Working With Multiple Mean of Representations: Knowledge Centered Teaching
During the Lines, Shapes and Spaces exercise, students had access to multiple means of representation of information and knowledge. There were vibrant, formal lessons involving the whole class, a variety of digital text, and verbal information. The most dynamic teaching space, however, occurred as the students engaged in active learning. It is the conceptual understanding, which involves an understanding of concepts, operations and relations, that is the focus of instruction, as opposed to the number and the different types of manipulatives. Teaching for mathematical proficiency requires a learning environment that provides a solid foundation of detailed knowledge and clarity about the core concepts, around which effective learning can take place.
Students working on Lines, Shapes and Spaces also had access and were encouraged to explore a wide range of authentic mathematical options to express their emerging understandings. Students were free to choose from a variety of digital and more conventional measuring instruments to calculate length, width, height and diameter. Several students purposely chose the most difficult route – they figured out the correct scale they must use to measure their shapes directly from their computer screens. Another student, in the diagram shown here chose to use a geoboard,
while another solved the problem using Geometer’s Sketchpad, as shown here.
Students then discussed how these two different expressions of the area of triangles were in fact, similar and connected. All students developed sophisticated, multiple ways to express solutions to problems. This principle lies at the heart of UDL in the mathematics classroom.
As students became more engaged with the topic of geometry, they also became more engaged with each other. Students’ conversations initially revolved around technical questions about using Geometer’s Sketchpad, and importing files. But as they became more confident in their abilities, talk quickly turned to the advancement of their mathematical understanding – students realized they could think their way through both the concepts, and the requirements needed to accomplish the task.
Initially, the students thought researchers were just being difficult when they were pushed to answer that question. But mathematics defines circles in a very precise way, and the question served as a jumping off point towards understanding that in math, names are much more than mere labels to memorize. This was an area where it was clear – because both special needs and regular students were asking the same questions – that the students had memorized such terms without acquiring the ability to reason mathematically with them. The ability to classify geometric forms is essential to descriptive geometry and mathematical proficiency.
There was another source of confusion among the students – they had been using the symbol pi since Grade 3, but four years later, after being introduced to variables in algebra, they found themselves on shaky ground. Is the number pi always the same? Researchers discovered here as well, there was a pervasive misunderstanding of some very fundamental ideas about mathematics.
It would be tempting to think of covering these topics again as back-tracking, but following the principles outlined in this research study, the discovery of areas that required some strengthening of conceptual understanding were welcomed. While no design process can anticipate when such misunderstandings will show themselves, the researchers’ way of constructing tasks, activities and assessments ensure that if students haven’t grasped key concepts, educators will notice. Problems with understanding therefore don’t become occasions to label students, and in fact, those who were coded as having special needs, started to question with confidence, and it became difficult to tell which students indeed had a learning disability.
