer would enter the picture, at least on one side of the equation. We had a lot of fun with that!  

Eigenvectors are a central topic for all the various subjects listed above, and a subject that is often given little attention at universities. Eigenvectors are usually in a college or university Linear Algebra syllabus, but quite often disinterested professors don't organize their courses well enough to give it due attention at the end of the semester. At the end of the semester we would bring to bear all the theory we learned, to give students a great understanding of such things. 

 

Forty Years of Teaching Math, Computer Science and Engineering Science at a Community College- Part XII- Physics

 

I wasn't at all surprised when Nick Jerla earned his Ph.D. in physics from SUNY Buffalo.  Nick is the person on the left of this photo. 

I first met Nick when he came to my office, asking for permission to take Linear Algebra, despite not having the prerequisites. Nick explained that he was trying to learn quantum mechanics, and he had run into a barrier with some terse linear algebra. I let him in.  Later he explained that he had failed 11th grade trig in high school.  I was skeptical of his abilities at that point.  Perhaps he was a dreamer who watched Good Will Hunting, and thought he was of the same caliber. 

In the actual Linear Algebra class Nick plowed through every topic, leaving no stone unturned. I knew at that point that he was something special. 

The last time I saw Nick was when he stopped by with a question about the 2-dimensional heat equation, and its application to a conduction problem on a disk. He had to solve one problem, and use the result to solve another. I explained what they wanted and he was good.  Nick then told me his transfer to U.B. was seamless.  His junior-level courses all picked up where we left off at NCCC.  That did not surprise me, either. 

The last chapter that I painstakingly added at the end of our calculus sequence develops the language of Maxwell's Equations, which govern the whole electro-magnetic universe. I developed my own problems that gave students a very deep understanding of various theorems in that chapter. I also added spherical coordinates to Calculus III, which are useful in quantum mechanics. In Differential Equations I added a plethora of applications of Newton's 2nd Law, including mechanical vibrations, and also circuits.  Various physics professors never seemed to explain these things well.  I must have "physics" written on my forehead, because through the years I answered more physics questions than I can count. Almost 40 years ago a student named Cheryl walked into my office and asked me a physics question.  Cheryl had pestered her physics professor and could not get to the bottom of some motion problem. That set in motion a series of questions that continued until my retirement. 

About 15 years ago my daughter took a C++ programming class at NCCC.  She was not a student at NCCC, so no one knew who she was. One day she was listening to a group of students behind her talking about physics.  The conversation went something like this.  Bill: if you have a physics question, ask Mr. T. Al: Yeah, he is real cool about it.  Really chill. Rachael: He can answer anything.  And he has a really simple way of explaining everything. 

One large problem I had Differential Equations students work on is J. J. Thompson's original determination of the ratio of the charge to the mass of an electron. In 1899 Thompson shot an electron beam into an electric and magnetic field, and it travelled in a cycloid path. That part of a cycloid looks somewhat like a quarter circle. Determining the ratio of e/m required solving a system of differential equations.  The students love this problem! It was an ingenious experiment, seeing how all the complex math played out. I also added some numerical methods that appear here and there throughout physics, including power series methods.  And then there were mixture problems, that perfectly matched the first junior-level Environmental Engineering course at U.B. Everything I added about physics applied to all the various branches of engineering, as well.  

I don't ever remember students complaining about Differential Equations being too difficult.  Students can accomplish some pretty amazing things, given the right circumstances.  Their confidence began with my confidence in them. 

I passed off my Differential Equations course to Tadeus Krupa back in 2013, when I got involved with teaching Engineering Science. Tad weaves everything together, as only he can do.  In the end he will start with 20 students, finish with 20, and the students will have received an excellent education, and are no worse for wear - all the while learning about 50% more content than when he and I took the class many years ago. 

Tadeus, by the way, almost did not get a job teaching at NCCC.  A shameless county legislator tried to put an unqualified candidate - a candidate who barely passed calculus - in his place when a search was done. Carolyn Goldberg, who was chairing the search, stood up to the county. That search was called off, under threat from the county. The very next year NCCC got a grant to replace a key faculty member, and the admin was going to spend the money on something else. Bulldog Tony Gullo called the admin out, and Tad was snuck into the department while the county legislature was looking the other way!  In the years since, the public has benefitted by having our graduates build rockets and stadium roofs, instead of receiving a substandard course taught by an unqualified instructor, with transfer students hanging on white knuckled, likely changing to a less quantitative curriculum.  Tad Krupa is unquestionably the best math teacher I have ever known, with no close second.

Very few people have the capacity and temperament to do what Nick did.  But, my point is that, for someone of his ability and interest,  it is possible to do what he did at NCCC, and to do it well.  Many more came before him, and did a similar thing according to their interest and ability. Nick was not a fluke.  We had countless students plug into all levels of STEM, successfully, and the math department played a pivotal role by removing many barriers, and propelling students forward to accomplish way beyond their wildest dreams. 

About 20 years ago I was invited to the wedding of Brian Milleville, who is now a math professor at ECC.  Brian teaches all their difficult courses, and does a magnificent job. One of his aunts asked how I knew Brian.  I told her I was his professor and advisor at NCCC.  She did not believe me at first.  She did not think it was even possible to take college math at NCCC. It was as if I stated that the Pope had become a Pentecostal.  I might just as well have stated that Sanborn really was Silicon Valley. 

When Brian got married he was in a Ph.D. program at Indiana University.  Brian had transferred from NCCC to U.B., and then was accepted into 6 Ph.D. programs for theoretical math.  He was ABD, but did not finish, instead taking a job at ECC and getting on with his life.

STEM at NCCC has been invisible to the public over the years.  Doc Kwitowski tried to advertise our programs and success stories during the Miller era, and was not allowed. It is one thing, after all, to give the public a brochure.  It is quite another to actually prove to the public that these programs work. In the 90's there was an NCCC commercial that was often played at area movie theaters. In that commercial a student making a clay pot was shown. I remember shrinking into my seat as fellow movie goers joked about going to NCCC to learn how to make clay pots. There is nothing wrong with fine arts, but during that era the public perception was that making clay pots was our forte, and that there was nothing more rigorous at NCCC than making clay pots. 

Dr. Cleveland would have gotten along with us, but she was not here long enough.  And Dr. Klyczek, well, it was no secret that all of our money went into advertising the NFCI.   Every time someone was hired at Delaware North there was a press release. 

At the time I did numerous searches on the NYS Department of Labor website, and the search term "Engineering" produced 500+ hits for a 25-mile radius of North Tonawanda, whereas search terms of "chef", "cook", etc. routinely returned about 10 hits.  Of the 10 or so hits, usually only one or two at most required a degree. 

In 2017 Kiplinger's ranked the top 10 undergraduate majors.  All but two were STEM fields, and the two that were not were health care-related, and required a lot of STEM classwork.  Kiplinger's also ranked the 10 worst, and the worst of the worst was Culinary. Their main reason was that very few culinary jobs require a college degree.  Kiplinger's recommended getting a business or accounting degree, and just working in a restaurant to get a handle on the business. 

I ran engineering forums for a number of years, inviting numerous former students back to speak. At one such forum, we had an engineer who designed the roof for the world cup soccer stadium in South Africa, an electrical engineer who was the plant manager at American Brass in Tonawanda, a mechanical engineer who built the wind tunnel at Calspan (not a former student) and a mechanical engineer who now works at SpaceX.  

Later on we had a roboticist who worked on a new autonomous vehicle manufacturing facility at BMW/Clemson, the senior marketing manager at Micky Thompson, a mechanical engineer who designs hunting bows, an environmental engineer who works for the DEC, a senior engineer who works for National Fuel gas,  a pharmaco-kineticist who worked for Merck, a VP of Analytics at M&T Bank, an actuary who is the senior underwriter for Magellan Health Care and an electrical engineer who works at a nuclear power plant. 

The engineer who designed the world cup stadium roof went on to also design the roof for SOFI Stadium, and also the LA Clippers stadium. He spoke to our students numerous times. He is the leading expert in this country for rubber membrane roofing systems.  He said other companies try to get into that space, but hire him as a consultant after they botch things up. 

None of this was newsworthy, of course.  Public Relations would never advertise these events, despite all the blurbs I put together each year for a press release. I assume PR had their orders from on high. In the end I transitioned these events to just having former students come speak directly to our students, outside the purview of the public and our administration.   Every single former student I asked to come and speak did, with the exception of one industrial engineer who wanted to, but had a conflict. 

I made the best of it.  And my students prospered.  It was quite satisfying doing all this with little institutional support. 

Forty Years of Teaching Math, Computer Science and Engineering Science at a Community College- Part XI - Teaching 9th Grade Math in a Lutheran School, Part II

 

My involvement in the Lutheran schools was beneficial to the math department at NCCC.  In 1999 academic affairs would make a move to offer college courses for credit in area high schools. The department called on me over and over again, since I thoroughly knew what was going on with k-12 math. 

In the spring of 2002 there was a public debacle with the Math A/Math B high school math curriculum.  Math A was a 1.5-year course that was required for graduation from high school.  Formerly students could either pass a 9th grade math regents exam with mastery, or pass a lower-level RCT exam with mastery. By requiring all students to pass the Math A exam, NYS was increasing the standards across the board, or so said the propagandists. 

Numerous students could not pass the Math A exam in the middle of 10th grade, so it could be taken again, as many times as required. By the spring of 2002 there was a log jam of students who were on their last chance.  70% of the state failed that exam. A public outcry followed. In the end the state put a huge scale factor, or curve, on the exam. A raw score of about 30% was scaled up to a passing grade of 65.  And a significant portion of the exam was multiple choice!

A statewide committee of math educators was formed, and colleague Carolyn Goldberg was asked to be on the committee.  She asked me if I was interested, and I declined. By then my wife and I were busy driving our four kids around to various activities.  I did offer my advice to Carolyn, and she took me up on that offer numerous times. A handful of my ideas made their way into the NYS high school math curriculum. 

The story Carolyn told me was that the large committee started with Kindergarten and worked their way up. By the time they got to high school most of the committee had stepped down, which left Carolyn and a clueless college professor who maintained that geometry proofs is what turns students on to math. My recollection is that Carolyn did the bulk of the work in putting together the whole high school curriculum.  How did I help out?  As one example, the new SAT exam was just developed, and the initial NYS draft was missing some of the new content, so I wrote what was missing on a piece of scrap paper, and off Carolyn went to Albany that weekend with the missing content.  I also told her to make 9th grade math "Integrated Algebra" and not just a random integration of various strands, and so it was. 

The new math curriculum was a vast improvement over the Math A/Math B debacle.  High school math teachers uniformly agreed with that opinion. Eventually it was superseded by Common Core, which was good, although according to the Thomas B. Fordham Institute, Common Core actually lessened our standards in NY, but raised the standards in southern schools.  

From 1996 to 2012 I taught 4-8 students per year in a Lutheran school, which was more than half of each 8th grade class. Both St. John and St. Peter had no admission standards, unlike elite private schools.  Their student population was roughly half from the congregation and half from without. In terms of native ability, students formed a pretty good cross section of the population. There were two differences worth mentioning.  (1) Special needs students were rare, and could get better services in the public school.  (2) Lutheran schools were likely to have students with parents who were more involved with their children's education.   So, comparisons with the public schools are difficult. 

I had a listing of all my students and what happened to them later, but that list was lost.  None the less, over half of my students went to college for a very quantitative STEM curriculum. Most of the STEM students majored in engineering (more than 2/3 of them). Of the whole lot, only one failed to graduate from college in the field in which they started. That surprised even myself. These difficult STEM fields have around 80% attrition, nationwide. 

Looking back I expected my students to do well in high school and beyond. Why?  Everything I taught them significantly amplified how much they learned in 10th-12th grade math, physics and chemistry.  And various branches of engineering are applied physics and applied math.  It was not obvious how to do this.  But, doing this was not as difficult as you may think.  It is easy, if you know how.