How We Get Our Students to Read the Text Before Class
(MAAOL Version)


Matt Boelkins
Grand Valley State University
Allendale, Michigan
boelkinm@gvsu.edu

Tommy Ratliff
Wheaton College
Norton, Massachusetts
tratliff@wheatonma.edu

Abstract: We describe an email-based approach to reading assignments that has been very effective in getting our students to read the text before class. The dramatic impact this approach has had on our courses is explained through sample assignments and student responses. We also share the results of seven semesters of student evaluations and address some implications of using these assignments.

1. Introduction

When students read the text before class, the fundamental nature of class meetings is changed. The students arrive familiar with basic concepts and definitions, providing more class time to address the major ideas and subtleties of the mathematics. In addition, the instructor is no longer viewed as the sole source of content for the course, and this encourages greater independence, and more lively interactions, among students. The challenge, of course, is getting students to consistently read the text before class for the entire semester.

Unfortunately, few of our students have experience reading a math text, and most treat the book as a reference to use after the professor has presented new material. To counter these habits, one approach is to simply give a reading assignment for each class meeting. In our experience, most students are unlikely to read consistently for the entire semester unless there is some form of direct evaluation to keep them accountable. Since any assessment during class interferes with the main goal of freeing class time to discuss mathematics, it is important that such a method use alternate means to promote the activity of reading. In this article, we describe how email-based reading assignments have transformed a broad range of our courses, including Introductory Statistics, Single and Multivariable Calculus, Linear Algebra, and Geometry.

2. Our Goals, both Big and Small

One of the challenges to learning mathematics is that understanding is often built in stages, and one's perspective deepens upon revisiting concepts a second, third, nth time. If class time may be spent on students' second exposure to basic terminology and elementary examples, then the class is able to get to deeper mathematics more quickly and in more detail. Indeed, this moves a class session from simply introductory lectures to a time when elementary ideas are clarified (as necessary) and expanded upon.

In addition, we strive in our courses to promote students' logical reasoning and writing skills. It is often a shock to first year mathematics students that the instructor would expect them to write (and in complete sentences!) about mathematical ideas. While one can encourage such activity on homework and exams, it is ideal to have as many different activities as possible in which to develop writing skills. By reading a mathematics textbook for content, as well as through responding to questions about the reading, we aim to raise the level of students' writing, along with improving their reading skills.

While these goals are broad and perhaps ambitious, our desires for individuals on a day-to-day basis are quite modest. We want the students to be familiar with past and upcoming terminology and to have a rough idea of the basic concepts from each section. If each student spends some time reading and preparing for class, then we believe that many of the bigger goals will be accomplished. Finally, we also desire to reward our students for their effort, while making sure that the approach to reading is perceived as reasonable by both student and instructor.

3. The Details of the Assignments

We place the reading assignments on a course webpage, usually in month or week long segments. This frees class time from announcing or distributing the assignments and makes the assignments conveniently available to students outside of class. The posting lists the specific section(s) to read, which parts should be emphasized, and which can be skipped, if any. There are also several basic questions that the student should be able to answer after completing the reading. The questions serve to focus the students' reading and give them feedback on their level of comprehension; students email their responses to the instructor before the following class meeting. This gives the instructor feedback on the level of the students' understanding before class and allows the instructor to make adjustments as necessary.

As an example, the following is an assignment from Calculus II; the course text was [1].

For February 17

Section 3.8 Inverse Trigonometric Functions and Their Derivatives

To read: All, but you can skip the section on Inverse Trigonometric Functions and the Unit Circle

Reading Questions:

  1. What is the domain of the function arccos(x)? Why?
  2. Why are we studying the inverse trig functions now?
  3. Find one antiderivative of 1/(1+x^2).
We have found that a binary grading scheme works well for the assignments: a student earns a 1 for sincerely attempting to answer the questions (independent of whether the answers are correct), or receives a 0 if no such attempt is made. In addition, the assignments count for 5% of a student's final grade in the course. This assessment method has several advantages. First, it emphasizes that a major point of the assignments is making an honest effort, and also reduces the stress that many students feel toward assignments in general. Further, this scheme makes the assessment of the assignments fairly easy for the instructor. For a class with 30 students, it takes approximately 20 minutes to read and record a given day's responses from the class. Another effective tactic has been to require the students to enter a specific subject line in their email messages. The instructor can then use an email filter to move messages with that subject line into a specific folder and generate an automatic response, letting the student know that the assignment has been received.

The student responses are always informative, and they often provide an excellent starting point for class discussion. We choose several of the best responses to each assignment and place them on a temporary webpage. By displaying these responses at the beginning of class, students can compare their own thoughts on the reading, as well as see the work of some peers. This activity sparks both questions and responses, often resulting in discussion of key subtleties in the material. By archiving these web pages, students are also able to view the responses after class at any point later in the term.

4. Sample Reading Questions and Student Responses

4.1 From Calculus II

In our calculus sequence, we do not cover inverse trigonometric functions until Calculus II. The sample assignment in Section 3 came after we had discussed numeric integration but before we had covered any techniques of antidifferentiation. The student responses that were displayed during class were:
  1. The domain of the arccos(x) is [-1,1], because the range of the cos (its inverse), is [-1,1]. A.V., First-Year
  2. We are studying inverse trig. functions now because by knowing the derivatives of these functions, we will be able to calculate more definite integrals using the FTC (Fundamental Theorem of Calculus). A.C., Sophomore
  3. One antiderivative of 1/(1+x^2) is arctan(x) + 3. M.K., First-Year
These answers all show that the students understand the fundamental issues raised by the questions. A.V.'s response shows an understanding of the relationship between the range of a function and the domain of its inverse. A.C. gives a nice justification for why we are introducing the inverse trigonometric functions at this point in the course, and M.K. demonstrates the important point that the antiderivative is not unique.

Obviously, not all students gave such precise answers to all questions. In fact, M.K. completely missed the motivation for studying the inverse trigonometric functions. However, most students' misunderstandings were minor and were cleared up at the beginning of the class. This allowed enough time in a 50 minute class to derive the derivatives of arcsin(x) and arctan(x) and to give the students 15 minutes of in-class work. Without knowing the students' level of understanding before class, it is highly unlikely that we could have accomplished as much in one class meeting. With no assessed reading assignment, more time would have been spent on introductory material and motivation. Assessing the reading in class would not only eat into class time but would also make it more difficult to adjust the class meeting based on the students' responses.

4.2 From Geometry

The following assignment from early in the semester centered on the introductory section to the study of Euclidean motions of the plane. While the material had a new geometric perspective to students, they should have been familiar with many of the basic ideas from prior courses. The course text was [3].
For Monday, January 24

Reading Assignment: Section 2.1 (all)

Reading Questions:

  1. What is the difference between a mapping and a function?
  2. Is every mapping a transformation? (Explain, including a description of a transformation.)
  3. Does every transformation have an inverse? Why or why not?
The following were among the student responses shared in class:

  1. Mapping means that every element a of A has a unique element b of B that is paired with a. A function is a set of ordered pairs (a,b) with no two different pairs having the same first element. Therefore, they have similar definitions. The main difference is that Mapping is the term used in geometry, rather than the term Function. M.M., Junior

  2. No every mapping is not a transformation. A transformation is when the (x,y) are altered or reversed in some way. It consists of one-to-one and onto functions. When you reverse the pairs, it does not always result in a mapping. Other than the reversing of pairs, a mapping is a transformation. S.S., Junior

  3. Every transformation has a unique inverse. Since a transformation is one-to-one and onto, it means that there is exactly one element in A that that matches with one element in B. So no matter if you are going to B from A or to A from B, there will always be a corresponding element in the second set. [It's kinda like "for every action, there is an equal and opposite reaction.''] L.S., Junior

M.M. shows here that he has good command of the basic ideas in question 1; not only are the definitions "similar,'' but in fact they are identical. This was the point of the question. Similarly, in question 3, L.S. demonstrates an understanding of the fact that all transformations are invertible. Her response includes a nice description of a one-to-one correspondence that students in class found a good explanation.

In question 2, however, S.S. reveals a less than complete understanding of the definition of a transformation. Such a response offers many opportunities in class: is there a difference in saying "every mapping is not a transformation'' and "not every mapping is a transformation''? The response includes some of the main ideas involving one-to-one and onto functions; the lesson is that sometimes an imperfect response can provide an excellent learning moment for the entire class, particularly if several students shared in the difficulty. All three responses enabled us to have a brief, but important, discussion of how important precise language is in mathematics.

In reviewing the reading responses to these three questions, it was clear before class that most students had a solid grasp of the material. A few short minutes at the start of class were used to make certain the terminology was clear to all, and from there we were able to quickly develop more in-depth ideas related to the geometric concepts we were studying with the Euclidean motions. Had class instead begun with the question "What is the definition of a function?'', followed by introducing the term "mapping'', and then "transformation,'' it is certain that a much more lengthy segment of time would have been devoted to elementary review.

4.3 General Remarks on Student Responses

We have observed several unexpected trends while reading our students' responses. First, students tend to be more verbose via email than they are in handwritten exercises. Certainly a part of this is the ease of editing and expanding their responses at the keyboard. Secondly, the lack of mathematical symbols in email is actually a large advantage since it forces the students to explain their thought process in prose. Finally, providing another regular mechanism for communication gives students who are typically quiet in class an outlet to express their insights and share them with the rest of the class when their email is displayed at the beginning of class.

5. Data from Student Responses to Supplementary Evaluations

In each class where this approach to reading assignments has been used, we have conducted a supplementary anonymous evaluation to gain further student feedback. The students were given four options
(1) Strongly disagree   (2) Disagree   (3) Agree   (4) Strongly agree
to respond to the statements:

  1. The reading assignments were helpful in understanding the course material.
  2. The reading assignments were useful in preparation for the class meetings.
  3. The reading questions were helpful in focussing my reading.
  4. I would have regularly read the text before class without the reading assignments.
Table 1. Mean Responses to Supplementary Evaluations
Term Course Q1
Understanding 		Q2
Preparation 		Q3
Focussing		 Q4
Read without
Spring 97 Calculus I 2.9 3.0 3.0 n/a
Fall 97 Calculus I 3.2 3.3 3.2 n/a
Calculus II 2.8 3.0 3.2 n/a
Multivariable 2.8 3.3 3.2 n/a
Spring 98 Calculus II 3.2 3.2 3.5 1.9
Fall 98 Calculus I 3.1 3.2 3.1 2.3
Linear 3.2 3.2 3.4 1.7
Multivariable 3.3 3.4 3.4 2.1
Spring 99 Calculus II 3.1 3.3 3.4 2.1
Fall 99 Calculus II 3.0 3.1 3.2 2.0
Linear 3.1 3.3 3.1 2.0
Spring 00 Intro Stats I 3.2 3.2 3.2 2.1
Intro Stats II 2.9 3.1 3.1 2.3
Geometry 3.4 3.5 3.5 1.9

Table 1 demonstrates that on average, students agree with the statements that the reading assignments were helpful in understanding course material, even moreso in preparing for class meetings, and likewise in helping them focus their reading. In addition, students generally disagreed with the statement "I would have regularly read the text without the assignments.'' This data supports what has been our consistent experience with this approach.

Not only did students believe that the reading assignments were a good idea, they actually did the reading! The first column of Table 2 shows the students' response to the question:

On average, how much time did you spend on each reading assignment?
(1) 0--15 mins   (2) 15--30 mins   (3) 30--45 mins   (4) 45--60 mins   (5) More than an hour

The latter two columns of Table 2 show the mean percent of respondents per assignment and the median percent of assignments completed per student. (We use the median to reduce the influence of the small number of outliers who completed few of the assignments.)

Table 2. Time per Assignment and Response Rates
Term Course Mean Time/
Student		 Mean Response/
Assignment (%) 		Median Completed/
Student (%)
Spring 97 Calculus I 2.5 82 86
Fall 97 Calculus I 1.9 74 88
Calculus II 1.8 78 88
Multivariable 2.2 73 70
Spring 98 Calculus II 2.0 82 88
Fall 98 Calculus I 2.0 80 89
Linear Alg 2.0 84 90
Multivariable 1.9 83 96
Spring 99 Calculus II 2.2 83 92
Fall 99 Calculus II 1.9 72 86
Linear Alg 2.0 75 86
Spring 00 Intro Stats I 2.6 82 83
Intro Stats II 2.8 82 92
Geometry 2.7 89 96

Overall, we observe that on average students spent about 30 minutes on a given reading assignment. In addition, consistently at least 75% of each class completed and responded to a particular set of questions. Moreover, the final column indicates that for most students, the vast majority of the overall collection of reading assignments was completed. These data, together with the student comments regarding their opinion that the exercises were effective, demonstrate the high level of student involvement in this activity, and make plausible our claims that the efficiency of class time was significantly improved. While we would prefer that every student complete every reading assignment, we consider the approach very successful when 80% of the students in an Introductory Statistics course spend, on average, more than 30 minutes reading the text before the material is discussed in class.

Finally, it is again students' own words that offer so much evidence of their satisfaction regarding these assignments:

"I firmly believe I would not have read as thoroughly and would not have been as prepared for class were it not for the reading questions. They weren't a big deal to complete at all, and I feel they were vital in my understanding of the course.'' -- Geometry

"I felt they were very helpful considering I tend to struggle with math courses. A very good idea!!'' -- Statistics

"Good stuff, helps to at least get a feel for the material before it is covered, allows a slightly faster pace.'' -- Linear Algebra

"I felt the reading questions made me concentrate more on what I was reading and (I) got more out of the reading than I otherwise would have.'' -- Calculus II

"They were quite helpful. But it was sometimes frustrating if I didn't understand the material to have to wait until class to finally see how to do it.'' -- Calculus II

The last quote demonstrates what we are striving for: students who are thinking about mathematics, working on mathematics, and cannot wait to get to class.

6. Other Issues/Potential Pitfalls

There are some start-up costs to be aware of when using these assignments. Writing the assignments can be a time-consuming affair, and we have found that it is easiest to write several weeks, or a month, of assignments at a time. This has required us to have our courses fairly well-organized to assign specific readings this far in advance. One advantage is that this has helped us keep a brisk pace in our courses and keep up with our initial syllabus.

Text selection is extremely important when using these assignments since the students will be reading the text as their first introduction to the course material. The students' perception of the readability of the text, as well as the choice of questions, can significantly affect their opinion of the efficacy of the assignments. If the questions are simplistic, then the students view the assignments as busy work; if the questions are too difficult, then they add to the frustration that many students feel when reading mathematics. Quite often, several semesters of minor adjustments are required to fine-tune the questions.

We also feel that it is important to recognize that these reading assignments add to the students' workload in the course. Since the assignments keep the students engaged with the course material on a nearly daily basis, they can serve a similar role to lengthy homework assignments. It is important that these reading tasks not simply be added to the list of things required of students, but that their addition is reasonably accomodated in an overall vision for expectations of students.

There are, of course, problems that can arise when an assignment is technology dependent, such as access to email, network outages, and student apprehension about using the technology. Since network problems will inevitably occur, we have told students that they can turn in their assignments on paper before class if they have trouble accessing email the night before the assignment is due. A bit of flexibility on the part of the instructor seems sufficient to handle these minor challenges.

7. Conclusion

We find the overall atmosphere in our classes exciting with this approach. Students read to learn mathematics . . . They explain their mathematical ideas in prose . . . Discussions become more lively . . . The instructor gets individual feedback on each student's understanding of concepts . . . Class time is spent more efficiently . . . Deeper mathematics is considered . . . Students even profess to like the assignments.

It sounds like everyone is winning! The approach has changed the fundamental way we direct our students in learning mathematics, and does so in a way with many important benefits. For all these reasons, we hope that other instructors will join us in the endeavor. The reader is encouraged to take a look at how an entire semester develops in this approach by visiting our courses on the World Wide Web at

http://acunix.wheatoncollege.edu/tratliff/
or
http://www2.gvsu.edu/~boelkinm.

References

[1] Ostebee, Arnold and Zorn, Paul. 1997. Calculus From Graphical, Numerical, and Symbolic Points of View, Volume II. Saunders College Publishing.

[2] Ratliff, Tommy. 1997 How I (Finally) Got My Calculus I Students To Read the Text, Innovative Teaching Exchange, on MAA Online.

[3] Smart, James. 1997. Modern Geometries, 5th edition. Brooks/Cole.

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