#38 Questions from prospective teachers

My Cal Poly colleague Anelise Sabbag recently asked me to meet with students in her undergraduate course for prospective teachers of statistics.  Most of the students in the class are majoring in Statistics, Mathematics, or Liberal Studies, the last of which is for students preparing to teach at the elementary or middle school level.

Prior to my visit, Anelise asked her students to pose a question to me about teaching statistics.  I was very impressed with the thoughtfulness of their questions, so much that I decided to write this blog post with some of my responses and reflections.  Breaking from this blog’s custom, questions in italics in this post were posed to me by these students.

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1. What is the hardest topic to teach in introductory statistics? 

This is a great question, one that I’ve thought about a lot.  My answer is: how the value of a sample statistic varies from sample to sample, if we were to repeatedly take random samples from a population. 

Of course, I could have expressed this answer with just two words: sampling distributions.  But while this two-word phrase provides a very handy shorthand for people who already understand the concept, I’m not convinced that using this term is helpful to students who are first learning the idea.

In fact, let me back up and split my answer into two parts: Before we can ask students to learn and understand sampling distributions, we need to begin with the more basic notion of sampling variability.  In other words, first we must help students recognize that the value of a sample statistic varies from sample to sample, before we tackle the more challenging* idea that this variability displays a predictable, long-run pattern.  That predictable, long-run pattern is what we mean by the term sampling distribution.

* This idea is not only challenging, it’s remarkable!  Isn’t it amazing that the long-run variability of a sample mean or a sample proportion turns out (in many circumstances, anyway) to follow a beautiful bell-shaped curve?!

Why is this topic so hard?  I suggest two reasons: First, it’s always difficult to comprehend a hypothetical: What would happen if …?  This hypothetical is central to many concepts in statistics, including probability, p-value, and confidence level, as well as sampling distribution.  Second, we’re asking students to think beyond a sample statistic (such as a mean or a proportion) as a straight-forward calculation that produces a number, to thinking of the statistic as a random variable that varies from sample to sample.  This is a very big cognitive step that requires a lot of careful thought*.

* An analogy from calculus is the large cognitive step from thinking of the slope of a tangent line to a curve at a point as a number, to then considering the slope of the tangent line to the curve at all points as a function.

What can be done to help students overcome their difficulties with this concept?  I will explore this question in a future post, but my one-word answer will come as no surprise: Simulate!

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2. What do math majors struggle with when studying statistics?

First, I want to emphasize that math majors, and other students who are comfortable with math, struggle with the same challenging concepts that other students do, such as sampling distributions.  I rely on simulations to introduce math majors to sampling distributions, just as with students who are less mathematically inclined*. 

* I also explore this concept in more mathematical ways with math majors.  For example, I lead them to determine the exact sampling distribution of a sample mean in a sample of size 2 or 3 from a small population or discrete probability distribution.

Math majors can also struggle with the fundamental ideas of uncertainty and variability.  Probabilistic thinking can provide a bit of a shock from the deterministic thinking with which they are likely more comfortable.  A related issue is tolerance for ambiguity, as math majors (and all students) can be uncomfortable with the lack of certainty associated with statistical conclusions.  In their statistics courses, students must learn to write conclusions such as “there is strong evidence that …” and “we can be very confident that …” and “the data reveal a general tendency that …”  These conclusions stand in stark contrast to the kind that might be more in the comfort zone for math majors, such as “the exact answer is …” and “we have therefore proven that …”

Speaking of writing, that’s another aspect of statistics courses that can frustrate some math majors.  Assessments in statistics courses often require students to write sentences, perhaps even paragraphs, rather than provide a single number as an answer.  These questions often begin with verbs – such as describe, compare, explain, justify, interpret – that might intimidate students who are more comfortable responding to prompts that begin with verbs such as calculate, derive, show, or even prove. 

Another potential source of frustration is that much of mathematics involves abstraction, whereas statistics depends heavily on context.

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3. How can teachers provide students with enough knowledge to prepare them to investigate good questions?

This question is a close cousin of one that Beth Chance and I are often asked by teachers who attend our workshops: How do you find time to include activities in class? 

I think many teachers under-estimate students’ ability to create their own understanding through well-designed learning activities.  I do not accept that teachers need to lecture on a topic, or have students watch a video or read a chapter on the topic, before they turn students loose on an activity.  The questions in the activities can lead students to new knowledge.  Necessary terminology and notation can be embedded in the activity.  Teachers can lead a discussion following the activity that reinforces key take-away lessons for students.

Here are three examples:

• The Gettysburg Address activity described in post #19 (here) is a long one that can take most or all of a 50-minute class session.  But this activity introduces students to many concepts, including sampling bias, random sampling, sampling variability, sampling distribution, and effect of sample size on sampling variability
• The Random Babies activity described in posts #17 and #18 (here and here) leads students to fundamental ideas of probability as a long-run proportion and expected value as a long-run average, along with topics such as sample space, equally likely outcomes, mutually exclusive events, and the complement rule.
• The simulation-based inference activities of posts #12 and #27 (here and here) enable students to discover the reasoning process of statistical inference, specifically hypothesis tests and p-values.  Teachers do not need to provide a multi-step outline for how to conduct a hypothesis test prior to engaging students with these activities.  They do not even need to define a null hypothesis or a p-value in advance.  Instead, teachers can introduce those terms after students have encountered the ideas in the context of real data from a genuine study.  

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4. What lessons have I learned from students?

I did not expect this question.  I think this is one of the best I’ve ever been asked.  This question truly caused me to pause and reflect. 

But I must admit that despite this considerable reflection, my answer is not the least bit clever or insightful.  Here’s my list of very basic things that I believe students value and teachers should prioritize:

• Show respect.
• Be organized.
• Make expectations clear.
• Provide timely feedback. 
• Stay humble. 

The first four items in this list are so obvious that they need no explanation.  About the last one: I like to believe that I have continued to learn more and more as time has gone by.  One thing I have surely learned is that there is so much that I don’t know.  I’m referring to the subject matter, and to how students learn, and everything else involved with teaching statistics.  I have also come to realize that my course is certainly not the center of my students’ world.  I also need to remember that no students will master every detail or retain every skill that they develop in my course.  It’s fine for me to set high expectations for my students, but I also need to keep my expectations reasonable.

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5. What advice do I offer to prospective teachers of statistics?

My #1 piece of advice is no secret, but first I’ll offer two other suggestions, which I hope are less predictable.

At the beginning of my teaching career, I learned a great deal about statistics, and formed a lot of ideas about how to teach students about statistical concepts, from reading textbooks for a basic course in statistical literacy: David Moore’s Statistics: Concepts and Controversies, Freedman, Pisani, and Purves’s Statistics, and Jessica Utts’s Seeing Through Statistics.  I have also enjoyed and learned a lot from books aimed at broader audiences that involve statistics and data.  Two examples on the history of statistics are David Salsburg’s The Lady Tasting Tea and Sharon Bertsch McGrayne’s The Theory That Would Not Die.  Examples from other fields include Freakonomics by Steven Levitt and Stephen Dubner and Thinking: Fast and Slow by Daniel Kahneman.  My first piece of advice is: Read non-technical books.

More than two decades ago, I invited Jim Bohan, a high school math teacher and math coordinator for his school district, to speak to students at my college who were considering whether to pursue math teaching as a career.  I’ll never forget Jim’s advice to these students: Don’t go into teaching because you love math; go into teaching because you love working with kids.  He reinforced his point by adding: When people ask me what I teach, I don’t say that I teach math; I say that I teach kids.  Jim’s message resonated with me and leads to my second piece of advice: Pay attention to the human side of teaching and learning.  

Now for the anti-climax …  The final piece of advice that I offered to the prospective teachers in Anelise’s class, the three words that I hoped to impress upon them more than any others, will surprise no one who is reading this blog*: Ask good questions! 

* If this is my first post that you’ve read, please consider reading post #1 (here) that provided an overview of this blog and my teaching philosophy. You can also find a convenient list of all posts (here).