Smart People, All of Them

People, need I say students, are smart in different ways.  They might be smart (expert at) playing a particular video game or shopping online or dog training or horse riding or the internal affairs of Poland or basketball or short story writing.  I am reminded of this and humbled by it, every time I get to know a new set of students.

The graphics below are my attempt at depicting this reality.  Consider each person (point in space) as an amalgam of traits.  Each trait is normally distributed among all people.  The length of the colored line represents the “accomplishment” for that particular trait, defined by direction and color.  The diversity of lengths of expertise is evident.

People Traits in Space

Closeup: People Traits in Space














Now we enclose each person (point) in a bubble with a two standard deviation radius.  We can see some trait spikes sticking out.  These represent an exceptional measurement, a ninety-fifth percentile of accomplishment.

They say everyone has a story. Also everyone has an exceptionality.


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My Worries

I worry that I sometimes waste my students’ time.  I know the standard lower levels of our math curriculum have antiquated parts. I need to  devise ways of delivering this required material in a modern context.

I worry that my students will embarrass themselves after they graduate and apply for their first job.  I worry that they will misspell or mispronounce a word or write incomplete sentences or reason sloppily or be ignorant of a common math concept.  Grammar counts when I grade written work.

I worry that I sometimes treat students like they are stupid.  They are not stupid.  They are just new to the material.  I mean to treat them like adults using adult vocabulary and adult reasoning and adult expectations.

In sum, curriculum, standards, respect – of course.

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“Sleep-Walking” Home

At the end of a day last week, I was deathly tired.  By that I mean I could hardly keep my eyes open.  I just wanted to lay down and sleep.  I could barely move.

And yet I needed to walk home – a mile uphill into the sun. It hurt to keep my eyes open so I started experimenting.  In a few blocks I had my method – close my eyes, walk eight steps, flash the eyes open to see the near terrain, walk eight steps, repeat.  I relaxed my face, kept my mind empty. Easy to do without any visual stimuli.  I arrived in, seemly, no time.  I had “sleep-walked” home.

Of course, I opened my eyes at intersections and upon hearing the approach of a car.  I was walking up unbusy residential streets with no sidewalks so no problem.

When I walk to work, I cross an empty parking lot.  A game I played was to walk with my eyes closed, taking as many steps as I could before I got scared or veered too far off course.  This required alertness to the kinesthetics. “Sleep-walking” eight steps at a time turned out to be unstressful, even relaxing.

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Bang for the Faculty Buck

As part of strategic planning, we are having discussions on campus about what our (college faculty’s) work will look in twenty years. One exercise was to list the percentage of time spent among various tasks. For many of us the percentages added up to over 100%. This lead me, in a moment of reverie, to the following ideas on efficiency and “bang for the faculty buck”.

One assignment – Three outcomes
I am working with a group of faculty (science, writing, math) that is considering this idea. What about assignments that have multiple outcomes? How about, say, a lab report that the science faculty could grade for science correctness, the writing faculty could grade for technical writing proficiency and the math faculty could grade for quantitative reasoning? Such an assignment would have efficiency benefits for the student – one paper for three classes and efficiency benefits for the faculty, if the logistics were not too bad, of just grading in the area of their expertise. I have seen syllabi that prohibit such “two-fer” projects but why not?
Outside experts – Collaboration among faculty
I have heard faculty complain that students in their classes cannot do simple quantitative reasoning tasks despite having taken our elementary statistics course (or not). What about having an expert, a math faculty statistician, visit the class, lecture on the issue at hand and even help design the assignment and evaluation instrument. Our librarians visit classes all over campus helping students with information literacy and I am sure there are other pockets of expertise on campus. This type of collaboration needs to be built into the faculty reward structure.
Time spent grading tests – Or not
I know of faculty members who carefully annotate tests and papers. And I have seen students place such papers immediately in the trash can. The faculty wasted their time and the student learned nothing further. Why not grade the papers just to determine the students’ grades with wrong or incomplete answers circled and points assigned according to a rubric? Then let the students correct the papers for an additional percentage. The faculty member will have to grade the papers twice but can be much more efficient each time. Other time saver possibilities. Record verbal comments as the paper is being graded to cut down on writing comments. If part of an answer is a Yes or No choice, have students circle the answer rather than write the answer somewhere on their paper – faster to grade.
There are commonalities of approach that occur within a discipline and between disciplines. In math for example there is Vedic Math developed (or rediscovered) in the early 1900’s which has 12 or 16 Sutras or common methods. Here is an example. The general scientific method would be common in biology, chemistry, and physics. Data analysis would be common to economics, political science, sociology, etc. I taught an efficient combined linear algebra and differential equations course – efficient because the topics had significant overlap – bases, linear combinations, independence and the like. The Common Core is designed with scaffolding that cycles upward through repeating basic concepts. If such commonalities were taught in a consistent manner, the faculty would have it easier and so would the students.
Variations – Common structure
This is a little bit in the math weeds, but often we teach how to approach a problem by working from simple to complex and then often give up on the complex. If we taught the logic of the problem with all the ramifications at the beginning, we would get more bang for the buck. Since I am currently teaching them, I am thinking about word, motion, and mixture problems.
Variations – Levels of learning
Again in the math weeds. The examples one works in class can be chosen so that we can talk to students of differing proficiencies at the same time. The basic algorithm that solves the type of problem would be what the average student needs. The variations of the problems would have lessons that the above average student can use and appreciate. This is a bit hard to articulate. I think I mean that the average student is most interested in mastering the method and the more proficient student can, with the aid of the instructor, get more context and nuance at the same time.

In conclusion
The ideas above are examples of collaboration outside of our disciplines, leveraging of expertise throughout the campus, consideration of the student learning value of a teacher’s time expenditures and leveraging commonalities. Some could be encouraged with an appropriate reward structure.

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Quick Joke?

My wife came home with

“The past, the present, and the future walked into a bar.

It was tense.”

I reposted

“.2, .3, and .4 walked into a bar.

They were tenths.”

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Rubrics as Data – Part III

It has been a long time since I have addressed this topic.  I was going to add, “for good reason.”  Is utter discouragement and temporarily quitting a good reason?  Anyway here is the saga.

I first worked my way steadily through Statistical Rethinking by Richard McElreath entering all the examples in R and learning about Bayesian modeling.  The examples seemed a little arcane and the golem metaphor a little off-putting but by the end (nearly the end)  I felt that I could build and explore simple Bayesian models.   I was empowered.   And what better way to use my newly-developed skills than to work on a rubric as data model.  I set about building the model as described in Section 11.1 Ordered Categorical Outcomes.  What could go wrong?

I made MAP (maximum a posteriori) and a stan (probabilistic programming modeling language) models using my example from the last posts – 3 1 scores, 13 2 scores, 15 3 scores, and 8 4 scores.  When I did the sampling from the models, I got the occasional nonsense which I attributed to having my cut points (the points separating the scores (logit) ) (technical details will be omitted) getting out of order.  Now what?  I quote Dr. McElreath’s text, “As always, in small sample contexts, you’ll have to think harder about priors.  Consider for example that we know alpha 1 < alpha 2, before we even see the data” (page 335).  So that was my problem.

How to fix them.  Before that, I need to know that I had “fixed” it. Here is how I graphed the sampled data from my models.

I sorted the samples by goodness-of-fit to the original data.  The distribution histogram is in the upper left corner. I moved down the sorted list so that I had 50 evenly spaced samples and plotted them on the large bar graph to give a sense of the variability.   I plotted the distribution for each score below the central bar graph as proof I could calculate prediction intervals.

I could now see if anything went wrong, like in this graphic for a small sample.

My object became to remove the odd looking distributions in the middle of the picture.  I began by making reasonable changes to the model.  Reasonable to me but, I eventually figured out, incomprehensible to the the shell that Dr. McElreath built to simplify his exposition.  This took a while.  I then plunged into the R literature.  This was difficult.  I ended up tracking through threads, reading non-answers and references manuals with few and poorly remarked examples.  In the end I decided to go with a purely stan model.  At this point I got discouraged.  I was fighting picky syntax and the only help (stackflow) always seemed to avoid direct answers.  The responders seemed more interested in criticizing the asker’s model or suggesting a better model.  Please just help us build the model no matter how wrong it is.  Also have a easily located place where we can ask about models not the code.  I got discouraged and quit.

A month ago, under the influence of excess coffee, I found the optimism to start again on the stan model.  It worked.  The use of the ordered data type did the trick and away I went.  Here is the result.

Now I can use posterior sampling to get prediction intervals.  For instance model in the graph above, The 95% prediction intervals are

My next step is to explore stratified sampling (say sample results from each major) using partial pooling.


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Roads Up Mountains

I was mountain biking up the 2060 switchbacks when I glanced up to the left.  The edge of my road loomed steeply.  “Wow, that’s high” I said to myself, imaging how hard it would be to scramble up the 120 feet of brush and rock with my bike on my back. I memorized a few landmarks and kept pedaling.  I was there in a short and easy time.  No surprise, that is why they build switched-back roads, to change a hard, nearly impossible ascent into a series of gentle inclines.

The analog to course planning holds.  We design a series of small incremental learning opportunities that, if followed, get our students up the mountain of knowledge.  Each step is “easy.”  Yet, requires effort.  If a student falters (stops pedaling) they stop or slide backwards.  By working steadily, they will achieve the goals of the course.  That is what the quizzes, homework, tests and projects are all about.

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