## by## Matthew DeLong, Taylor University## and## Dale Winter, University of Michigan |

One day Alice came to a fork in the road and saw a Cheshire cat in a tree.

"Which road do I take?" she asked.

His response was a question: "Where do you want to go?"

"I don't know," Alice answered.

"Then," said the cat, "it doesn't matter."

- Lewis Carroll, in "Alice in Wonderland."

Alice is a girl with a serious problem. She knows what she has to do (choose a path to continue her journey) but she doesn't know how to recognize her destination when (and if) she gets there. Like Alice, college mathematics instructors choose paths of instruction when they plan their lessons - choices intended to help their students learn and understand mathematical ideas. Although many instructors face their classes with beautifully clear mathematical expositions, they (also like Alice) often do not have equally clear ideas for what the resulting mathematical understanding will look like when (and if) their students attain it.

Although conscientious instructors know that effective teaching is the result of careful preparation, planning is often more focused on the content to be covered than on the outcomes of instruction. By contrast, explicit student-learning objectives, used to guide lesson planning, implementation, and assessment, can provide instructors with clear guides to instruction and learning, and standards with which to measure students' progress.

Unlike most professors, K-12 teachers are almost universally trained in the use of systematic planning methods based on the formulation and use of explicit learning objectives. Many of these methods have their origins in the work of Tyler [6]. More recently lesson planning methods for college teaching have also appeared that incorporate the formulation of student learning objectives as an integral part of the process [1]. Furthermore, the current emphasis on continuous quality improvement included in many models for assessment pushes towards the use of objectives.

While the phrase "student-learning objective" is used differently by various authors, we will employ the specific definition given in our recent article [2] and following Farrell and Farmer [4, p. 196]. To do so we first differentiate between goals and objectives. Goals are large-scale, overarching ends of a process of instruction and learning. Objectives are the discrete educational steps that learners take in pursuit of educational goals. Objectives are much more specific, concrete, limited and connected with specific areas of subject matter than are goals. Student-learning objectives (SLOs) [2] consist of three components:

1. A description of the observable student behavior that will result from instruction.

2. A characterization, description or definition of the conditions under which students will exhibit this behavior.

3. A characterization, description or definition of the performance standards that will be judged to represent success for the student.

The following are some examples of SLOs.

* Given a data set and some clues as to which quantity is the input and which is the output [condition], students are able to draw a set of axes [behavior] that will comfortably accommodate all of the data points [standard].

* Given a quadratic inequality [condition], students are able to use a graph, algebra and the quadratic formula [behavior] accurately to solve the inequality (or to determine that the inequality has no solutions) [standard].

* Given the equation of a vector field and a list of possible graphs for the vector field [condition], the students should be able to correctly [standard] identify which graph is produced by that equation [behavior].

SLOs are formulated during the planning phase of instruction. SLOs can be developed from existing descriptions of the mathematical content of the course, or directly from an instructor's content knowledge [3]. Once developed, SLOs can be used to guide lesson planning [1]. We strongly advocate that instructors communicate their SLOs to the students, either before or after instruction. Finally, SLOs can be used to guide the development of assessments and evaluations.

In our article [2], we provide a lengthy list of potential benefits to students, instructors, and departments from the formulation and use of explicit SLOs. A brief sample of some of these potential benefits includes the following.

Writing SLOs...

· provides a definitive set of standards for relevance and appropriateness that the instructor can use as a guide for selecting learning activities, assignments, and assessments.

· can give a much more realistic impression of how much material can be covered.

· helps to determine which learning outcomes will be the most vigorously pursued in the course.

· can (when communicated to the students prior to assessments) help students get more out of their preparation for quizzes, tests and examinations.

· may help the students to take more responsibility for their own learning.

· can help students to judge the effectiveness and relevance of their own learning and study habits and activities.

We have used SLOs in undergraduate courses ranging from College Algebra through PDEs. We have used them in courses taught individually and in large, multi-section courses. The following shows some sample comments from end-of-semester evaluations in courses we have designed and taught using SLOs.

· I felt the harder I worked, the more my grade would reflect the effort.

· The tests are fair and actually reflect what we were supposed to learn, a first in my experience at [Prestigious University].

· You are by far the most organized and goal oriented professor I know

· Instead of memorizing formulas I have actually learned concepts and gained an understanding of math.

· The amount of thinking is immeasurable and once I stopped being scared, I developed a clear/concise way of approaching problems

The use of SLOs can have a particularly positive impact on under-prepared and math-anxious students. For these students, a required mathematics class can be a risky and intimidating obstacle that stands between them and their goal. Perhaps based on previous experiences [5], these students can believe that success or failure in a mathematics course is due to factors beyond their control [7]. The logical conclusion for the students is that they can do nothing to prevent failure. Formulation of SLOs, communication of SLOs to the students, and the use of SLOs as a basis for creating assessments can help this situation. By basing the course on explicit SLOs, instructors provide students with a concrete way to take responsibility for their own learning. SLOs represent a path to success that consists of steps that are each sufficiently explicit and self-contained that they are not intimidating. Students can perceive success in the course as the result of something they can control-the amount of time and effort that they devote to accomplishing the SLOs. The first two student comments (above) seem to echo this sentiment.

Although SLOs provide many benefits, some instructors may be resistant to using them. One obvious potential downside is the additional planning time that it takes to put them together. In spite of the fact that writing SLOs for a lesson adds a step to the lesson planning process, this step can pay off in future time saved. The clarity that SLOs add to the planning process can streamline lesson preparation. In addition, writing assessments becomes much more straightforward with SLOs as a guide. The timesavings from these and other benefits can offset the time spent developing the SLOs. Moreover, further timesavings can accrue when SLOs are reused in future iterations of a course, as the SLOs themselves need not be redeveloped each time.

The most persistent challenge to using SLOs from some instructors may stem from a fear of "dumbing down" a course. We hope that the last two student comments quoted above help to allay this concern. We believe that rather than dumbing down a course, SLOs can actually help "dumb it up." That is, SLOs can enable instructors and students alike to focus on the intended outcomes of instruction. Doing so can reduce fear and guesswork, and ultimately free students to think, learn, and understand.

Please see our articles for more information on the advantages of SLOs [2], the development of SLOs [3], and the use of SLOs in lesson planning [1].

References

2. DeLong, M., D. Winter and C. Yackel. 2005. Student Learning Objectives and Mathematics Teaching. PRIMUS. To appear.

3. DeLong, M., D. Winter and C. Yackel. 2005. Mental Maps and Learning Objectives: The FAST-SLO Algorithm for Creating Student-

Learning Objectives. PRIMUS. To appear.

Learning Objectives. PRIMUS. To appear.

4. Farrell, M. A. and W. A. Farmer. 1988.Secondary Mathematics Instruction: An Integrated Approach. Providence, RI: Janson.

5. Jackson, C. D. and R. J. Leffingwell. 1999. The Role of Instructors in Creating Math Anxiety in Students from Kindergarten through

College. Mathematics Teacher, 92(7): 583-586.

College. Mathematics Teacher, 92(7): 583-586.

6. Tyler, R.W. 1949. Basic Principles of Curriculum and Instruction. Chicago, IL: University of Chicago Press.

7. Weiner, B. 1985. An attribution theory of achievement motivation and emotion. Psychological Review, 92(6): 548-573.

Matthew DeLong (mtdelong@tayloru.edu, Department of Mathematics, Taylor University, 236 W. Reade Avenue, Upland, IN 46989) is an associate professor of mathematics at Taylor University. In addition to thinking about teaching and mathematics, Matt thoroughly enjoys spending time with his wife, son and daughter, directing his church choir, and acting in community theater productions. Soli Deo Gloria. Dale Winter (amanitav@umich.edu, Department of Mathematics, University of Michigan, Ann Arbor, MI 48109) is an assistant professor at the University of Michigan where he helps to direct the introductory program. He has also taught at Harvard University, Duke University, Bowling Green State University and the University of Auckland. His dissertation focused on mathematical methods in general relativity and cosmology. In addition to his professional interests in mathematics and education, he enjoys the novels of Primo Levi, military history, marine

biology and evolutionary psychology.

The Innovative Teaching Exchange is edited by Bonnie Gold.