As I attempt to teach myself something about stochastic calculus, I have been reading a great many articles, and several textbooks, on the subject. It has left me with the distinct notion that mathematicians hate to spill the plot too early in the story. Each proof builds upon clues that have been scattered throughout the text, just as a writer might have the butler passing down a hallway in the second chapter for no apparent reason. Lemmas and sub-theorems that, to all outward appearances, are wholly unrelated to the general theme begin to appear. Then slowly, sometimes painfully, these logical manipulations are pulled together as the proof draws to a conclusion. But even so, the mathematician is reluctant to come out and say, “the butler did it.” Rather, phrases like “it is clearly obvious,” and “it is easily proven” are used to inform the reader that the point of mathematics is the mental challenge of figuring out how the pieces fit together.
There is no enjoyment in reading a novel that lays out the entire plot in the first paragraph. So a plot is simply a literary construct used by authors to evoke emotion, just as sculptors do with statues, and dancers with physical movement. Likewise, spelling out every mathematical truism and trick would make a proof lengthy and boring, with no pleasure left to be savored, right? So I think that mathematicians must write proofs (whether intentionally or not) in a manner intended to allow other mathematicians to sense the thrill of unraveling a logistic knot.
Sometimes I enjoy this artistry. Today I do not. 🙁
Ever hear of the “Marshmallow Challenge?” Small teams of individuals are given the following assignment: use twenty sticks of uncooked spaghetti, one yard of masking tape, and one yard of string to construct the tallest possible free-standing structure that supports the weight of a marshmallow. Most people assume that since a marshmallow doesn’t weigh much, it shouldn’t significantly affect the support structure. Of course, even a small mass can produce structural failure when placed atop a long unsupported column.
So what profession does best at this task? According to Tom Wujec, a Fellow at software company Autodesk, the tallest structures are built by engineers and architects. They consistently outperform similar teams of lawyers, business school students, or corporate managers. This is not an unanticipated result, as we expect our engineers to know something about static structures. However, it is rather surprising to learn that youngsters, even kindergarten students, do far better than most adults—kids are simply not afraid to repeatedly fail as they search for an approach that works. (You may discover more about this learning exercise at MarshmallowChallenge.com).
Two insights come from this anecdotal report of group behavior. First, that engineers have been trained to think in a manner that is distinctly different from those in other professions. Second, that repeated rounds of prototyping and evaluation may be an effective means for dealing with the messy, unstructured, uncertain problems that engineers frequently encounter.
Happened to stumble across Mango Languages last night. It seems to be a nicely constructed site for language instruction. Once upon a time I worked for a firm headquartered in Germany, and had taken some company-sponsored language lessons, so I poked around in the first-level German module. One of the things I noticed right away was that translation was required in both directions. First, I was asked to translate from English to German. Then, after practicing a phrase, I was asked to translate from German back to English. This was easy for the first couple of phrases, but became increasingly difficult as I had to juggle more and more words in my head. I gave up halfway through the lesson as it was getting late, and my head was starting to hurt. However, someone fluent in both languages has obviously mastered the translation going in either direction.
All engineers deal at some level with the language of their particular engineering specialty, as well as the language of mathematics. Fluency in math provides magnificent tools for creating and analyzing new engineering methods. However, my recent experience as a graduate student (as well as two decades as a design engineer) leads me to believe that the emphasis is too heavily focused on the making the translation from physical reality into the language of mathematics. Once the math domain has been entered, there is little effort to move back into the physical domain. This seems to me a great oversight, as I consider engineers to be those who function primarily in the physical realm, rather than the mathematical domain. I love the beauty of mathematics, but I fear that the current generation of engineers may lack much substantive understanding of how to convert mathematical results into an enhanced understanding of physical realities. Many of my classmates fail to “see” how a problem solution relates to any real-world situation.
On the other hand, I frequently find myself struggling for hours trying to make the connection between a solution and its physical meaning. Often times the available textbooks and reference materials made it sound as though this connection should be immediately clear to the reader. It drives me crazy when technical material makes no concession for those of us who are not yet be completely fluent in the “language.” Since learning is a lifelong process, all of us should be constantly entering domains to which we have not been previously exposed. While the use of a particular method or technique may be plain-as-day to its creator, it’s often not nearly so obvious to those of us struggling to acquire an understanding of the new concept. So I suppose this post should serve as a reminder for myself to keep looking for better ways to describe technical concepts in a manner that novices can comprehend, and that experts will still find insightful. I’m not sure that it can be done at the same time, or via the same communication method, but surely there has to be a better way.
While skimming through a year-old post over at edtechpost, I noticed the following reflection:
So I will forgive you if you ignore me from here on out as a perennial dimwit when I tell you that it took me this long to ‘get’ how crucial narrative and storytelling are to everything we are doing, be it learning online, connecting, weaving one’s online presence, blogging…
What really caught my eye was the phrase “narrative and storytelling.” Why are these factors not more frequently incorporated into the teaching of technical issues? While sitting through long lectures that cover intricate mathematical development, I often long to hear more about the context in which the methodology was developed.
What problem drove the development of a new approach? These equations don’t just drop out of the heavens! Aspiring engineers need to understand that effective problem solving is within their grasp; that “correct” solutions are not just found in dusty old reference texts. Novel methods are driven by persistence and hard work—this reality is rarely emphasized.
How long did it take to create, prove, and document the approach? It’s easy to get frustrated when the development of a new method hits repeated roadblocks. There needs to be some understanding of the hundreds (or thousands) of hours that are often spent in developing a new solution. Even though a proof can be sketched out in two minutes, the path from problem statement to solution is usually not intuitively obvious.
Finally, a pet peeve of mine: All contributions are made by real people with real lives, not mystical figures existing beyond the earthly realm. Even if there is no time for biographical sketches of these individuals, what is the correct pronunciation of their names? Most engineers finally figure out that “Euler” is “oy-ler,” not “you-ler.” However, I’ve sat through many lectures where a theorem author is identified on the overhead slide, but their name is never mentioned aloud. And rarely have I heard any emphasis on correct pronunciation. This small detail seems central to allowing engineers to properly communicate with others in the language of mathematics, as well as providing some sense of human involvement. By the way, I often refer to the Mathematics Pronunciation Guide. How else would I learn that “Stieltjes” is pronounced “steel-tyuhs?”
Having taught college courses many moons ago, I am well aware that trying to incorporate contextual material into lectures means even more work for already overstretched professors or lecturers. However, I’ve come to the decision that it’s better to master a few topics than to be aware of many. A good story makes any topic easier to remember, and also promotes a richer understanding of the material.
This past week I had the opportunity to attend an excellent workshop presented by Jeffrey Karpicke, who heads up the Memory and Cognition Lab at Purdue University. He makes the case that students don’t know how to study effectively. As a result, they spend hours and hours “laboring in vain”—performing tasks that produce absolutely no learning. This is understandable, he says, as the only guideline that most college students are given is that they should spend 2 to 3 hours “studying” for every hour they spend in class. Without additional guidance, they tend to confuse comprehension with actual learning.
What constitutes studying in the minds of college students? More than 80% of the students that Dr. Karpicke and his colleagues surveyed listed “rereading notes or textbook” as a study method, and more than half identified it as their preferred strategy (see Karpicke, Butler & Roediger, Memory, 2009). Unfortunately, rereading has absolutely no benefit; time spent rereading material is wasted (see Callender & McDaniel, Contemporary Ed Psych, 2009). So students are losing precious time on ineffective study methods. Dr. Karpicke tells his students that they can substantially improve their learning by spending 45 minutes each week performing three easy steps:
Step 1: Collect and Organize — Encourage students to organize their grasp on a subject by creating their own outlines or self-study guides. This is something I have always done in my studies, but only 24% of students surveyed by Dr. Karpicke carry out this process.
Step 2: Assess Comprehension — Students often confuse familiarity with comprehension. To get beyond this, students should try explaining concepts in the absence of notes or textbooks. When presented with central topics or keywords, they should examine what comes to mind. If they are unable to produce an accurate response, or are unsure of an answer, then this is an area where they need to “study.” This step is carried out by only 13% of the students surveyed by Dr. Karpicke.
Step 3: Practice Retrieval — Long term learning requires more than comprehension; repeated retrievals of the material are required to ensure that the information is not forgotten. Have students pull out a blank sheet of paper and write down everything they can remember about the topic. Encourage students to use flash cards. Knowledge retrieval is often seen as “neutral,” a process that does not modify or alter the learning process. But Dr. Karpicke claims that the act of retrieval itself produces learning. (see Roediger & Karpicke, Psychological Science, 2006) What percentage of students carry out self-testing? Less than 11%.
Most scary is that students who utilize rereading as a study strategy are over-confident about their learning, while those that practice active retrieval are under-confident. Making a rough estimate from the graphs that Dr. Karpicke displayed, it appeared that rereaders are about 20% more confident in their “judgment of learning” than are self-testers, but perform about 35% worse when tested. Thus, it may be quite useful to discuss potential test questions during lectures, so that students begin to assess their comprehension, and have appropriate lines of thought for self-testing. One of the workshop participants stated that he has students submit potential test questions as part of their homework; this forces them into the beginning stages of self-assessment.
How much do we really remember from our college days? Some big concepts, for sure, but many of the small details are quickly lost. For a humorous look at this topic, watch Father Guido Sarducci’s Five Minute University.