THE DEAN'S COLUMN
Motivating Instruction: for Mathematics and Other Subjects
By Dean Alfred S. Posamentier, Ph.D.
Introduction
One of the most important aspects of classroom instruction is the way you motivate the students in your class to be receptive (enthusiastically) to the topic of a lesson. It would appear that geometry, because of its visual nature, would readily generate interest among students. Unfortunately, this is not always the case. Much of the course deals with proving theorems and then applying these theorems to artificial problems. Students interested in mathematics in general will probably be excited by this, as they will be by almost any mathematics activity. As an effective teacher you should focus your attention on the less interested students.
To motivate students is to channel their interests to the specific topic to be learned. In this chapter we will consider some techniques that can be used to motivate secondary school students in mathematics. Specifically, ten different techniques are presented, and each is illustrated by examples from algebra and geometry. (Note that the technique is the important part to remember. The examples are provided merely to help understand the techniques.)
What Is Motivation?
How to motivate students to learn is at the crux of your concerns when preparing to teach a lesson. If students can be made to be interested and receptive learners, then the rest of the teaching process becomes significantly easier and profoundly more effective. We will examine two categories of motivation, namely, extrinsic and intrinsic. Extrinsic motivation usually takes place outside the learner’s control. These are usually produced in the environment within which the student learns, and, to a large extent, are controlled by you, the teacher. Intrinsic motivators occur within the learner, and must be considered when planning your lessons.
When thinking of ways to “make a student want to learn” what you are about to teach, certain extrinsic methods of motivation come immediately to mind. Extrinsic motivation involves rewards that occur outside the learner’s control. This may include token economic rewards for good performance, peer acceptance of good performance, avoidance of “punishment” by performing well, praise for good work, and so on. Extrinsic methods are effective for students in varying forms. Students’ earlier rearing and environment have much to do with their adaptation of commonly accepted extrinsic motivators.
However, many students demonstrate intrinsic goals in their desire to understand a topic or concept (taskrelated), to outperform others (egorelated), or to impress others (socialrelated). The last goal straddles the fence between being an intrinsic and an extrinsic goal.
In a more structured form, intrinsic motivators tend to conform to the following basic types:

The Learner Wants to Develop Competencies. Students are often much more eager to do a challenging problem than one that is routine. It is not uncommon to see students beginning their homework assignment with the “challenge for experts” problem, even if the time spent on this prevents them from completing their routine work.

The Learner Is Curious about Novel Events and Activities. It is a natural human trait to seek out unusual situations or challenges that can be conquered by existing skills and knowledge and thereby provide a feeling of competence. When the learner’s curiosity about unusual stimuli is piqued, it becomes a form of motivation.

The Learner Has a Need to Feel Autonomous. The desire to act on something as a result of one’s own volition is often a motivating factor in the general learning process. To determine for oneself what is to be learned, as opposed to feeling that learning is being done to satisfy someone else or to get some sort of extrinsic reward, is another basic human need.
The teacher’s task is to understand the basic motives already present in the learners and to capitalize on these. The teacher can then manipulate this knowledge of students’ motives to maximize the effectiveness of the teaching process. Often, this manipulation can result in some rather artificial situations, contrived specifically to exploit a learner’s motives in order to generate a genuine interest in a topic. This is eminently fair and highly desirable!
With these basic concepts in mind, there are specific techniques, which might be expanded, embellished, adapted to the teacher’s personality, and, above all, made appropriate for the learner’s level of ability and environment. These are the strategies that should be taken to the classroom on a regular basis.
1. Indicate a Void in Students’ Knowledge
Students usually have a natural desire to complete their knowledge of a topic. This motivational technique involves making students aware of a void in their knowledge and capitalizes on their desire to learn more. For instance, you may present a few simple exercises involving familiar situations followed by exercises involving unfamiliar situations on the same topic. Or you may mention (or demonstrate) to your class how the topic to be presented will complete their knowledge about a particular part of mathematics. The more dramatically you do this, the more effective the motivation. Often, guiding students to discover this void in their knowledge is effective.
2. Discovering a Pattern
Setting up a contrived situation that leads students to “discovering” a pattern can often be quite motivating, as students take pleasure in finding and then “owning” an idea.
3. Show a Sequential Achievement
Closely related to the preceding technique is that of having students appreciate a logical sequence of concepts. This differs from the previous method in that it depends on students’ desires to increase, but not complete, their knowledge. A chart may be useful in applying this method of motivation.
4. Present a Challenge
When students are challenged intellectually, they react with enthusiasm. Great care must be taken in selecting the challenge. The problem (if that is the type of challenge used) must not only definitely lead into the lesson, but it must also be within reach of the students’ abilities. A challenge should be short and not complex. It should not be so engrossing that it detracts from the intended lesson. This would certainly defeat the purpose for which this challenge was intended. Thus, challenges providing motivation for one class may not do so for another. Teacher judgment is most important here.
5. Entice the class with a “geewhiz” amazing mathematical result.
To motivate basic belief in probability, a very effect motivation is to discuss with the class the famous “Birthday Problem.” Its amazing (and we dare say, unbelievable) result will have the class in awe.
6. Indicate the Usefulness of a Topic
Here a practical application is introduced at the beginning of a lesson. The applications selected should be of genuine interest to the class. Once again the applications chosen should be brief and not too complicated so that they motivate the lesson rather than detract from it. Student interest must be considered carefully when selecting an application. Remember, usefulness is appropriate only when a student has a prior knowledge of the topic involving the application.
7. Use Recreational Mathematics
Recreational mathematics consists of puzzles, games, paradoxes, or facilities. In addition to being selected for their specific motivational gain, these devices must be brief and simple. A student should realize the “recreation” without much effort in order for this technique to be effective.
8. Tell a Pertinent Story
A story of a historical event or of a contrived situation can motivate students. All too often teachers, already knowing the story they are about to tell and eager to get into the “meat” of the lesson, rush through the story. Such a hurried presentation minimizes the potential effectiveness the story may have as a motivational device. Thus, a carefully prepared method of presentation of a story for motivating a lesson is almost as important as the content of the story itself.
9. Get Students Actively Involved in Justifying Mathematical Curiosities
One of the more effective techniques for motivating students is to attempt actively to justify a pertinent mathematical curiosity. The students should be comfortably familiar with the mathematical curiosity before you “challenge” them to justify it. Although this could consume more time than may be normally allotted for a motivational activity, to proceed with a justification before sufficient exposure has been achieved would be counterproductive.
10. TeacherMade or Commercially Prepared Materials
Here motivation can be achieved by presenting the class with concrete material of an unusual nature. This may include teachermade materials, such as models of geometric shapes, geo strips, or specifically prepared overhead transparencies, or practical “tools” that illustrate a specific geometric principle. Some fine commercially prepared materials are available, ranging from geometric models to films of various kinds. Materials selected should be reviewed carefully and their presentation carefully planned so as to motivate students for the lesson and not to detract attention from it.#
Dr. Alfred Posamentier is Dean of the School of Education at City College of NY, author of over 40 Mathematics books, including: Math Wonders to Inspire Teachers and Students (ASCD, 2003) and The Fabulous Fibonacci Numbers (Prometheus, 2007), and member of the NYS Mathematics Standards Committee.