Playful Math: Doodling to Aleph_null

I believe most people who really get mathematics are the ones who see the frivolity in much mathematical construction not as a flaw, but as a strength ( I am teasing my profession here). And sometimes presenting mathematics in playful ways is precisely the best way to expose deep meaning.

With a hat-tip to Engineering Innovation (@JHU_EI), a high school summer program in the Whiting School of Engineering here at Hopkins, I am reposting a video from one of their recent tweets.

Doodling in Math Class: Infinite Elephants

Have fun!

Oh, and BTW, we mathematicians tend to associate letters from other alphabets to important constants and concepts in our work. Aleph (the first Hebrew letter) is commonly used for measures of infinity. Aleph_null, or Aleph with the subscript zero, is used to denote a kind of infinity called countable infinity, and denotes the size of a set of objects that can be placed in a one-to-one correspondence with the natural numbers 1,2,3,... Jus'sayin'....

Math (not quite) in the Media... College not producing?

A new study has come out which seems to conclude that a college education may not be worth the many hundreds of thousands of dollars it is costing these days. Here is an article from the Huffington Post detailing some of the points coming from the data. I have not read the study. The article, I have, and am less than impressed.

My take: While there may be quite an important trend uncovered by this survey, I believe (from my position) that some of the assumptions may not be fully vetted.

45% of Students Don't Learn Much in College

The article starts off well enough (emphasis mine):

A new study provides disturbing answers to questions about how much students actually learn in college – for many, not much – and has inflamed a debate about the value of an American higher education.

The research of more than 2,300 undergraduates found 45 percent of students show no significant improvement in the key measures of critical thinking, complex reasoning and writing by the end of their sophomore years.

However, this part bothers me:

One problem is that students just aren't asked to do much, according to findings in a new book, "Academically Adrift: Limited Learning on College Campuses." Half of students did not take a single course requiring 20 pages of writing during their prior semester, and one-third did not take a single course requiring even 40 pages of reading per week.

Where is the metric measuring critical thinking and complex reasoning OUTSIDE of a writing class? Mathematics and physical science-based courses are prime venues for the development of rational and analytic reasoning, no? How come no mention of non-writing-based courses in the ENTIRE article? Perhaps no data from these courses is in the study?

More from the article:

Three of the five classes [Julia Rheinecker, a freshman at the University of Missouri,] took... were in massive lecture halls with several hundred students. And Rheinecker said she was required to complete at least 20 pages of writing in only one of those classes.

I am very dubious about any direct link between college class size and either the difficulty of a course or its perceived lack of rigor. Large-lecture courses can be quite challenging and yet every bit as personal and interactive as small seminar-type classes. I doubt this study had any focus on class size at all. So this part is not relevant, IMHO.

And finally, some additional conclusions in the study:

_Students who studied alone, read and wrote more, attended more selective schools and majored in traditional arts and sciences majors posted greater learning gains.

_Social engagement generally does not help student performance. Students who spent more time studying with peers showed diminishing growth and students who spent more time in the Greek system had decreased rates of learning, while activities such as working off campus, participating in campus clubs and volunteering did not impact learning.

Knowing something about how math is taught and learned at the university level, I am hard-pressed to believe any data showing that studying alone is better than studying with peers. On this point, I can write volumes! And there are data and programs (Think PLTL, for example) to support the opposite conclusion. In fact, we have a local PLTL program here at Hopkins, and our data do not support this study's conclusion in this point.

Overall, I see this study as being a bit alarmist. Critical studies on the effectiveness of college education are absolutely necessary. I will read this study. But I am already biased due to this article. Oh well....

Undergraduate Research Opportunity!

I just got a solicitation for a great summer program in research for undergraduates. Click on the flier to see the details. It is the Research in Industrial Projects for Students (RIPS) program at the Institute for Pure and Applied Mathematics (IPAM) at UCLA. Top-notch, IPAM is, and I am sure the opportunity is quite competitive. But the program is a good one, and will make for a great credential (not to mention the experience!). Check it out and let me know if you plan to apply: The deadline is February 10. Plenty of time to get your act together, no?

Mathematics Study Tips - Finals Time!!!

I have posted many of my own tips on studying in mathematics classes here in this blog (see the tag for Study Tips). So when I happened on this article on the online newsroom Huffington Post, I clicked through.... What I saw sounds remarkably familiar, and ALL good advice. Give it a read:

12 Ways to Ace Your Finals

Pay attention to the study groups part, the adding stress to your studying to reduce stress in the exams, and such. Good luck all!

Math in the Media - Me??

I received a request recently to do an interview for the student-run newspaper here at Hopkins, called the Newsletter. It came as part of a series they do called "Things I have learned" and features short interviews here with professors to learn more about who they are, why they do what they do and where they see their future taking them.

I got contacted by a student Emily Nadelmann. Here is the article. It was fun, and gave me a chance to talk a bit about how I see my profession, my study and my interacitons with it. Give it a read. The title is

Professor discusses the artistic aspect of mathematics and teaching

Math in the Media - Shock Therapy as a study technique??

Just ran across this report:

Electric brain stimulation can improve math skills

Read the article. An Oxford study has found that low levels of electrical stimulation of the brain (read: shock therapy?!!?) can improve math assessment skills. And these skills last for at least for a long time (at least half a year??).

I can already see the ill-advised do-it-yourself home implementations of this kind of study technique. Geez....

I can already see this giving other study aids a run for their money, no?

Mathematics Study Tips - Pre- (and post??) pare for lectures

Hey again all. Some time ago, I started writing down what I consider are good tips and practices for studying and getting through mathematics courses at the university level. Click on the "Study Tips" tag to see those. here is another I find myself often saying to struggling students:

To make most of the lectures (the backbone of the course, no?), why not prepare for them. To really make them an integral part of your course experience, why not post-pare for them also!? What do I mean...?

It is commonly accepted that it is easier to remember something when you hear it more than once. Maybe then it is placed in more than one position in your brain, with different associations triggering its location. Maybe it still sits in one location but is better interconnected with other triggers. Maybe I have no idea.

But it would certainly help greatly to not only know what is coming up in the next lecture (check the syllabus online before the meeting), but also to have spent some time in the book on the material before the show. No ?

Preparing for the Lecture

If you know the next lecture is on ( a hypothetical) Section 4.6, say, why not spend 10-15 minutes before the lecture reading through that section? You don't have to understand completely what you are reading. Some of it will make sense, some not. But at least you will be exposed to terms, concepts, boxed items, examples, BEFORE you enter the class to actually see the lecture develop the ideas. This way, the stuff you did understand on first reading will be cemented by the lecture development. You can even relax a bit on this part of the lecture.

When you get to the part of the lecture that comes from material in the book that mystified you, you can then spend some precious attention time focused. Pretty efficient, huh? We, as instructors, really do like to follow the book in many of our courses. Books are written in an organized fashion. We may embellish the material, but the core usually comes from the book.

Your notes in a lecture are an important facet of your eventual understanding. See here for some tips on taking notes.

Post-paring for the lecture

After the lecture is where the fun really starts! Here is my idea for good practice....

Grab a small block of time (another 15 minutes), in a quiet place and free from distractions, somewhere between an hour and three hours after the lecture. It won't take long. Go over your notes slowly and carefully an in your mind, relive the lecture, re-listening to the instructor and imagining the lecture hour unfold. As you f0llow your notes, you will remember things that were said that you did not write down (or finish writing down). Write them down now. You will see examples half-finished. Finish them now. You will references to the book. check them and make a mental note of them, You will make connections that you did not make before. Note them in your notes. You will see thing in your notes that STILL mystify you. Mark them (in red?) with a big question mark. Make a note to yourself to go to your TA or the instructor and ask specifically about these question marks (much of this part is also talked about in my installment of this series on Notes. Take a look there also).

All of this is called "completing the experience". It is a way to make your notes of the lectures a complete and central account of the course, as they should be. And it is a form of studying that WILL pay big dividends as the course progresses.

Try this for a few weeks. It really is a minimal effort for the benefit it offers. You will see improvement.

In bocca al lupo!