UDL

Multiple Means of Representation

The pervasiveness of the typical script for teaching mathematics – involving the set order of reviewing previous material, demonstrating how to solve problems, practicing the demonstrated problem, and correcting seatwork before assigning homework – provides context as potential for UDL in the classroom is explored.

Universal Design for Learning principles – which provides for multiple means of representation, expression and engagement in the classroom – are discussed here, in a mathematical context.

 

Multiple Means of Expression

Representation refers to the ways teachers organize or formulate content for classroom instruction, in order to present key ideas and concepts to students. Making math accessible hinges on the ability of teachers to create and find appropriate representations. The ways in which mathematical ideas are represented are fundamental to how people understand and use those ideas. It’s important to note, however, that simply providing more representation or models of concepts isn’t enough – textbooks are an example of this criticism. What matters most is the teacher’s ability to determine the appropriateness and equivalence of each representation.

Using this concept as a guide, researchers worked with students as they progressed through various problems. As a result, misconceptions were addressed as they arose, students were able to receive specific, dynamic feedback, and different representations were provided if a student was having unproductive difficulty with what they were using. This not only helped engage students and advance their ability to reason mathematically, but it helped determine the classroom’s next learning steps.

By the second week of the study, the Grade 7 math students were starting to realize the subject means much more than getting the right answer, right away – they were learning the process towards solving a problem is just as important as getting the right answer. Being good at math had come to include collaborating, building on and challenging other ideas. Students were also encouraged to embark on their own path to understanding, by using pattern blocks, computer programs and geoboards to demonstrate their work. It’s interesting to note that had the researchers insisted everyone use only their geometry sets, students with learning disabilities who had difficulty using those tools would have been mistakenly labeled as having difficulty understanding the concept itself.

Students usually don’t have the opportunity to express their thoughts and ideas in multiple ways. In fact, they’re usually presented with one way to solve a problem, and their job is to practice that procedure until it becomes routine. When skills are learned without understanding, they become isolated bits of knowledge. Learning new topics becomes more difficult because there is no network of previously learned concepts and skills to build from. As a result, students may come to believe that even slightly different problems require different procedures. They also fail to apply what they’ve learned in the classroom to the outside world.

This was not the case in this group of students. Questions of how they got to the answer were not dreaded, but became an invitation for further learning, for the benefit of everyone in the classroom.

 

Multiple Means of Engagement

By using multiple means of engagement, researchers were able to tap into the students’ interests, offer appropriate challenges, and increase their overall motivation to learn as much as they could about geometry.

Two significant benefits of engaging students in multiple ways towards mathematical proficiency were evident– there was increased enthusiasm among the students for ambiguity and uncertainty, and students were also willing to spend extra time exploring a question or idea. In fact, once the Grade 7 teacher decided to increase the amount of time the students would spend working on math, there weren’t any complaints – even when it cut into their free time. Cognitively demanding tasks were created for the students, students were told about the concepts they were to learn, and sufficient time was given so they could engage and spend time on the tasks. The use of digital media also effectively engaged the students, and offered more of a challenge and flexibility in their learning.