Advice for New and Student Lecturers on Probability and Statistics

Michael D. Larsen
Iowa State University

Journal of Statistics Education Volume 14, Number 1 (2006), www.amstat.org/publications/jse/v14n1/zacharopoulou.html

Copyright © 2006 by Michael D. Larsen, all rights reserved. This text may be freely shared among individuals, but it may not be republished in any medium without express written consent from the authors and advance notification of the editor.


Key Words:Key Words: Active learning; Contrasts; Problem Solving; Statistical Reasoning; Student Participation; Teaching Methods.

Abstract

Lecture is a common presentation style that gives instructors a lot of control over topics and time allocation, but can limit active student participation and learning. This article presents some ideas to increase the level of student involvement in lecture. The examples and suggestions are based on the author’s experience as a senior lecturer for four years observing and mentoring graduate student instructors. The ideas can be used to modify or augment current plans and preparations to increase student participation. The ideas and examples will be useful as enhancements to current efforts to teach probability and statistics. Most suggestions will not take much class time and can be integrated smoothly into current preparations.

1. Introduction

Elements of probability and statistics are taught to students at all educational levels, from grade school (e.g., Siegel 2004) to high school (e.g., Peck 2002, College Entrance Examination Board 2004) to college ( Loftsgaarden and Watkins 1998). Reform of curriculum at the undergraduate level (Moore 1997, Garfield, Hogg, Schau, and Whittinghill 2002, and references therein) and development of programs in high schools emphasize active learning and experiential learning with data (Moore 2000). An ever-increasing number of high-quality resources, including textbooks, teacher guides, websites (e.g., Carver and Peters 2004, Larsen 2004c), and computational tools are available to enable active learning of the subject. Increasing active learning when teaching probability and statistics can be challenging for a number of reasons, including inertia, inexperience, personality, and planning.

Lecture, at all levels, is a very popular presentation style. There are reasons why it is so as will be elaborated later. Some styles of lecture, however, do not encourage student participation and critical thinking in class. This article provides some examples and suggestions for increasing active learning within the lecture format.

The comments in this article are based on the author’s experience as a Senior Lecturer from 1999 to 2003 at The University of Chicago during which time he observed numerous graduate student instructors in class and in practice sessions (microteaching; Derek Bok Center 2002) and wrote notes that were discussed with students. The author also has been influenced by his own teaching experiences at the master’s and undergraduate level, including supervising course assistants (1 to 12 at a time, enrollments 13 to 205 students) and teaching parallel sessions with graduate student and new faculty instructors. The author has interacted with high school instructors as an exam reader and consultant (1999-2003) and summer weeklong institute leader (2001, 2002) for the College Board’s AP statistics exam.

The article was written after reviewing notes from observations of and interactions with graduate students at The University of Chicago. The suggestions, therefore, should be particularly appropriate for new and adapting college instructors, including graduate student instructors. They should also be relevant for many high school teachers who deliver lectures. Hopefully middle school teachers will find the examples and ideas of some use, as will researchers planning experimental comparisons of teaching and learning styles. No true experiment was conducted. Evaluations of graduate instructors and practices are based on the author’s experience without outside confirmation. Grade distributions and student evaluations are confounded with instructor characteristics and are not used to rate instructors or practices.

Section 2 discusses active learning and lecturing. Section 3 presents examples involving problem contrasts and questions. Problem contrasts are created when a problem is asked in similar, but different ways. Section 4 illustrates the value of outlines and writing more details while lecturing. Section 5 focuses on pictures and diagrams for problem solving and comments on modes of presentation. Examples are presented throughout these sections. A summary is given in Section 6.

2. Lecturing and Active Learning

Not all lecture styles lead to active learning, and it may be that no lecture style can promote active learning as much as do other forms of presentation. However, by incorporating the suggestions in this paper, instructors can give lectures that offer enhanced opportunities for active learning by students without significantly changing lesson plans or time allocation in class. Section 2.1 discusses the meaning of active learning. Section 2.2 reviews reasons for lecturing.

2.1. Active Learning

Active learning is concerned with what students do in and out of class and how they gain insight into main ideas, technical details, and intuition. It is contrasted to passive learning in which information is presented to students who are expected to absorb it through contemplation and memorization of notes and textbook material. Students can learn actively by working problems for themselves, thinking about concepts to form their own summaries and statements, and explaining and discussing ideas with others (see, e.g., Garfield 1993). Working in small groups, learning through interactive case studies, and in-class problem solving are examples of active learning strategies. Homework assignments that require students to provide explanation in their own words also encourage active learning by forcing students to do more than copy phrases. Schwartz and Martin (2004) evaluated active learning in a ninth-grade class and found pedagogical advantages to discovery-based instruction. Kvam (2000) found advantages in terms of retention for average students in a college-level engineering statistics course. Magel (1996) and Lackritz (1997), among others, have discussed participation in large introductory statistics classes.

A traditional lecture format in which students take notes based on a presentation by an instructor is considered passive. Although students write notes, it is believed that most students do not think critically while writing. Some students do not even capture whole ideas or explanations. Rather, they copy what is written on the blackboard, but omit many explanations given verbally. An instructor might ask a question now and then. If students expect the instructor to provide the answer, however, then the pause after a question mainly acts as a chance to catch up with writing or to rest. Thus, a student in a classroom situation in which an instructor “gives knowledge” to students through lecture generally is not in an active learning situation. Since active learning generally is a desirable goal due to its positive impacts on retention and understanding, why, one can ask, lecture?

2.2. Why Lecture?

Lecturing is a very common form of presentation (Garfield, et al. 2002), because the instructor strictly controls the organization of and time allocated to topics. The instructor can prepare everything ahead of class and include interesting (to the instructor) examples. The material can be technically challenging and detailed. A well-prepared lecture is unlikely to be judged unprofessional. Members of many groups, as evidenced by the frequency of lecturing, find the advantages of lecturing compelling. These groups, based on the author’s experience, include graduate students with little teaching experience and possibly English as a second language, high school teachers or college or university instructors (including mathematics instructors) new to teaching probability and statistics, experienced instructors who are used to a traditional lecture format, instructors short of time to prepare participatory activities, and others for whom interacting with but not necessarily lecturing to large groups is intimidating. In other words, lecturing is attractive to many people who teach. Undergraduate students generally have encountered lecture and it is not unexpected. Even if one minimizes the amount of time spent in traditional lecture-style presentation, it is necessary and convenient now and then to give demonstrations and explanations to a class.

The examples and ideas of Section 3, Section 4 and Section 5 were suggested to and tried by graduate student instructors who had the tendency to prefer a lecture organization for their classes. Most instructors who made an effort were able to successfully incorporate some of the ideas into their teaching preparations and usual presentations and maintain desired time allocations to topics. Other sources for general advice on lecturing can be found in Knight (2002), Derek Bok Center (2004), and Mosteller (1980).

3. Problem Contrasts and Asking Students Questions

For many graduate student instructors at The University of Chicago from 1999 to 2003, lecture was the default mode of presentation in classes and problem sessions. Certainly these students were not unique in this regard, and there were some exceptions to the rule. For most graduate student instructors, expanding the use of problem contrasts and planning the use of questions helped lecture be more effective and involve more student participation without requiring a major change in lecture planning and preparation. The author presented previously some of the examples below (Larsen 2003) and has discussed some of them with high school teachers of statistics.

3.1. Problem Contrasts

Problem contrast can be very effective in helping students learn how to use and interpret formulas by examining results of alternative scenarios. It also can provide students a basis in experience for making general statements about what influences results. In terms of time planning in a course, effective use of contrasts allows one to present more problems in a shorter period of time than otherwise might be possible. At a minimum, students rehearse computations based on formulas. Ideally, the instructor will engage the students in a discussion of how and why results change when the problem details change. The instructor also could allow students a small amount of time (3-5 minutes) to work individually or in small groups, perhaps based on where students are sitting, on multiple versions of problem contrasts. A representative of each group could report an answer to the class. The following examples in five areas of introductory statistics illustrate the idea of problem contrasts. Further examples can be found in Larsen (2004a) and Larsen (2003).

3.2. Asking Questions

The second area in which most student instructors can improve is in terms of asking questions of students. Depending on personality, some first-time instructors naturally ask many questions, but in the author’s experience most do not. The basic practice of asking if students have questions is often problematic, because not enough time is allowed for students to think about lecture material and formulate a question. The tendency to move quickly through material and to cover a lot before asking for a question also contributes to the failure to elicit questions from students. Planning questions means deciding when in lecture and what to ask students. Waiting too long before asking a first question in a class tends to increase the reluctance of the students to participate. Waiting a long time between questions tends to increase the inertia against speaking in class. At the same time, it is necessary to provide adequate background and introduction to an example so that the students can formulate reasonable answers. It also is necessary to accomplish the goals of a class period, so there is a limit to the number of questions to be asked. In the author’s experience, instructors usually ask too few questions. A general or very sophisticated question will elicit little response. A very easy question also will decrease motivation to participate. In order to prepare students for participation, the instructor might need to tell students that a question is going to be asked and they will be expected to think about it and answer. Below are four examples of questions that many student instructors were able to use in their lectures.

4. Writing More Details and Making Outlines

What should one write on the blackboard (or whiteboard or overhead) in lecture? Perhaps the image of an instructor with back to the audience scribbling away talking to the blackboard comes to mind. Based on observing new graduate student instructors in probability and statistics, a more common image is a lecturer pontificating from a set of notes and writing little on the board while students have difficulty keeping focused. An active learning approach that asks students questions and involves them in calculations, explorations, and activities decreases the tendency toward to the first image. The second image can be overcome by writing more details on the board. Writing more details and using outlines in introductory statistics lectures can increase student participation by making the purpose of lecture and examples clearer without taking up too much time in class. Using contrast and asking planned questions can be combined readily with these suggestions. In the author’s opinion, the graduate students want to appear knowledgeable and precise and therefore prepare notes. Planning what to write and being organized can enhance the students’ perception of the instructor.

4.1. Writing More Details

4.2. Making Outlines

In addition to writing less than could be profitably written in some examples, most student lecturers do not present an outline of topics at the beginning of a lecture. Student instructors are not alone in omitting outlines. Many lecturers also do not make the structure of the class topics explicit as lecture progresses. Both of these practices, an initial and an evolving outline, can increase clarity, help students take notes, and emphasize the purpose of discussing a particular topic. Outlines can be written on a black/whiteboard before class. Topics can be “checked off” as they are covered in class. Here are some sample outlines that could have been used to describe lectures observed at Chicago:


Outline Example 1 Outline Example 2
1. Two-way tables I. Pr(A or B)
2. Chi square statistic II. Pr(A and B)
3. Chi square test III. Independent versus Mutually Exclusive
4. Chi square distribution
Outline Example 3 Outline Example 4
A. Student’s t distribution i. Average of a sum
B. Confidence interval for the mean ii. Standard error of a sum
C. Matched pairs study


The outlines are short, highlight main points, and do not present details. In the first example, one could combine lines 2-4 into either two lines or one line. All four connect to the purpose of the lecture. It would not take much time to write such an outline and also the relevant section number of the textbook on the board at the start of class.

An evolving outline provides verbal and visual punctuation that keep students focused on the current topic. It helps students relate what is happening in class to the larger outline, or bigger picture. Winston (1999) emphasizes the value of making the structure of a talk explicit. Students tend to write in their notes what is written on the board. As in any audience, students in a class do not pay attention the whole time. At the end of class, students should be able to accurately state the main focus of the lecture and remember some impression of what happened. Of course students have to study outside of class to better learn details, but an evolving outline coordinated with an initial outline can help lecture become a better learning experience.

Say the lecture topic is the Binomial probability distribution. A lecture title could be “The Binomial Distribution.” An initial outline might list three topics: A. probabilities; B. mean and standard deviation; and C. normal approximations. The actual topics discussed during class could be enumerated as they occur as follows: 1. model and assumptions; 2. probabilities; 3. expectation; 4. variance and standard deviation; 5. – the sample proportion; 6. normal approximations to probabilities; and 7. assumptions, revisited. Under 2., students could perform calculations under different values of n and p and summarize the contrasts. Under 4., in addition to the formulas for variance and standard deviation, students could construct bar charts of Binomial probabilities: for what values of p is the distribution skew and symmetric? Under 6., students can apply normal approximations in situations with large n, with small n (the approximation is not good), with a correction for continuity, and for the sample proportion. The main purpose of the lecture, the Binomial distribution, remains in sight. The subtopics are given names, and students can appreciate that they are the topics of interest to a statistician.

Graduate student lecturers tended not to present outlines or a coherent outline throughout lectures. Some did upon encouragement provide labels to sections and emphasize main themes. In the author’s opinion, the outlines and section titles increased the sense of organization in their classes. The organization and punctuation is what is important here, rather than numbering in a particular style. Certainly the author observed students in the classes copying the outlines and topic headers into their notes.

The combination of writing a little more and providing some sort of outline do not have to make lecture substantially longer, but can greatly increase student understanding as lecture progresses. The use of contrast and planned questions within a clear structure can help students be more aware of what is happening and why and to participate more readily.

5. Pictures and Alternative Modes of Presentation

Pictures and diagrams can help students learn to solve problems effectively and should be standard tools for many types of problems. Graduate student lecturers tend to use few visual aids, especially when using modern technologies that make it difficult to write a lot and draw diagrams in class.

5.1. Pictures and Diagrams for Problem Solving

Pictures and diagrams can be very helpful in problem solving in introductory probability and statistics. Most lecturers introduce some standard pictures, because they realize they are useful for explaining solutions and current textbooks (e.g., Peck, Olsen, and Devore 2001, McCabe, Moore, and Yates 1999) include them. Based on the author’s observations, however, students are not consistently encouraged to use pictures as problem-solving tools. Diezmann and English (2001) also advocate the use of pictures and diagrams in teaching and reasoning about probability and statistics. The examples below suggest diagrams and pictures that can be produced quickly and effectively in lecture to help students learn to solve problems in class.