A good test

- has students emulating reasoning patterns.
- has students supplying reasons for steps.
- has students grabbing the best tool for the job.
- has students display what they learned doing their homework.
- has students combining, not concatenating, concepts.

On a first level, having effective websites and transparent simple procedures for registering for classes, changing classes, paying bills, etc. would reduce a student’s intellectual load and save them time – time that would be better spent studying. This is a known problem, often unaddressed. Just check any random universities website with a specific question in mind.

The second level, entails the intellectual load imposed by varying academic standards across campus. Often students have to think through what is acceptable in different courses – typed or handwritten, polished or rough, spelled correctly or who knows, all steps shown or the important ones or none. One gets the sense that students sometimes are just testing for the particular course’s standards or worst, disregarding said standards – still a choice. If the faculty, even by department, spoke as one voice, students’ intellectual load, and for that matter stress, would decrease.

There is a third level that I am exploring this term. I teach college algebra to a cohort of students that are taking both introductory biology and chemistry. I have reordered my curriculum so that I am going over the math that I know they will be using at the same time in their labs and on their exams. For instance, teaching straight line math early, using significant digits and decimal calculations early and consistently, and making up examples that come directly from their labs using the same units. If one thinks about it, most math students in precalculus and higher will not be math majors but will be STEM majors. This sharing of common problems among the disciplines can only help reduce a college student’s intellectual load. Looking over biology and chemistry labs and seeing the keys for their exams have given me good examples and changed the way I discuss issues in class. No longer just “You will see this technique in calculus class” but “The behavior of this graph will be important in your spectrometer lab”. I now know, even, that mixture problems in algebra II are much more important than the usually phrased motion applications.

I also see the glimmer of a fourth level. Thinking as such is not usually taught on campus, at least, in the sense of Thinking in Bets by Annie Duke and How to Think by Alan Jacobs. These thinking skills and sensibilities apply to all disciplines and also to life. Using their structure to teach introductory critical thinking and problem solving in all programs would reduce a student’s intellectual load. We know that skills often don’t transfer across disciplines but they might if we work at it.

]]>At best, students leave Stat 101 able to define and use p-values. They understand sampling variability and the role of the central limit theorem. They can correctly state confidence interval and hypothesis testing conclusion by the end of the course. They can use a calculator and/or spreadsheet and/or statistical software to do tests and can dutifully interpret p-values. For most, a few months later, all they will remember is, p<.05 is “good.”

Thus, Stat 101 students finish their course having learned an increasingly rejected statistical paradigm which will propagate into their science labs, advanced statistic courses in the social sciences and business, into their capstone papers and beyond into who knows what after they graduate. We are placing the vast majority of our students in a “statistical” hole.

So here is my plea for somebody somewhere to do something. Not that people haven’t tried, and failed by their lights. My interpretation of such efforts is that the authors tried for too much – include too much explanation, too much background, too complicated of models, too complex model checking.

Take a look at an elementary statistics textbook such that it is. The explanations are limited. The types of questions/problems are simple and the subsequent analysis can be written in a sentence. This is appropriate for these students. Why not, instead of a p-value conclusion, have a modeling conclusion based on model sampling. Stat 101 students can just as easily be taught to state conclusion properly in modeling language as in p-value language. Since most students treat calculators or statistical software as a black box anyway, using provided p-value results or model sampling results will make no difference to them. The situations are simple – inferring population parameters from a sample, comparing two populations from samples, using normal or binomial distributions. These simple problems cover most of the cases they will confront in their college career.

So my plea – someone, somewhere write a elementary statistics textbook for non-majors using Bayesian models. Get it underwritten by a major publisher. Most of the content will be like the traditional text. The changes will be just in how the problems are worked out using Bayesian capable software and how the conclusions are stated. this would be good start.

Note: No links in this post. It is just my general sense of the field. I am not an expert in the field of statistics. I have avoided teaching elementary statistics for many years now because I disagreed with the content. I could do this because I am department chair and past retirement age.

]]>One: U.S. academics are shaken by the current political situation. They are looking to their research for solace, hope, and/or solutions. Creativity as it relates to wisdom and ethics is now a hot topic. The social-cultural aspects of creativity are being described.

Two: I found many presentations tinged with bravery and sadness. Bravery because many of the presenters were at the beginning of their professional life and putting a significant part of their work out there for critical questioning (Though most people were nice.). And sad because this consuming work was dismissed with a desultory question or two and a polite clapping. Many times I wanted to engage them, as the interesting people they are, in a detailed conversation, really out of curiosity, but I was off to the next section. And anyway, attention from me would not have enhanced their career or research.

]]>- Creativity can be taught.
- Creativity can be measured.
- Creativity is good.
- Creativity is trans-disciplinary.
- Creativity is thinking.
- Creativity produces artifacts.
- Creativity is resisted in college.
- Creativity involves empathy.
- Creativity needs permission.
- Creativity can be embedded in a story.
- Creativity requires overcoming fear.
- Creativity requires discipline.
- Creativity takes time.
- Creativity is hard to transfer.
- Creativity is a process.
- Creativity is domain specific.

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My perception is that that one idea generates many other ideas – one realization generates many other realizations. The implication is that we will run out of resources before we run out of ideas. I notice this in science and math. One research paper generates many possible paths for future research. There is not enough money or people to follow all the leads. Too many paths are not taken and it will only get worse. I say worse because some of those ideas could save the world.

]]>The idea is that the measured underlying behavior could be ideally rated on a continuous scale and that our rubric criteria divides the scale into intervals. Thus, in these figures the underlying behavior is depicted as the blue normal curve and our rubric divides the number line into intervals by establishing cut-points, the black dots.

The histogram thus represents the frequencies in each interval. As can be seen, the rubric assigning D, C, B and A could break up the line unevenly in many different ways. The question becomes, can we recover the parameters of the blue line from the given histogram? The answer will be yes – if we can tolerate some error.

My research scheme was therefore, after setting a normal curve (mean [*mu*] and standard deviation [*sd*]), to define a set of cut-points and generate a histogram. Then use the cut-points to get numerical values representing the categories. Use the R function, *fitdistr()*, to recover *mu* and *sd*. Repeat 10,000 times. Change the cut-points and try again.

Since the scale is arbitrary, I chose it to be the standard deviation of the given normal distribution which fixed to be one. I used a *mu* of 2.5 since I have been addressing a rubric scale of 1, 2, 3 and 4. The problem became: How does varying the cut-points affect the recovery of the mean? The answer was that we can get fairly close.

The key was devising a way to attach numerical values to each category. Let *d1*, *d2* and *d3* be the cut-points. Then the numerical value of the two inner bars was assigned to be *(d1+d2)/2* and (*d2+d3)/2*. Define *le* as the “limit” of the left end bar. In point of fact, the continuum extends infinitely in the left direction but I set it to be *d1-sd*(1.5)*, one and a half standard deviations to the left of *d1*. The value for the leftmost bar became *(le+d1)/2*. Similarly I set *re* to be *d3+sd*(1.5)*, and the value for the rightmost bar to be *(d3+re)/2**.*

Other details. The histogram represents a sample size of 100. I used the R function *fitdistr()* to find a parameters for a normal distribution since I knew where the data came from. For that reason I did not do any normality checks. *fitdistr()* actually gives error bounds but I preferred to just do 10,000 samples.

The following table has the results of the experiments. Note that even with a fairly wide variation of cut-points, we still get close to the underlying mu of 2.5.

From the table, it looks like a reasonable scheme to compute a mean for the underlying distribution on an arbitrary scale could be devised. First devise a set of cut-points using a scale that you want for the standard deviation. Calculate representative numbers for each category. Pass the information to *fitdistr()* and report *mu* with the error. For example here is the result for some data I have been looking at.

The cut-points were calculated assuming the expected results that 70% of our seniors would lie in the two rightmost categories and that 7.5% would be in each of the outermost categories. When simulated this yielded a *mu* of 2.56. In other words the crude method I used to calculate values for the categories yielded a slight plus average. From the table, I think it is fair to say that in cases 4, 6 and 7 our students did not meet our expectations and in case 5 they exceeded our expectations.

In sum, this exploration turned out to be a pleasant diversion from my intention to model rubric data. At best the results offer a way of quantifying the results with one number that has more justification than just taking an average of rubric numbers.

A note on the method. I built my R program to read and write from a CVS file. This allowed me to set the experimental values on a line with comments and get back on the same line the results including the file name of the associated graph.

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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.

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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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.

]]>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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