Constructed Response in Mathematics, Part IV The Role of Vocabulary Nelson Palmer, FCPS Math Curriculum Specialist, PK - 5

Imagine a classroom…The curriculum is mathematically rich, offering students opportunities to learn important mathematical concepts and procedures with understanding…Students are flexible and resourceful problem solvers…Orally and in writing, students communicate their ideas and results effectively. They value mathematics and engage actively in learning it. NCTM Principles & Standards 2000

Communication is a two-directional activity. It involves the input of information (in this case, reading and understanding the problem) and the output of information (in this case, explanation of procedure or justification). One of the most important considerations related to this communication is that these two activities are directly linked and must be viewed as interrelated. The benefit to this realization is that the strategies suggested for successfully reading and solving a constructed response problem directly support the ultimate communication of one’s thinking. Mathematics terminology plays an important role in both of these aspects of communication.

Reading and Understanding the Problem A written math problem (whether presented in words or Vocabulary symbols) is a form of informational text and thus the The more words you know, the more you can do with strategies recommended for making meaning of language…Learning new words involves three things: informational text should be applied in math. Knowing collecting new words, using your new words, and and/or making meaning of math vocabulary is critical for exercising your word skills. How do people improve their vocabulary? How do successful comprehension. This means that students vocabularies grow? You learn words naturally – by should be expected to make meaning of text involving listening, by reading, and by talking. But you can also grade-level mathematical vocabulary and subsequently use learn words by learning about context clues, word parts, it in their oral and written communication regarding their and word tools, such as a dictionary. reasoning, thus making it part of their owned vocabulary. Reader’s Handbook: A Student Guide for Reading and Learning Strategies for Making Meaning Math problems present unique challenges when math terms are not owned by the reader. The Reader’s Handbook identifies strategies for decoding unknown terms in a text - context clues, word parts, and word tools. These should be reinforced in the math classroom. One must keep in mind that constructed response problems have few words and thus provide few context clues that might help with making meaning. Additionally many math terms used provide little clue from their word parts. Furthermore, math problems fail to provide the typical text features that often support young readers as they work to make meaning of other forms of informational text. Word tools like glossaries can serve to assist the reader in making meaning of unfamiliar terms, but unfortunately these, too, can pose challenges in interpretation of meaning.

Visualization An important strategy that needs to be available to students in making Numbers and pictures are both meaning of text is visualization. This is especially true in reading ways of making ideas easier to mathematics. Visualizing is the creation of visual representation in the understand. So it’s not mind’s eye. For the purpose of solving math problems, this visualization is surprising that drawing a problem and visualizing it in enhanced when it can be seen by the human eye. Physical objects often your mind can help you see it. provide the best means for representing math problems because they, in fact, Reader’s Handbook: A Student Guide can be physically manipulated to correspond to the actions presented in the for Reading and Learning problem. But, in the absence of physical objects, one should look to other methods. Visual representations can take many forms – pictures, labels, diagrams, charts, graphs, outlines, flowcharts, and even symbolic representations. 1 2/7/2007 Linking Reading and Writing As mentioned before, reading and writing are processes that are interdependent. Representing math problems visually (this visualization can occur in the mind) provides a graphic representation that can be used as the prewriting to assist with explaining student’s mathematical thinking in writing. Equally important is that during reading, students need a mechanism for gathering important information and ideas. One such device used during reading is a visual representation of the information provided to support the reader’s ability to set up and solve a problem. Just as authors of expository text often provide pictures to support comprehension, when visual representations are provided along with student’s writing, they further support the understanding of the reader.

Speaking and Writing Two important considerations exist regarding the importance of mathematical vocabulary in communicating one’s thinking. The first consideration relates to the accuracy and efficiency of one’s response. The second relates to the impression made to the reader. Vocabulary Development in Mathematics Accuracy and Efficiency Students of mathematics should be treated as Mathematical vocabulary (as is the case with all mathematicians and behave as such. Mathematicians coin words to describe common occurrences in vocabulary) comes from the need for clarity and mathematics. efficiency in language. As an example, consider the This suggests that mathematical vocabulary should be term “symmetrical”. In order to describe the human learned (given to students) as the need arises to describe face, one might say “if I draw a line down the middle, a mathematical occurrence. Thus, when a concept (like the part on the left is the same as the part on the right”. symmetry) is acquired, the term can be ‘coined’ to describe it. This description is very wordy but, even more In this way, the student defines the term even before importantly; it lacks clarity in that the word “middle” he/she knows the term. This approach leads to true allows for misinterpretation (“somewhere in the ownership of math vocabulary.

Vocabulary Development in Mathematics - middle”). A more efficient and accurate description is “the NCTM It is important to avoid a premature rush to impose human face is symmetrical”. Originally, the word “symmetry” formal mathematical language; students need to was coined because of a recurring need to describe objects in develop an appreciation of the need for precise the world that have this property. definitions and for the communicative power of conventional mathematical terms by first Thus, the role of vocabulary in constructed response is one of communicating in their own words. Allowing students to grapple with their ideas and develop accuracy and efficiency. It is certainly possible to accurately their own informal means of expressing them can respond to a mathematical question without using be an effective way to foster engagement and mathematical terminology. Mathematical terminology simply ownership. affords the communicator more efficiency (and sometimes NCTM Principles & Standards 2000 accuracy) in communication. Strength of Voice Successful communication of information involves many Six Traits of Expository Writing considerations. For the purpose of constructed response, the Ideas: The topic, focus, and details make student needs to focus more on their ideas, word choice, and the essay truly memorable. voice. The traits of word choice and voice are extremely Organization: The organization makes the essay important and interrelated. The student must be perceived as easy to read. Voice: The writer’s voice sounds confident and strong in their understanding. Mathematical confident, knowledgeable, and language, when used appropriately, allows the student to enthusiastic. demonstrate his/her knowledge of the concept being addressed. Word Choice: The word choice makes the essay It is a voice of strength and confidence that helps to persuade clear, informative, and interesting. the reader that the student’s reasoning is valid. (Note: although Sentence Fluency: The sentences flow smoothly and will hold the reader’s interest. organization, sentence fluency and conventions are important Conventions: Mastery of conventions adds style to aspects of communicating, they are not required for the essay. successfully responding to constructed response problems.) Write Source: A Book for Writing, Thinking, and Learning 2 2/7/2007