This month, I want to say more about a tool that I use to get my students to read the textbook and think about the upcoming lesson before I teach it, a tool a colleague of mine has named the Reading Reflection. He borrowed the idea from me, and I got it from Tommy Ratliff at Wheaton College. We have each done our own modifications to suit our needs. What I find most surprising is how simple and easy and yet effective it is.
Tommy describes the basic idea in his article, "How I (Finally) Got My Calculus I Students to Read the Text" [1], which is one of four articles on teaching students to read mathematics, included among a collection of excellent articles with great teaching ideas on the MAA website Innovative Teaching Exchange [2]. The site is maintained by Bonnie Gold. You can get to it via the MAA's Professional Development webpage.
The basic idea for the Reading Reflection is as follows: For every class for which there is an appropriate section of the text, students are asked to answer three questions about that section before class. The first two are reading specific, and the third is generic: "What was the most important point in this section?" I conclude by asking: "If there was anything that you found confusing, please describe it". As an example, in our 100-level Discrete Math course, before beginning our study of modular arithmetic and Diophantine equations, students were told to read Chapter 4 of Coutinho's The Mathematics of Ciphers [3]. My chapter-specific questions were
- Using the fact that 7^12 = 1 (mod 13), find the smallest positive integer that is congruent to 7^98 (mod 13).
- What is a Diophantine equation?
Of the 21 students in the class, 9 answered question 1 correctly, telling me that either they had learned the basics of modular arithmetic before my class or had picked up enough by reading this chapter that they had a good understanding of modular multiplication. The wrong answers that I got told me a lot about where students were confused about the nature of modular arithmetic or were struggling with exponents.
Question 2 was particularly helpful because Coutinho's definition of a Diophantine equation is buried within a paragraph of prose, and most of the students had missed the fact that a solution to a Diophantine equation must consist of integers (or rational numbers). I went into class knowing that that was a point I needed to clarify.
Asking students to identify the main point of the reading is very enlightening, because they will often focus on something that I consider tangential. It alerts me to when the main point of the reading has not been clear.
For the Reading Reflection, students are not penalized for wrong answers. The Reading Reflections also do not earn a lot of credit. Collectively, they count for 5% of the final grade, and to avoid cries for make-up opportunities, students are allowed to miss a prescribed few without penalty. Despite this, I have been very pleased at how carefully students read the assigned chapter or section in order to try to get the correct answer. There are times I can go into class knowing that my students already have a good grasp of the new idea we will be exploring. When they are confused, I enter class with some understanding of where the confusion lies. It is a great tool for improving my effectiveness in the classroom.
In his article, Tommy Ratliff describes how he had tried to get his students to do advance reading by giving a short quiz at the beginning of class. That failed on two fronts: the "short" quiz wound up taking more and more class time, and, because students had no idea what they would be asked, they had often failed to read for the ideas that he considered to be important. The Reading Reflections give some shape to this advance reading, and taking it seriously appears to make a real difference in student performance.
My colleague in Geology, Karl Wirth, who borrowed this idea from me, has run a correlation of his mid-term exam scores against final grades and compared it to the correlation of Reading Reflection scores against final grades (see graphs below). One expects a high positive correlation since both are components of the final grade. Revealingly, even though mid-term exam scores are a much larger piece of the final grade, the score on Reading Reflections had a higher correlation (R^2 = 0.71) than did the average on the mid-term exams (R^2 = 0.44). While this says nothing about causation, it is highly suggestive.
In view of how useful these Reading Reflections are, they are very easy to collect, grade, and use. I require that they come in at least an hour before class, so that I have time to read them before class and adjust my lesson plan accordingly. Tommy Ratliff suggested having students email them with a prescribed subject line so that they can be filtered and set aside to be read en masse. I did this initially, and it works well, but we now use Moodle courseware, and that makes it even simpler. Students simply record their answers as an online text assignment. Moodle collects the answers for me, and a single click enables me to give credit or not to each student.
David Bressoud
is DeWitt Wallace Professor of Mathematics at Macalester College in St. Paul,
Minnesota, and president-elect of the MAA. You can reach him at bressoud@macalester.edu.
This column does not reflect an official position of the MAA.