The second principle of Universal Design for Learning focuses on providing students with multiple means of representation. This principle recognizes the fact that there is great variation in students’ strengths, needs, and preferences related to the perception, internalization, and comprehension of content. The image to the left shows the guidelines and checkpoints that educators use to eliminate or minimize barriers that may prevent their students from perceiving and understanding information. Once again, the guidelines are structured in ascending order of complexity from bottom to top.
Gadanidis (2015) writes that “coding offers new ways of experiencing, representing, and investigating mathematics concepts and relationships” (p. 162). Many students may find that they are able to understand mathematical ideas presented in this form than in other more traditional means. For this reason alone, the use of programming is a good addition to the mathematics classroom. However, a closer examination reveals that there are deeper ties between computational thinking and the UDL guidelines of representation.
Provide options for perception
In 1980, long before modern brain-imaging technology was created, Seymour Papert referred the existence of different regions of the brain. Furthermore, he posited that school math travels along one “brain route” between different regions, and that if that particular route was not available in some students, then alternate routes had to be harnessed (p. 46). To him, this is akin to how sign language works. The use of sign language allows many people to communicate even though their “speech organs” are unavailable to them. Papert claims that we can, through the use of computational thinking and programming, enable students to learn mathematics without accessing the traditional “brain route” taken by school mathematics (pp. 46-47).
What Papert was really talking about was offering students options for perception. Cognitive science tells us that there are many pathways in the brain along which information can travel from the sensory organs to regions where it can be processed. Learner variation is, in part, the result of the fact that the availabilities of these pathways very from person to person. One form of representation may work for a small percentage of students, but no single representation will work for all. When teachers add computational means of representation to their repertoires, they give their students another option for perceiving mathematical information.
Papert (1980) takes the idea of perception even further by outlining how some types of knowledge are best represented, and therefore most easily perceived, in specific ways. Some types of information are most clearly represented through action. To rely on words, symbols, or diagrams to represent these ideas is to disadvantage the student by offering a less than optimal portrayal that unduly taxes the student’s powers of perception. He refers specifically to complex ideas that are difficult to learn because students cannot experience them. Newtonian motion is an example. Apart from being a passenger of a car that is sliding on ice, it is very difficult to mimic this sensation. While some people have had this experience, most have probably not been in the state of mind at the time to conduct a mathematical analysis of the situation. The mathematics behind Newtonian motion are often learned through equations, but many students have difficulty perceiving the concepts through this medium. Within a computer programming environment, students can use computational thinking skills and concepts to develop simulations that will allow them to see and experience Newtonian motion first hand. This opportunity to perceive the mathematics will allow them to make greater sense of the equations that may follow.
Gadanidis (2015) writes that “coding offers new ways of experiencing, representing, and investigating mathematics concepts and relationships” (p. 162). Many students may find that they are able to understand mathematical ideas presented in this form than in other more traditional means. For this reason alone, the use of programming is a good addition to the mathematics classroom. However, a closer examination reveals that there are deeper ties between computational thinking and the UDL guidelines of representation.
Provide options for perception
In 1980, long before modern brain-imaging technology was created, Seymour Papert referred the existence of different regions of the brain. Furthermore, he posited that school math travels along one “brain route” between different regions, and that if that particular route was not available in some students, then alternate routes had to be harnessed (p. 46). To him, this is akin to how sign language works. The use of sign language allows many people to communicate even though their “speech organs” are unavailable to them. Papert claims that we can, through the use of computational thinking and programming, enable students to learn mathematics without accessing the traditional “brain route” taken by school mathematics (pp. 46-47).
What Papert was really talking about was offering students options for perception. Cognitive science tells us that there are many pathways in the brain along which information can travel from the sensory organs to regions where it can be processed. Learner variation is, in part, the result of the fact that the availabilities of these pathways very from person to person. One form of representation may work for a small percentage of students, but no single representation will work for all. When teachers add computational means of representation to their repertoires, they give their students another option for perceiving mathematical information.
Papert (1980) takes the idea of perception even further by outlining how some types of knowledge are best represented, and therefore most easily perceived, in specific ways. Some types of information are most clearly represented through action. To rely on words, symbols, or diagrams to represent these ideas is to disadvantage the student by offering a less than optimal portrayal that unduly taxes the student’s powers of perception. He refers specifically to complex ideas that are difficult to learn because students cannot experience them. Newtonian motion is an example. Apart from being a passenger of a car that is sliding on ice, it is very difficult to mimic this sensation. While some people have had this experience, most have probably not been in the state of mind at the time to conduct a mathematical analysis of the situation. The mathematics behind Newtonian motion are often learned through equations, but many students have difficulty perceiving the concepts through this medium. Within a computer programming environment, students can use computational thinking skills and concepts to develop simulations that will allow them to see and experience Newtonian motion first hand. This opportunity to perceive the mathematics will allow them to make greater sense of the equations that may follow.
Provide options for language, mathematical expressions, and symbols
In order for students to be able to access any representation of content, they need appropriate support when interacting with language, symbols, diagrams, video, and any other facet of the representation. Learning mathematics in a CT environment has the potential to support students in this manner. There are a wide range of programming languages that teachers can use to represent mathematical concepts. Many of these are highly syntactical in nature (for example, C++ and Java) while those termed as visual programming languages (for example, Scratch and Alice) are very much less so. This latter type allows “students to focus on the logic and structures involved in programming rather than worrying about the mechanics of writing programs” (Kelleher & Pausch, 2005, p. 131 in Lye & Koh, 2014, p. 53). Computational programs of this type do not clarify syntax as much as they eliminate it. The “drag and drop” block nature of these programs allows students to focus on computational concepts and skills as they investigate mathematics. Once students have developed proficiency enough that these skills and concepts no longer require intense focus, they are better positioned to move onto the more complex languages as they will have greater capacity to focus on the syntax.
Papert (1980) claims that modern language often fails to effectively describe some phenomena whereas computational thinking often offers more precision. He uses the example of juggling to show how a programming language can use computational ideas to create a succinct yet accurate explanation of a skill. Computer science, he says, has become expert in the area of highly descriptive languages because the audience for whom computer scientists write, the computers, require very high levels of precision. Unlike humans, computers will not infer meaning or fill in blanks. They will only do what is asked of them, and if their instructions are not sufficiently precise or descriptive, they will not perform as desired. Therefore, computer programming languages are, as Papert says, highly descriptive. Many students would benefit from the inclusion of such precision in the classroom, and so computational languages could very well be an aspect of UDL implementation.
In order for students to be able to access any representation of content, they need appropriate support when interacting with language, symbols, diagrams, video, and any other facet of the representation. Learning mathematics in a CT environment has the potential to support students in this manner. There are a wide range of programming languages that teachers can use to represent mathematical concepts. Many of these are highly syntactical in nature (for example, C++ and Java) while those termed as visual programming languages (for example, Scratch and Alice) are very much less so. This latter type allows “students to focus on the logic and structures involved in programming rather than worrying about the mechanics of writing programs” (Kelleher & Pausch, 2005, p. 131 in Lye & Koh, 2014, p. 53). Computational programs of this type do not clarify syntax as much as they eliminate it. The “drag and drop” block nature of these programs allows students to focus on computational concepts and skills as they investigate mathematics. Once students have developed proficiency enough that these skills and concepts no longer require intense focus, they are better positioned to move onto the more complex languages as they will have greater capacity to focus on the syntax.
Papert (1980) claims that modern language often fails to effectively describe some phenomena whereas computational thinking often offers more precision. He uses the example of juggling to show how a programming language can use computational ideas to create a succinct yet accurate explanation of a skill. Computer science, he says, has become expert in the area of highly descriptive languages because the audience for whom computer scientists write, the computers, require very high levels of precision. Unlike humans, computers will not infer meaning or fill in blanks. They will only do what is asked of them, and if their instructions are not sufficiently precise or descriptive, they will not perform as desired. Therefore, computer programming languages are, as Papert says, highly descriptive. Many students would benefit from the inclusion of such precision in the classroom, and so computational languages could very well be an aspect of UDL implementation.
Provide options for comprehension
Many of the checkpoints under this guideline relate directly to the computational skills that are described on the “What is CT?” page of this website. The ideas of highlighting patterns, making generalizations, and determining big ideas are connected to the computational skill of abstraction. Information processing brings up the computational skill category of data practices. By engaging students in these computational thinking skills, educators provide them with options for comprehension.
Grover and Pea (2013), in their description of how a student might enter into computational thinking, refer to the “use-modify-create” progression (p. 40). This entry is gentle. The student first uses programs without modification. During this step, he/she is given background knowledge of a concept or skill. He/She then begins modifying existing programs and observing the effects of the changes. He/She is testing new knowledge and skills to learn more about them. Finally, the student creates his/her own programs from scratch, thereby providing evidence of what he/she has learned.
Computational thinking affords students a unique opportunity involving comprehension. Students often have intuitions about things, but sometimes their intuitions may seem to betray them as things do not work as they think they should. For example, a child may see the direction that a boat takes in response to the movement of its rudder as counter-intuitive, or not understand why 35 + 35 does not equal 610 (Papert, 1980). In some cases, the error is superficial and the repair is simple. However, in other cases, the student achieves no resolution no matter how much he/she considers the problem. At times like these, the student is in danger of foregoing true understanding and latching on to a procedure. So, the student may add 35 + 35 and get 70, but never really understand why this answer is correct. Thinking computationally, either in a programming environment or with the use of manipulatives, allows the student to create external models of his/her problems. These models are easier to work with than internal wonderings. By tinkering with them, a student can update his/her intuitive knowledge and develop a stronger understanding of a mathematical concept (Papert, 1980). It is important to realize that some key computational skills are at play in this process. First, the student uses the skill of abstraction to develop the model of the problem or concept. As well, the entire process is a debugging of his/her intuition. Just as a student might test and debug a computer program, he/she can enact the same skills upon his/her own understanding.
Many of the checkpoints under this guideline relate directly to the computational skills that are described on the “What is CT?” page of this website. The ideas of highlighting patterns, making generalizations, and determining big ideas are connected to the computational skill of abstraction. Information processing brings up the computational skill category of data practices. By engaging students in these computational thinking skills, educators provide them with options for comprehension.
Grover and Pea (2013), in their description of how a student might enter into computational thinking, refer to the “use-modify-create” progression (p. 40). This entry is gentle. The student first uses programs without modification. During this step, he/she is given background knowledge of a concept or skill. He/She then begins modifying existing programs and observing the effects of the changes. He/She is testing new knowledge and skills to learn more about them. Finally, the student creates his/her own programs from scratch, thereby providing evidence of what he/she has learned.
Computational thinking affords students a unique opportunity involving comprehension. Students often have intuitions about things, but sometimes their intuitions may seem to betray them as things do not work as they think they should. For example, a child may see the direction that a boat takes in response to the movement of its rudder as counter-intuitive, or not understand why 35 + 35 does not equal 610 (Papert, 1980). In some cases, the error is superficial and the repair is simple. However, in other cases, the student achieves no resolution no matter how much he/she considers the problem. At times like these, the student is in danger of foregoing true understanding and latching on to a procedure. So, the student may add 35 + 35 and get 70, but never really understand why this answer is correct. Thinking computationally, either in a programming environment or with the use of manipulatives, allows the student to create external models of his/her problems. These models are easier to work with than internal wonderings. By tinkering with them, a student can update his/her intuitive knowledge and develop a stronger understanding of a mathematical concept (Papert, 1980). It is important to realize that some key computational skills are at play in this process. First, the student uses the skill of abstraction to develop the model of the problem or concept. As well, the entire process is a debugging of his/her intuition. Just as a student might test and debug a computer program, he/she can enact the same skills upon his/her own understanding.
References
Gadanidis, G. (2015). Coding as a Trojan Horse for mathematics education reform. Journal of Computers in Mathematics and Science Teaching, 34(2), 155-173.
Grover, S., & Pea, R. (2013). Computational Thinking in K-12: A Review of the State of the Field. Educational Researcher, 42(1), 38-43. doi:10.3102/0013189x12463051
Lye, S. Y., & Koh, J. H. (2014). Review on teaching and learning of computational thinking through programming: What is next for K-12? Computers in Human Behavior, 41, 51-61. doi:10.1016/j.chb.2014.09.012
Meyer, A., Rose, D. H., & Gordon, D. T. (2014). Universal design for learning: theory and practice. Wakefield, MA: CAST Professional Publishing.
Papert, S. (1980). Mindstorms: children, computers, and powerful ideas. New York: Basicbooks.
Gadanidis, G. (2015). Coding as a Trojan Horse for mathematics education reform. Journal of Computers in Mathematics and Science Teaching, 34(2), 155-173.
Grover, S., & Pea, R. (2013). Computational Thinking in K-12: A Review of the State of the Field. Educational Researcher, 42(1), 38-43. doi:10.3102/0013189x12463051
Lye, S. Y., & Koh, J. H. (2014). Review on teaching and learning of computational thinking through programming: What is next for K-12? Computers in Human Behavior, 41, 51-61. doi:10.1016/j.chb.2014.09.012
Meyer, A., Rose, D. H., & Gordon, D. T. (2014). Universal design for learning: theory and practice. Wakefield, MA: CAST Professional Publishing.
Papert, S. (1980). Mindstorms: children, computers, and powerful ideas. New York: Basicbooks.