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Professional Development SeriesVolume 12

Motivating Middle School Students: The Critical Part of Lesson Planning in Mathematics

by
Alfred S. Posamentier, Ph.D
Dean, School of Education
The City College of The City University of New York

Inspiring students to learn is the cornerstone of successful teaching. A teacher's skill in engaging a class in its opening moments can set the tone for an entire lesson and contribute to its success. Regardless of the approach-whole class, small groups, or individuals-a key planning objective is to determine ways to draw students in at the outset. Teachers should strive to develop motivating activities that will not only introduce the lesson, but also hold students' attention throughout the class period.

Positive learning environments make the best use of students' attitudes, abilities, and experiences. Teachers can create a successful learning environment by crafting activities that appeal to students motivated in one of two ways: extrinsically or intrinsically.

Extrinsic motivation stimulates action in pursuit of tangible rewards or set goals. Sometimes extrinsic methods of motivation may work well. These methods include: grades, charts with personal goals, competition, contextualizing tasks that relate to students' experiences, economic rewards for good performance, peer acceptance of good performance, avoidance of "punishment" by performing well, and praise for good work (Guild and Garger, 1998). Extrinsic methods are effective for students in varying forms: they often demonstrate extrinsic goals in their desire to understand a topic or concept (task-related), to outperform others (egorelated), or to impress others (social-related).1

Intrinsic motivation-learning for its own sake-results from the internal drives already present in learners, such as the following:

  • curiosity
  • the learner's need to understand his or her immediate environment
  • a need to acquire a more complete understanding of a topic or subject
  • a need to improve one's position
  • a need to be entertained

This last drive may be affected by a teacher's classroom behavior, the content of material, or the style in which it is presented.

Sources of Motivation

Innate Curiosity

It is a natural human trait to seek out challenges that can be conquered by using existing skills and knowledge, resulting in a feeling of competence (Wolters, 2004). Often, a student's interest is stimulated when his or her curiosity is piqued. Teachers can whet students' curiosity by bringing, for example, an unusual item to class-a large ball to demonstrate a geometric principle or to explore Earth's properties.



1 The socially-related goal can apply to both extrinsic and intrinsic motivation.

A teacher might also stir curiosity by demonstrating a mathematical trick with an unusual result that prompts students to wonder why and how this trick works. Try this "trick": invite students to pick a number (x), then multiply it by 2 [2x]. Then have them add 9 to that [2x + 9], add in the original number [3x + 9], and divide by 3 [x + 3]. To this students will add 4 [x + 7], and as a final step, subtract the original number (7). No matter what original number is chosen, the final result will always be 7.


Invite students to try several numbers to see that this "trick" does, in fact, work every time. Algebra will then reveal the "trick."

A Choice to Learn

The desire to act on something as a result of one's own volition is often a motivating factor in the general learning process. Students will be more motivated if they can determine for themselves what is to be learned, rather than learning merely to satisfy someone else or to attain an extrinsic reward (Reeve, 2006).

There are motivational activities to support autonomy and encourage students to want to learn (Reeve, 2006). To do this, a teacher can provide a problem that students may not know how to solve, such as factoring a binomial or a trinomial. Students can be reminded that every math skill they develop is based on knowledge and strategies they already know. In this case, the teacher can suggest that they factor two- and three-digit sums and differences to find the strategies they already know. Then students can be encouraged to use these known strategies to factor the binomial or trinomial.

Desire for Challenge

Some students are more eager to do a challenging problem than a routine one.

It is not uncommon to see this type of student begin homework assignments with the most challenging problem. If a test has an "extra credit" item, these students might tackle it before looking at the remainder of the test, even if the time spent on the item prevents them from completing the required portion.

This drive to know more seems to go hand in hand with achievement. According to Gottfried, "Academic intrinsic motivation was found to be significantly and positively correlated with children's school achievement and perceptions of academic competence and negatively correlated with academic anxiety" (Gottfried, 1985).

Helping students achieve intrinsic motivation contributes to the positive learning environment. Here, teachers can introduce the lesson with an example that will challenge these students. In addition, the lesson presentation can be followed with an enriched problem to further allow students to develop their competencies.

SOURCES OF MOTIVATION FOR A LESSON INTRODUCTION
  • Innate Curiosity
  • A Choice to Learn
  • Desire for Challenge
  • Need for Acceptance

Need for Acceptance

Students' social needs, particularly by middle school, influence their relationship with the teacher, as well as with peers, thereby affecting how students learn. For instance, students tend to seek the approval of teachers, and tend to view higher grades as marks of approval, and conversely, lower grades as disapproval (Fehr, 2007).

Students also develop a need to perform well in front of their peers (Wolters, 2004). A teacher's awareness of these developing attitudes is integral in planning effective motivation and positive interactions with students.

Techniques for Engaging Students

Active engagement helps students to make knowledge their own and enables them to think about new situations. There are many techniques the teacher can use in the mathematics classroom to engage students, including:

Find Patterns

Patterns are inherent in mathematics; they are fundamental to algebra. Selected properly and presented enthusiastically, patterns can be an effective device for generating a concept and stimulating students' interest.

For example, consider a lesson on multiplying two negative numbers. This concept can be introduced by presenting a pattern similar to the one below by having students predict the next three numbers-in this case, 5, 10, and 15. Students can be asked to specify a general rule about this pattern and then complete a table to demonstrate the rule.

Factor 1 X Factor 2 = Product
-5 X 3 = -15
-5 X 2 = -10
-5 X 1 = -5
-5 X 0 = 0
-5 X -1 = ?
-5 X -2 = ?
-5 X -3 = ?

Teachers can use the table as a basis for a discussion on the pattern in the "Product" column, engaging them as active learners. The teacher can guide students' thinking through modeling or questioning to help them understand the underlying rule and justify it through common usage: "two negatives imply a positive."

For example, "I will not withdraw (two negatives) money from the bank." To further engage students, the pattern can be extended- in this example, to -15, -10, -5, 0. Students then work to predict the next three numbers.

A similar table can lead students to generalize that the product of two negative numbers is a positive number. They may also discover that their responses are to be addressed as the focus of the next lesson, thus increasing anticipation and interest. In this activity, the chart might appeal to field-dependent learners, who, states Whitefield (1985), "love to graph, map, illustrate, draw, role-play, create charts, invent games, make things, etc."

Patterns can also bring students closer to having a clearer understanding of negative exponents. Presented with a pattern such as 81, 27, 9, 3, students can be asked to predict what comes next. Students can also be asked to consider the first four powers of 3 in reverse order (from 34 to 31) and be guided to see the pattern as dividing by 3 each time to get the next number:

34 = 81
33 = 27
32 = 9
31 = 3

Extending this pattern of subtracting 1 from the exponent and dividing the value by 3, the sequence continues:


Some students may quickly see that they can continue the pattern by subtracting 1 from the exponent and dividing the preceding value by 3. This example can be a good lead-in to a deeper discussion of negative exponents.

The above examples show how patterns can serve as motivators if selected strategically and presented appropriately.2 In addition, this type of problem solving might appeal to students who need the support of specific teacher directions, concrete solutions, and clear instructions on what they are expected to learn.

Present a Challenge

A challenge can be rewarding, particularly for those students who see a challenge as a way to become a better thinker, a better problem solver. For students for whom challenges might invoke anxiety, offering a guided challenge can instill a sense of support and confidence.

For example, a teacher might say, "I'll show you that 2 = 1," and write the following proof.

  1. Let a=b
  2. Multiply both sides by a.
    a2=ab
  3. Subtract b2 from both sides.
    a2-b2=ab-b2
  4. Factor.
    (a+b)(a-b)=b(a-b)
  5. Divide both sides by (a-b).
    (a+b)=b
  6. Since a=b, then
    2b=b
  7. Divide both sides by b.
    2=1


2 A word of caution: there are patterns that appear to lead in one way but do not necessarily follow that anticipated direction. Teachers must be careful to select those that will not lead the class into an ambiguous situation. One example is the sequence: 1, 2, 4, 8, 16, ..., which can follow at least two perfectly correct mathematical patterns: 1, 2, 4, 8, 16, 32, 64, 128, ... or 1, 2, 4, 8, 16, 31, 57, 99, .... To find out more about this kind of problem see: 101+ Great Ideas for Introducing Key Concepts in Mathematics by Alfred S. Posamentier and Herbert A. Hauptman (Thousand Oaks, CA: Corwin Press, 2006). In addition, the National Mathematics Advisory Panel (2008) found that patterns are not emphasized in high-achieving countries. Because the prominence given to patterns in PreK-8 in this country is not supported by comparative analyses of curricula or mathematical considerations (Wu, 2007), the Panel strongly recommended that "algebra" problems involving patterns should be greatly reduced in state and NAEP assessments, textbooks, and curriculum expectations (p.59).

The students can then be challenged to find the error (Posamentier, Letourneau, Quinn, 2009). If they cannot see it, point out in step 5 that the original equation shows that a and b are equal terms. In step 5, then, (a - b) = 0. Division by zero is undefined, or simply not permitted! At this point, the rest of the proof falls apart.

Two important goals can be accomplished using the above example: (1) it can provide a powerful illustration of the role of definitions, or rules, in mathematics and (2) introduces the concept that division by zero cannot be permitted because doing so leads to contradictions such as 2 = 1. Such a challenge might inspire students to think critically, challenge their assumptions, and engage with mathematics in new ways.

Challenge Student's Thinking

Another form of motivation can be described as enticing students with "amazing and unexpected" aspects of mathematics (Sobel, Maletsky, 1988). This may include, as a leadin, showing the class some counterintuitive mathematical results.


For example, when introducing concepts in probability, a teacher can show the totally unexpected results of the birthday problem. To do this, a teacher would have the class announce their birth dates aloud one at a time and ask the remaining students in the class to raise their hand if they hear their own birthday mentioned.

Students will observe that in a room of 30 people, the probability of two of them sharing the same birthday is about 70%-far greater than anyone would intuitively expect it to be. To further add to the "amazing and unexpected" aspect, a teacher could tell students that among the first 34 American presidents, two had the same birthday of November 2-the 11th president, James K. Polk, and Warren G. Harding, the 29th president (Posamentier, Letourneau, Quinn, 2009).

Engaging students in discussions about mathematics can "promote their active sense-making" (Jansen, 2006). It can also foster a sense of community, contributing not only to students' motivation, but also to their success (Lewis, Schaps, and Watson, 1996).

Connect to the Real World

Pointing out a topic's usefulness can also be motivating (Boyer, 2002). For example, students might be asked how they would measure the height of the Empire State Building. A discussion of applying trigonometry, rather than dropping a tape measure down from the top of the structure (though possible, highly impractical), would enhance students' appreciation of trigonometry's real-world applications (Posamentier, Letourneau, Quinn, 2009). It might help students to see this problem exemplified in words, numbers, and pictures on the board, overhead, or interactive whiteboard.

Tell a Story

A well-told story can be motivating, can reduce math anxiety by activating theimagination, and can provide a relatable connection from the story's context to the topic (Schiro, 2004). Storytelling also appeals to the nature of field-dependent students, as it puts mathematics in a framework that is realistic or that relates to students' lives (Whitefield, 1995).

For example, there is a well-known story about one of the greatest mathematicians, Carl Friedrich Gauss (1777-1855). When he was in elementary school, Gauss found the sum of the first 100 natural numbers much faster than his teacher anticipated he would. Rather than add the numbers 1 + 2 + 3 + ... + 98 + 99 + 100 in the order in which they are written, he decided to add them in pairs:

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101
.
.
.
50 + 51 = 101

Then he multiplied 101 by 50 (the number of pairs) and found the product 5,050. After providing this relevant anecdote to the class, the teacher can use the procedure to help the students generate the formula for the sum of an arithmetic sequence. The style in which this story is presented is key-not as a rush to get to the derivation of the formula, but as story with interesting embellishments.

TECHNIQUES FOR ENGAGING STUDENTS
  • Find Patterns
  • Present a Challenge
  • Challenge Students' Thinking
  • Connect to the Real World
  • Tell a Story
  • Make Math Fun
  • Discuss Surprising Relationships
  • Present the Unknown
  • Integrate Technology

Make Math Fun

Often, a recreational feature of mathematics that is related to a topic can be motivating. For example, a teacher might inspire an algebraic discussion of number properties by asking each student to select a three-digit number in which the hundreds digit and the ones digit are not the same (for example, 847). The students then write their selected number in reverse order (748) and subtract the lesser number from the greater number (847 - 748 = 99). Having them reverse the digits in the result (099 to 990) and add these two numbers (099 + 990) yields a result they could share with the class.

Those who did not make an arithmetic error should all have arrived at the same answer: 1,089-no matter the number with which they started. Students may be amazed at this unusual number characteristic, and motivated to determine why it works; at the same time, the teacher has produced a thought-provoking introduction to an algebraic investigation.

Discuss Surprising Relationships

The teacher can further cultivate students' interest by discussing the many surprising relationships in mathematics. These curiosities may pique the interest of students and motivate them to determine why they are true. Here, a teacher can have students draw any quadrilateral: the purpose to show the same result among varied shapes. The class can be instructed to join the midpoints of the sides and to observe each other's drawings. In every case, they will get a parallelogram. This can lead to a discussion of the properties of parallelograms.

Present the Unknown

A motivational technique for more advanced learners can be to make them aware of a lack or void in their knowledge of a subject. Students should be given an opportunity to discover this lack on their own.

An activity for presenting the unknown can be found in trigonometry. Students begin their study of trigonometry with the various trigonometric ratios (sine, cosine, tangent) on the right triangle. Eventually they will be led to consider angles of measure greater than 90º. To spark interest in this topic, a teacher might have students find values such as: sin 30º, cos 60º, and cos 120º.

Students familiar with the 30-60-90 triangle will likely be able to figure out the first two values fairly easily. The third value may confuse some students, since they are unfamiliar with trigonometric functions of angles that measure more than 90º. Students will realize that they can find the trigonometric functions of acute angles, but not of obtuse angles. In this sense, students might draw on their curiosity and be motivated to extend their knowledge to find the answer.

The technique of presenting the unknown can also be employed in geometry when introducing the measures of angles having vertices outside a given circle. To illustrate this, suppose students have learned the relationships between the measures of arcs of a circle and the measure of an angle (whose rays subtend these arcs) with its vertex in or on the circle, but not outside the circle.

The teacher might present the class with the examples like the following, asking for the value of x:


Because students are familiar with how to find the value for x in the first two diagrams, their confidence may grow. Then the knowledge may set in that they cannot find the measure of the angle formed by two secants intersecting outside the circle. If asked (appropriately) what they would like to learn during the ensuing lesson, they would likely ask to learn how to find the measure of an angle formed outside the circle. This shows they have been motivated.

Integrate Technology

Many teachers have begun employing presentation stations, interactive whiteboards, and other technologies to present lesson introductions and engaging activities like those above. These can provide or activate important background knowledge, as well as stimulate student interest. Such tools can make technology-based teaching resources such as virtual manipulatives, videos, animations, and software tools available (with oversight) to an entire class.

Studies report that students' attitudes toward learning improve when technology is used in instruction (Silvin-Kachala 1998, Kulik 1994). The illustration mentioned earlier, where the midpoints of a quadrilateral are joined to form a parallelogram, can be very dramatically demonstrated with geometric software such as Geometer's SketchpadTM.

Selecting a Motivational Activity

Careful selection of a motivational beginning is the most creative, if not perhaps also the most difficult aspect of planning a lesson. Some helpful guidelines for selecting and presenting a motivational activity include the following:

  • Brevity
    So as to allow time for teaching the content of the lesson, teachers should keep the length of the motivational activity to a minimum.
  • Focus
    The motivational activity should not become the lesson. It should be a means to an end, not an end in itself.
  • Appropriateness
    The motivational activity should match the students' level of ability and interest.
  • Resourcefulness
    The motivational activity should draw on interest already present in the learner.
  • Transparency
    The motivational activity should clearly connect to the content of the lesson, as well as reveal the lesson's goal. Success here will determine how effective the motivational activity was.

Conclusion

Conclusion A teacher can use a host of motivational activities that have the ability to invigorate the first few moments of a lesson-that critical time when students' attention and interest might be won or lost. Using activities like those discussed here, a teacher can capture his or her students not only in those first moments, but also throughout the lesson, making learning a joy, not a chore. The rewards can be boundless.

About the Author


Alfred S. Posamentier, Ph.D., Dean of the School of Education and Professor of Mathematics Education at The City College of the City University of New York, is the author of more than 40 mathematics books for teachers, secondary and elementary students, and general readership. He is also a frequent commentator in newspapers on topics relating to education. Dr. Posamentier has been a frequent speaker at National Council of Teachers of Mathematics (NCTM) conventions and has been a long-term reviewer of new publications for The Mathematics Teacher journal. Dr. Posamentier is also co-author of Sadlier-Oxford's Progress in Mathematics program for Grades K-8.

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