This is a Primary 3 class composed of 16 male and 21 female students. All 37 students have been identified as middle-progress to high-progress learners, with academic scores in the highest percentage (top 37 out of 106 P3 students). Mathematic factors that were measured include computational skills, mathematical language skills, and logical thinking abilities. The classroom teacher determined this class makeup due to formative assessments and daily observations. 

I planned a lesson for the beginning unit on "angles" for Primary 3 students. The one hour lesson included the pedagogical approach C-V and V-P-A. Table 1 includes the content in this lesson. 

After the lesson, I gave the students a survey to see what students value in mathematics learning. The survey also asked students' opinions on using technology to support their mathematics learning. 

This is the first time students were exposed to the concept of angles. There was no prerequisite knowledge of identifying or forming angles. 

Students gained many practical experiences, such as forming angles using paper and AngLeg manipulatives and identifying angles in the classroom and environment. The key objective was to have students see that angles are all around them and an angle can be formed with a certain amount of turning. By the end of the first lesson, students were able to:

• Associate an angle with a certain amount of turning (Big Idea of Measure)

• Identify angles in the classroom and environment

• Form a right angle using a mix of manipulatives (paper, AngLegs, virtual Geoboard)

Why is the C-P-A approach particularly useful for introducing angles in Geometry?

The C-P-A approach encouraged students to develop initial awareness of angles, specifically right angles, through purposeful concrete experiences. Once students were secure in their understanding of how angles were formed, they moved towards pictorial based problem solving (homework worksheet).  

I used a mixed media approach to demonstrate the idea of right angles because I wanted students to have multiple entry points in discovering right angles:

• In paper (familiar, life-relevant object found in nature)

• In AngLegs (new concrete manipulative)

• In virtual Geoboards

During the concrete activity, students showed high enjoyment and interest when discussing their reasoning with peers because they were able to link new, abstract information (a right angle) to already solidified networks of knowledge. Students were engaged through a relevant object in their daily lives (a piece of paper) and applied their knowledge and skills to make sense of the abstract concept of right angles. Then, they connected their understanding of right angles in the paper folding activity to the formation of right angles in the concrete manipulative, AngLegs. Afterwards, they took the AngLegs to reason, communicate, and justify right angles around the classroom. For metacognition, students had the opportunity to communicate with their peers and reflect/elaborate on their problem-solving processes of finding right angles around the classroom. 

Each medium played a critical role in student understanding and discovery, and students built new understandings at each step that bridged the three media. In the end, students used AngLegs to communicate and demonstrate their reasoning of right angles within their virtual Geoboard designs and their pictorial based homework worksheet. The versatility of the mixed-media design of this lesson encouraged students to expand their learning environment beyond the classroom and in the environment (Objective 2).  

Benefits of the C-V and V-P-A approach?

I had students create a real-life object/idea with one or multiple right angles on virtual Geoboards. Students came up with many creative ideas, such as a house, a shirt, a tank, and a cube. This activity leveraged students’ diverse interests, talents, and perspectives and steered them toward the learning objective of identifying and justifying angles, specifically right angles, in the classroom and environment (concept visualization). If possible, I would encourage teachers to plan for a longer period of exploration (30 minutes minimum) and discourse so every student can get the opportunity to express their ideas, reasoning, and thinking behind their creations. 

a. Effectiveness

Students were only able to construct polygons on virtual 11x11 Geoboards. In addition, the virtual Geoboards encouraged students to be creative and have fun in meaning-making. There was student choice and student-driven discovery based on individualized needs (e.g. drawing simple polygons vs. more complicated designs). 

b. Efficiency

The Geoboard was accessed quickly on the iPads by typing in one code. There was no logistical time spent distributing and retrieving physical Geoboards and rubber bands. In addition, the construction of the polygons was less time-consuming since there was no fine motor skill requirement to apply/take off rubber bands. Students used supplies appropriately and focused on the task at hand (instead of playing around with physical rubber bands).

The intuitive interface of the virtual geoboard required a low cognitive load so students focused on the problem task. 

The virtual Geoboard allowed students to create dynamic angles, in which students could drag, sweep, and rotate a line to make an angle. Many students created and highlighted right angles in different orientations, rather than the typical horizontal “L” angle orientation. Students were able to demonstrate a variety of right angles and gain a deeper understanding of angle measures, hopefully leading to connections between later angle concepts (i.e. acute and obtuse angles). 

Most importantly, the virtual Geoboard had embedded annotation and shading tools for the students to use alongside mathematical language to justify their choices. The application encouraged students to get immediate feedback from peers and the teacher. The teacher was also able to share student examples and creations to the entire class with a simple push button.