Chris Aldrich

A lovely little experiment in visual pedagogy in mathematics! We could certainly use more of these types of simple "explorations" particularly for younger math students who may not always be good at visualizing material. These seem reminiscent to me of Wilson Rugh's <a href="http://pages.jh.edu/~signals/">java applets</a> for digital signal processing years ago.

Some of this pedagogy argument about slides likely fits better into the area of the flipped classroom versus non-flipped. Though slides may be great (particularly if students have copies of them beforehand to take notes on as is often done commercially by many biology textbooks which have 2-4 slides down the left side of the page and note space on the right hand side) for mathematics I much prefer having notes in <a href="http://refer.livescribe.com/a/clk/2G3Md">Livescribe</a> .pdf format with embedded audio. This allows a student to not only relisten to the entire lecture, where often a lot of complicated material (especially in higher level abstract courses) is said very quickly and isn't able to be written down even by some of the best transcriptionists with shorthand (and how many students know this anymore?), but they can skip around to parts they need more work on. As an example, download this <a href="https://www.dropbox.com/s/7g4rekbuzzr8nfz/Lie%20Groups-Lecture%203.pdf?dl=0">.pdf file from a Lie Groups lecture</a> and open it up in the most recent version of Adobe Reader to be able to see the visual and audio functionality.

The particular value often comes in seeing problems worked out in full detail with discussion, which some students need much more of than others. One of my favorite lecture experiences almost a decade ago was hearing an audible gasp from the classroom of undergraduates when Sol Golomb (USC) stopped in the middle of a complicated lecture on combinatorics and worked out a 4th grade simple long division problem on the board to get a final solution. Half the class had pulled out a calculator to get the answer and he'd finished before most of them had the first number input. The class learned more in that one minute of example than most did the entire lecture. Most of the best math is done in the "exploration" and practice than in a stripped down lecture as is highlighted in Ben Orlin's recent post: <a href="https://mathwithbaddrawings.com/2016/05/04/the-essence-of-mathematics-in-one-beatles-song/">The Essence of Mathematics</a>. While it's always lovely seeing theoretical mathematics as if it seemingly sprang fully formed from the head of Zeus, many students should see how the proverbial sausage (or laws) are actually made, especially at an earlier age.

Brief reply to: Is majoring in liberal arts a mistake for students? https://medium.com/@vkhosla/is-majoring-in-liberal-arts-a-mistake-for-students-fd9d20c8532e#.d64awbm87

What magisterial sounding pontification! Sadly, it’s not much different than the early philosophies of Socrates and Plato or many of the other early progenitors of the humanities and liberal arts. I get the impression that the author hasn’t read much philosophy and has not much grounding in the liberal arts. While I agree with the spirit in which the piece is written, I find it deplorable that there aren’t what should be obligatory mentions of words like trivium, quadrivium, or philosophy, but rather the corpus of work in which the author seems steeped is that of only modern day authors of popular science (Pinker, Gladwell, Kahneman, et. al.) who have some interesting viewpoints, but ones which require at least a grounding in the liberal arts to pick apart. Several times Khosla demeans the liberal arts and uses the repeated example that a reader should be able to pick apart and think critically about articles in The Economist. To do this requires a knowledge of logic and rhetoric which are two of the pillars of what? — yes, the liberal arts!

He also seems unaware of big movements within the humanities and sciences like Bill Gates and David Christian’s Big History Project which are going a long way towards providing a more balanced education in history, economics, physics, chemistry, biology and evolution. I find here, no prima facie evidence of his knowledge of Thomas Kuhn or Karl Popper, which might help win me to his argument. In all, aside from the passing references to one or two recent works, this entire argument is not much different from many that could have been written at the beginning of the industrial revolution. How blind so many must be to seemingly think there’s something new here.

Most appalling to me here is that the author doesn’t seem to give even a passing nod or small wink to C.P. Snow or “The Two Cultures” [http://boffosocko.com/2013/11/28/two-cultures/]which, at heart, is really the substance of his entire argument, he’s just blind to it’s existence.

Yes, we certainly need more emphasis on the quadrivium portion of the liberal arts, and in particular mathematics and critical thinking which seem to have been left by the wayside. It is deplorable that the highest extent of mathematics that 99% of college students are exposed to terminates in the 17th century for the most part. Sadly, many college students are left without the ability to think critically and deeply, not to mention the hordes of students in America who barely make it through high school and don’t attend college. One also only needs to skim through recent issues of Nature [http://www.nature.com/news/reproducibility-a-tragedy-of-errors-1.19264], one of the world’s most pre-eminent scientific journals to discover that a multitude of advanced researchers with Ph.D.s lack the ability to properly design scientific experiments or evaluate the simple statistical analyses to reach the correct conclusions. What does this mean for readers of The Economist who aren’t even presented with any actual data and are supposed to be able to think critically about a writer’s hidden assumptions.

Yes, we need far, far more, but alas, this poor article only touches the tip of the issue and it sadly only does so with less than half of the picture.