This is a Primary 3 class composed of 11 male and 8 female students. All 19 students have been identified as low-progress learners, with academic scores in the bottom 15-20% of their class. Mathematic factors that were measured include computational skills, mathematical language skills, and logical thinking abilities. Due to this, the school has created a smaller class size with differentiated pedagogy, alternative assessment methods, and adapted textbooks and materials. However, students are not "taught less" or with simplified materials; the curriculum and syllabus requirements are the same as the high-progress and middle-progress students are taught with equal rigor. Their classroom teacher highlighted the importance of creating a classroom environment with opportunities for active participation, concrete experiences, and positive feedback and encouragement. 

I planned a lesson for the beginning unit on "bar graphs" for Primary 3 students. The one and a half hour lesson included the pedagogical approach C-V and V-P-A. Table 1 includes the content in this lesson. Table 2 includes the pre-requisite knowledge that students should have from Primary 2. 

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 reading and interpreting data from bar graphs. Bar graphs represent information (data) in a visual representation (graph). They show the relationship between two or more sets of data. The students have some prerequisite knowledge of reading and interpreting data from picture graphs. 

For the first lesson, I focused on the vertical bar graph. The horizontal bar graph will be built-up in later lessons. By the end of the first lesson, students were able to: 

• read and interpret data from bar graphs using mathematical language

• compare data from bar graphs

• make overall conclusions about the purpose of bar graphs

Why is the C-P-A approach particularly useful for low-progress students? 

The C-P-A approach facilitates a visual approach to teaching and learning. It enables the teacher and students to make mathematical ideas visible and accessible. It is important for students to use concrete manipulatives, such as post-its, to 1) explain the relationship between picture graphs and bar graphs and 2) compare two or more values by physically taking post-its on and off.

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

The C-V and V-P-A approach ensures that the bars can be moved and manipulated like a physical model. The bars also provide a visual image like a pictorial model. Virtual manipulatives add another layer, especially when students got the experience of dragging the unifix cubes one-by-one. 

a. Effectiveness

The unifix cubes automatically snapped together and snapped apart (instead of requiring students to draw or erase a separate box for each segment of a bar). In addition, the virtual unifix cubes ensured that each column in the bar graph had equal width (instead of requiring students to measure with a ruler to draw columns with equal width). The annotation of the height of the bars on the virtual model corresponded to clear values. 

b. Efficiency

The virtual manipulatives were accessed quickly on the iPads by typing in one code. There was no logistical time spent distributing and retrieving concrete unifix cubes. During the lesson, students were able to construct the virtual model quickly and efficiently compared to the concrete model of individually constructing and connecting post-its and then drawing a literal bar around the boxes to represent the final total value.