Review: Calculus (Kline)

Review: Calculus (Kline)

Calculus: An Intuitive and Physical Approach by Morris Kline

My rating: 5 of 5 stars

Morris Kline’s book, Mathematics for the Nonmathematician, is my favorite book on the discipline. Kline showed an amazing ability to explain mathematical concepts intuitively, and to situate them within a sensible human context. In his hands, math was not simply a series of equations or deductive proofs, but an integral aspect of our civilization: a crucial tool in our species’ attempt to understand and manipulate the world. The book changed my view of the subject.

So when I found that Kline had written a book on the calculus, I knew that I had to read it. Calculus represents the furthest I have ever gone with mathematics in my formal schooling. By the time that I graduated high school, I was a problem-solving machine—with so many rules of algebra, trigonometry, derivation, and integration memorized that I could breeze through simple exercises. Yet this was a merely mechanical understanding. I was like a well-trained dog, obeying orders without comprehension; and this was apparent whenever I had to do any problems that required deeper thinking.

In time I lost even this, leaving me feeling like any ordinary mathematical ignoramus. My remedial education has been slow and painful. This was my primary object in reading this book: to revive whatever atrophied mathematical skills lay dormant, and to at least recover the level of ability I had in high school. Kline’s text was perfect for this purpose. His educational philosophy suits me. Rather than explain the calculus using formal proofs, he first tries to shape the student’s intuition. He does this through a variety of examples, informal arguments, and graphic representation, allowing the learner to get a “feel” for the math before attempting a rigorous definition.

He justifies his procedure in the introduction:

Rigor undoubtedly refines the intuition but does not supplant it. . . . Before one can appreciate a precise formulation of a concept or theorem, he must know what idea is being formulated and what exceptions or pitfalls the wording is trying to avoid. Hence he must be able to call upon a wealth of experience acquired before tackling the rigorous formulation.

This rings true to my experience. In my first semester of university, when I thought that I was going to study chemistry, I took an introductory calculus course. It was divided into lectures with the professor and smaller “recitation” classes with a graduate student. In the lectures, the professor would inevitably take the class through long proofs, while the grad student would show us how to solve the problems in the recitatives. I inevitably found the professor’s proofs to be pointless, and soon decided to avoid them altogether, since they confused me rather than aided me. I got an A-minus in the class.

Though Kline forgoes the rigor one would expect in formal mathematics, this book is no breezy read. It is a proper textbook, designed to be used in a two-semester introductory course, complete with hundreds of exercises. And as fitting for such a purpose, this book is dry. Gone are the fascinating historical tidbits and gentle presentation of Kline’s book on popular mathematics. This book is meant for students of engineering and the sciences—students who need to know how to solve problems correctly, or planes will crash and buildings will collapse. But Kline is an excellent teacher in this context, too, and explains each concept clearly and concisely. It was often surprisingly easy to follow along.

The exercises are excellent as well, designed to progress in difficulty, and more importantly to encourage independent thinking. Rather than simply solving problems by rote, Kline encourages the student to apply the concepts creatively and in new contexts. Now, I admit that the sheer amount of exercises taxed my patience and interest. I wanted a refreshment, and Kline gave me a four-course meal. Still, I made sure to do at least a couple problems per section, to check whether I was actually understanding the basic idea. It helped immensely to have the solutions manual, which you can download from Dover’s website.

In the end, I am very glad to have read this book. Admittedly this tome did dominate my summer—as I plowed through its chapters for hours each day, trying to finish the book before the start of the next school year—and I undoubtedly tried to read it far too quickly. Yet even though I spent a huge portion of my time with this book scratching my head, getting questions wrong, it did help to restore a sense of intellectual confidence. Now I know for sure that I am still at least as smart as I was at age 18.

And the subject, if often tedious, is fascinating. Learning any branch of mathematics can be intensely satisfying. Each area interlocks with and builds upon the other, forming a marvelous theoretical edifice. And in the case of the calculus, this abstract structure contains the tools needed to analyze the concrete world—and that is the beauty of math.

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Review: Tools for Teaching

Review: Tools for Teaching

Tools for Teaching: Discipline, Instruction, Motivation.  Primary Prevention of Classroom Discipline ProblemsTools for Teaching: Discipline, Instruction, Motivation. Primary Prevention of Classroom Discipline Problems by Fredric H. Jones
My rating: 5 of 5 stars

Have you ever looked at the work kids turn in these days and wondered, “What will happen to this country in the next 50 years?” When you watch Larry sharpen his pencil, you know that the future is in good hands. It’s inspirational.

Last year I switched from teaching adults to teaching teenagers. Though I’m still teaching English, the job could hardly be more different. With adults, I could focus entirely on content; my students were mature, intelligent, and motivated, so I could think exclusively about what to teach them, and how. With kids, I am dealing with a classroom full of energetic, distracted, unruly, loud, and sometimes obnoxious humans whose main motivation is not to fail the upcoming exam. They’re not there because they want to be, and they would always inevitably rather be doing something else.

This probably makes me sound jaded and disenchanted (and I hasten to add that I actually have a lot more fun teaching kids, and my students are great, I swear!); but the fact is inescapable: when you’re teaching in a school setting, you need to worry about classroom management. Either you will control the kids, or they will control you.

It is the hope of every beginning teacher, myself included, to manage through instruction. We all begin with the same dream: to create lessons so dynamic, so enriching, so brilliant, and to teach with such charisma and compassion, that misbehavior isn’t a problem. But this doesn’t work, for two obvious reasons. For one, we don’t have unlimited control of the curriculum; to the contrary, our room to maneuver is often quite limited. And even with complete autonomy, having interesting lessons would be no guarantee of participation or attention, since it only takes one bored student to disrupt, and only one disruption to derail a lesson.

Even if you’re Socrates, disruptions will happen. When they do, in the absence of any plan, you will end up falling back on your instincts. The problem is that your instincts are probably bad. I know this well, both from experience and observation. Our impulsive reaction is usually to nag, to argue, to preach, to bargain, to threaten, to cajole—in other words, to flap our mouths in futility until we finally get angry, snap, yell, and then repeat the process.

But no amount of nagging creates a motivated classroom; and no amount of speeches—about the value of education, the importance of respect, or the relevance of the lesson to one’s future—will produce interested and engaged students. In short, our instinctual response is inefficient, ineffective, and stressful for both teacher and students. (Again, I know this both from experience and observation.)

Some strategies are therefore needed to keep the kids settled and on task. And since teachers are chronically overworked as it is—the endless grading and planning, not to mention the physical strain of standing in front of classes all day—these strategies must be neither too complex nor too expensive. To the contrary, they must be relatively straightforward to implement, and they must save time in the long run.

This is where Fred Jones comes in. Fred Jones is the Isaac Newton of classroom management. This book is nothing less than a fully worked out strategy for controlling a room full of young people. This system, according to him, is the result of many hundreds of hours of observing effective and ineffective teachers, trying to analyze what the “natural” teachers did right and the “unnatural” teachers wrong, and to put it all together into a system. And it really is systematic: every part fits into every part, interlocking like the gears of a bicycle.

This makes the book somewhat difficult to summarize, since it is not a bag of tricks to add to your repertoire. Indeed, its main limitation—especially for me, since I’m just assistant who goes from class to class—is that his strategies cannot be implemented piecemeal. They work together, or they don’t work. As a pedagogical nomad who merely helps out, I am not really in a position to put this book into practice, so I cannot personally vouch for it.

Despite this, Jones manages to be utterly convincing. The book is so full of anecdotes, insights, and explanations that were immediately familiar that it seemed as if he was spying on my own classrooms. Unlike so many books on education, which offer ringing phrases and high-minded idealism, this book deals with the nitty-gritty reality of being a teacher: the challenges, frustrations, and the stress.

The main challenge of classroom management—the problem that dwarfs all others—is to eliminate talking to neighbors. Kids like to talk, and they will talk: when they’re supposed to be listening, when they should be working, whenever they think they can get away with it. This is only natural. And with the conventional classroom approach—standing in the front and lecturing, snarling whenever the kids in the back are too loud—talking to neighbors is inevitable, since the teacher is physically distant, and the kids have nothing else to do.

Jones begins by suggesting board work: an activity that each student must start at the beginning of class, something handed out or written on the board, to eliminate the usual chaos that attends the beginning of the lesson. He then goes into detail about how the classroom should be arranged: with large avenues to the teacher can quickly move around. Movement is key, because the most important factor that determines goofing off is physical proximity to the teacher. (This seems certainly less true in Spain, where people are more comfortable with limited personal space, but I imagine it’s quite true in the United States.)

This leads to the lesson. Jones advocates a pedagogical approach that only requires the teacher to talk for five minutes or less at a time. Break down the lesson into chunks, using visual aids for easy understanding, and then immediately follow every concept with an activity. When the kids are working, the teacher is to move around the classroom, helping, checking, and managing behavior, while being sure not to spend too much time with the students he calls “helpless handraisers”—the students who inevitably raise their hands and say they don’t understand. (To be clear, he isn’t saying to ignore these students, but to resist the impulse to re-teach the whole lesson with your back turned to the rest of the class.)

This leads to one of the main limitation of Jones’s method: it works better for math and science than for the humanities. I don’t see how literature or history can be broken down into these five-minute chunks without destroying the content altogether. Jones suggests frequent writing exercises, which I certainly approve of, but it is also hard for me to imagine teaching a lesson about the Spanish Reconquest, for example, without a lengthy lecture. Maybe this is just due to lack of imagination on my part.

When it comes to disruptions, Jones’s advice is refreshingly physical. The first challenge is remaining calm. When you’re standing in front of a crowd, and some kids are chuckling in the back, or worse, talking back to you, your adrenaline immediately begins to flow. Your heart races, and you feel a tense anxiety grip your chest, intermediate between panic and rage. Before doing anything, you must calm down. Jones suggests learning how to relax yourself by breathing deeply. You need to be in control of your emotions to respond effectively.

Then, Jones follows this with a long section on body language. The way we hold our bodies signals a lot about our intentions and our resolve. Confidence and timidity are things we all intuitively perceive just from looking at the way someone holds herself. How do you turn around and face the offending students with conviction? How do you signal that you are taking the disruption seriously? And how do you avoid seeming noncommittal or unserious?

One of the most brilliant sections in this book, I thought, was on dealing with backtalk. Backtalk can be anything, but as Jones points out, it usually takes a very limited number of forms. Denial is probably the most common; in Spanish, this translates to “Pero, ¡no he hecho nada!” Then there is blaming; the student points her finger at her neighbor, and says “But, she asked me a question!” And then there is misdirection, when the offending student says, “But, I don’t understand!” as if they were in a busy intellectual debate. I see all these on a daily basis. The classic mistake to make in these situations is to engage the student—to argue, to nag, or to scold, or to take their claim that they “don’t understand” at face value. Be calm, stay quiet, and if they keep talking move towards them. Talking back yourself only puts you on the same level.

The penultimate section of the book deals with what Jones calls Preferred Activity Time, or PAT. This is an academic activity that the students want to do, and will work for. It is not a reward to hold over their heads, or something to punish the students with by taking it away, but something the teacher gives to the class, with the opportunity for them to earn more through good behavior. This acts as an additional incentive system to stay on task and well behaved.

The book ends with a note on what Jones calls “the backup system,” which consists of the official punishments, like suspension and detention, that the school system inflicts on misbehaving kids. As Jones repeatedly says, this backup system has been in place for generations, and yet it has always been ineffective. The same small number of repeat offenders account for the vast majority of these reprimands; obviously it is not an successful deterrent. Sometimes the backup system is unavoidable, however, and he has some wise words on how to use it when needed.

Now, if you’ve been following along so far, you’ll have noticed that this book is behaviorist. Its ideas are based on control, on incentive systems, on input and output. As a model of human behavior, I think behaviorism is far too simplistic to be accurate, and so I’m somewhat uncomfortable thinking of classroom management in this way. Furthermore, there are moments, I admit, when the job of teaching in a public school feels more like working in a prison than the glorious pursuit of knowledge. Your job is to keep the kids in a room, keep them quiet and seated, and to keep them busy—at least, that’s how it feels at times. And Jones’s whole system can perhaps legitimately be accused of perpetuating this incarceration model of education.

But teachers have the choice of working within an imperfect system or not working. The question of the ideal educational model is entirely different from the question this book addresses: how to effectively teach in the current educational paradigm. Jones’s approach is clear-eyed, thorough, intelligent, insightful, and eminently practical, and for that reason I think he has done a great thing. Teaching, after all, is too difficult a job, and too important a job, to do with only idealism and instinct as tools.

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