KEYWORDS: mathematics, education, elementary, language, corpus linguistics.

ABSTRACT: This paper examines the language of mathematics as reflected in lesson plans created by elementary school teachers (grades K-6) in the state of Alabama using a corpus linguistics methodology. Corpora in this study come for the language teachers intended to use in published lesson plans which are housed in a state sanctioned and vetted website. The researchers found that teachers use more sub technical terms than technical terms in their geometry lesson plans. The reasons for this use of more simplistic language are not clear. However, the researchers assert that it is important for teachers to be cognizant of the potential misunderstandings that children may have in mathematics lessons due to simplistic language use.

The language of mathematics is a topic that has come to the forefront of mathematics education over the last decade (Schleppegrell, 2007; de Freitas & Zolkower, 2009; Ng & Rao, 2010). This trend, it is argued, has been ushered in part with the implementation of Common Core Mathematical Practices, which have increased awareness of how language impacts the teaching of mathematics (NGACBP& CCSSO, 2010). Many educators believe that mathematics is a universal language. Some even go so far as to suggest that mathematics is language free (Walkerdine, 1988) and culture free (Burton, 1994). This is contrary to the National Council of Supervisors of Mathematics (NCSM, 2009) position statement that contends: “Mathematics is neither value free nor culture free, but a product of human activity. Thus race, class, culture and language play a key role in its teaching and learning” (p.1). To inform this debate, we argue that it is necessary to understand the role and scope of language in a context of mathematics education at various developmental levels. The purpose of this study was to examine the mathematical vocabulary used in lesson plans by elementary school teachers (grades K-6) to determine ways in which their planned language in mathematics lessons transmits or hinders students’ mathematics register.

“Language skills have become increasingly important in mathematics classrooms” (Pierce & Fontaine, 2009, p. 239). Klibanoff, Levine, Huttenlocher, Vasilyeva, & Hedges (2006), argued that acquiring the language of mathematics is important to the acquisition of mathematical concepts. Campbell and Rowan (1997) suggested that “language has the power to help children organize and link their partial understandings as they integrate and develop mathematical concepts” (p. 63). Furthermore, Adams (2003) discussed that for students across all grade levels, weakness in their mathematics ability is often due in part to the obstacles they face in focusing on symbols as they attempt to read the “language of mathematics.” He also suggested, “a knower of mathematics is a doer of mathematics, and a doer of mathematics is a reader of mathematics” (2003, p. 794). This further supports the view that language is clearly and deeply embedded in mathematics education.

Within the context of language of mathematics, further examination of the vocabulary of mathematics is important. Thompson and Rubenstein (2000) stated, “educators need to remember that vocabulary learning and mathematical understanding are intertwined” (p. 569). Mathematical vocabulary can be difficult to understand, due to the fact that most children do not have many opportunities to talk about mathematics (NCTM, 2000; Cobb, Wood & Yackel, 1993). To be an effective communicator of mathematics, one must understand the nuances of the vocabulary. Many mathematical words have very different meanings in everyday usage, such as net or plane, which can make mathematics confusing to children (Rubenstein & Thompson, 2002). Studies reveal there is a relationship between a robust mathematical vocabulary and an understanding of mathematical concepts. Miller (1993) indicates that often students are unable to define mathematical terms on their own. Furthermore, research addressed this problem by stating “without an understanding of the vocabulary that is used routinely in mathematics instruction, textbooks and word problems, students are handicapped in their efforts to learn mathematics” (p. 312). Marzano, Pickering, and Pollock (2001) echoed Miller’s sentiment and went so far as to encourage focused instruction in vocabulary acquisition by suggesting “instruction on words that are critical to new content produces… powerful learning. The effects of vocabulary instruction are even more powerful when the words selected are those that students most likely will encounter when they learn new content” (p. 127).

In an effort to address the issues surrounding learning mathematical jargon, Monroe and Panchyshyn (1995) divided it into four different vocabulary types: technical, sub-technical, general and symbolic. They also offered suggestions for teaching each vocabulary type. Technical vocabulary includes words that have a precise mathematical denotation (Pierce & Fontaine, 2009). They are often difficult to express in common language. Examples of technical vocabulary would be polyhedron and equilateral. Sub-technical vocabulary includes words that have different meanings according to the context in which they are used. Examples of sub-technical vocabulary would be table and shape. These words have a specific mathematical context but can be easily used in a variety of settings, topics or contexts. General vocabulary is words that are common in students’ everyday readings. They are not necessarily related to mathematical concepts, but can be used to describe them. Symbolic vocabulary is words or non-alphabetic symbols that are considered abstract and difficult to define. Examples of symbolic vocabulary would be π or ∑. This paper adopts Monroe and Panchyshyn’s (1995) mathematical vocabulary taxonomy as an a priori means of classifying the words that are used to describe mathematical concepts in K-6 lesson plans.

Theoretical Framework and Context of the Study

This study uses a sociocultural theoretical framework. Central to Vygotsky's (1981) position on the social nature of learning is the belief that the study of language and thought cannot be separated. Therefore, it is through internalized tool use that higher order thinking skills are developed. While language and thought are separate processes, they are interdependent and their individual study would be fruitless (Bakhurst, 1991). In this sense, language is a tool of thinking and the primary means by which higher order thinking skills are developed. This means that learning in general and, for the purposes of this paper, the learning of math, cannot be investigated without an analysis of the language that mediates mathematical practice.

This study examined the use of mathematical language in elementary geometry lesson plans ranging from kindergarten through sixth grade in the state of Alabama in the US. To examine mathematical language use, it was essential to select a mathematical concept that has substantial language. Therefore, geometry was chosen due to the large amount of vocabulary associated with its instruction. The lessons were retrieved from the Alabama Learning Exchange website (http://alex.state.al.us), which is managed by the Alabama State Department of Education. All of the lessons on the website have been reviewed by peers as well as by the Alabama State Department of Education prior to their publication. Furthermore, the lessons all follow the same format. That is to say, they include sections that detail local and national standards, lesson objectives, materials needed, preparation, procedures, and assessment. For the purpose of our analysis, only the procedure section of each lesson plan was examined for language use.

In the state of Alabama, there are 48,615 teachers, of that number, 56.3% hold a Master’s degree or above. 78% are Caucasian. 20% are African-American and the remaining 2% belong to other ethnic groups. The state website ensures that lesson plans come from educators in all regions of the state and that there are equal representations of these educators. The researchers used all available geometry lessons in grades kindergarten through sixth in this study.

Alabama has consistently been considered a low achieving state on national 4th and 8th grade mathematics assessments (National Center for Education Statistics, 2009). Since 2000, Alabama 4th graders (9 year-olds) have scored on average 10 to 17 points below the national average, which makes the state consistently in the bottom 10%. Considering the achievement level of the state, the researchers were curious as to what type of vocabulary the teachers were using in lesson plans. Examining lesson plans, contributes to our understanding of what mathematical language teachers used when planning mathematics instruction and allowed for the exploration of the following research question: What mathematical vocabulary words do kindergarten through sixth grade teachers utilize in geometry lesson plans?

Methodology

Researchers used a corpus linguistics methodology to investigate the language of mathematics in this study. A corpus is a “collection of linguistic data, either written texts or a transcription of recorded speech, which can be used as a starting point of linguistic description” (Crystal, 2002, p.112). No matter if the language being studied is written or spoken, the corpus used to study it must be representative. That is to say it should board enough sample so that it encompasses the language being studied. Therefore, a corpus is not a random amalgamation of language. Instead it has been meaningfully and purposefully assembled from authentic oral or written texts with the aim of representing a particular language or language subset.

From the sociocultural paradigm, corpus linguistics is a particularly strong methodology to study language because it does not privilege a generalizable experience. Instead it investigates the actual language that is used in naturalistic settings. Furthermore, the corpus created for the purposes of this study can be considered to be ecologically valid as defined by Van Lier (2004) because it considers the context which the language takes, examines the language through a critical lens, is longitudinally descriptive and is emic in its perspective.

Data Analysis

Analysis of the data began with the text of the procedure section of the lesson plans saved as plain text files. Then data was analyzed in a concordance program called AntConc; a free, downloadable software available from (http://www.antlab.sci.waseda.ac.jp/software.html). This software created a list of the most commonly used words in the corpus. As was expected, the most prevalent words were function words such as “the,” “to,” “a,” and “student”. Since these terms are not mathematical in nature, these words were excluded from analysis. Words lists of mathematical terms were printed. First, the researchers separately hand tagged instances of technical and sub-technical vocabulary. In turn, these word lists were organized into graphic displays of data (Miles and Huberman, 1994). Generic and symbolic vocabulary, as defined by Monroe and Panchyshyn (1995), were not included. Generic vocabulary was not included because we judged it to not be germane to our discussion. Symbolic vocabulary was not included because no instances of it were found in the corpus.

Upon completion of the individual tagging, the researchers compared their word lists. When there were instances where the word usage, the researchers examined the word was used in context of the lesson plan. From this context, the different functions and their instances of the occurrence were able to be separated. Inter-rater reliability concerning the hand tagged words and their classification as either technical or sub-technical was calculated by dividing the number of agreements by the total number of units included in the sample (Miles & Huberman, 1994). The inter-rater reliability calculation yielded a 92% agreement in sample for the technical mathematics vocabulary and 89% in the samples for sub-technical mathematics vocabulary. In the instances when there was no agreement, the researchers discussed each word to determine if the term was to be considered as part of the technical or sub-technical vocabulary lists, or if the term was to be removed from the sample. In order to examine to the question “what mathematical vocabulary words do kindergarten through sixth grade teachers utilize in geometry lesson plans?” the researchers conducted word frequency analysis.

Findings

Through the analysis, two types of mathematical vocabulary words became apparent. These terms were classified according to the criteria put forth by Monroe and Panchyshyn (1995) as either technical or sub-technical vocabulary. Technical vocabulary was defined as words that only have one specific meaning. Whereas sub-technical terms were defined as words that had multiple meanings. The percentage of occurrences of technical vocabulary in the K-6 geometry lesson is displayed in Table 1.

Table 1. Technical vocabulary

Several of the most frequently occurring vocabulary words are names of common shapes. Interestingly, polygon was second most commonly used word. The following example illustrates the way that the word polygon was used in the context of a third grade (eight year-olds) lesson plan:

Have the class review polygon shapes, and ask them to name each polygon. As each polygon appears have the class tell how many sides each polygon has and how to measure its perimeter. Explain that perimeter can be measured using different units, depending upon the size of the polygon. Examples: inch, centimeter, yards, miles, etc.

The use of polygon in this lesson is setting the context for a lesson on perimeter. While the concept of perimeter is considered a geometric measurement, the use of polygon in the context is vital to the teaching of this lesson. This example demonstrates the interconnectivity of not only mathematics, but also its vocabulary.

The example above for polygon is consistent with Monroe and Panchyshyn (1995) definition of technical vocabulary. These vocabulary words have specific meanings that are paramount in mathematical understanding. Although only one example was given, the provided sample was consistent with the context in which the remaining vocabulary terms were used in lesson plans across the grade levels.

While the technical vocabulary had multiple terms, the analysis resulted in only six sub technical vocabulary terms. Although the number of sub-technical terms is small, their use is more frequent. The percentage of sub-technical vocabulary in the K-6 geometry lesson is displayed in Table 2.

Table 2. Sub technical vocabulary

Shape was the most frequently used geometry vocabulary word (technical and sub-technical) in the entire corpus. This is not surprising considering the term shape is found in the state geometry standards for every grade level except for third and sixth. However, the context in which shape was used in the lesson plans was quite varied. The following statement illustrates the way that the word shape was used in a second grade (7 year-olds) lesson plan.

Tell the students that you are going to hold up some incomplete patterns of geometric shapes. In each instance, one shape is missing. They should identify the part of the pattern that is missing, and then create that shape with their ropes. Hold up one pattern on a poster paper at a time, such as: circle, rectangle, square; circle, rectangle, ______ . The students should create a square. To verbally reinforce what they have created, ask the students to say the name of the shape.

This sample from a second grade lesson plan illustrates how teachers demonstrate more than one concept in a mathematics lesson. While this lesson uses geometric terms, the focus of the lesson is algebraic in nature.

The use of the term shape in a sixth grade (11 year-olds) lesson plan was quite different. The intent of this lesson was to have the children determine the volume and surface area of different size and shape packages that are going to be shipped. The students were to calculate the cost for shipping of each package: “Students will create the models of each package and complete the cost analysis spreadsheet that they have created using a spreadsheet software. The spreadsheet will include the volume and surface area of each shape.” This geometry lesson included several different mathematical content ideas. The second grade lesson plan used shape in a generic fashion to describe geometric figures. The sixth grade lesson plan used the term shape to describe three dimensional, more complex figures. These two examples showed how different the use of the term shape can be, which is to be expected with sub-technical vocabulary.

The analysis of technical and sub technical vocabulary terms provide researchers and teachers with an understanding of which terms teachers are selecting when writing lesson plans. This information can help guide and refine mathematics instruction in the elementary school.

Implications

Understanding the vocabulary a teacher uses in lesson planning is important, because it lays the foundation to examine their actual vocabulary use when they deliver the lesson. One finding from this study is that teachers predominately use technical vocabulary. Furthermore, the findings of the sub technical vocabulary terms have implications for teachers. The high frequency of sub technical terms in the lesson plans is important for teachers to be aware because as Rubenstein & Thompson (2002) explain the multiple meanings of mathematics vocabulary can be confusing for children. With such a high frequency of occurrences of sub technical mathematics vocabulary, teachers need to be cognizant of the potential misunderstandings that children may have in mathematics lessons.

Both the technical and sub technical word frequency lists revealed that teachers seem to favor simplistic vocabulary. For instance, in technical vocabulary teachers overwhelmingly use the words triangle and polygon, meanwhile they avoided using more complex terms such as decagon or octahedron. The same can be said of teachers’ word choice in terms of a sub technical vocabulary. The reasons for this are not clear. Perhaps it is because they possess an incomplete mastery of geometric content knowledge or maybe they tend to use less complex language because they judge the use of formal mathematics vocabulary to be developmentally inappropriate for their students.

This study makes an important methodological contribution to the field by broadening the application of corpus linguistics to mathematics education. The analysis of the language that teachers plan on using in their geometry lessons provides a quantifiable way of investigating the language of mathematics. Furthermore, the authors argue that corpus linguistics can be used as a powerful, investigatory tool for language no matter the content area.

Bibliography

1. Adams, T. L. (2003). Reading mathematics: More than words can say. The Reading Teacher. 56(8): 786-795.
2. Bakhurst, D. (1991). Consciousness and revolution in Soviet philosophy. Cambridge University Press.
3. Burton, L. (1994). Forward to the Twenty-First Century? Teaching Mathematics Applications,13, 1-4.
4. Campbell, P. F. & Rowan, T. E. (1997). Teacher Questions + Student Language + Diversity = Mathematical Power National Council of Teachers of Mathematics Yearbook. (p.60-70) Reston VA: NCTM.
5. Cobb, P., Wood, T., & Yackel, E. (1993). Discourse, mathematical thinking, and classroom practice. In N. Minick, E. Forman, & A. Stone (Eds.), Education and mind: Institutional, social, and developmental processes (pp. 91-119). New York: Oxford University Press.
6. Crystal, D. (2003). A dictionary of linguistics & phonetics. Oxford: Blackwell.
7. De Freitas, E. & Zolkower, B. (2009). Using social semiotics to prepare teachers to teach for social justice. Journal of Mathematics Teacher Education, 12(3), 187-203.
8. Klibanoff, R., Levine, S., Huttenlocher, J., Vasilyeva, M., & Hedges, L. (2006). Preschool children’s mathematical knowledge: The effect of teacher “math talk”. Developmental Psychology, 42, 59-69.
9. Marzano, R. J., Pickering, D. J., & Pollock, J. E. (2001). Classroom instruction that works: Research-based strategies for increasing student achievement. Alexandria, VA: Association for Supervision and Curriculum Development.
10. Miller, D. L. (1993). Making the connection with language. Arithmetic Teacher, 40(6), 311–316.
11. Miles, M. B., & Huberman, A. M. (1994). Qualitative data analysis: An expanded sourcebook. Sage: London.
12. Monroe E. & Panchyshyn, R. (1995), Vocabulary considerations for teaching mathematics. Childhood Education, 72(2), 80-83.
13. National Center for Educational Statistics (2009). The Nation’s Report Card. Retrieved from http://nces.ed.gov/nationsreportcard/
14. National Council of Supervisors of Mathematics (NCSM). (2009). Improving Student Achievement in Mathematics by Addressing the Needs of English Language Learners. (NCSM Improving Student Achievement Series, No. 6). Reston, VA: Author.
15. National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core Standards for Mathematics. Washington, D.C.: Authors.
16. Ng, N. N. S., & Rao, N. (2010). Chinese number words, culture, and mathematics learning. Review of Educational Research, 80(2), 180-206.
17. Pierce, M. E. & Fontaine, L. M. (2009). Designing vocabulary instruction in mathematics. The Reading Teacher, 63(3), 239-243.
18. Rubenstein, R. & Thompson, D. (2000). Learning mathematics vocabulary: Potential pitfalls and instructional strategies. Mathematics Teacher, 93, 568-574.
19. Rubenstein, R.N. & Thompson, D.R. (2002). Understanding and supporting children's mathematical vocabulary development. Teaching Children Mathematics, 9(2), 107–112.
20. Schleppegrell, M. J. (2007). The linguistic challenges of mathematics teaching and learning: a Research review. Reading & Writing Quarterly, 23(2), 139-159.
21. Van Lier, L. (2004). The ecology and semiotics of language learning: A sociocultural perspective. New York: Springer.
22. Vygotsky, L.S. (1981). The genesis of higher mental functions. In J.V. Wertsch (Eds.) The Concept of Activity in Soviet Psychology (pp.144-188). Armonk, NY: M.E. Sharpe.
23. Walkerdine, V. (1988). The mastery of reason: Cognitive development and the production of rationality. Routledge, Chapman & Hall: Incorporated.