Before the semester starts … I’m playing with pictures!

I am sure I’m not alone in wanting to use the time between semesters to make adjustments to what I am teaching or how I am teaching it. By now, you probably recognize that I am a fan of learning about new pedagogical techniques. I am dedicated to helping students to truly understanding the concepts of statistics. Often, having visuals when you teach is useful for students.

I use the chalk and a board (OK, more like 8 boards that move). I draw a lot of pictures. However, a mathematics professor (who is both a great colleague and friend) has been bugging me about using Mathematic in addition to chalk (a delivery system she also loves).

With Mathematica, it is my hope that I will not only be able to present my students with a visual image of certain concepts during class time (like how a normal distribution changes when the size of the standard deviation gets larger or smaller) but by making these demonstrations available electronically to students for them to explore these concepts on their own, I am hoping students will gain a greater conceptual understanding of critical statistical concepts.

Mathematica is a software package, that among other things, provides demonstrations of statistical concepts. Each demonstration was designed by an instructor. For it to be published, it is my understanding that it goes through a rigorous peer-review process. As such, if it’s printed for use, you know it will work. The down side is that your university would have to pay for a subscription to Mathematica for the demonstrations to be useful. http://www.wolfram.com/solutions/education/higher-education/uses-for-education.html

As I stated last week, in my list of resolutions, my goal is to find five different demonstrations this semester. Why five? It seemed like a reasonable number … not too challenging.

This was really easier than I anticipated. I started by indentifying the concepts that would most benefit from being able to visualize and manipulate variables. Then I visited the Mathematica web site and searched the topics. Each search yielded anywhere from 5 to 25 demonstrations, some were appropriate, others weren’t. I looked through the demonstrations and selected the ones I liked.

Here are the concepts and the demonstrations I identified as being potentually useful this semester.

(1) The Normal Distribution, where students get to input mu and sigma, would make a nice visual demonstration.

http://demonstrations.wolfram.com/TheNormalDistribution/

This Normal Distribution also shows the area under the curve (i.e., you can manipulate the z-score)

http://demonstrations.wolfram.com/AreaOfANormalDistribution/

(2) Another good demonstration would be the Sampling Distribution of the Means, where students can see the impact of changing mu, sigma, or sample size on its shape.

http://demonstrations.wolfram.com/SamplingDistributionOfTheSampleMean/

I’m also going to throw in a demonstration on the Central Limit Theorem, as how can we talk about the Sampling Distribution of the Means without mentioning the Central Limit Theorem?

http://demonstrations.wolfram.com/TheCentralLimitTheorem/

(3) Of course, what changes in the Sampling Distribution of the mean is the standard error, thus showing how a standard error changes due to changes in the sample size and/or variability makes a great deal of sense. I was really hoping that a demonstration on the standard error would already be available, unfortunately, it doesn’t seem to be. A similar concept is the confidence interval, though even with this demonstration the writer of the Mathematica code for this demonstration did not include how variability (i.e., standard deviation) impacts the size of the “margin of error.” However, it still could be a useful demonstration.

http://demonstrations.wolfram.com/ConfidenceIntervalsConfidenceLevelSampleSizeAndMarginOfError/

Though not as clean looking at the one above, this demonstration also includes the size of the standard deviation. http://demonstrations.wolfram.com/ConfidenceIntervalExploration/

I would expect that the two demonstrations would be necessary for student to get a richer understanding of confidence intervals.

That having been said, I believe that two new Mathematica Demonstrations are in order … one dealing with the size of the standard error based on changes in sample size and variability and a possibily a new CI demonstration that merges the best of these two demonstrations.

(4) The effects of the sample size and population variance on hypothesis testing with the t-test seems like a great visual demonstration.

http://demonstrations.wolfram.com/HypothesisTestsAboutAPopulationMean/

(5) How changes in the variables impact correlation’s (depending on how they are calculated) should be useful for my students.

http://demonstrations.wolfram.com/CorrelationAndRegressionExplorer/

(6) Those of you who know me, are probably not surprised that I can’t just stop at 5 examples for this first semester … so here is a great demonstration on Power. Though I can get students to define power, and identify threats to power, I am never fully certain that they truly get the beauty (and hassle) of power. This demonstration may help.

http://demonstrations.wolfram.com/ThePowerOfATestConcerningTheMeanOfANormalPopulation/

Of course, without proper instruction during class time and an accompanying explanation following class instruction, these demonstrations may end up being little more than pretty pictures to students.

In a few weeks, especially after I actually try these demonstrations with my students, I will provide for you the information I attached with the demonstrations as well as feedback as to what worked and what didn’t. After all … anyone who has taught long enough knows, even the best planned lessons and demonstrations some times flop.

Though not specifically having to do with teaching statistics … I found a nice article at Chronicle of Higher Education on Iphones, Blackberries, etc … and apps that could help professors. The attendance and learning students’ names apps look promising. http://chronicle.com/article/College-20-6-Top-Smartphone/125764/

I look forward to hearing from any of you who have used Mathematica Demonstrations (or others) during class and for homework.

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Teaching hypothesis, design, analysis & inference as one thing

“So, the question I would like to pose for the sages and anyone else interested in commenting … for a first semester undergraduate applied statistics class … what are the most critical student learning outcomes that have to be mastered?”

First let me just comment on the blog Bonnie just posted today: I think her list of core concepts is excellent, and I agree that those concepts (all of which have to do with the ever-present error inherent in all our observations and measurements) should certainly be taught in the introductory statistics course.  Nevertheless, let me introduce a different perspective.

When I first saw the question posed by Bonnie  (reproduced above) I thought the answer would be an easy one to write. It turns out it is not quite so easy. My problem is that I see all the parts of the application of statistics as parts of an integrated whole. So my answer will appear to be a daunting one.

I hope that students can take away an appreciation (mastery would be too much to ask at this level) of how we use data to make inferences about the behavior under study. My typical homework problems were not, except for some initial ones, about calculations: finding means, t or F values, and p values. Rather an experimenter’s hypotheses would be stated along with how she collected the data to test those hypotheses, and (relatively simple) data would then be listed. The question posed was: what can you conclude from these data, and especially what can you conclude about the hypotheses? The appropriate way to answer such problems was to present the means and to interpret what the pattern tells us, with the statistical test of significance to guide us as to which differences we could assume due to the independent variable.

I understand that this is asking a lot of the students, but just getting statistics from data sets bores the heck out of me, and I don’t see why it would not be equally boring to the students. A few weeks into the semester we would be into the Analysis of Variance (Keppel’s book does a wonderful job facilitating early introduction of AOV), and the course especially emphasized factorial designs in which interpretation of patterns of means with the assistance of significance testing becomes, for me at least, most challenging and most interesting. The logic of the interplay of hypothesis, design, data, statistical analysis and inference is to me all one thing.

Such an integrated concept, satisfying to me, may or may not be an asset when applied to teaching the first undergraduate course in statistics.

 

Core Statistical Concepts

I have been spending the week thinking about what I consider to be the “core concepts” that need to be covered in an applied statistics class, be it in psychology, health, business, or education. However, before I post my personal thoughts, I felt it necessary to see what other applied statisticians had to say. In my search, I found http://www.statlit.org/pdf/2004McKenzieASA.pdf . This work was conducted by John McKenzie (2004), Conveying the Core Concepts, is from the Proceedings of the ASA Section on Statistical Education, pages 2755-2757.

In reading what  McKenzie, and several other professors of applied statistics identified as the core concepts in statistics, I must say … I concur. Listed below are the core concepts in applied statistics … the information that, in my opinion, simply has to be covered regardless of illness, snow days, or anything else that could interrupt a professors’ teaching schedule.

Variability: Students cannot understand the purpose of statistics unless they get the concept of variability. Within this, we can further talk about variability due to chance and variability due to effect. Including in the discussion of variability should be the difference between systematic and random variability. I would have to say that not a class period goes by without me spending at least a little time on helping students to focus on issues of variability (especially variability due to the individual differences of the subjects who just happen to be in our sample). 

Randomness: Though I would see randomness and variability as being part of the same large concept, McKenzie’s work identified the concept of randomness as not only separate from variability but also critical for students to master.

Sampling Distribution: Along with Hypothesis Testing, the teaching of sampling distribution is considered to be one of the most complicated to teach.  I would concur, which is why I spend an entire class period just on a single activity with M&M’s to demonstrate the concept of sampling distribution. (Please see a prior blog entry for details on this tactile activity).

Hypothesis Testing: The sages and I spent the month of October and much of November discussing whether Hypothesis Testing is critical and if so, how to best tackle the teaching of this complex topic. Not surprising, McKenzie identified the teaching of hypothesis testing as being one of the two most difficult concepts to teach in applied statistics (the other being sampling distribution). Though there may be several published articles on hypothesis testing no longer being a critical concept to teach, the individuals who were surveyed for McKenzie’s work, certainly consider it to be a critical concepts.

Data Collection Methods: Though I have said to my students more times that I can count, “the quality of our statistics is limited by the quality of our sample,” I must admit to being a bit surprised that this was considered critical by others, especially since when I look at many undergraduate statistics textbooks, data collection methods are barely mentioned. Kiess and Green’s (2010) Statistical Concept for the Behavioral Sciences, 4/e, certainly tackles the issue of data collection methods.

Association vs. Causality: This core concept makes me smile, as often when I meet someone for the first time, and they ask me what I do … my response is often met with one of two comments … “Oh, I hated statistics” or “Correlation does not mean causation.” It’s kind of like me recalling how to greet a person in German, a class that I had for three years, and yet recall so little. We, as applied statisticians, certainly engrave this concept into the minds of our students, but I’m sure most of you are like me, hoping student get more than a “pat phrase” out of our classes.

 Significance (Statistical vs. Practical): This is a critical concept in applied statistics and one that is probably not mentioned in theoretical statistics classes. Sure, we delineate a mark in which we have to say … these results are too extreme for us to attribute them to “chance” … but just because we found a statistically significant difference, doesn’t mean it’s a difference that truly matters. In applied statistics, it’s not enough to understand how statistical significance works, but to be able to interpret the results to determine practical difference. I must admit to not covering this core concept to the same extent I cover the others.

As I think of other “critical concepts” they tend to be a bit more specific and fall under the larger concepts listed above (e.g., understanding what a standard deviation can tell us, clearly falls under the concept of variability. I invite all of you, to comment on what concepts, if any, are missing from this list.