Fostering a Student’s Abstraction of the Relationship Between Parallelogram and Trapezoid Within Quadrilateral Hierarchy

Figures & data

Table

Level 0 (VH0)	Learners pay attention to the part of a shape’s visual characteristics but cannot identify common shapes.
Level 1 (VH1)	Learners recognize geometric figures by their appearance as visual wholes and ignore their properties. Visual perception and prototypes dominate learners’ thinking when identifying geometric figures. For these learners, for example, a rectangle is a rectangle because it looks like a window, or parallelograms are not rectangles because they are slanted.
Level 2 (VH2)	Learners identify and characterize geometric figures by their geometric properties and spatial relations between their parts. At this level, visual appearance is overshadowed by geometric properties. Students’ relational conceptualizations may range from visual informal reasoning (e.g., a square has four sides) to formal property-based reasoning (e.g., all sides and angles are equal on a square). The properties of geometric figures and the figure classes are independent. For VH2 thinkers, a square is a square because it has four equal sides and four right angles, not because it looks like a picture or is an item previously labeled as a square, as in VH1. Again, for these thinkers, a rectangle has two pairs of congruent sides with right angles, but it differs from a square.
Level 3 (VH3)	Learners understand that the properties of geometric figures are interrelated, and so are the classes of geometric figures. They can now classify figures hierarchically and justify that hierarchy. For this level thinker, a square is a rectangle because the property of ‘two pairs of congruent sides’ includes the property of ‘four congruent sides.’ Note that VH3 can be at the empirical level (e.g., empirically concluding “if a shape has one property it has another” (Battista, 2008, p. 852)), at the componential analysis level (e.g., a quadrilateral with four right angles must have equal opposite sides), at the logical inference level (e.g., mentally operating on property statements rather than images via logical inferences), or at the hierarchical classification level (e.g., justifying hierarchy through logical inferences).
Level 4 (VH4)	Learners can develop proofs using necessary and sufficient conditions and understand deduction.
Level 5 (VH5)	Learners can explore different geometries (e.g., spherical, Euclidean) and compare different deductive systems.

Table 1. Description of the task sequence development in the pilot study.

Pilot Trials	Tasks	Brief list of our reflections during task development
Trial 1 (Applied to Participant #1–1 session only)	1. Free experiment with shape makers – The student was given some shape makers (square, rectangle, trapezoid, deltoid, rhombus, parallelogram, a generic quadrangle that allows to make isosceles trapezoid) and was asked to do free experimentation using the dragging feature.	− Deemed useful in terms of helping the student use the dragging feature and learn about how to work with DGE − No focus on properties but some discussion about what shape makers make what part of the given picture. − Decided to consider Free Experiment and Warm-up Tasks as a single task based on student feedback and our analysis − Decided to drop deltoid from the list of shape makers since it was not in the national mathematics curriculum and was going to make things even more complicated.
	2. Warm-up Task – The student was given a picture of a shape collection and asked what parts could be made with square, rectangle, rhombus, parallelogram, trapezoid, deltoid, and quadrangle that allows isosceles trapezoid.
Trial 2 (Applied to Participant #2 and Participant #3–4 sessions per participant)	1. Warm-up Task – without deltoid	− The Guess-and-check Activity allowed students to investigate shape makers and their draggable features. Therefore, we decided to drop the Game. − Guess-and-check Activity fell short of helping the student understand that a given shape maker represented the whole class of shapes it could make. We decided to expand the set of shapes and design the Grouping Activity for the next trial.
	2. Guess-and-check Activity – Students were to guess whether the given drawings could be made by each of the given shape makers. The drawings had sets of square and rectangle; square-rhombus; square-rectangle-rhombus, and square-rectangle-rhombus-parallelogram, respectively.
	3. Game – Students were given a picture consisting of different shapes and asked which of the given shape makers could make which part of the given picture.
Trial 3 (Applied to Participant #4–5 sessions in total)	1. Warm-up Task – as in Trial 1 but without deltoid	− Guess-and-check Activity now includes square-rectangle-parallelogram, square-rhombus-parallelogram − Not including angle measures and side lengths caused problems and misled students. − We decided to engage students in an investigation of shapes followed by an investigation of certain measures (e.g., interior angles, side and diagonal lengths, angle between diagonals).
	2. Guess-and-check Activity – with square-rectangle; square-rhombus; square-rectangle-parallelogram, square-rhombus-parallelogram, and square-rectangle-rhombus-parallelogram.
	3. Grouping Activity – Students were asked to group the given shapes on paper (squares, rectangles, parallelograms with different sizes, shapes, and orientations) into three groups based on the properties they have seen so far.
	4. Naming relations as child-parent-grandparent
Trial 4 (Applied to Participant #5 and Participant #6–6 sessions per participant)	1. Warm-up Task as in Trial 3	− We realized that the rectangle maker used in Battista’s design gets bigger or smaller proportionally when dragged from some vertices, which misled students. So we decided to change the construction of the rectangle maker, so the dragging gave different size rectangles. − We integrated the “checking the groups on GSP” option into the Grouping Activity, but it led students to focus on what they saw on screen rather than making a property-based analysis. We then decided to drop the GSP-checking part of this activity. − Focusing students on interrelations using the child-parent-grandparent model seemed problematic as it did not foster property-based conjecturing and argumentation. We decided to remove it and focus students on affordances-constraints of shape families from the beginning (e.g., What constraints allow a parallelogram to be limited to a rectangle? What affordances allow a rectangle to become a parallelogram?). − Including a quadrangle that allowed drawing isosceles trapezoid confused students as it required them to return to the square and build up the hierarchy again. So, we reduced the number of quadrilaterals to five (square, rectangle, rhombus, parallelogram, trapezoid)
	2. Guess-and-check Activity as in Trial 3
	3. Shape makers with measures (side lengths, interior angles, diagonal lengths, angle between diagonals)
	4. Grouping Activity on paper + allowing students to check their groups on GSP
	5. Naming relations as child-parent-grandparent
Trial 5 (Applied to Participant #7 and Participant #8–7 sessions per participant)	1. Warm-up Task – with square, rectangle, rhombus, parallelogram, and trapezoid.	− Working with affordance-constraint dichotomy worked well in establishing the hierarchy − Grouping Activity seemed to push students back to VH2 rather than operating at VH3. We decided to drop Grouping Activity. − Guess-and-check Activity now includes rectangle-rhombus, square-rectangle-rhombus-parallelogram-trapezoid and worked well. − Task 3 needs to be supported with another activity to help students reflect on some properties satisfying the others. We decided to generate riddles in addition to Task 3. − When adding trapezoid to the hierarchy directly, students had difficulty conceptualizing it in relation to other shapes. We decided to have students investigate parallelogram-trapezoid relation and then make a connection to the hierarchy.
	2. Guess-and-check Activity – with square-rectangle; square-rhombus; square-rectangle-parallelogram, square-rhombus-parallelogram, rectangle-rhombus, square-rectangle-rhombus-parallelogram-trapezoid.
	Shape makers with measures (side lengths, interior angles, diagonal lengths, angle between diagonals) as in Section 3
	3. Grouping Activity on paper only
	4. Certain Questioning – (e.g., What constraints make a rectangle a square? What affordances make a square a rectangle? Can a square be part of the rectangle family? How so?) – very close to Task 3 in Table 2

Table 2. Description of the task sequence applied in the main study.

	Warm-Up Task: Students were initially given a collection of shape makers to drag and do a free investigation. The shape makers could be dragged from at least two of its vertices in all the tasks. Then students were given the same shape makers to fit in each picture as seen on the lower part of the left column. The purpose of the Warm-up Task was to have students investigate the shape makers and see how they could be dragged from the vertices to fit into the given picture made up of a combination of square, rhombus, and trapezoid. As students did these, they started to think about the given shape makers’ operating conditions. At this stage, there was no expectation on the students’ part to generalize common characteristics or think about the properties of the given geometric figures. These initial activities were designed to serve them in learning about the use of shape makers and the main features of GSP. This part is very similar to Battista’s design.
	Task 1: Students were given a single shape maker (e.g., square maker) and a collection of figures from the same family (e.g., squares of different sizes and orientations) and from a most immediate successor family (e.g., rectangles of different sizes and orientations) and then did the following: Predicted whether the first given shape maker (e.g., square maker) could make all the given geometric figures, Checked their predictions using the given shape maker (e.g., square maker) in GSP, Investigated why the given shape maker (e.g., square maker) could (not) make figures by focusing on the affordances or constraints of the shape maker using dragging, using the teacher’s guidance, and probing. Students were then given a second shape maker from a most immediate successor family (e.g., rectangle maker) and a collection of figures from the same family (e.g., rectangles of different sizes and orientations), and the subset family (e.g., squares of varying size and orientations). Students were then asked to go through the above steps for the second family in addition to: Focusing on the relationship between the two families (e.g., squares versus rectangles) based on constraints and affordances under the teacher’s guidance and probing. (Note that when comparing parallelograms to trapezoids, students were given both shape makers together and a collection of shapes from both families. This is because by the time students developed parallelograms, they only had trapezoid left in the sequence to investigate.) Once such a process was completed, students were engaged in Task 2 before exploring an immediate successor family. The purpose of Task 1 was to have students simultaneously (and tacitly) compare one quadrilateral to its immediate successor in a hierarchy as they focused on the spatial structure of the given quadrilateral. Such operation aimed to help students consider the affordances and constraints of the given quadrilateral and its successor based on their properties. Note that the students work under teacher guidance and probing throughout their engagement with this task.
	Task 2: Students were given previously investigated shape makers (e.g., square and rectangle makers) on separate screens. By dragging and considering the measurements revealed on the screen, they were asked to investigate the given family’s properties (e.g., squares and then rectangles, respectively) regarding side lengths, interior angles, diagonals, and parallelism and to make notes of those. They went through this process for each shape maker in order (e.g., first square maker and then rectangle maker). The purpose of Task 2 was to help students investigate the geometric properties of a quadrilateral set, and its immediate successor quadrilateral set considering measurements. Note that Task 2 is not treated in isolation from Task 1. Students’ engagement with and discussions of Task 2 were constantly related to the results of students’ experiences with Task 1. In this sense, Task 1 and Task 2 went together.
What constraints make a rectangle a square? What affordances make a square a rectangle? Can a square be part of the rectangle family? How so?	Task 3: Once students had enough experience with geometric properties of the quadrilaterals (e.g., squares and rectangles) given in both Task 1 and Task 2, they were then asked to compare the properties of two sets of quadrilaterals in tandem to decide which set of properties completely include the other. The purpose of Task 3 was to enable students to complete the geometric structuring of the affordances and constraints of the given quadrilateral set compared to its predecessor set based on their properties regarding side lengths, interior angles, diagonals, and parallelism. This is the stage in which students were encouraged to heavily reflect on their experience with Task 1 and Task 2 and on the interrelations, in the absence of DGE, on paper-pencil environment.
Riddles: My opposite sides are congruent (equal), but they can all be made congruent (equal) since I am flexible. When my vertices (corners) are dragged, my diagonals always stay congruent. Who am I? Please think of all the possibilities and color the appropriate shapes that fit this description!	Closing Task: Students were given some riddles about each quadrilateral set with or without a collection of figures (e.g., the riddle on the left column is intended for the rectangles set) and asked to identify the quadrilateral set that fits the description given in the riddles on paper-pencil environment. The purpose of the Closing Activity was two-fold: (1) to help them reflect on properties of quadrilateral sets at a higher level; (2) to assess their understanding of interrelationships to see a fit with VH3. This part is also very similar to Battista’s design.

Figure 1. The sequence in which the quadrilaterals were investigated (“Par.” means Parallelogram).

Table 3. Difference between Shape Makers and our task design.

Shape Makers	Our Design
The activities use all shape makers at once to investigate the properties of all shapes: “Which Shape Maker can make the least number of different types of shapes?” (Battista, 2012, p. 153)	The activities involve shape-family pairs or triplets sequentially rather than all at once (using a bottom-up approach to help them think about asymmetry and transitivity relations).
The focus is on properties in the following sense: 1. Can every shape made by the Square Maker also be made by the Rectangle maker? Explain why. 2. Can every shape made by the Rectangle Maker also be made by the Square Maker? Explain why. 3. Why can the Rectangle Maker be used to make squares? 4. What property or rule could be added to the Parallelogram Maker to make it a Rectangle Maker? (Battista, 2012, pp. 189–190)	Focusing on inclusion relations using affordance-constraint dichotomy for each shape-family pair and allowing students to think about shape families with the affordances and the constraints of their properties: “What affordance does the parallelogram family have that the rectangle family does not?”
Using language such as “every rectangle is also a parallelogram” or “every parallelogram is also a rectangle” (Battista, 2012, p. 193)	Using language such as “rectangle family is part of parallelogram family” or “parallelogram family includes rectangle family.”
Mixing spatial features and measurement Rectangle makers proportionally gets bigger/smaller from certain vertices in GSP	Moving from investigations with spatial features to measurement No proportional expansion of shape makers.

Figure 2. The relation between parallelogram and trapezoid – Data Chunk 1 (Task 1 – Time: 00:00–03:10 minutes).

Figure 3. Zooming in on the geometric properties – Data Chunk 2 (Task 2 – Time: 08:30–9:35 minutes).

Figure 4. Considering parallelism as an affordance – Data Chunk 3 (Task 3 – Time: 16:00–18:40 minutes).

Figure 5. Inclusivity of trapezoid family – Data Chunk 4 (Task 3 – Time: 18:50–20:15 minutes).

Figure 6. Efe’s representation of quadrilateral hierarchy (from Teaching Session 7 – Time: 25:30 minutes).

Table

Section 1	Section 2	Section 3
1. Which of the following shapes is/are square?	1. What is a square? What are its most basic properties?	1. Is a square a trapezium?
		2. Is a rhombus a rectangle?
2. Which of the following shapes is/are rectangle?	2. What is a rectangle? What are its properties?	3. Is a rhombus a trapezium?
		4. Is a rectangle a parallelogram?
3. Which of the following shapes is/are rhombus?	3. What is a rhombus? What are its properties?	We asked the following two additional questions in post-interview only: 5. Is a trapezium a parallelogram? 6. Is a parallelogram a square?
4. Which of the following shapes is/are parallelogram?	4. If you wanted to tell a friend about the parallelogram, how would you do it? What properties must a shape have for it to be a parallelogram?
5. Which of the following shapes is/are trapezium?	5. What is a trapezium? What are its properties?

Note that the students went through all the questions in Section 1, Section 2, and Section 3, respectively.

Battista, M. T. (2008). Development of the shape makers geometry microworld. In G. W. Blume & M. K. Heid (Eds.), Research on technology and the teaching and learning of mathematics: Cases and perspectives (Vol. 2, pp. 131–156). Information Age Publishing.

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Battista, M. T. (2012). Shape makers: Developing geometric reasoning in middle school with the Geometer’s Sketchpad. Key Curriculum Press.

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