Literacy in Mathematics
One of the best ways to help my students build on their mathematical literacy skills is to provide them with the opportunity to use vocabulary, reading, and writing strategies in which they can demonstrate their understanding of mathematics. According to Vacca, Vacca, and Mraz (2011) students must understand how to use reading, writing, talking, and viewing in class to help them become literate in any content area. By integrating thinking and learning processes into my mathematics classroom, I am able to help my students not only communicate mathematically but better understand what they are learning as well.
Vocabulary
The National Council of Teachers of Mathematics’ (NCTM) standards (2005) explain that students need to be challenged to communicate their thinking and be precise in their use of mathematical language. Today, language skills have become increasingly important in the mathematics classroom and it is necessary that students have the opportunity to build on their technical mathematical language. Vacca, Vacca, & Mraz (2011) define technical vocabulary as words that are unique to specific content areas. While the words are usually unfamiliar to students at first, they are necessary for disciplinary learning and thinking, and need to be taught to students in mathematics so they can be successful problem solvers. In order for my students to become mathematically literate, they need to have the guidance from me in learning the important definitions of mathematical language, how to use the language correctly, and how to work with the technical vocabulary they may come across.
In a 7th and 8th grade AIS classroom where I substituted long term, it was necessary that I made it a point to recognize vocabulary as an important aspect of our classroom. In order to build on my students technical vocabulary, I needed to provide them with different strategies they could use to not just build on their technical vocabulary but help them to remember the vocabulary words and how they can be used in their math classroom. One strategy I used was a word wall. On the word wall, I put together different words my students would see in their math classroom and hung them around the room for the students to use as a guide if they ever got stuck on a word. When I created the word wall, I told the students that we would add to the word wall throughout the rest of the year and the vocabulary would be easily accessible to students each and every day. In one of my 8th grade classes, I gave the students a worksheet in which they had to solve equations. The directions on the worksheet included words such as “integers” “coefficient” and “variables.” One student in particular came up to me with his worksheet and told me that he didn’t understand what the directions were asking him to do. Another student shouted out to him that he should look up on the word wall to see if any of the words were up there. On our word wall was each of the words he was confused about and he was able to use the word wall to help him dissect the question being asked. The student was able to figure out the question and told me that although he had not thought of it on his own, using the word wall was a great way to help him to figure out what the question was asking and helping him determine how he could solve it.
In this word wall I used in my AIS classroom, I printed words offline and provided visuals with the words. Instead of using the visuals to help students remember the words I could provide them with the actual definitions as well. While visuals can be beneficial to some students, it may be beneficial to include the definitions for other students. Depending on my class, I will determine which type of definition would be best. In my future classes, I also think a way I could improve the word wall would be to collaborate with my students and each day as we come across a new word, have students write the words on the wall. If I give students the opportunity to add to the word wall and write words themselves it could motivate them and help them to learn their technical vocabulary a bit more easily.
In order for my students to build on their technical vocabulary, they also need to be willing to participate in vocabulary activities. In addition to a word wall, when I was doing field work in an Algebra classroom I taught this lesson on translating verbal and mathematical expressions. The technical vocabulary in this topic was extremely important because students needed to use verbal phrases and synonyms to translate a verbal phrase into an algebraic one and vice versa. It was important the students knew the key mathematical terms and what operations they stood for. Before I started the lesson, I put this chart on the board of the symbols and words for addition, subtraction, multiplication, and division. I asked the students for different phrases for each word and as a class I filled in the chart on the SmartBoard. This chart helped the students to build on their technical vocabulary because it provided them with an opportunity to take words they know (i.e., addition, subtraction, multiplication, division) and find other mathematical ways to express those words.
At the end of the lesson, students played a game called “I have, who has.” In the game, the cards contained phrases involving the technical vocabulary we discussed in the beginning of class. The first person would start and read the front of their card. The classmate who had the matching word or problem would raise their hand and read their answer. The student would then flip their card over and read the back of their card for someone else to match with. The game continued on until each person had read and matched their cards.
This lesson built on students’ technical vocabulary skills and helped them to not just learn and remember new synonyms for familiar words, but take those new terms and apply them to mathematics. In the future, I would want to print copies of the chart for the students so they can use at as a reference for the rest of the year. The key synonyms for adding, subtracting, multiplying, and dividing are technical terms the students will see throughout the entire year of Algebra and even into the later years of mathematics. Students should be able to use it as a guide and have it firsthand so they can go back to it when need be.
In my classroom, while it is necessary for students to become familiar with the mathematical terminology associated with my lessons, they cannot just learn the definitions by simply looking up the word in a textbook. Greenwood (2002) states that while looking up the word, defining it, and using it is something that occurs daily in classrooms, it does not necessarily give students enough information to allow for complete understanding of the word. In order for students to receive a true understanding of the word, I must provide them with complex processes of integrating the words with ideas that exist in the learning. One way to build on and understanding of vocabulary is through the Frayer model. The Frayer model contains different components in which students can identify characteristics and facts about the concept, create examples and nonexamples, and write a definition for the concept using their own words.
In my class, I would use this vocabulary strategy to help students understand important concepts that can be rather confusing. For example, I would provide this Frayer model to an Algebra class when working with rational and irrational numbers. With this model, students are able to not just look up the definitions but rather take the definitions, put them in their own words, and find important characteristics of the word as well as some examples and nonexamples. Depending on the class and whether or not students have seen this strategy before, I would determine if this strategy would be an independent one or a cooperative one. I would not use the Frayer Model as a form of assessment until my students have become familiar with the strategy. Once students have learned how to utilize it, then I could begin to assess students understanding. It would be great to provide students the opportunity to share their work with the rest of the class, and allow them to continue to add to the Frayer Model during the rest of the unit.
Another vocabulary strategy I would use in my future classroom is the LINCing strategy. The vocabulary LINCing strategy is a cognitive tool for helping students making connections between the meaning of new words and prior knowledge (Ellis, 1998). The LINCing strategy is used to promote understanding and recall important terms and the essential characteristics of that term. The LINCs strategy is made up of five major components. L: List the word and its definition. I: identify a reminding word. N: note a LINCing story. C: create a LINCing picture. S: self-test. Students are given a blank chart and asked to fill in each component in the chart. Each component provides students with different helpful strategies to help them remember and learn the new word. An advantage to using the LINCing strategy is that is combines both auditory and visual learning in one strategy. I used the LINCing strategy a few times during fieldwork when working with students with disabilities. It provides opportunity for students to create reminders and make connections for new words. This provides more students with opportunity for success.
If I were to use this strategy in my classroom in an Algebra classroom, I would begin by explaining the five components to my students. I would provide them this poster I created so they could have easy access to the steps until they become familiar with the strategy. I would then provide students with a teacher-constructed example I have created such as this to ensure my students have a complete understanding of the strategy. I would not only give students an example of one already done, I would do two or three others as an entire class so they get the gist of it. I would try to teach this strategy in the beginning of the year so students are familiar with it quickly. If I was teaching a unit in Algebra on equations, I would begin the lesson by providing students with the important terms that will be addressed during the lesson, and let them know that they will be using the LINCs strategy to help them learn and recall the meanings of those terms. After I have gone over the terms, students will work independently to construct LINCs memory devices either in class or for homework. An advantage of LINCs is that the teacher can decide how they wants it to be used in their classroom. If he/she wants it to be done as independent work, that is what they will assign. If the teacher thinks it would be more beneficial for a specific topic to be completed in pairs, then the students can work together. Depending on my classroom and the topic I am teacher the way my students use the LINCing strategy will vary.
Reading and Writing
With the implementation of the CCSS (2010) it is not enough for students to simply answer the problems given to them on assessments. When completing assessments, students are now required to explain their thinking through writing. The NCTM (2005) Communication Strand specifically states that students need to be communicating the results of their work orally or in writing with clear conviction, and not just “procedural descriptions or summaries.” In my classroom, so they can become proficient at explaining their reasonings. According to Vacca, Vacca, & Mraz (2011), academic journals help create a context for learning in which students can interact with the information they are learning and make personal connections as well as clarify ideas, issues, and concepts. In any math classroom, teachers can include exploratory writing activities, summaries, letters, student-constructed word problems and theorem definitions, descriptions of mathematical processes, calculations and solutions to problems, and feelings about the course (Vacca, Vacca, & Mraz, 2011). The content of math journals in my classroom can vary.
For instance, during a Geometry leave replacement, I provided my students with multiple prompts on a daily basis to develop their literacy skills. I gave opportunities to explain steps and solutions to different mathematical problems. In practice for their final, I gave my students practice writing prompts in which they did not have to find an answer to a problem, but rather write down what they would do to get the answer. One practice prompt was that they had to explain the steps they would use in order to get the correct answer. I explained to my students that with this activity they would be able to demonstrate their mathematical thinking as well as their method of solving a problem (Kostos & Shin, 2010). I had one student in particular who was an avid believer in mental math. It was very easy for him to come up with answers in his mind without writing anything down. When he was given this assignment, he told me that he did not know how to write out how he got the answer. I explained to this student that math communication is important in our class and that he needed to not just have the ability to solve problems, but the ability to explain them as well. Together, we discussed the problem and talked about the different steps he did take to solve the problem mathematically. After we had a conversation, he was ultimately able to put together some sentences explaining how he got his answer.
In this same geometry class, another example of a prompt I gave my students was when they were doing proofs. Usually, proofs are written in statement/reason columns and are just written in numbered form. For one week, I required all my students to write their proofs in paragraph form where they had to use the same theorems but write them out in sentences. The advantage of a paragraph proof is that the students had a chance to explain their reasoning in their own words. Some students, for example this student, actually preferred a paragraph proof to a two column proof. One student even told me she would continue most of her proofs in paragraph form when she was given the opportunity because she felt that she was able to explain herself better. She said that when she is using the two column proofs she leaves information out with the paragraph proof she is able to write more down. For regularity, I think it is beneficial to throw a question on a unit test/quiz in which the students would either need to explain how to get an answer or why an answer was incorrect. By doing this, my students would be able to show they have an understanding of the concept by being able to explain their reasoning and the steps needed to solve the problem.
Vacca, Vacca, and Mraz (2011) discuss that teachers who use academic journals in their classrooms are encouraging students to use every day, expressive language in which they can write about what they are learning in a way that encourages the use of technical vocabulary and talk to explore ideas during discussion. In my classroom, journal entries are used frequently and take place regularly in order for them to be effective. In addition to these, I would provide a geometry class with a reflective essay which they could include in their journals. This type of writing would allow my students to reflect on their school year as a whole. I think it would be beneficial for both me and my students to provide a reflective essay similar to this one at the end of each quarter rather than one big one at the end of the year. This would give students more of an opportunity to write down their feelings toward the course.
Along with providing students the opportunity to explain their understanding of the material, I also provide my students with activities in which they can use different prompts I have created to take a different approach in their writing. For example, I can provide them with a RAFT assignment. The RAFT method, developed by Nancy Vandervanter, offers students a way to gain a better understanding of content topics and subjects as they write (Santa et al, 1988). It is an acronym for role, audience, format, and topic. A RAFT enhances understanding of a topic and allows students to demonstrate their understanding of the information through a nontraditional format. RAFT writing is used for students to rework, apply, and extend their understandings of newly acquired information (Santa & Havens, 1995). It gives students the freedom to pick different roles and express different opinions and facts from unique perspectives.
In my math classroom, I would use RAFT to allow my students to find different ways to be creative yet informative about the information they have learned. This is an example of a RAFT I would assign to my algebra students. It provides students with many different options and would allow them to use creativity to express their mathematical understanding. During a unit on algebraic expressions and equations, I would use this RAFT so my students could explain what they know about the topic and elaborate on the topic in a fun way. With this assignment, I would provide my students with a rubric such as this so they knew I was interested in seeing that they must have both content and creativity in their writing.
Literacy is an important component of a mathematics classroom and is a topic that needs to be incorporated in a math class to help students communicate mathematically. As a teacher, I need to understand the importance of literacy in order to help my students become skillful with their mathematical language. With the mathematics standards provided to teachers today, it is necessary to give opportunities for students daily to improve on their reading, writing, and vocabulary skills in mathematics. By creating journal entry prompts, using vocabulary strategy models, and providing students with different writing activities, I am helping my students to build on their technical vocabulary as well as their mathematical communication skills.
Standard
2: Knowledge of Mathematics |
||
Aspects |
Explanation |
Ways
I Met Standard |
Have
a deep and broad understanding of the concepts, principles, and techniques |
Accomplished
mathematics teachers use the knowledge to inform curricular goals and shape
their instruction and assessment. They guide students to apply their prior
knowledge about solutions to future problems they may see. |
By
providing my students with a word wall, I am allowing them to get a deep and
broad understanding of the important concepts they come across in their math
class. The word wall is easily accessible for students and I can use it to
help my future planning. If I notice students are struggling with certain
words, I can put them on the word wall and take the time to explain them. Using
the Frayer Model, and addition, subtraction,
multiplication, and division chart, students are able to connect prior
knowledge to newly acquired knowledge. They can write what they know down and
use that information to help guide their learning. |
Know
the ways of thinking, talking, and writing about mathematics |
Accomplished
mathematics teachers know ways of thinking, talking, and writing about
mathematics and share those experiences with their students in order to help
students develop the ability to think mathematically and communicate
correctly both verbally and in writing. |
Using
journal entries, such as explanations of problems, paragraph proofs, and
reflective essays, students are able to communicate mathematically in
writing. This helps me to ensure the students understand the information well
enough and can communicate it back to me as their teacher In addition to journal entries, providing
students with a RAFT assignment allows them to communicate specific ideas
through nontraditional writing. |
Standard
6: Ways of Thinking Mathematically |
||
Aspect |
Explanation |
Ways
I Met Standard |
Develop
their abilities as well as their students’ abilities to reason and think
mathematically |
Accomplished
mathematics teachers are aware of the concepts, principles, procedures, and
reasoning processes that are involved in mathematics. They think
mathematically by representing, modeling, proving, experimenting
conjecturing, classifying, visualizing, and computing. |
·
When I model these strategies for my students, I
am showing them that I am aware of the important concepts and procedures
involved in mathematics. ·
Through RAFT, I am helping students determine
which way they feel would be the best way to represent their knowledge. ·
Through paragraph proofs, students are able to
make conjectures and prove their mathematical reasonings. |
Make
students aware of strategies for solving problems |
Accomplished
mathematics teachers are proficient in solving problems as well as making
students aware of different strategies for solving problems. They make
strategic decisions in the classroom when students have to explore unfamiliar
concepts. |
·
Teaching all of these strategies to my students
allows me to help them become aware and familiar with different learning
strategies. When students are learning new topics, I can use the frayer model and LINCs strategy to help them understand
the material better. ·
When students have learned the material and need
practice and reinforcement on it, if they are struggling they have to word
wall around the classroom to help them. If they are familiar with the topic
and have a good understanding of it, they are able to use the “I have who
has” game to improve their skills. |
Develop
their abilities as well as their students’ abilities to justify and
communicate solutions |
Accomplished
mathematics teachers develop their abilities to formulate and solve problems,
to justify and communicate solutions, and to question and extend their
conclusions into a deep level of understanding. |
·
Through vocabulary strategies such as Frayer Model and LINCs, students are able to develop
their abilities to solve problems and extend their thinking into a deeper
level of understanding. They are able to activate prior knowledge and use
these strategies to think outside of the box but connecting examples and nonexamples, reminding words, and stories to the new
vocabulary they are learning. ·
Through journal entries in my classroom, students
are able to communicate their solutions to problems but explaining in writing
the processes they will take in order to solve the problem. |
Resources
Ellis, E.S. (1998). Visual and auditory
LINCS to background knowledge: A key for learning new terms. Retrieved from
http://www.st.cr.k12.ia.us/reading/CALvocabEllisLINCS.pdf
Greenwood, S. C. (2002). Making words matter: Vocabulary
study in the content areas. 75(5), 259-263.
Kostos, K., & Shin, E. (2010). Using math journals to enhance
second graders' communication of mathematical thinking. Early Childhood
Education Journal, 38(3), 223-231.
National Council of Teachers of Mathematics (2005). Principles
and standards for
school
mathematics. Reston, VA: The National
Council of Teachers of Mathematics, Inc.
NYSED (2012). NYS common core P-12 learning standards. Retrieved
from
http://engageny.org/resource/new-york-state-p-12-common-core-learning-standards/
National
Governors Association Center for Best Practices & Council of Chief State
School Officers. (2010). Common Core State Standards for Mathematics.
Washington, DC: Authors.
Santa, C., &
Havens, L. (1995). Creating independence through student-owned strategies:
Project CRISS. Dubuque, IA: Kendall Hunt.
Santa, M.,
Havens, L., Nelson, M., Danner, M., Scalf, L., & Scalf, J. (1988). Content
reading including study systems: Reading, writing, and studying across the
curriculum. Dubugue, IA: Kendall/Hunt
Swanson,
M., & Parrott, M. (2013). Linking literacy and mathematics: The support for
common core standards for mathematical practice. Retrieved from http://files.eric.ed.gov/fulltext/ED539526.pdf
Vacca, R. T.,
Vacca, J. L., & Mraz, M. (2011). Content
area reading: Literacy and learning across the curriculum. (10th ed.). Boston, MA: Pearson
Education, Inc.
Yates, P. H., Cuthrell, K., &
Rose, M. (2011). Out of the room and into the hall: Making content word walls
work. Clearing House: A journal of Educational Strategies, Issues and Ideas,
84(1), 31-36. Retrieved from
http://www.informaworld.com.online.library.marist.edu/openurl?genre=article&id=doi:10.1080/00098655.2010.496810.