Study tip for the day: Quality vs. Quantity

People often ask me how many hours one must spend outside of class per hour inside of class to succeed in a university-level math course. I hesitate to answer. The reason is that the question is ill-posed. To really understand the logical structure of mathematical ideas, how they work and fit together, why they exist and work the way they do, one must really spend the time to dig into every nuance of the idea. To do this on a research level, a mathematician understands that she must isolate herself from outside distractions for sufficient periods of time to fully explore the structure of a new idea. The isolation-booth manner is as vital to the process as deep, dreamless sleep is to the health of a person. I call the process of getting to the level where one can focus exclusively on the task at hand without distraction as "going deep". It is kind of a meditation-type thing, and only really works when one practices it regularly. It can be dangerous, though, as the day-to-day tasks tend to get neglected. But it is quality time for understanding complicated mathematics, and easily outdoes simple quantity time when the latter is filled with noise and attention-taking "shiny objects" (distractions).

It's loads more productive to spend an hour in an isolation booth environment focusing solely on your mathematics work instead of 3, 4, or 6+ hours poring over books and notes while checking your phone, listening to music or chatting with friends (or potential friends). Distractions keep you from "going deep" and really digging into the conceptual and logical structure of the math you are doing. And if you do not allow yourself to go deep to really get a concept or idea, you wind up simply memorizing facts and patterns. While this may work for problems just like the ones you have seen, the minute a problem of a different form comes up, you will be lost.

Hence, there are no good guidelines in the form of "6 hours of study per hour in lecture". Studying is a personal thing, and the studying environment matters. If you refuse to allow yourself NOT to fully understand a new concept, then any and all time spent in the pursuit of full understanding is worth the effort. To do it right, allow yourself the ability to "go deep". Then you minimize the time spent to only the quality time.

Some people brag about their ability to multi-task. To me, the ability to mono-task is the lost art in society.

How to Learn by Lewis Carroll

I have recently been talking to a student about the whole idea of how one learns, especially at the university level. This student is thinking of starting a club of Hopkins students dedicated to discussing the theory and practice behind how one learns. Really a great idea. I will advise and keep you posted. For now, I give you Lewis Carroll's (Of Alice in Wonderland fame) ideas for learning. Enjoy! What is that old saying: Much truth is said in jest....

How to Learn by Lewis Carroll

1. Begin at the beginning, and do not allow yourself to gratify mere idle curiosity by dipping into the book, here and there. This would very likely lead to your throwing it aside, with the remark `This is much too hard for me!’, and thus losing the chance of adding a very large item to your stock of mental delights . . .

2. Don’t begin any fresh Chapter, or Section, until you are certain that you thoroughly understand the whole book up to that point and that you have worked, correctly, most if not all of the examples which have been set . . . Otherwise, you will find your state of puzzlement get worse and worse as you proceed till you give up the whole thing in utter disgust.

3. When you come to a passage you don’t understand, read it again: if you still don’t understand it, read it again: if you fail, even after three readings, very likely your brain is getting a little tired In that case, put the book away, and take to other occupations, and next day, when you come to it fresh, you will very likely find that it is quite easy.

4. If possible, find some genial friend, who will read the book along with you, and will talk over the difficulties with you. Talking is a wonderful smoother-over of difficulties. When I come upon anything—in Logic or in any other hard subject—that entirely puzzles me, I find it a capital plan to talk it over, aloud, even when I am all alone. One can explain things so clearly to one’s self! And then you know, one is so patient with one’s self: one never gets irritated at one’s own stupidity!

If, dear Reader, you will faithfully observe these Rules, and give my little book a really fair trial, I promise you, most confidently, that you will find Symbolic Logic to be one of the most, not the most, fascinating of mental recreations! …

Mental recreation is a thing that we all of us need for our mental health. Symbolic Logic will give you clearness of thought—the ability to see your way through a puzzle—the habit of arranging your ideas in an orderly and get-at-able form—and, more valuable than all, the power to detect fallacies, and to tear to pieces the flimsy illogical arguments, which you will continually encounter in books, in newspapers, in speeches, and even in sermons, and which so easily delude those who have never taken the trouble to master this fascinating Art. Try it. That is all I ask of you!

Advice for an independently-learning pre-college student.

Back again, I am....

Here was a question I received recently:

Hello, ... I recently finished my sophomore [year in high school].... I have, for the past year, learned mathematics independently, taking trig, pre-calculus, calculus BC, and multi-variable calculus on my own. For the first two courses, I used an online provider. The third, I took an AP test to demonstrate that I have sufficiently learned the material so that I might receive credit for it when I go to college.

For the last course, multi-variable calculus, I would like to find a way to gain either recognition (such that I would not have to take the class in college) or credit for the course before I enter college. This is not the only class that I will have taken independently for which I cannot take an AP exam to be granted recognition. Also, I plan on starting other math courses independently (Linear algebra, differential calculus, etc.), so there will be multiple classes which I will have learned, but nothing to show for them.

Is there a way, through Johns Hopkins, I could acquire either credit for having learned college-level courses independently? If not through Johns Hopkins, do you know of a way to do this using different means?

A good question, as there are many pre-college out there struggling to quench their thirst for mathematical knowledge amidst a dry, arid environment void of opportunity. My reply:

While I like your initiative, and value your capabilities, I am wondering why you are trying to burn through all of this material at such a high speed. The AP exams, while a nice system for providing advanced training in mathematics to pre-university students, do not really measure proficiency in calculus. Rather, they measure your ability to apply proper techniques to appropriate problem types. While this is helpful, it is not really what mathematics is all about.

In your case, looking for opportunities outside high school for advanced training (as you are doing through self-study) is a good idea. But simply relying on an online course or a book and a standard exam may wind up giving you a false indication of your true knowledge base in these subjects. And if you foundation is not strong in basic subjects, you may find yourself faltering later on at the higher levels.

Some questions: (1) Do you have a mentor at your high school, or nearby, a math instructor, or mathematician to help guide you through your self studies? Someone who can see your "path" from above while you walk it is very important to your training. (2) Is there a goal in your life, which provides the reason for going from trigonometry to vector-calculus and beyond in a single year? These are beautiful subjects full of amazing insight and deep conceptual meaning. Burning through them at top speed is really selling the individual topics short. This is like driving through a safari park at 80 miles an hour. You have done the park, but have you really spent time learning about the animals. (3) Have you looked at simply taking courses on these topics at your local university, one at a time, and with live instruction? Even at the community college level, there are very good instructors whose lectures in class and conversations outside of class can be extremely helpful in seeing more then the techniques.

Yes, we here at Hopkins have many ways of evaluating the proper level for students to start at their first semester here. And we are committed to ensuring that students are not taking courses they are clearly too advanced to take. Acknowledging a students proficiency in a mathematics course may not always involves credits for the course (maybe just a waiver), but most of our evaluation involves some sort of comprehensive documentation of prior training, and not just an exam. Exams are not usually very good indicators of real understanding.

I hope this helps. Good luck in your training.

Spirit, initiative and resourcefulness are primary qualities of the budding scholar. Having and/or finding a mentor or guide is absolutely fundamental (even Harry Potter wouldn't have made it on his own!) And taking your time to digest what you are learning always leads to "better nutrition", no?

Surviving university-level Math?

I gave a orientation talk this fall to incoming freshmen. It was part of a set of Academic Interest Panels, designed to facilitate the transition from a student's previous life to university life. My contribution was to make them aware of a problem I see with incoming freshmen in their first math class here at Hopkins: That what they expect for math at this level, in terms of workload, focus, level of rigor, expectations of the student's as well as that of the instructors is really very different from the reality. And the transition shock that sometimes results can doom a student's chances in that first class. The talk was entitled

Thriving in University-level Mathematics

It became more than just a warning, however. I wound up giving lots of advice on how to study, how to treat the course and its components, the role of the lectures, recitations, homework, the instructor the TA, etc.

The step from secondary school math to what we offer is quite large.... easy to trip on, so to speak.

Click on the title link to see a PDF of the slides. It's worth a look, I believe. Enjoy.

New Advice for Incoming Freshmen....

A new item under the category of Incoming Freshmen advice has been worked out, and an announcement here is worth the post. In short,

Who should be taking the honors versions of our mathematics courses?

A lot of questions have come up among individuals about the role of the honors versions of our service courses, and who is really qualified to take them. We have updated our advice page:

http://www.mathematics.jhu.edu/new/undergrad/placement/

Before the change, the advice page recommended the honors version of multivariable calculus, 110.211 Honors Multivariable Calculus, to anyone with a score of 5 on the Advanced Placement BC-level exam. While this score certainly opens up access to the course, really the focus and intent of the course is different from that of the regular version 110.202 Calculus III.

The honors version, like all of our honors versions, is really a course in "mathematics taught the way mathematicians would really like to teach mathematics" (my quote). It is a highly theoretic versions of the standard curriculum, focusing to a large extent, on the underlying theory of a topic and focusing less on the applications and techniques. It is a great course for budding mathematics majors and those who aspire to learn mathematics in a more formal way. In fact, it is a great course to use as a bridge to higher level mathematics, and we encourage our mathematics majors to take the honors versions of all of the courses where we offer such a version.

On the other hand, the honors versions of our courses are not really for someone who simply wants to have the title "honors" on their transcript. Nor are they for students who are not interested in gaining a deep understanding of why topics like calculus are so foundational to higher level understanding of all mathematical modeling.

We have found that many students were jumping directly into this course (and the other honors courses) and having to reassess their choice after a couple of weeks into the semester. Many of these students found themselves switching "down" to the regular version of the course. Not a good way to start one's career here at Hopkins, no?

With this new advice page, we hope to better inform students of our intent, as well our offerings in courses. We always welcome any and ALL commentary of our curriculum, and strongly encourage questions about our programs.

And for ALL of the incoming freshmen out there, welcome to Hopkins. My door is always open!

Beyond Calculus II: Honors Vector Calculus or not...

A common question I often get from our very ambitious undergraduates focuses on a choice of vector calculus classes we offer. Vector Calculus, Calculus III, and Multivariable Calculus are all names for the same basic study of the properties of functions of more than one independent variable. This material is required for most engineering disciplines, as well as mathematics, and most of the natural science majors here at Hopkins. Since the techniques and material makes a lot more sense once students have studied most of the properties of functions of one independent variable, this course naturally follows the Calculus I and II sequence.

We have two flavors of vector calculus here at Hopkins:

110.202 Calculus III and 110.211 Honors Multivariable Calculus

The basic question is; Which should I take?

The basic answer is: depends....

Both of these courses fulfill the same requirements for all majors and minors that require multivariable calculus. Both can serve as prerequisite courses for any higher level course that requires multivariable calculus. Both cover the same basic material over the length of one semester, and run from the basic notions of vectors, matrices and the real space R^n through notions of continuous and differentiable functions of more than one independent variable, ending the basic material with the final major theorems tying together major aspects of the course: Green's, Stokes' and Gauss' Theorems.

The major difference between these two courses is one of focus. 110.202 Calculus III is more of a standard Calculus course, developing a blend of theoretical background on the nature of functions of more than one independent variable and the actual calculations involved in solving problems pertaining to this material. 110.211 Honors Multivariable Calculus, on the other hand, spend much more time on the theoretical nature of the material, digging deeper into the "why" aspects of calculus instead of "how things work". Students in the latter will develop a better understanding of content like the Inverse and the Implicit Function Theorems, and learn better how to analyze functions and problems that are not so straightforward. Furthermore, the honors version goes a bit beyond Gauss' Theorem and 110.202, with an introduction to differential forms, and a basic development of a generalized unified theory of the latter three theorems entitled "generalized Stokes'". Both courses are a challenge, but the latter is more so.

Students getting a BC score of 5 (or a 110.109 grade of B+ or better) can be encouraged to take this version if they are so inclined. Students with less strong scores should stay in 110.202, or at least should inquire further with the Math Department before registering for the honors version. In either case, while 110.211 is indeed a great course in vector calculus, taught the way mathematicians really want to teach a math course, it should be understood that the course will be quite a serious challenge.

Course sizes typically run over 100 easily for each lecture of 110.202, with about 4 recitation sections of 25 each. In contrast, 110.211 runs with about 40 students in 2 recitation sections.

Though always self-selected, students are usually quite enthusiastic about the honors version. it is also great fun to teach!

Hope this helps....

Chicken or Egg or ...: Which comes first? CalcIII, LinAlg, or DiffEq?

If I can point to a set of questions that are asked most often of the Math Department, one definitely on the list is the following:

Is it better to take Calculus III or Linear Algebra first?

Throw Differential Equations into the mix, and you get a branching of one's math career into three distinct paths. All three of these courses, at least here at Hopkins, have a full year of single variable calculus as a prerequisite; necessary for technique as well as theory in the case of multivariable calculus (Calculus III) and Differential Equations, and necessary for a sufficient level of "mathematical maturity" in the case of Linear Algebra.

But for many majors, and interests, one must take courses in and well understand two, if not all three, of these topics. So what order makes the most sense?

Its a good question. It turns out, it is not really important....

I am starting a new series about these and other courses under the tag and title "Beyond Calculus II". In this series, I will explain better the idea and focus of these three (and other) courses taken after a full year of calculus is achieved. Here at Hopkins, a large population of our students start their tenure here at this level.

For now, though, let's stick with the topic above. To start:

There is neither multivariable calculus nor differential equations in linear algebra, yet there is a bit of linear algebra in both of the others. In contrast, linear algebra is a more mature course, sometimes requiring more in the way of expanding one's frame of reference mathematically than the other two.

That said, we actually took a look at performance among students who took the two courses 110.202 Calculus III and 110.201 Linear Algebra back to back over a two year period (there are quite a few of them). I will pass on the details of this study, but we found that there was no real preferred order to these two, at least as far as ultimate grades go.

Couple this with the fact that any linear algebra found in either calculus III or differential equations is essentially covered within the courses, and any of the three may be taken in any real order. Hence preference for time slots, professors, and/or friends in the course may be of higher priority in your choice than content.

And one last note, our course 110.302 Differential Equations, fairly standard in content with most sophomore-level courses at American universities, is a course in ordinary differential equations (involving functions of one independent variable, in contrast with partial differential equations). One can easily describe this course as Calculus II.5 (weird notation, hih?). It can be viewed as the proper successor to Calculus II, rather than Calculus III. Just sayin....