Unlocking the Mystery of Mathematics: Give Vocabulary Instruction a Chance

by Bizzie Cors

Let me begin by setting the scene: It was late fall last year. Sitting down one evening with a stack of honors geometry tests staring at me, I knew I was in for several long hours of grading. I began penning the same words on many of the papers, “Please, read the directions,” “Is this what the question is asking?” and “You need to remember characteristics of parallel lines.” I had just spent the past three weeks teaching about congruent triangles. These tests represented what I had seen over the past seven years: the same misconceptions, the same errors, the same number of students who “got it,” and the same number of students who were already lost. It seemed that most students were not recalling the concepts or terms from the previous chapters. It was only November, and yet here it was again—another year of many students believing that geometry was a subject of memorization and mystery.

I walked into class the next day with my speech in mind about how “you have to study more and make more effort in class,” a speech I was going to give before I handed back the tests. But before class began—and before I had a chance to make my speech—a student approached me and asked, “Can I throw away all the information from the congruent triangle chapter now?” This young lady was one of my top students; surely, I thought, she understands the importance of keeping all this math work to refer to later. I looked at her and replied, “Why would you ask that?” Politely she answered, “I don’t need that information anymore since we will be on the next chapter today.” I stared at her, and I could not respond. One of my top students, and yet she hadn’t made the connection of how to construct meaning and understanding in mathematics. Okay, I jettisoned the speech. It was time to ask myself some important questions:

Why didn’t she think she’d have to refer to the information in this chapter again?
Do the students have a clear understanding of the purpose of the daily lesson?
How can students make connections and keep their current background knowledge in math accessible?

After days of thinking, reading, and soliciting colleagues’ input, it was time to develop a new approach. I was overwhelmed. Deciding what to do was not going to be an easy process. I needed to break the learning process down to the foundational level of understanding for all students. I needed to level the playing field for all learners.

With a distinctively large gap in the understanding of geometry that already existed among students, what could I do that could begin to have an impact on the learning of every student? The answer was vocabulary instruction. True, I had already asked the students to write the definition of different terms, draw pictures, write in their own words the meanings, and supply synonyms.  What I found out was that these techniques didn’t work well. Yet, wasn’t I using researched-based best practices to instruct vocabulary acquisition? Even though I was making the kids “do” vocabulary work, they were studying the words in isolation; they were not learning them in a meaningful context or making authentic connections. This is when I got the idea to teach vocabulary as an integral part of every chapter, not just as a sidebar or an add-on.

I understood that students needed to construct meaning for all vocabulary terms and connect to prior knowledge as well as to new concepts and algorithms. This then led me to the realization that I would have to create a process to teach vocabulary development. What would I need to provide in the classroom? I sketched out the issues that had to be addressed:

• The vocabulary must be constantly visible.
• Each strategy has to be explicitly modeled and practiced.
• As part of the process of defining terms, students need to constantly revise their definitions until each student has created the meaning.
• Students have to be taught questioning techniques on how to explore the meaning of the words and the mathematics involved.
• Students need to discuss, revise, and build upon the pictures they’ve made of each term.
• Students also need to summarize and reflect on how the terms connect to solving, proving, and evaluating math problems.

Sticky-Note Chain Process

The method I chose to use involves the liberal use of sticky notes in before-, during-, and after-reading strategies.

Before Reading: I select the key vocabulary words before the reading of each section. The students write each word on sticky notes.

During Reading: The students read the section in the textbook and follow the procedures below. On a sticky note for each vocabulary word, they should:

• Write any signal words that can help define the given words. Formal definitions of the word are not to be written.
• Write any word parts that may help in understanding the word. Include any pictures.
• Write any previously defined concepts or terms related to the word.
• Write the problems in the homework that use the word. (I have them write just the problem numbers; we reference them when we go over the lesson to develop a pattern of key vocabulary words and to open discussion.)

After Reading: At the beginning of class, the students—working in groups of three—discuss the information collected on their sticky notes. They are to follow these procedures:

• Each student reads aloud the information on his or her sticky note without comment or discussion from others in the group.
• Each student answers the questions and follows the directions:
• Are any of the key words connected? If so, place the sticky note in a chain. Draw an arrow to each word, and write the connection along the arrow.

Example: Congruent would be connected to Equal by the phrase “compares like quantities.”
• In the homework problems, did I need to know the definition of the key word to understand the method used in solving the problems? If yes, list the specific problems. How do the directions for the problems help define the word? Write all information discussed on the sticky note for that word.
• How do the key words relate to the title of the chapter?
• Is there anything missing that would help me understand the key word?

Discussing their responses with one another, the group members continue to use this process of writing, adding, and moving sticky notes throughout the chapter. As the instruction of the chapter proceeds, each student continues to monitor his or her sticky-note chain and make revisions. There are no set answers at this point.

At the end of the chapter, one day is devoted to the evaluation and synthesis of the terms. This day the students, still working in their small groups, place their sticky-note chain on a large white paper with the chapter heading written at the top. They begin to place the sticky notes in “chunks” according to how they believe the terms relate. During this time, there is a lot of discussion and formative assessment taking place as I walk around the room. The process takes about twenty minutes.

I then hand out a formal definition of each key word to every student. The students review the definitions and add anything to their own definitions that clarifies the meanings. The concluding piece is assigned as homework. For their assignment, the students draw on their work throughout the chapter to write their own example of a problem that relates to or directly uses the vocabulary word—one problem per key vocabulary word.

For the final product, each group puts together a graphic organizer complete with sticky notes, connections, and problems. For each chapter, the graphic organizer looks different, and I’m always surprised at the connections the students have made through this constructivist approach to learning vocabulary.

The need to be flexible while guiding the students is critical to this process. In the beginning, the process is time consuming, but as the students develop the skill of asking questions and exploring the meaning of the terms, the quality of time on task increases. I no longer lecture and repeat definitions and concepts, but teach the application and process of mathematics.

Outcomes

This process was different and sometimes uncomfortable for me as a traditional classroom teacher of mathematics. But the positive outcomes surpassed the struggle of implementation. This process achieved the following:

• Vocabulary in mathematics became an integral piece for all students in creating understanding and meaning in mathematics.
• The students made authentic connections from one section to another using the vocabulary to clarify, solve, prove, and evaluate problems in mathematics.
• The sticky-note chain created a systematic approach that made visible the ongoing process of acquiring vocabulary understanding.
• Students began to ask questions to connect conceptual understandings of mathematics. An example: “I think scale factor and slope must have something in common.” This statement was made by a student reviewing her sticky notes on the two terms. When I demonstrated the equations, students immediately were able to relate the two concepts.
• Solving word problems became fun and less difficult for most students. When asked why this wasn’t a struggle, a student responded, “I have read so much math, I know what to look for now.”

In conclusion, when I taught math in the traditional mode, I was focused on covering the curriculum and providing the students with the knowledge and strategies to solve problems. This meant I did all the work, rather than actively involving the students in their learning. Changing my approach to instead offer students an opportunity to develop their own meaning opened a new way to provide the information and structure for all learners. This is a process that has proved to stimulate and encourage all learners and help them know that mathematics is a process and not magic. The results have been great!

Bizzie Cors is a teacher in the Wooster City Schools. She has been a teacher of mathematics and computer technology, grades 7–12, for 22 years. She earned a B.A. in mathematics from the College of Wooster and currently is finishing her master's degree in integrating technology into the curriculum.

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