7-Piece Mosaic Puzzle Tasks

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This is something I first read about when I was researching the Van Hieles’ work on geometric thinking while I was working on my specialist degree. The most powerful part of this research came from Developing Geometric Thinking through Activities That Begin with Play. “Play” is what really grabbed my attention but I really got interested when I realized the 7-piece puzzle was not a set of tangrams. This puzzle has more shapes that are different (see below), provides an opportunity for students make connections between the shapes, their properties, and area, and the activities – as with tangrams – start with play.

The Scenario:

The puzzle offers multiple opportunities for, in this case, young children to begin to explore shapes, their attributes, and properties.

The Exploration(s):

What can we do with these pieces? This open question intrigues students and encourages them to play and create. Students may even make a house or a dog. Children should be provided with plenty of time for free play and sharing their creations. This type of play allows teachers to observe how children use the pieces and to informally assess how they think and talk about shapes. During this time, students may also use two pieces to make another piece. When they do, ask them to find all of the pieces that can be made from two other pieces. When they do that, ask them to find the one piece that can be made from three pieces.
How many different shapes can be made with two pieces? Students select two pieces and search for all of the different shapes they can make by joining them with sides that match (The shapes can be different from the other pieces in this puzzle).
What if? If a student asks this, run with it. If not, you can subtly suggest it. What if?… is a powerful question that keeps students thinking and engages their creativity. What if we used three pieces instead of two? How many shapes could we make then? Try pieces 1, 2, and 5. Then try it with pieces 1, 2, and 5 with the number side down. Can you make all of the same shapes? Which shapes are “flippable?”
Puzzle Cards. These puzzle cards (see below) were adapted from some activities within the article mentioned above. There are many more puzzles and ideas mentioned in the article that are worth checking out.

The Math (and the Pedagogy):

The explorations above progress through the first three of the five levels of geometric thinking introduced by Dina and Pierre van Hiele in 1957:

Level 1 – Visualization – At the visual level of thinking, figures are judged by their appearance. We say, “It is a square. I know that it is one because I see it is.” Children might say, “It is a rectangle because it looks like a box.” (van Hiele, 1999). Orientations of shapes can be difficult for students at this level. For example, triangles may only be triangles if the “point” is on top.

Level 2 – Analysis – Students start to use vocabulary relating to properties. Size and orientation are no longer an issue because students begin to focus on specific properties of a shape. The properties of shapes are not related at this level. Students do not make connections between the properties of a shape, but they can identify different properties of a given shape.

Level 3 – Abstraction – This level is sometimes referred to as Informal Deduction. At this level, students begin to see the importance of properties in categorizing shapes. The properties are not seen in isolation anymore because students are beginning to see the relationships between properties. Instead, the properties are ordered logically, and the relationships existing among the properties are recognized (Mistretta, 2000).

Level 4 – Deduction – This level is characterized by the idea of a formal geometric proof. A person at this level can construct proofs, not just memorize them. Students at this level can see the possibility of developing a proof in more than one way.

Level 5 – Rigor – At this level, students understand the way how mathematical systems are established. They are able to use all types of proofs and may begin to become more creative in their proofs. They understand Euclidean and non-Euclidean geometry and the differences between these two systems.

The van Hieles also introduced five phases of learning to help students progress from one level to the next. Instruction should be designed strategically using these phases of learning.

Information/Inquiry

Teachers observe students to assess prior knowledge through inquiry; questions are raised.
Teachers learn students’ prior knowledge and students gain direction; Goals are set.
Ideas are introduced for students to think about.
Level-specific vocabulary is introduced when appropriate.

Directed Orientation

Activities should reveal geometric structures characterized by the level.
Tasks and materials should make the geometric concept very evident.
Tasks should be short and elicit specific responses.
Students actively explore and physical objects and shapes.

Explication

The teacher’s role is minimal but questioning is used to draw out student thinking.
The teacher facilitates students’ descriptions of concepts from informal to more precise mathematical vocabulary during discussion, but only after they can articulate the concept in their own words.

Free Orientation

Students find their own ways of solving problems.
The teacher’s role is to select appropriate problems with multiple-solution pathways and to encourage students to reflect, share, and elaborate.

Integration

The emphasis is on students understanding mathematical structures.
The students review and summarize what they have learned.
Ideally, the new level of thought replaces the old, and students can proceed to the next van Hiele level. If not, the students need to recycle through all or several of the phases using new tasks.

Five phases of Learning adapted from Ohio Department of Education, 2024

References:

Ohio Department of Education (2024). van Hiele model of geometric thinking, ohio department of education. https://education.ohio.gov/getattachment/Topics/Learning-in-Ohio/Mathematics/Model-Curricula-in-Mathematics/Van-Hiele-Model-of-Geometric-Thinking.pdf.aspx?lang=en-US/1000

van Hiele, P. M. (1999). Developing Geometric Thinking through Activities That Begin with Play. Teaching Children Mathematics, 5(6), 310–316.