Since we are all so good, why don’t we sing together?

All students [singing]: I say s-h-a-p-e-s… I say s-h-a-p-e-s … That is all

around us! (See Figure 4.82).

All students: And that was Shape Idol!

[Applause]

Voyeur - New/Wonderful/Surprising

This DMP offers conceptual mathematical surprises because it explores (a) connections between representations of geometric shapes and (b) connection between representations of shapes and everyday objects.

In the first song, the students contrast some properties of squares, pentagons, and hexagons through the lyrics “Hey hexagon, I am straight up playa. The arms go up to me, isn’t

that better? Things I hate, pentagons aren’t nice. Four vertices like way up tight.” In typical

explorations, students see shapes in isolation. In this song, students are exploring some connections between three different types of polygons.

In song two, the triangle is portrayed as a “strong” shape. This is an interesting and non- typical aspect of seeing triangles in elementary school. The “strength” in this case refers to the fact that in order to deform a triangle, one has to change its angles and, to change the angle the

Figure 4.80: Madame N

Figure 4.81: Student 1 and

Student 2

lengths of its sides has to be changed necessarily. That is, in order to deform a triangle, one needs to change the lengths of its sides. “Triangles are the strongest shape because they have fixed angles. The angles do not collapse like the angles of a square do. Triangles are used to make a truss - the strongest architectural support.”

(http://answers.askkids.com/Weird_Science/why_is_the_triangle_the_strongest_shape). In the third song, students mention the fact that shapes can be explored in terms of their relationships, not in isolation. They sing: “They are coming over. They are all together. They are

a family of shapes and they have been that way forever… We’ll always stick together, and that’s never gonna change.” In the last song, students mention that representations of shapes may be

found in the world about them. They sing: “I say s-h-a-p-e-s…That is all around us!”

Potentially, the ideas and connections explored in the four songs of this DMP offer to the audience ways of seeing the new and wonder in mathematics because mathematical objects are not seen individually or isolated, they are explored in relation to other objects and in relation to the world.

Voyeur - Sense-Making

In this DMP students do not present arguments to support the ideas they are exploring. In the first song, the hip-hop style makes it difficult to understand the words the student is singing. Probably, the audience can only understand exactly what the student is singing after listening to

the song several times.23 Although students are seeking to explore connections between a square,

a pentagon, and a hexagon, there is no explanation about how the connections are made. In the skit, the student is introduced, says the she will sing a song about squares and she sings: “Hey

hexagon, I am straight up playa. The arms go up to me, isn’t that better? Things I hate, pentagons aren’t nice. Four vertices like way up tight.” Like in DMP #2 Geometrical Idol and

DMP #3 Shape Songs, students do not present justifications and explanations about the insights posed in the lyrics. Students could have explored similarities and differences between the shapes. They could have, for instance, explored the fact that squares, pentagons, and hexagons are all polygons and they can be seen as formed by triangles. In contrast, they have different number of

23 _{From a voyeuristic point of view (Boorstin, 1990), it would be interesting if DMPs based on songs}

sides and different number of angles. Thus, they could have explored the question: are the angles of these shapes congruent?

In the second song, students explore the fact that there are three types of triangles (equilateral, isosceles, and scalene) and they are the “strongest shapes.” Like in the DMP #8 2D

Land, students are exploring thus the notion of representation of triangles as everyday objects,

specifically regarding notions in architecture, that is, representations of triangles as a shape used to construct roof structures and the structural rigidity of the shape. Students could have provided more explanation about why triangles are strong shapes. They could have compared triangles to shapes such as parallelograms and justified that triangles are strong because to change their angles it is necessary to change the length of its sides.

In the third song students suggest that shapes should not been seen separately or in isolation. They sing: “[there] are a family of shapes.” Students probably intend to explore the notion that a set of shapes can be classified in the same category (similar to DMP #5 Little

Quad’s Quest), or that one shape fits in other shapes (like the idea explored in the DMP #4 Triangles), or that one shape many be a special case of other (similar to the idea explored in the

DMP #13 Square Trial). However, again, students do not present explanation or examples to justify or support what they intend to explore. They could be more explicit in the lyrics

(presenting more information) or they could have used visual elements (e.g., posters) to support their ideas, as is done in DMP#17 Square Based Pyramid, for instance.

In the last song, students mention that representations of shapes may be found in the world about them. They sing: “I say s-h-a-p-e-s … That is all around us!” However, students do not offer examples of how representations of shape can be found in the world. DMP #2

Geometrical Idol, for instance, mentions that bee hives can be seen as representations of

hexagons. In DMP #5 Little Quad’s Quest, students relate, for instance, a representation of a rectangle to a wide TV screen and a representation of a rhombus to a diamond.

The use of gestures is also important in some moments of the DMP. In the introduction, for instance, the presenters/judges say “Welcome to Shape Idol!” and they make a synchronized and rotated body movement that offers to the audience a sense of symmetry (see Figure 4.83 a-b).

Vicarious Emotion

Differently from DMP #2 Geometric Idol, DMP #3 Shape Songs, DMP #7 Math Facts

Show, DMP #9 We are the Polygons, DMP #11 Sphere on the Lose, DMP #13 Square Trial, and

DMP #14 Pointacula, students are not taking on roles as geometric shapes in this DMP. They are playing roles as singers who sing about shapes. However, some mathematical emotions are explored when students (a) portray the triangles as strong shapes (song two) and (b) explore a sense of belonging by saying “they are all together. They are a family of shapes and they have

been that way forever.”

The DMP also uses close-ups on the actors’ facial expressions. The audience may vicariously feel what the actors are feeling, but some of these feelings are not necessarily mathematical emotions, that is, is this case, emotions related to the mathematical properties of the shapes. Some of the emotions in this skit, like in DMP #2 Geometric Idol, refer to a “sense of achievement” by having performed well the song in the conquest. Although it is not a

mathematical emotion, it is an emotion that may be significant to the mathematical activity, since

achievement and/or recognition may be an aspect that may encourage students on getting

engaged in future and further mathematical experiences.

Visceral Sensation

This DMP may offer some visceral sensations to the audience because it explores how different shapes fit each other and that shapes fit in the world around the students. However, as mentioned in the section on Voyeur - Sense-Making of this case, there is no explanation to support these connections. The DMP is also based on soundtracks, which is a fundamental characteristic of the visceral eye (Boorstin, 1990).

Figure 4.83 a-b: Students’ gestures involving symmetry.

DMP #17: Square Base Pyramid

Table 4.24: Square Base Pyramid

URL: http://www.edu.uwo.ca/mathscene/geometry/geo13.html Strands and

Content:

Strand: Geometry and Spatial Sense

Content: Properties of a square base pyramid Format: Video.

Time length: 2:30.

The Arts: Musical: Skit and Songs

Participants: Three students and a guitar player Setting: Performed in a classroom

Info: None

Description

Students play a skit to explore properties of a square base pyramid such as number of vertices, faces, and edges. Two students play roles as archaeologist exploring a pyramid and one student plays the role of Cleopatra’s mummy, who explains the properties of a square base pyramid. Following, I present a transcription of the DMP:

Myrtle: Hey! Look that pyramid. Let’s go see what is inside. Student 2: Ok…Myrtle! (See Figure 4.84).

Myrtle: What?