Universal Design for Learning
Multiple Means of Representation
Multiple means of representation has to do with the “what” of learning and how we gather facts and categorize what we see, hear, and read (CAST, 2011). As teachers, our classrooms are full of students with unique and diverse learning styles. Because of this, it is crucial to provide options for perceptions and reach students individual recognition networks. One aspect of multiple means of representation suggested by CAST (2011) is that teachers provide many opportunities for manipulating with or working with different types of language and symbols. For example, in a unit on the number system, students are presented with vocabulary and symbols that they need to be able to recognize. Students should already be familiar with the different types of numbers in the real number system, so they would fill in a concept map with the different types of numbers. Eventually, students would discover the characteristics of performing operations with each type of number which will help them later in the unit. Students can review the vocabulary and make a concepts map to display the vocabulary and use that information to learn how operations with real numbers are performed. In this particular concept map, I would look for some examples students could use to help them remember and understand each type of number. With the real number system, many types of numbers are components of other types of numbers. Students would need to recognize this. On an end of unit assessment, I would ask students a couple of questions about which numbers are components of other numbers to show the concept map was successful.
Learners vary in their facility with different forms of representation (CAST, 2011). While examples of definitions may clarify concepts for one student, it actually may be more confusing for another. It is important in my classroom that I provide alternative representations not only for accessibility, but for clarity and comprehensibility across all learners. When using a concept map on number systems, I can give students the opportunity to use technology to build on their vocabulary as well. In the concept map they just give examples, however, with this website mathematics glossary created by Bruce Simmons, students can look up actual mathematical terms and concepts and learn them through text, images, and examples relating to real world applications. This online dictionary uses multiple means of representation to help students decode mathematics notation.
Tomlinson (2009) discusses that learning is shaped by a variety of factors including different learning styles. Another aspect of multiple means of representation is the ability to provide students with a variety of ways to learn the same concepts. In order to meet the needs of all my students, I must allow students multiple ways to explore the same topic so each individual has the opportunity to learn the way that best fits their needs. After doing a brief review of the number systems using the concept map or online dictionary from above, the students watch a video “The Number Line Dance” created by Alex Kajitani which is a rap of the different integer rules for adding and subtracting integers (MathRaps, 2009). After the students watch the video, they can actually get up with the whole class, learn the words to the song and act out the number line dance like they do in the video. After the video, the students are given a number line on paper in which they can they can perform the operations discussed in the video and discover how they work.
Along with the video, I would like to give students songs or riddles they can use to help them remember the different rules or formulas in mathematics. In an algebra classroom, I would teach my students this song to help them remember the rules for adding and subtracting integers. When I substitute in math classes and students have to add and subtract integers, I always ask them if they have heard this song. Most of the time, they have not, so I sing it for them in hopes they will want to learn it to help them solve certain problems in math. If I had my own classroom, I would try to teach my students songs like this to help them connect to certain topics. It would be great to post the songs throughout the room for students to see daily to help them remember certain rules.
In addition, over the next few class periods students can participate in different activities to help build on their skills of adding and subtracting integers. One type of activitiy that would get students moving around the classroom would be the “Human number line activity” where they can physically get up and use their bodies to add and subtract different integers. There would be a number line taped to the floor and students would be given specific numbers and operations to perform by walking up and down then number lines (depending on whether they are positive or negative numbers). Along with this type of activity, students who are visual learners could use integer chips as manipulatives to help them visualize the addition and subtraction of integers.
In addition to the human number line activity, another type of activity I would use is a game. This game cups and counters could be used for solving equations using addition and subtraction properties of equality. It is a hands on game that allows students to work together to solve problems quickly and accurately. This game allows students to use their knowledge of solving equations and have a friendly competition with their classmates. The students break into groups of three and each group member rotates jobs. Each student will have to create a problem, solve a problem, and check a problem before the group can get a point. The students would have to work together with their teammates but complete their job independently. When allowing students to compete with games in class, I would have to observe my students and ensure they are playing in a friendly manner and staying focused on the task at hand.
Multiple means of expression
Multiple means of expression is defined as the “how” of learning and is concerned with planning and performing tasks (CAST, 2011). In a classroom full of diverse learners, I need to provide my students with the opportunities to express what they know in multiple ways. This action and expression require a great deal of strategic networks, and require practice and organizations from the learner. One of the best ways to accomplish multiple means of expression would be to provide students with the opportunity for choice to express what they have learned. A choice board offers students a way to make decisions about what they will do in order to meet class requirements.
Although I have never used a choice board, I think that if I was teaching a unit on integers I would want to end the unit with one similar to this so that students could decide how to express what they have learned. The choice board is designed to target different learning styles and allows students to use creativity and a variety of different techniques that they might not traditionally express in a mathematics classroom. With this choice board, I would be able to direct the process, and the students would be able to take control of their learning by making their own choices (Tienken, n.d). My students would select the choices which are most appealing to them, while I ensure they are still able to tackle key ides and use the key skills central to the topic of integers. This choice board would show me that they have learned to rules and concepts of integers and but would also allow them to express their knowledge in way that best fits their learning styles.
Another strategy that is great for teachers to use when wanting to give students the opportunity to express their knowledge is a RAFT. A RAFT strategy is an acronym for role, audience, format, and topic. It offers students a way to express the material they have just learned in a format of their choice which helps students to gain a better understanding of the content topics as they write (Santa et al, 1988). Students are given a list of topics to choose from and use their knowledge to explain information about that topic. They are required to pick a role they want to play, an audience they wish to speak to, and the format in which they want to use to try and get their point across. Here is an example of a RAFT I would give m students during a unit on integers. Since I have never used a RAFT in a classroom, when I do I will create a rubric for my students so they are aware of my expectations for the rubric. My students will know that while the activity may be a fun way to express what they know it will still be graded and calculated into their average.
Another way students can express what they have learned is through communication and discussions. With the implementation of the Common Core State Standards, it is becoming more important for students to be able to express their answers to problems in different forms. By providing students with the opportunity to not only solve the problem but to explain in writing as well, I am giving them multiple ways to express themselves and I am able to use the information to determine whether or not they fully understood the whole concept. For example, some students may be better at explaining how to solve a problem than at actually solving the problem. If I saw a student was getting an answer wrong on a test, without having the students write an explanation for how to solve that problem I might immediately think the student is unaware on how to solve it. In reality, the student might know how to solve the problem but may be struggling with the mathematical computations of it. For example, in a unit on solving equations, I could give my students some problems like these in which they are not required to get the answer, but rather explain the steps they need to take to get there. This type of problem helps me to see what it is my students understand about the topic and what they are struggling with. It also helps my students by allowing them to express their knowledge in writing rather than through mathematical computations.
This can also be used in geometry classroom. When I was doing a long term leave in a geometry class we completed a unit on analytic proofs where the students had to use their mathematical skills to prove that a rectangle is a parallelogram. These types of questions require many different computations and I noticed that some students struggled with getting every computation correct because there were so many steps along the way. They would get lost in the “6 step proof” which would then make their final answers incorrect. When I noticed this, I wanted to provide students with the opportunity to tell me that they did actually know how solve the problem. I asked the students to take out a piece of paper and gave them a new analytic proof to solve. This time, I told my students that instead of solving the proof, they could choose to explain the steps they would take to solve it and what the process would look like. It was their choice, and they could choose whichever way they felt would be best to express their knowledge. To my surprise, a lot of students chose the explanation. When I asked them why, one student told me that she is better at expressing her knowledge through words than through mathematical computations. She said that she has the vocabulary and knows the steps she needs to take but when it comes to applying that to the mathematics she makes mistakes that cause her to get the answer incorrect. When I saw how well students took to explaining their answers, I actually decided to add an extra problem similar to this one to their final.
Multiple means of engagement
Multiple means of engagement is the “why” of learning and is a part of UDL that stimulates interest and motivation of my students (Cast, 2011). In order to reach the affective networks of my students I need to challenge, excite, and interested them in order to get them engaged and keep them motivated in my classroom. The unit on integers includes many different activities that provide multiple means of engagement to students. Through games, songs, technology, concept maps, and different activities, students attention can be grabbed which can help them to stay engaged in learning longer. To end the unit on integers, it I would provide students with the choice board from above. This would motivate a variety of learners because it would allow them to choose how to represent the information in a way that is most interesting to them. For example, a student who may be musically inclined might choose to create a song or rap about the integer rules, while a student who is artistically inclined may prefer to create a poster. In addition, students who are motivated by using technology could create a PowerPoint while students who are interested in writing could write a book. Offering students’ choices can develop self-determination, pride, and accomplishment, as well as increase the degree to which students feel connected to learning (Cast, 2011).
According to Cast (2011), individuals are engaged by information and activities that are relevant and valuable to their interests and goals. Individuals are rarely interested in information or activities that have no relevance or value to them. In an educational setting, one important way to get students interested is to demonstrate the material through authentic, meaningful, fun activities. For example, many students participate in sports, clubs, or hobbies in which they have a competitive side. Playing games in math is also an effective way to get those students motivated and keep them engaged. In the unit on integers, students could play a few different games to keep them motivated and engaged while at the same time increase their understanding of the topic.
In a 7th grade math classroom I substituted in, I presented students with an integer card game that they could use to practice addition and subtraction of integers. The students were given a deck of playing cards. The red cards represented negative numbers and the black cards represented positive numbers. To start the game the students put the cards in a pile in the middle of their desks face down and one player picks up a card. The next player picks up another card and adds that card to the first card. The game ends when someone shows a card that when added to the stack, results in a sum of zero. This type of game motivated students and showed that they are able to have fun with mathematics. While I was walking around the room, I noticed the students were getting into the game and not just solving the problems for themselves but they were also solving the problems with their group members to ensure they were not cheating. In the future, it would be beneficial to have students write down the numbers they pick up to show they have simplified the integers correctly. I would collect the sheet and go through each computation to see how students are doing with their integer rules. I could use their work and count it as a quiz grade so that students know their computations are of meaning to them and they are not just playing a game to get out of mathematics class. If students are successfully computing integers correctly and getting all their problems right, I will know they have learned the rules and learned how to apply them to integers.
Due to the diversity of our students, Tomlinson (1999) discusses that importance of engagement and how it is essential for a great class and occurs when a lesson captures students’ imaginations and grabs their curiosity. In order to engage all learners, I need to ensure that the techniques I use will keep engagement in my classroom and are differentiated. Along with this, fostering collaboration and communication in a classroom can significantly increase the available support for sustained classroom engagement (Cast, 2011). After substituting in many geometry classes, one topic I have noticed that students struggle with was that of proofs. They struggled not just because the proofs were challenging to them, but because there was a lot of information that went into the proofs that the students needed to remember and understand. If I were to teach another geometry class, when I notice students are becoming discouraged with proofs I will find a few different ways to grab their curiosity while still keeping them engaged in the learning. One thing I would do is have students create posters with partners on the different properties of quadrilaterals. Each student would be assigned a shape and they would have to create a poster that included the shape, its proper definition, and all the properties that make up the shape. In the posters, the students would have to be creative and use pictures to relay the information. In the end, the posters could be hung up around the classroom for all classes to see. Using these posters, students would be able to not only think about definitions and properties of quadrilaterals but show their knowledge in a creative way.
As teachers, we need to be able to address individual differences in our students’ recognition, strategic, or affective networks. The Principles of UDL are designed around the idea that all students are able to learn when they are provided with differentiated instruction. When I give students multiple means of representation, expression, and engagement, they will excel in my classroom and will be able to individually become successful. By taking the time to present information and content in different ways, differentiate the way students express what they know, and stimulate interest and motivation for learning; I am able to embrace the different learning styles of my students and show them that their unique differences are accepted in my classroom (Cast, 2011).
Standard
3: Knowledge of Students |
||
Aspects |
Explanation |
Ways I
Met Standard |
Knowledge
of dispositions students bring to and develop in the classroom |
Accomplished
mathematics teachers are aware of dispositions students bring to the class.
Such attitudes may include math anxiety, fear of failure, confidence in doing
mathematics, perseverance, and valuing mathematics. These teachers construct
lessons and activities that build on positive attitudes. |
By
building a variety of activities to teach each lesson, I have can cover the
range of needs in my classroom and can improve on the struggling math
students. For students who are visual learners, they can use the concept map
or computer to help build their understanding of concepts. For students who
prefer auditory learning, I provide them with songs and riddles they can use
to help them remember certain rules. Lastly, for students who enjoy hands on
learning, playing games and well as hands on activities gets them engaged and
motivated. |
Knowledge
of the different ways students learn mathematics |
Accomplished
mathematics teachers are aware of the different ways in which students learn
mathematics and are sensitive to how students with difference strengths,
interests, and ways of learning come to understand mathematics. |
By
teaching the same concept in multiple ways through concept maps, the use of
technology, songs, riddles, instructional strategies, and games with
manipulatives, I am able to provide each student in a diverse group the means
to learn the way that fits them. |
Identify
individual students strengths |
Accomplished
mathematics teachers identify strengths, interests, and experiences
particular students bring to the mathematics classroom. |
By
using a choice, I would be able to acknowledge students strengths and
interests by allowing them to choose a way to present the information they
have learned. Through both the choice board and the RAFT assignment, students
are able to choose the way they feel would help them to best express their
knowledge and understanding of the material they were taught. These
strategies provide students with the opportunity to take tradition classroom
concepts and express them in a nontraditional way. |
Standard
4: Knowledge of the Practice of Teaching |
||
Aspects |
Explanation |
Ways I
Met Standard |
Knowledge
of pedagogy, mathematics, and student learning. |
Accomplished
teachers use their knowledge of pedagogy, mathematics, and student learning
to inform curriculum decisions. |
By
using the choice board, I am able to connect mathematics to other
disciplines. |
Commitment
of Mathematics Learner |
·
Accomplished
mathematics teachers stimulate and facilitate student learning by using a
wide range of practices. ·
Accomplished
mathematics teachers foster learning by choice designed to motivate students ·
Accomplished
mathematics teachers recognize that mathematical vocabulary must be used in
the context of the learner |
·
By
teaching the same concept in multiple ways, (i.e., concept maps, songs,
hands-on activities, technology) I am able to modify classroom plans and
activities in response to student needs, interests, and unexpected opportunities
for learning. ·
By
using choice boards and RAFT, I provide imaginative examples, problems, and
situations designed to interest and motivate students and illuminate
important ideas. ·
By
using concept maps as well as online dictionaries, I am able to provide
auditory and visual cues for students who may lack one or the other. |
Resources
Brand, S. T., & Dalton, E. M.
(2012). Universal design for learning: Cognitive theory into practice for
facilitating comprehension in early literacy. The Forum on Public Policy.
CAST. (2011). Universal
design for learning guidelines version 2.0. Wakefield,
MA. Author. Retrieved from www.udlcenter.org.
Simmons,
B. (2011, March 24). Mathwords. Mathwords.
Retrieved, from http://www.mathwords.com/
Tienken, C. H. [Web log message]. Retrieved
from http://www.keansburg.k12.nj.us/cms/lib02/NJ01001933/Centricity/Domain/163/Choice-boards-menus-presentation.pdf
Tomlinson,
C.A. (1999). The differentiated classroom: Responding to the
needs of all learners. Alexandria, VA: Association for Supervisions
and Curriculum Development.
MathRaps, (2009, February 1). The rappin’ mathematicial: The number
line dance. Retrieved from https://www.youtube.com/watch?v=6EWq9EZmIKg