EFFECTIVE STRATEGIES FOR TEACHING CONTENT VOCABULARY IN MATHEMATICS By Courtney Taylor A Thesis Submitted in Partial Fulfillment Of the Requirements for the Degree of Master of Science in Education Department of Language, Learning and Leadership At the State University of New York at Fredonia Fredonia, New York May 2009 __________________________________________________________ Dr. Barbara MalletteDr. Anna Thibodeau Thesis AdvisorDiscipline Chairperson College of EducationCollege of Education ___________________________ Dr. Christine Givner Dean of the College of Education __________________________ Dr. Kevin P. Kearns Associate Vice President for Graduate Studies & Research State University of New York at Fredonia Department of Language, Learning and Leadership CERTIFICATION OF THESIS WORK We, the undersigned, certify that this thesis by Courtney Taylor, EFFECTIVE STRATEGIES FOR TEACHING CONTENT VOCABULARY IN MATHEMATICS candidate for the Degree of Master of Science in Education, LITERACY EDUCATION: 5-12, is acceptable in form and content and demonstrates a satisfactory knowledge of the field covered by this thesis. ______________________________ _______________________ Thesis Director: Date Department of Language, Learning, and Leadership ______________________________ ________________________ Chair: Dr. Anna Thibodeau Date Department of Language, Learning, and Leadership STATEMENT BY AUTHOR This Master’s Thesis has been submitted in partial fulfillment of requirements for a Master of Science in Education, LITERACY EDUCATION: 5-12 at State University of New York at Fredonia and is deposited in the University Library to be made available to borrowers under rules of the Library.
Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the copyright holder. Copyright ( Courtney Taylor 2009 SIGNED: ______________________ ABSTRACT Table of Contents Chapter One: Introduction1 Background1 Aims of the Project3 Rationale4 Literature Reviewed6 Terminology6 Chapter Summary7 Chapter 2: Literature Review8 Theoretical Foundation5 Teaching and Learning Vocabulary9
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Content Vocabulary13 Mathematical Vocabulary14 Instructional Methods and Strategies18 Strategies24 Chapter Summary31 Chapter 3: Effective Strategies for Teaching Content Vocabulary in Mathematics33 Overview of Curriculum Project33 Goals and Objectives33 Format of Handbook34 Intended Audience34 Dissemination34 Chapter Summary35 Chapter 4: Links to IRA, NCTM and NYS Standards36 International Reading Standards44 National Council of Teachers of Mathematics Standards34 New York State Mathematics Standards34 Chapter Summary35 Chapter 5: Discussion45 Chapter Summary45
References46 Appendix A50 Chapter 1 Introduction As students enter the intermediate and middle grades, they are required to make a switch from primarily fictional storybooks to informational and content related texts. Often students are not given much guidance on how to read such texts. Of the content areas, mathematics is often acknowledged as some of the most difficult content area reading material. Schell (1982) claims mathematical reading has “more concepts per word, per sentence, and per paragraph, than any other area” (p. 544). Because of the difficulty of athematical content, instruction in the language specific to mathematics is necessary for success in understanding concepts specific to mathematics. Background In my own personal experience as a middle school and high school student, I continually felt anxious when it came to math class. Expressing my thoughts about a problem and how I solved it, whether verbally or written, was difficult for me. Because of this difficulty, I was resigned to the idea that I would never be a good math student. I felt completely disconnected from all mathematical concepts, mainly because I did not understand the material in the mathematics textbook.
Now, after teaching a small academic intervention services class in mathematics, I have come to realize that part of that past anxiety was due to my lack of understanding that a language specific to math existed. Today many of my middle-school students consistently moan when dealing with mathematics. When given word problems, their moans increase. Through my courses to become a reading teacher and through the experience I have gained in teaching struggling readers, I have been given the tools I need to encourage students to gain a general understanding of a text.
However, based on my own experiences and my current teaching situation, I wanted to explore how a teacher can help students construct meaning specifically from a mathematics textbook in order to better prepare these students for high stakes testing. Statement of the Problem Due to the No Child Left Behind Act, high stakes tests have been implemented at all grade levels for all content areas. Oftentimes, teachers are forced into “teaching to the test” instruction where students are driven to the textbook in order to cover all material relevant to the high-stakes test.
As a result, vocabulary knowledge becomes essential in order for students to understand what they read in their textbooks. In 2000, the National Reading Panel (NRP) submitted a report entitled The Report of the National Reading Panel: Teaching Children to Read” to the U. S. Senate. The report contains research-based knowledge on reading instruction. One entire section of the report includes research on vocabulary instruction. As stated by the National Reading Panel (2000), a reader’s “reading vocabulary” is crucial to reading comprehension.
Vocabulary skills are essential in order for students to independently learn new vocabulary in the content areas. In the area of mathematics (also referred to as “math”), the content specific vocabulary consists of new concept words and familiar words used in new ways. For example, many of my own students found the statement “evaluate the expression” difficult to understand. The words evaluate and expression mean have two very different connotations in English Language Arts classes. In math, the term “evaluate” essentially means to solve the “expression,” or problem.
This confusion between the two terms can be especially difficult for students who encounter formal mathematics for the first time, usually as students in middle school (i. e. , grades six through eight). Most of the difficulties students have with mathematics do not lie in the actual solving of math problems, but in the understanding of the language specific to the content (Krussel, 1998). Therefore, the problem lies in identifying what type of vocabulary strategies are successful in increasing student knowledge of the language of mathematics.
Teaching content vocabulary can remove potential difficulties in reading comprehension and can aid students in acquiring the language of a content area (Vacca & Vacca, 2008). If students are not familiar with words they meet in texts, they will, without a doubt, have trouble comprehending what they have read. Vacca and Vacca (2008) claim, “the more experience students have with unfamiliar words and the more exposure they have to them, the more meaningful the words will become” (p. 142). The question therefore becomes, what does research say are effective strategies for teaching content vocabulary in math?
Aims of the Project The aim of this project is the creation of a handbook for teachers at the middle school level entitled Effective Strategies for Teaching Content Vocabulary in Mathematics. Information collected from a wide-range review of the literature will be combined to offer teachers the most current research base available for implementing various vocabulary strategies into regular instruction in order to enrich the regular mathematics curriculum and foster mastery of mathematical vocabulary. Rationale
Vacca and Vacca (2008) create an analogy between vocabulary and content reading to fingerprints and the human hand. Just as a fingerprint is specific to an individual, so too is content area language specific to the content area. Reading for content inundates the content area curriculum, accounting for 75 to 90 percent of the learning that takes place in content area classes (Gunning, 2005). Teachers in content areas outside of English Language Arts understand that the vocabulary, or the “language,” of their content area needs to be taught.
Unfortunately, due to state standards as well as high-stakes tests that emphasize content over language, vocabulary instruction is, in my experience, reduced to methods that involve students in looking up vocabulary in the dictionary, defining the word, and creating sentences using the assigned words. This type of vocabulary instruction has the potential to create a disconnect in the vocabulary from the subject matter rather than providing a way for students to connect their prior knowledge to vocabulary found in texts. The language of mathematics can often be confusing.
Many mathematical words and symbols are unusual, some have content-unique meanings, and many are used in multiple ways. For example, when given the word sum, many of my students confuse this word with some. Another example of this confusion is in the word square, as in the shape, and square as in a number multiplied by itself. Vocabulary and symbols can carry equivalent meanings, such as the symbol “+” which can mean sum, increase, positive, more, and, add, group, or combine. Symbols, terms, and phrases are vital for communicating mathematical ideas; becoming fluent in this language is essential for students to learn mathematics.
The National Council of Teachers of Mathematics (1989), or NCTM, believes that future success in math is reliant on mathematical literacy. There are five goals that the NCTM has established to help develop students’ mathematics literacy. The five goals include: learn to value mathematics, become confident in own mathematics ability, become a mathematics problem solver, learn to communicate mathematically, and learn to reason mathematically. NCTM (2005) stresses in its Communication Standard that teaching from pre-kindergarten to twelfth grade “should incorporate strategies for using mathematical language to express mathematical ideas ccurately” (p. 402). In English Language Arts classes, new vocabulary can be integrated into conversations, but using mathematics language in this manner is much more difficult. Therefore, math teachers should find methods and instructional practices that will be effective in teaching students to communicate using the language of math. In order for students to become proficient readers they must develop a variety of skills for text comprehension. One such skill is vocabulary. Scientific research confirms that vocabulary instruction should be performed directly, indirectly, explicitly, and systematically (NRP, 2000).
Direct-Reading Instruction (Kameenui, Simmons, Chard, & Dickson, 1997) that incorporates the delivery and design of the interactive model of instruction is more apt to enhance students’ success on reading assignments than typical instructional strategies that include practice only, the use of basal reading material strategies, and only non-Direct-Instruction methods. Indirect vocabulary learning can include the learning of words through conversation, by being read to, and independent reading. Direct vocabulary learning is specific word instruction and word learning instruction.
Combining indirect vocabulary learning and direct vocabulary learning can result in students increasing their vocabulary and becoming more successful in reading comprehension (NRP, 2000). Literature to be Reviewed A range of literature bases, including online databases, journals, and books, will be utilized in order to obtain a comprehensive analysis of the research related the teaching of vocabulary in the content areas. A preliminary examination of the literature reveals that the number of empirical studies related to the teaching of content area vocabulary is limited.
In addition, these studies tend to examine social studies and science almost exclusively; indicating a gap in the literature related to the content area of mathematics. As a result, my search was broadened to include other key words, different age levels, and more databases in order to better build my research base. Terminology In order to aid in understanding the research found within this project, definitions of the following terms are provided. The first two terms related to the various reading models developed by theorists. They are the top-down reading model and the bottom-up reading model.
The bottom- up reading model suggests a part-to-whole process of a text. Those who support a bottom-up reading model believe a reader needs to identify the features of letters, combine the letters to recognize spelling patterns in order to identify words, and then move on to sentences, paragraphs, and text comprehension. The top-down reading model is a reading model that stresses what the reading brings to the text he/she encounters. Supporters of the top-down reading model believe that reading is driven by meaning and progresses the opposite of bottom-up in a whole to part fashion.
Direct and indirect instruction are also terms to that require definitions. Direct instruction for the purposes of this project includes lessons in how to learn vocabulary independently as well as instruction in specific words. Indirect instruction includes learning vocabulary through indirect methods such as conversations and being read to. Summary Once students enter the intermediate and middle grades, they exchange learning primarily from fictional texts to learning from informational textbooks. However, students are not always provided the training needed to read and understand textbooks.
Math is often acknowledged as one of the more difficult content area texts to understand. Furthermore, mathematical language is in general difficult to understand mainly because mathematical terms have multiple meanings, are related to multiple concepts, and can be represented sometimes by symbols rather than words. Therefore, many of the problems math teachers encounter do not relate to the actual act of solving mathematical problems, rather they result from the language math texts utilize when explaining the problem. The NCTM believes that success in mathematics is reliant on attainment of mathematical literacy.
One method of succeeding in mathematical literacy is to learn one of the smallest parts of literacy learning: vocabulary. Instruction in the language specific to mathematics is vital for students to understand the concepts within the math curriculum. Through the use of the vocabulary strategies within my handbook, Effective Strategies for Teaching Content Vocabulary in Mathematics, it is anticipated that math students will be able to construct meaning from their math texts. Chapter 2: Literature Review The following chapter examines the literature dedicated to teaching vocabulary in the content area of mathematics.
Included in this chapter is information about teaching and learning vocabulary, content vocabulary, and mathematical vocabulary. The chapter concludes with research regarding instructional methods and strategies as well as explanations of the various strategies researched. Summaries of studies related to the topic of vocabulary instruction will be presented, with attention on teaching and learning vocabulary at the middle school level. In addition, the teaching and learning of vocabulary in the content areas, particularly mathematics, will be addressed.
Finally, studies related to specific vocabulary instructional methods and strategies are featured. Also included in this chapter is the theoretical support for vocabulary instructional methods. Theoretical Underpinnings As discussed by Stanovich (1980), reading instruction can be approached from an interactive model perspective. The interactive model, a combination of the top-down reading model and the bottom-up reading model “assumes that a pattern is synthesized based on information provided simultaneously from several knowledge sources” (p. 35).
Just as reading instruction should be taught using the interactive model, so should vocabulary instruction, a large component of reading instruction. A second theoretical basis for vocabulary instruction would be a balanced literacy approach to instruction. A balanced literacy approach to instruction requires the educator to make choices each and every day about the most profitable way to aid each student in becoming a better reader and writer. This approach allows the teacher to keep his/her instructional methods flexible. A balanced approach is often viewed as essential because not all students learn in exactly the same way.
A final theory that supports the use of vocabulary instruction the content area of mathematics is the theory of visual literacy. Visual literacy is the “ability to construct meaning from visual images” (Giorgis, Johnson, Bonomo, Colbert, & et al, 1999). Visual literacy involves interpreting as well as producing images to effectively communicate information. The vocabulary strategies presented in this project use visual literacy frequently in that students both interpret information from visual formats as well as create images that communicate vocabulary information. Teaching and Learning Vocabulary
Vocabulary and Reading Comprehension Within a language, the words are the basic elements of a language that influence thinking and communication (Vygotsky, 1986). Ouellette’s (2006) study focused on a sample of 60 fourth grade students and their skills in decoding, visual word recognition and reading comprehension, as well as receptive and expressive single-word vocabulary to assess the students’ depth of vocabulary knowledge. In respect to reading comprehension, Ouellette hypothesized that vocabulary depth is vital, as the breadth of a student’s word knowledge can facilitate or restrict reading comprehension.
The students were given various assessments in order to determine nonverbal intelligence, oral vocabulary, decoding skills, visual word recognition skills, and reading comprehension. According to the data analysis, there are two points that must be considered when teaching any content area. The first is that reading involves decoding and visual word recognition, as well as comprehension. The second is that oral vocabulary includes the breadth and depth of knowledge. ADD ACTUAL DATA Oullette found that reading comprehension was related to both the student’s breadth and depth of vocabulary knowledge.
ADD ACTUAL DATA While decoding and visual word recognition play roles in reading comprehension, there also seems to be an important role for oral language beyond simply visually recognizing a word. Many of the vocabulary assessments suggest WHAT VOCABULARY ASSESSMENTS that phonological issues are less relevant than semantic knowledge and organization. Therefore, the connections between depth of vocabulary knowledge and reading comprehension are far more credible and widely accepted.
For this reason, reading instruction must incorporate these reading skills WHAT READING SKILLS and attend to increasing the number of words in a student’s vocabulary and the extent of the word knowledge of the vocabulary. Frequency of Vocabulary Instruction Scott, Jamieson-Noel, and Asselin (2003) performed an observational study that examined when, where, how often, and how effectively vocabulary instruction took place in 23 upper-elementary classrooms in Canada. Their data revealed that based on the 308 hours of observation, only 6% of instructional time was dedicated to the students’ advancement in vocabulary knowledge; only 1. % of instructional time was spent on vocabulary in subjects other than language arts. The researchers’ also found that the few instances of vocabulary instruction occurring in content area classrooms were less than outstanding. As stated by Scott, Jamieson-Noel, and Asselin, “different types of words demand different types of instruction” (2003, p. 283). The most noteworthy fact found within this study is that vocabulary instruction is often not present in content area classrooms as frequently as it should be. Teachers are facing more diverse students who need to learn the language of the specific contents in order to succeed in school.
However, very few teachers spend the time needed to aid students in creating personalized connections with vocabulary so that students can perform well in the content area classrooms. With more time spent on instructing students in vocabulary learning strategies, perhaps the students’ comprehension of textbooks will increase. Contextually Based Instruction versus Dictionary Based Instruction In an article published in Preventing School Failure, Monroe and Orme (2002) offer two methods for teaching vocabulary: meaningful context and direct teaching.
Monroe is a professor of mathematics education at Brigham Young University in Utah, and Orme is a former third grade teacher who at the time the article was published was pursuing a master’s degree in education at Brigham Young. Monroe and Orme assert, “an important component in language learning is learning vocabulary” (p. 140). Likewise, an important element in understanding the language of mathematics is learning the vocabulary. In order for students to learn mathematical vocabulary, it is vital that they learn the context of words.
According to the National Council of Teachers of Mathematics (1991), instruction that supports teacher and student communication will provide contexts for learning mathematical language. While the use of context is helpful, it is by no means sufficient as the only method for teaching mathematical vocabulary. The context may not always be rich enough to provide a definition. Therefore, direct teaching of mathematical vocabulary is necessary as well. Dictionary driven direct instruction often does not provide enough information for students to create an accurate meaning of a word.
Using graphic organizers are a form of direct instruction that can be successful in teaching accurate word meaning. A graphic organizer can more fully represent concepts and relationships between concepts while also providing a visual. Monroe and Orme (2002) emphasize that a combined approach is most beneficial. A study performed by Kossak (2007) compared the performances of academically at-risk middle school students when learning vocabulary using traditional dictionary definition strategy contrasted with a non-traditional visual strategy.
Eighty-one of the 86 students at a middle school in the second largest school district in the United States participated in the study. Her focus was on five middle school reading classes for at-risk learners. The same teacher taught all of the students involved. Fifty-three percent of the participants were male, while 47% were female. The ethnic makeup of the participants was 42% Black, 35% Hispanic, 19% Caucasian, while the remaining 1% were of other nationalities. A pre-test was given to all students to assess their knowledge of a list of ten vocabulary terms. Students were presented the ten vocabulary terms to study.
The words were divided randomly into two groups, with half of the words being defined by a dictionary and the other half defined with a visual. After the post-test, students knew an average of 3 words on the whole list. They knew an average of two words out of the five using the dictionary approach, and four out of the five words using the visual definition approach. These data demonstrate that using visual definitions clearly produces more understanding of vocabulary terms than the traditional dictionary driven method of instruction. Nelson & Stage (2007) chose to focus on learning vocabulary through context.
They too felt that dictionary driven instruction of vocabulary does not aid students in fully understanding new words and concepts. The purpose of Nelson and Stage’s study was to assess the effects of contextually-based multiple meaning vocabulary instruction on the knowledge of vocabulary and reading comprehension of students. The participants of this study were 283 third and fifth grade students enrolled in a midwestern public school system. The students were assigned to two groups based on their initial vocabulary and comprehension achievement on the pre-test Gates-MacGinitie Reading Test.
Students in the condition group received contextually based multiple-meaning instruction on 36 target vocabulary words. Day one instruction consisted of activating students’ prior knowledge of words. On day two, students were taught the word history of each target word. Then a word meaning map activity was conducted where students matched the related words from day one with the meanings of the target words in a graphic organizer format. After this, the students completed a definition activity for each multiple meaning word. The students in the non-condition group received instruction as the teacher would normally provide them.
According to the data collected (post-test), students who received the contextually based instruction of vocabulary words showed significant gains in vocabulary knowledge. Students with below average initial vocabulary and comprehension ability that were instructed in the contextually based multiple meaning vocabulary method showed significant gains in their knowledge of vocabulary. SUPPORT A follow up Newman-Kuels post hoc test showed that the students in the low achieving group were “more likely to show improvements in their vocabulary skills than those that were in the average to high group” (Nelson & Stage, 2007, p. 4). In terms of the teachers’ involved perception of the strategy’s effectiveness, the teachers consistently rated contextually based multiple meaning vocabulary instruction high. According to the teachers, they reported that the contextually based multiple meaning vocabulary instruction provided their students with a new challenge and aided students in learning key vocabulary and comprehension skills. Additionally, students were very responsive to the lessons. Therefore, Nelson and Stage assert that contextually-based multiple eaning vocabulary instruction can produce positive outcomes for low achieving third and fifth grade students. Not only did the students show gains in vocabulary and comprehension knowledge, but the teachers also found the instructional method worthy of classroom use. While the primary focus of this thesis is on middle school students, the procedures and results of this study can certainly be adapted and applied to students in sixth through eighth grade. Content Vocabulary Content area vocabulary acquisition is important for middle school students, as during these three years of school, students encounter many new words and concepts.
In a study performed by Keel, Slaton, and Blackhurst (2001), the effects of the constant-time-delay (CTD) procedure on the learning of students with learning disabilities in a small-group instruction arrangement were compared. The purpose of their study was to determine if having students copy the visual model of a vocabulary word before responding would be more effective in promoting word reading and spelling of words. Four students from three resource rooms and three students from a non-graded, multi-age regular education classroom participated in the study.
One small group instruction session took place each day. Eight target words were taught using two variations of the constant time delay procedure, four words for each condition. For each condition, two types of attention responses were compared: group (everyone writes) versus individual (only a target student writes) copying of the target word on whiteboards. The results of the study duplicate previous research on the constant time delay procedure in small group instruction arrangements. All of the students involved learned to read and spell words that were assigned to other students during the study.
The results indicated that the group condition (everyone writes) was more effective in increasing learning of non-target words than was the individual (target student writes) condition. Three of the students in the Everybody Writes condition maintained a 100% correct on observational words taught. Eleven students maintained an 80% correct on observational words taught during the Everybody Writes condition. Conversely, no students maintained a 100% correct on observational words taught during the Target Student Writes Only condition.
Keel and colleagues found that the Everybody Writes condition resulted in increased level of performance among students with disabilities. In summary, incorporating whole group responding in small group settings is useful in increasing learning of content area vocabulary. In relation to Keel, Slaton, and Blackhurst (2001), Malone and McLaughlin (1997) also investigated the effects of small group instruction on vocabulary learning. They chose to compare reciprocal peer tutoring with a group contingency to a traditional vocabulary program in a regular middle school classroom.
The participants in the study were 20 seventh grade students and 12 eighth grade students who attended a private parochial school. The students received vocabulary instruction in an ABAB and BABA design. When one group of students was receiving traditional vocabulary instruction, words and dictionary definitions, the other group was participating in reciprocal peer tutoring, using vocabulary words and flashcards. In order to measure the effects of the two forms of vocabulary instruction, the students were given a vocabulary quiz.
For the first quarter, the eighth grade students were taught using the reciprocal teaching method and scored an average of 90. 88% on the vocabulary quiz. The seventh graders taught in the traditional method scored a mean of 78. 32%. During the second quarter, the eighth graders were taught using the traditional method and scored an average of 84. 74%. The reciprocal peer tutoring seventh graders scored an average of 88. 79%. Overall, reciprocal peer tutoring was shown to be effective in increasing the quiz grades of the students.
During the traditional vocabulary instruction sessions, the students spent more time correcting the homework and reviewing vocabulary words and less time actively studying the words than in the reciprocal peer tutoring sessions. In regard to the teaching of mathematical vocabulary, middle school students could benefit from the reciprocal peer tutoring method of instruction. At this age, students often find success in socially constructed learning experiences. Thus, the multiple encounters with vocabulary words and their peers could result in increased knowledge of various mathematical vocabulary.
Additional support for the use of reciprocal peer tutoring can be found in an article by Laurice Joseph in the journal The Reading Teacher (2006). In her article, Joseph describes a technique, the Incremental Rehearsal Technique for using flashcards in learning content vocabulary. The Incremental Rehearsal Technique is a drill procedure where unknown content words are interspersed with known content words. In her experience, students enjoyed this activity because they were successful with some of the words while struggling with other words, as opposed to being unsuccessful with all of the words.
Thus an increase student in confidence and motivation to complete the procedure resulted. This direct approach to vocabulary instruction can be integrated into daily instruction as a warm up or way to wrap up a lesson. Focusing on vocabulary development during reading instruction can lead to a greater ability to infer word meaning as well as a greater ability to comprehend. Similarly, as comprehension skills increase, so does the ability to infer the meaning of new words from context therefore increasing known vocabulary words.
Twyman, McCleery, and Tindal (2006) investigated the effects of two instructional strategies on two groups of middle school students being taught a unit on U. S. colonial history. The purpose of their study was to contribute to the growing research that has investigated concept-based instruction for the promotion of student content knowledge and problem solving skills. The study was conducted at a suburban middle school in a Pacific Northwest public school district of 18,000 students. The school was of average size (just over 500 students) and average in socioeconomic status.
The students met for 21 class sessions. Classes met for 46 minutes a day for five weeks. Students in the experimental group were explicitly taught concepts. Previously taught concepts were reviewed at the beginning of each class period. After an instructional activity, students completed a notes sheet filling in examples and explanations of the concept taught. The students in the control group were taught in a more traditional, textbook-driven approach to instruction, including reading part of the textbook, individual reading of the textbook, and the completion of comprehension questions.
The results of the study show that in the realm of vocabulary, students receiving the concept based instruction (CBI) performed better on the three vocabulary tests and essay tasks that were administered than those not receiving CBI. For instance, students in the control group performed sufficiently on the final vocabulary assignment but only 13% of their responses on the final essay task presented conceptually based supporting details. However, 83% of the students in the experiment group included conceptually based supporting details. According to Twyman et al. 2006), this result may be due to the transformation of content words into concepts. Students were able to “hook” word definitions into a framework helping the students to create deeper conceptual meanings of the words. While students in both groups performed well on the vocabulary tests, the students in the control group performed lower on the final performance task (essay task). Thirteen percent of the control groups’ responses included conceptually focused supporting details. While the study was performed with a social studies class, the results of this study can certainly be modified for a math classroom.
Twyman, McCleery, and Tindal (2006) believe that their study provides support for an emphasis on teaching concepts within a content area. Since social studies teachers teach from texts that are heavy on reading skills and memorization of facts, CBI is a teaching method that can replace the traditional modes of teaching content. The results of this study validate the need for teaching students content concepts rather than rote memorization of terms, theories, procedures, and formulas for solving problems.
In relation to the content area of mathematics, organizing the content material in a concept-based manner can aid students in uncovering connections of concepts implicitly found in the textbook. The clearer the connections can be made, the easier it will be for students to acquire and utilize information to solve problems and justify their answers to the problems. In a study performed by Fore, Boon, and Lowrie (2007), the effectiveness of two types of vocabulary instructional methods on the learning of content-area vocabulary words of six middle school students with learning disabilities was investigated.
The study aimed to answer two research questions. The first question was, “will the concept model of vocabulary instruction lead to a larger increase in the number of vocabulary questions answered correctly than the definition/sentences writing model for students with learning disabilities at the secondary level? ” (Fore, Boon, & Lowrie, 2007, p. 56). The second question was “will students like learning vocabulary words more with the concept model compared to the definition/sentence writing model? ” (Fore, Boon, & Lowrie, 2007, p. 56).
The study was conducted in a middle school of 2,300 students. The participants included in this study were six seventh-grade students, age 12 to 13 years. All students were classified as having a learning disability and were able to read at the third grade level. One session of 20 minutes of vocabulary instruction was held twice a week during the normal class period. During each session, five vocabulary words were given to each student. The students were instructed using a vocabulary definition model the first four weeks, and then instructed using a concept model the final four weeks.
The concept model of instruction utilized a diagram that included the vocabulary word, the definition and characteristics of the word, as well as examples and non-examples. The words given were mathematical terms. Students were then tested on the vocabulary learned. Results indicated an increase in all six students’ vocabulary knowledge when instructed using the concept model. While the participants in the study were few and they each had learning disabilities, the results can be applied to a larger body of students and those without learning disabilities.
Further support for the use of the concept model of instruction is provided by Christine Renne (2004) in an article published by the Teaching Children Mathematics journal published by the National Council of Teachers of Mathematics. In her article, Renne explains that she investigated how to give students experiences that required them to describe and compare basic characteristics of squares and rectangles, or move beyond their simple definitions of the two.
In order to do this, she needed to develop the students’ conceptual understanding and vocabulary necessary so that a common language could be utilized. Renne chose to use composite comparison charts to aid students in determining similarities and differences between squares and rectangles. What she found was that through the use of the chart, students were better able to justify and explain their answer to the question “Is a square a rectangle? ” They were able to provide concrete examples as well as solid evidence as to why they answered yes or no.
This evidence shows that students need to develop mathematical vocabulary in order to establish connections between the relationships of math concepts. As with Capraro and Joffrion (2006), Harmon (1998) considered vocabulary learning a complex task that occurs in multiple settings. Harmon maintains that these settings range from minor occurrences in oral and written contexts to direct instruction. Learning from context plays a substantial role in a student’s acquisition of vocabulary. However, direct instruction also plays a critical role in vocabulary achievement.
In a study performed by Harmon, vocabulary learning opportunities were explored in a seventh grade reading program. The purpose of her study was to examine the explicit and implicit actions of the teacher as well as the student response to vocabulary teaching and learning. The study was conducted at a middle school in San Antonio, Texas. Through interviews, observations, and transcriptions of taped class sessions, guiding questions and responses generated important information regarding vocabulary teaching and learning.
Throughout the vocabulary teaching and learning, students engaged in a variety of literacy assignments with a focus on critical reading and problem solving. After examining the data, Harmon found that literature-based reading programs can be rich in opportunities for vocabulary teaching and learning at the middle school level Through analyzing observations, interviews, and transcripts, Harmon discovered that students were involved in variety of literacy activities where word learning opportunities were consistently situated.
The teacher highlighted words in pre-reading discussions, students used these same words in writing assignments, and words were practiced in small group discussions as well as during Sustained Silent Reading. In book discussion groups the Vocabulary Enricher (literature circle role), as well as other group members, provided social opportunities for students to engage in word learning and observe word learning strategies from their peers. Outside of individual and small group opportunities for word learning, Harmon also observed whole-class word earning where learning was teacher-directed as well as peer-directed in that students were exposed to strategies of their classmates. SUPPORT While vocabulary learning was not the main focus in the program, students engaged in activities that provided for word learning that enhanced their overall reading comprehension. The teacher was able to incorporate vocabulary instruction into her literature instruction, thus providing students with multiple opportunities to interact with new vocabulary.
This study’s results are particularly significant in that it shows students can learn vocabulary, more importantly content vocabulary, through experiencing it through literature. Mathematical Vocabulary One of the expectations of the National Council of Teachers of Mathematics (NCTM) within the Algebra Strand of the Middle School Standards is that students use symbolic algebra to represent and solve linear equations. As a result, middle school students must develop representational skills in order to be successful with linear equations.
Conceptual understanding of mathematics is the understanding of ideas and concepts that connect mathematical constructs (Capraro & Joffrion, 2006) and can be compared to reading comprehension, or making sense of what is read. Understanding the meaning of the words is necessary in order to read in mathematics. As students learn math, they must also learn the meaning of new words that are not part of their every day oral vocabulary, and words that have different meanings from what they already know (Raiker, 2000). In many cases, students learn the meaning of words indirectly, through experiences and conversation.
In their investigation, Capraro and Joffrion (2006) examined curriculum materials, factors affecting student learning, and professional development support for teachers of middle school math. The participants in the study were 668 middle school students in 25 classrooms in two separate states. Students were given pre- and post tests to determine the necessity of procedural or conceptual knowledge in order to answer the question. What Capraro and Joffrion (2006) found was that students were able to apply prior knowledge in a new situation if their conceptual understanding had been developed.
In the post-test, students received a combined score on three test items. The first two items were multiple choice. The multiple choice questions were analyzed for the answers the students gave as well as the students’ explanations for their answers. One of the multiple choice questions asked students to choose an expression that could be used to show the number of rows if there were n girls total and each row contained 6 Girl Scouts. The correct answer was choice D—n/6. Of those who answered, only 33. 5% answered correctly, while 37. % answered choice C—6n, the most popular distracter. In explaining their answers, the students who answered C, according to the researchers, may have had better procedural skills. One student explained, “6 girls times n rows would give him the number of girl scouts” (p. 160) Another stated, “n girls…6 girls in a row that would be 1, 2, 3, 4, 5, 6…(making marks), multiply 6 times the rows” (p. 160) The application of n changed as this student explained her reasoning, and was incorrect according to the problem because n identified the number of Girl Scouts.
However, through the student’s verbalization, she demonstrated the correct answer because she used n to signify the rows in choice C. Instruction in translating written representations (words) to symbolic representations (algebra) relies on procedural knowledge. Typically a teacher would give students a list of words that represent different operations (+, -, x, / ). When students translate word for word they need to be reminded that “less than” and “more than” need to be reversed[ADD MORE to EXPLAIN]. The word “each” can be attributed to both multiplication and division.
In answering the second multiple-choice question, the students’ translation of the word “each” may have contributed to the students’ incorrect answers and more importantly, their confusion. Most of the students chose to multiply when the question required division of the total rows by 6 girls. When required to take the conceptual information and translate it to a procedure, the students typically failed. Procedural knowledge is very limited because students are only required to repeat the solution steps similar for questions answered previously to answer a math problem.
Capraro and Joffion conclude that without conceptual understanding, procedural knowledge means very little. Connecting with mathematical concepts and vocabulary words make math more meaningful and memorable, and thus more powerful. According to Adams, Thangata, and King (2005), students at all grade levels need opportunities to define math vocabulary in ways that make sense to them. Self developed definitions result in mental ownership of the vocabulary, not just the memorization of terms given by the teacher.
The authors suggest helping students build their mathematical voices by providing written and oral prompts focused on particular words and language issues that are a challenge to students. They also suggests giving students opportunities to explain themselves if their responses to assignments seem unusual. This way the teacher can determine where the student has misunderstood a word or concept. Most importantly, students need the opportunity to voice their understanding and misunderstanding of mathematical words by writing, drawing, acting, and speaking, giving vocabulary instruction a multi-sensory style.
Just like Adams, Thangata, King (2005), and Carter and Deen (2006) investigated multi-faceted vocabulary instruction. Carter and Dean explored the incorporation of reading strategies in mathematics lessons. Fourteen students attended a three-week summer intervention program at a southern university with the purpose of increasing mathematical understanding. The purpose of the study was to investigate whether math teachers include reading strategies for decoding, vocabulary, and comprehension in their lessons and to explain how these strategies help students understand math concepts.
Analysis of the data collected was qualitative with the purpose of determining if teachers incorporated reading instruction in their math lessons. Taped lessons, along with notebooks containing the lessons, and student work were collected. The tapes showed that when math texts were read aloud, three of the instructors read to the students, three prompted the students to read, and the other two instructors used both techniques. Most of the reading instruction during this clinic was focused on vocabulary building.
Seven of the eight instructors used vocabulary strategies seventy times. Of all three areas of reading investigated, decoding occurred the least often, 70 of the 101 instances of reading instruction were focused on vocabulary, while the remaining 29 instances of reading instruction were focused on comprehension strategies. The researchers use these data to state that the responsibility of increasing students’ reading abilities does not lie solely on the reading teacher, but should also be a priority of the math teacher.
Capraro and Capraro (2006) analyzed how one teacher used children’s literature to enhance middle school geometry lessons. Specifically, they wanted to answer the following two questions: how does the integration of geometry-centric children’s literature influence students’ understanding of geometry and their performance on content and non-content specific measures of mathematics ability, and how does geometry-centric children’s literature, read aloud by the teacher, impact mathematical communication?
The participants in this study were sixth grade students of average SES and enrolled in one of three teachers’ classes. The three classes were split into one story group and two non-story groups. Students were given three pre- and three posttests in a multiple choice format in order to discover the impact of literature on their math learning. The story Sir Cumference and the Dragon of Pi (1999) was the literature used during the geometry lesson in the story group. Overall the students in the story group performed better than the students in the non-story groups on the post-test involving geometry.
Of those in the story group, ninety-three percent scored 84% or higher on the post-test. During interviews with students in the story group the students were able to explain pi, the relationship between circumference and diameter, and explain why pi is used when finding the area of a circle. Capraro and Capraro (2006) maintain that content-related storybooks can be used as an introduction to build interest, create anticipation, focus the lesson, or as a culminating activity.
Instruction in mathematical vocabulary is often overlooked as essential to the curriculum because math teachers are focused mainly on the content. However, as shown through the studies previously discussed, exchanging some of the focus on content for vocabulary instruction may indeed reduce a number of the difficulties students have with math texts as well as tests. Through the use of various instructional methods and strategies, math teachers can begin to increase student understanding of mathematical language.
Instructional Methods and Strategies Boulware-Gooden, Carreker, Thornhil, and Joshi (2007) conducted a study that investigated the effectiveness of systematic direct instruction of multiple metacognitive strategies designed to assist students in comprehending expository text. The investigation was designed to determine the effect of the metacognitive strategies on vocabulary learning. The participants in the study were 119 third grade students in two urban elementary schools across six third grade classrooms.
One of the schools was the intervention school while the other school was the comparison school. The students were given pretests before the study and a posttest at the end. Students in each school were instructed for 30 minutes a day in reading comprehension for 25 days. The lessons in the intervention school provided metacognitive strategies. There were five parts to each lesson: Introduction, Vocabulary, Read the story, Summary, and Questions. The comparison school students also began each lesson with an introduction and students were introduced to the same vocabulary words.
However, the comparison students were not instructed using vocabulary webs as the intervention school students were. The comparison students used traditional dictionary-driven methods to determine word meaning. The results of the posttests showed significant improvement among the intervention students compared to the comparison students. The students in the intervention school saw an increase of 40% greater than students who merely wrote the vocabulary word and created a sentence using the word.
Reading comprehension gains were also found in the intervention school. The intervention school was 20% higher than the control school. One significance of this study in relation to mathematical vocabulary learning is the metacognitive strategy used for vocabulary learning was the semantic web as well as the multiple meaning web. Both of these graphic organizer strategies were found to have a deeper understanding of the words taught. Students in the intervention school were required to produce synonyms, antonyms, and other related words in their webs.
The use of the webs created a visual representation of the students’ metacognition, as well as the word’s meaning and increased the students’ conceptual understanding over the traditional memorizing the definition and use in a sentence method. This study is influential in the strategies available for teaching mathematical vocabulary. Due to the amount of words with multiple meanings as well as new vocabulary encountered in mathematical texts, semantic maps as well as multiple meaning maps can be valuable tools for teaching mathematical vocabulary and concepts.
In relation to Boulware-Gooden, Carreker, Thornhil, and Joshi’s study (2007), William Rupley and William Dee Nichols (2005) published an article in Reading & Writing Quarterly focused on strategies for creating connections between previously learned vocabulary words and new words and how relationships can form among words. In their article, various research supported instructional activities are discussed as ways to create connections between known words to new words.
Strategies such as concept wheels, semantic word maps, concepts of definitions, webbing, and semantic feature analysis are tools included in the article as ways to expand students’ vocabularies and supporting the active processing of new vocabulary. Semantic word mapping is discussed at length as a way to incorporate the many guiding principles of teaching vocabulary, such as activating/building background knowledge, encouraging questioning characteristics of words and creating visual connections between words.
Rupley and Nichols (2005) describe a semantic map “as a diagram that groups related concepts through the use of a graphic organizer and allows the learner to visually display the connections between concepts” (p. 9). According to Rupley and Nichols (2005), using a semantic map before reading can give struggling readers a foundation that will support their comprehension of a mathematical text. The map helps free struggling readers of text restrictions that are caused by not understanding words in the text.
In an article published by the Journal of Adolescent & Adult Literacy, Catherine Rosenbaum (2001) uses the previously discussed study by Harmon (1998) to provide an example of using word mapping of vocabulary. Harmon lists eight techniques that clarify word meanings for students: synonyms, brief descriptions, examples and non-examples, rephrasing, repetition, associations, and unique expression. Rosenbaum (2001) took these eight techniques and created a word map combining the techniques. According to Rosenbaum, the map “provides a framework and…satisfied all the criteria for effective vocabulary instruction” (p. 45).
Stahl (1986) recommended giving both context and definitions when teaching vocabulary. Rosenbaum’s word map does just this. She suggests a widespread use of her word map. In a mathematics classroom, her map can provide students with a rich, deeper understanding of unknown words by providing the definitions within the context of math. Similar to the word map is the keyword method of learning vocabulary. In a study conducted by Fritz, Morris, Acton, Voelkel, and Etkind (2007), retrieval practice and the keyword mnemonic strategies were compared when learning foreign vocabulary. The study was separated in to two experiments.
The purpose of the first experiment was to compare the effectiveness of the keyword method with the retrieval practice. The two strategies were compared using rote rehearsal as a control condition. The participants’ performances were tested at the end of a practice session and following a three-day delay. Forty-five individuals ranging in age from 19 to 35 participated in the experiment. Fifteen participants were separated into three groups, key word, retrieval practice, and rote rehearsal. The participants listened to three different tapes, depending on the group they were placed in.
The retrieval practice tape consisted of the presentation of the vocabulary word and practice events (“the French for hedgehog is herrison;” “the French is herrison, what is the English? “). The rote rehearsal tape consisted of all 15 English words and translations being presented 10 times a 5 second intervals. The keyword method tape contained an English word, its translation (‘”The French for hedgehog is herrison”), followed by instructions on how to connect the English word with the foreign word by creating an image (Imagine your hairy son looks like a hedgehog’). Following the listening of the tapes, the participants were tested.
The results indicated that the retrieval practice and the keyword method improved performance over the rote rehearsal condition. –use actual data to support this finding–After the three-day delay, participants in the first two groups still outperformed those in the rote rehearsal group in recalling the English words. –use actual data to support this finding–Retrieval practice as well as keyword methods are powerful techniques of learning vocabulary. While the focus of this proposal is on mathematical vocabulary, the study is applicable in that it focuses on rote memorization versus a deeper study of vocabulary terms.
The previous studies concentrate on one of the five components of reading instruction: vocabulary instruction. Research studies focused on vocabulary instruction, content area vocabulary, and ultimately on the instruction of content area words in mathematics lay the foundation for providing mathematics teachers with strategies effective in expanding depth of vocabulary for middle school students. Strategies The following section includes the various strategies found effective in teaching content vocabulary in mathematics. Each strategy is described. The procedure for each strategy is also provided.
A model of the strategy is also given. Key Word Method The Keyword Method is a mnemonics technique for learning vocabulary. A student can be taught to create a visual image that connects a vocabulary term with a more familiar word that is similar in pronunciation and structure. An example of this vocabulary strategy can be seen in Figure 1 below. Figure 1 Word Wall A word wall is a display of words that are related. It can be used as a tool to teach concepts, especially mathematical concepts. Word walls are typically displayed on a wall in a large format.
However, students can have personal word walls of their own on paper as well. The walls can be sed for many classroom activities. To implement this strategy a teacher must determine the vocabulary words the students need to know for a concept or unit of study. Then the words need to be printed or copied in large letters for posting on a wall or bulletin board. The teacher can choose to post all of the words on the wall prior to beginning the unit, ro as the students encounter them. However it is essential for the teacher to discuss the word and why it is being included on the wall. The words on the wall should be eviewed regularly and the students’ attention should be drawn to the wall when included words are used. An example of a word call can be seen in Figure 2 below. The word wall can include just a piece of Figure 2, for example the “Addition” section, or it can consist of all of the words related to the concept. Figure 2 [pic]Word Sort A word sort is a simple individual or small group activity. Word sorts involve giving students a list of vocabulary related to a concept or unit of study. Students examine the terms for meaning and characteristics and then “sort” the list into categories.
The sorting allows students to connect prior knowledge to new information related to a concept. There are two types of word sorts available. The first is a Closed Word Sort. This type of word sort is when the teacher gives the students the categories in which to sort the words. The second is an Open Word Sort. This type of sort is when the teacher allows the students to determine for themselves how to sort the words given. A typical word sort for middle school students should contain no more than 15 words. The teacher should instruct the students to either perform a Closed Sort or an Open Sort.
Following the sorting, a class discussion should be completed so students can defend their sorting and explain why they chose to sort the words in the manner they did. An example of a mathematical vocabulary word sort can be seen in Figure 3 below. Figure 3 Frayer Model A Frayer Model is an activity based on word categorization that helps students develop an understanding of a concept. The student must provide the definition of a word or concept, characteristics or facts about the word/concept, and examples as well as non-examples of the concept.
Because there are many concepts in mathematics, students can easily become confused because of the similarities and differences between concepts. The Frayer Model provides students with a chance to what the specific concept being learned is and what it is not (National Center for Reading First Technical Assistance). Figure 4 Semantic Map A Semantic Map, or sometimes called a word map is a graphic illustration created to draw together and connect facts, ideas, concepts, as well as words.
The strategy is an “effective strategy for activating prior content knowledge” (Barton, Heidema, Jordan, 2002, p. 25). Using this strategy, a student can associate word meanings and make connections between background knowledge and new information. A word web is particularly helpful to visual learners because it creates a visual organizer of the relationships between abstract concepts and the words related to the concepts. In addition, Semantic Maps can be used as an anticipation guide or as a way to pre-assess students’ vocabulary and concept knowledge of a particular unit.
Semantic Maps can also be used as a way to gauge student learning from a lesson (Rupley & Nichols, 2005). An example of a Semantic Map can be seen below in Figure 5 below. Figure 5 Concept of Definition Map A concept map, or sometimes referred to as a semantic map, can be used for many purpose with most types of texts. The concept map strategy can be used as a pre-reading activity to teach difficult concepts or vocabulary that will be introduced in mathematical texts. A concept map is another visual display of the relationships between new words and words the student already knows.
Using a concept map well aid the student in gaining a better understanding of the content or concepts found in the text prior to reading. The map can also be added to after reading. An example of a Concept of Definition Map can be seen below in Figure 6 below. Figure 6 Semantic Feature Analysis A Semantic Feature Analysis (SFA) involves the student in completing a table that describes the characteristics that distinguish the elements included, be it words or concepts. SFA is a procedure that aids students in combining new information with background it helps students understand the elationships among the meanings of words and concepts (Bryant, Ugel, Thompson, & Hamff, 1999). New vocabulary or words are organized in a column and characteristics (or semantic features) are shown in a row moving horizontal in a row. An example of a Semantic Feature Analysis can bee seen in Figure 7 below. Figure 7 In this section, specific vocabulary strategies were explained. Along with each explanation was included a visual of each. Through the incorporation of these strategies, mathematics students could improve their vocabulary knowledge and improve their overall success in mathematics.
Chapter Summary Chapter 2 presented the research basis for the teacher handbook Effective Strategies for Teaching Content Vocabulary in Mathematics. Included in the chapter was information regarding the theoretical background related to vocabulary instruction. Also included in this chapter was an exploration of the teaching and learning of vocabulary as a whole. Following this section an examination of research focused on content vocabulary and then mathematical vocabulary exclusively was explored.
Finally, instructional methods and strategies as well as specific strategies included in my teacher handbook were described, examples of each strategy were provided. This chapter provided a foundation and rationale for the implementation of the strategies listed in my teacher’s handbook. Chapter 3 Curriculum Project This chapter will provide a brief ove