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(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.

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.

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.

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

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.For February 17Section 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:

- What is the domain of the function arccos(x)? Why?
- Why are we studying the inverse trig functions now?
- Find one antiderivative of 1/(1+x^2).

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.

- The domain of the arccos(x) is [-1,1], because the range of the
cos (its inverse), is [-1,1].
*A.V., First-Year* - 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* - One antiderivative of 1/(1+x^2) is arctan(x) + 3.
*M.K., First-Year*

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.

The following were among the student responses shared in class:For Monday, January 24Reading Assignment: Section 2.1 (all)

Reading Questions:

- What is the difference between a mapping and a function?
- Is every mapping a transformation? (Explain, including a description of a transformation.)
- Does every transformation have an inverse? Why or why not?

- 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* - 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* - 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*

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.

- The reading assignments were helpful in understanding the course material.
- The reading assignments were useful in preparation for the class meetings.
- The reading
*questions*were helpful in focussing my reading. - I would have regularly read the text before class without the reading assignments.

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:

(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.)

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.'' --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.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

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.

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

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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