Book Review-A Mathematician's Lament By Paul Lockhart

In the forward to this monograph. Kevin Devlin of Stanford University, a well renowned mathematician, tells the story of how Paul Lockhart, someone who had given up his career as a research mathematician to devote himself to the mission of improving K-12 mathematical education, turned an earnest but obscure essay into a resounding statement.

The first genesis of this book is as a 25-page document that was passed around in the mathematical education circles. It became a sensation because many felt that Paul Lockhart had hit the nail on the head with his observations; observations and beliefs that resonated with mathematics educators; indeed, he struck a very sensitive nerve. As this document was passed around, it became a clarion call to mathematicians, mathematics teachers, and anyone who has a passion for how mathematics is taught.

A Mathematicians Lament is short and compact. Paul Lockhart had a lot to say, and he says it with urgency and alarm. Part I of  the book is the lamentation, he goes into everything that he feels is wrong with mathematical education. He makes his argument progressively starting with a discussion on mathematics and culture,  then a discussion on mathematics in the school, a dive into the national mathematics curriculum — a chapter in which he was unsparing in his criticism. In the last chapter in Part I, Lockhart zeroes in on a well-known and well reviled target: high school geometry.  Lockhart gave it a subtitle: Instrument of the Devil. This is his coup de grâce, his pronouncement on the abysmal state of mathematics education in the United States.

He expounds on the insidious practice of limiting mathematics education to just computation, while emphasizing the mechanical and uninspiring practice of training skills without giving the students a vision of what true mathematics is. We don’t give the students enough credit for being perspicacious enough to sense the immutable and deep beauty of mathematics. We don’t give the allure of the mathematical abstraction enough credit for being able to inspire and elicit  passion from the students; we think that the average student could not fathom the depths of meaning of mathematics; and that the student can only appreciate mathematics in its most utilitarian and unimaginative incarnation. It is an insult to the students and to mathematics.

As an engineer by training, I managed to survive my formal mathematics training with my love of mathematics intact, even though I knew my talent for theoretical math is limited.  I recognize all the stated pitfalls and shortcomings of how mathematics is taught because I had experienced it firsthand.

Although I  appreciate the beauty of mathematics, as I had aspired to be an applied mathematician; unfortunately, I had made a mess of the higher math that I took as a grad student in engineering, I didn’t have the patience nor the curiosity to sustain my interest because I was studying to gain a degree rather than studying for the love of a discipline. I was resigned to take enough applied math to help me become an engineer even though I was always curious about doing pure mathematics. Even as I have  resigned myself to the fact that I won't ever be a pure mathematician nor  even be a good applied mathematician, I have come to appreciate and love the subject.

In the second part of A Mathematician’s Lament — titled Exultation — Paul Lockhart made his elevator speech  to  anyone and everyone reading about the beauty of mathematics. He assiduously avoided the equations, a smart decision in my estimate. He dealt with mathematics as a holistic entity. He is much more eloquent in stating his case than I will ever be, so I will let the reader  read the book rather than dilute his passion and his narrative.

He discusses the common sensical instinctive aspect of  mathematics. There are crude but effective sketches about the points that he wanted to make, adding to the intuitive charm of the narrative.  He refrains from delving into the dreaded and unwelcoming geometry that he wrote about in Part I;  he uses simple sketches to ease the reader into mathematical thinking.

When he hit his stride talking about mathematics, it is a beauteous expression of passion he speaks of the raw beauty of mathematics that makes it so attractive, intoxicating,  and habit forming for so many. It is as if  mathematics is some kind of addiction. And to mathematicians that I know, and to a much lesser degree to me, that addiction is very real.

The second part of the monograph reminds me of the passion exuded by another book written by a mathematician. Francis Su wrote Mathematics for Human Flourishing, (Su 2020). I reviewed it in 2020. (https://polymathtobe.blogspot.com/2020/02/book-review-mathematics-for-human.html) Prof Su had the advantage of having a book to make his point about the allure of mathematics. It is perhaps a good companion book to buttress the second part of the argument.

The monograph is an extended essay identifying the problems with the way we have taught mathematics, how the math that is taught is contrary to what the mathematics lovers love about mathematics; what mathematics is in the eyes of those that are knowledgeable in the art; while  proposing in broad strokes what need to be done to change that paradigm. It is a timely and necessary clarion call to our society and our educators that we are irresponsibly squandering our opportunity to educate our society in the art of thinking, questioning, and creating. It is an attempt to reverse the trend, and more broadly, it is a valiant attempt to convince a math deficient public that they are missing the boat, and our society will suffer.

I hope that this is not just preaching to the choir, but the obstacles to universal understanding of the importance of the subject is quite high. I hope that Paul Lockhart is not too late.

Works Cited

Su, Francis. Mathematics for Human Flourishing. Yale: Yale University Press, 2020.

 

 

 

 

Volleyball Coaching Life-On the Experienced Retiring

 A friend of mine told me in the middle of the volleyball season that we should be seeing many experienced and well-known volleyball coaches calling it quits at the end of this season. He said in very strong terms that many people have just had it and are burnt out and they don't feel like they can do the job correctly, or as they perceive to be doing it correctly.

A person posted a list on  VolleyTalk along with his opinion:

Tom Hilbert, Hugh McCutcheon, Marc Rosen, Joe Sagula, Dave Rubio are the big names, but Chad Callihan also retired from Wyoming. Suzie Fritz, Kent Miller, Mike Sealy and Ray Gooden may not have left on their own terms but they are certainly well-known coaches with good records and are they trying to get back in?
Add in Shelton Collier from D2, Jenny McDowell from D3 and Bob Bertucci, who left the men's game this past offseason. Last year we lost Russ Rose, Cathy George, and Fran Flory...
There's a lot of significant people who aren't in the game anymore. There may be more (there always are) but I don't think we need to speculate about who else might leave - we've lost more than enough people to call this prophecy "true."

We can surmise that they are also tired of the grind required in order to keep up the pressure: from themselves, from their staff, from the players, from the administration, from their own families, from the parents of the players, and from the general fan base. According to the Jobs Thread on VolleyTalk, at the end of the 2022 season there are 34 Division One head coaching openings, 22 Division Two, 216 Division 3, 37 NAIA and 9 JUCO. Of course not all of those are people quitting, some are coaches getting fired, some are coaches who are transitioning to new jobs and other coaches are moving up in the food chain.  

We know we've gone through three difficult years of transitions coming rapidly, especially within the volleyball/athletic landscape, one may call the confluence of events the perfect storm. There is the COVID pandemic, and the way the powers that be chose to deal with the great unknown. There is the NCAA’s decision to give those athletes that are affected extra years of eligibility; which no one would say in the name of fairness, is a bad decision. The problem is the way they chose to ignore the unintended consequences on the rosters, scholarship limits, and effects on future recruiting classes. The combination of the COVID eligibility and the transfer portal  made a difficult situation nearly impossible.

As I reflect upon the realities of college volleyball,  I — inevitably — started to draw a comparison to my other life as an engineer. I start to think about the ebb and flow of careers and the relative attributes between the younger and older employees in the work force.

When I was working as a mid-career engineer, i.e. I wasn't wide eyed, innocent, and willing to try everything just because I could; nor was I decrepit, stubbornly set in my ways,  and cynical because I've seen it all, done it all, and knowing that there's nothing new under the sun. A newly hired HR person, who was fresh from the campuses stepped in front of the assembled workers at the work site and declared that the company needed to get rid of all the older workers and hire newly minted engineers because they are the ones with the bright new ideas, the initiative to break new grounds, and get things done. She followed up with the thought that all the older workers do is take up space and bring down morale with their skepticism, and they don't really add any value to the organization.

I took offense to that, as did most of my more experienced colleagues. There is an old joke/story which tries to make light of the bias for the new and young. A machine started to malfunction at a manufacturing plant. The workers and managers were at a loss, and nobody knew how to fix it. The head of the plant got in contact with the old manufacturing engineer who's been retired many years. He arrives unceremoniously, looks around the machine, asks for a hammer, and  whacks the machine three times in the middle of a panel with the hammer. Lo and behold, the machine sputters to life and starts to work smoothly anew. The retiree walks away saying: “you guys will be getting a bill from me”. A few days later, the bill arrives with a single number: $10,001. The corporate CFO blows his top, and calls the retiree screaming: I don't understand why I should be paying you, this is an outrageous amount of money when all you did was whack the machine three times with our hammer. I need an itemized bill. Another bill arrives a few days later, it's itemized. It says: $1: Hitting the machine three times, $10,000 for knowing where to hit the machine. And that, to me, is the essence of having experience: knowing how to use the experience to diagnose the problems, knowing why the machines were not working, and how to fix it.

Realizing, of course, that not all experience is great or useful. My friend Al’s favorite saying was: do you have 20 years’ worth of experience, or did you just have one year of experience 20 times? I have known many people who are very limited in what they have experienced, what they have observed, and what their experience can do for them in application. But, when it comes down to it, I would rather have experience because experience is not something that can be learned or memorized from a book. Experience is sentient information, connected to learned knowledge filtered through the prior experience into more knowledge.

The advantage that the inexperienced have is that they are not cynical because they have no pre-conceived notions. In their minds, there are no highs too high and lows too low because they have no experiential reference. The main mission of young engineers, actually all engineers, are to wreak havoc, create problems, propose outrageous solutions, experience as much failure as they can, but also to learn from them; that is how one accumulates experience, rather than having the same experience repeatedly.  

The spirit of making mistakes, experiencing failures with every challenge is the first thing that dies as the engineer gets older. The hard limits set on the inexperienced engineers come from both the employer’s conservatism, they are after all, paying for those failures; far more insidious is the engineer’s tendency to self-censor in deference to the employer’s anticipated response to failures. Not all engineers wither under the employer’s expectations, some are able to keep their curiosity and spirit of fresh perspectives alive.  

The  chief engineer and a bunch of other engineers at one of the companies I work for were sitting around having a beer after a really good technical discussion, I asked him: how much money do you think you've cost the company over the years by your failures or errors in judgement? He ponders the question a bit and he says: $2 million. We were stunned by the number and started questioning him about how he got to that number. He replied: if I didn't cost the company $2 million, then I wouldn’t be trying hard enough  and I wouldn’t be contributing enough with all of my ability, because failures and mistakes comes with the art of engineering.

This is what young engineers bring to the party from the beginning. They should have no fear; because there will be enough experienced engineers around to check their ideas and avoid the truly outrageous and predictable failures and mistakes. Of course, corporate managers and the engineer’s natural proclivity towards being safe and conservative can and most often does kill the spirit of invention that are the most desirable trait for the newly minted engineers. Management will often whack the new young engineers on the nose with a figurative and proverbial rolled-up newspaper when they make any mistakes, which effectively cures them of any initiative and creativity.

Reality of course demands something in the middle: engineers with creativity, courage of their convictions,  and the spirit of invention while also having experience and the ability to think critically to avoid making the nonsensical, predictable, and egregious errors.

As I am on the downside of my engineering career. I have been taking a 20/20 hindsight at my career. Regrets and lamentations are the natural reaction while looking back at the painful mistakes, miscues, and missteps that are vivid in my memory. There is no sense in crying over spilt milk. I judged myself as being too deferential as a young engineer and too cynical as an older engineer. I deferred to my experienced and elder colleagues when it comes to ideas that are untested and original, and I self-censored those same type of ideas later on in my career because: ”That will never work.” I was fearful of exposing myself and suffering from the  vagaries of corporate politics.

I have rediscovered my motivation to be aggressive, to try things, and to make mistakes after I moved on from the corporate world. Why did I? Why am I so much more open to it? The main motivation, as far as I can figure it, is that I no longer have that sword of Damocles of being employed in the long-term hanging over my head. Partly because the subjects of my study are topics that I find curious. The general feeling is that if I happen to contribute, so be it.

Getting back to experience, I look  at the amount of experience that are cycling out of volleyball coaching from the people who are retiring, getting out of the game — either voluntarily or forcibly — and I think about the significant, deep, and broad body of knowledge that will be disappearing from the state of the game; a pool of knowledge that can’t and possibly won't ever be taught or  repeated because the knowledge is  resident within that person. I also think about the parallel loss of knowledge in engineering that disappears with engineers that are made redundant and retired. I wonder if all that is lost forever, whether all the knowledge lost throughout history is also lost forever. Do we lose a volume of knowledge that is an equivalent of the library of Alexandria with every generation of retirees? 

Of course, not all the experiential losses are equally valuable for posterity. Engineering and coaching evolve and mostly improve as the arts advance over time. Analytical methods are improved upon by numerical methods bolstered by computational capabilities. Coaching methods are improved upon by the integration of psychology, human mechanics, cognitive sciences, and motor learning into known paradigms. The question is whether the body of knowledge lost is valuable.

Why talk about this? Well, it just occurs to me that we are losing knowledge and wisdom in the form of intangible, undocumented, and unorganized experience. I am sure that many of those who have left the game, both in volleyball and engineering, will be documenting some of their knowledge in the future. Hugh McCutcheon already has a new book published recently in conjunction with his transition from coaching players to coaching coaches. That is not the knowledge that I am concerned with; my concern is with the intangible knowledge, the extemporaneous decision-making basis of the retired coaches and engineers.

It seems that we are in the midst of a changing of the guard, coaches who are of a certain generation are leaving the game, realizing that coaching volleyball at the highest level of competition in the US is a young person’s game. While younger coaches are getting their opportunities to sit in the big chair, it is the natural course of human endeavors. My recognition of the ebb and flow does not mean that I cannot lament and wonder about the loss of wisdom, just as I lament and wonder about the loss of wisdom in engineering as I watched my mentors and icons retire and pass away.

My hope is that we take advantage of these coaches and engineers, to use them as references, sounding boards, and resource. My hope is that we actively seek out that wisdom rather than waiting for them to volunteer their wisdom.

Given our society’s current preoccupation with Artificial Intelligence, I hope we are perspicacious enough to indulge and invest in natural intelligence that are the retired coaches and engineers.