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.