## Introduction

### Statement of Problem

Educational articulacy and lucidity focus on the effects of mathematical vocabularies on students’ achievements. This is inclusive of poorly performing students as well as those with learning complications and disabilities. It is observable that most students, especially those with English as not their first language, continue to demonstrate weaknesses in mathematical computation as well as problem-solving. The observed linguistic limitations contribute to lowering success in math. There are lower grades recorded amongst these students. This complicates the possibility of these students earning their diplomas. Therefore, there is an increasing significance in isolating basic components within mathematics understanding (Anderson & Little, 2004). Additionally, there is also a need to recognize the appropriate approaches and methodologies to instruct and teach these students to assist the process of improving the notable lower performance within these groups of students.

Most investigations have indicated that there exists a powerful correlation between vocabulary and mathematical comprehension as an important component for the understanding of this critical subject amongst the students (Scruggs & Mastropieri, 2000). Vocabulary instruction including mnemonic methodologies has persistently led to a considerable elevation in the learning as well as retention for the learners that depict disabilities (Burton & Morgan, 2000). This notable trend has also been recognized or realized within the nondisabled peers relative to other identifiable or applicable approaches. The basic question of important investigations and studies in this context, therefore, is to find out how mathematics vocabulary may be taught by instructors. In addition, there should be a great focus on identifying which strategies that students can apply in learning the mathematics vocabulary.

Several studies in the past have majored and discovered the particular literacy methods that can be applied by instructors and teachers in the instruction and instruction of math vocabulary (Scruggs & Mastropieri, 2000). These investigations have also majored in the ways of strengthening the conviction that literacy remains vital when students get to know the critical skills and comprehend the mathematics vocabulary. As a mathematics teacher, I have a deep understanding of the basic factors that cause poor students performance in mathematics. The vocabulary applicable in math remains potentially one of these basic reasons. The students suffer difficulty in the application of the concise language of mathematics (Stone, 2007). This is observed in both verbal as well as written explanations. Due to this, I have wondered about the required modifications I may execute within the classroom for transforming the situation. This interest is, therefore, the basis of my intended research on this topic.

There is an increasing need to apply critical research and investigative approaches to enhance the process of applying appropriate mathematical language in the teaching of mathematics (Scruggs & Mastropieri, 2000). This research initiative considers students with English as not their first language. A critical review of other past studies indicates the disparities that are eminent between the proposed study and the previous ones. This is, particularly, with the consideration of the level of objective and focus of the research. Recent investigations within this topic have mostly focused on concerns other than distinct vocabulary issues. In most cases, the investigations have focused on concerns like the readability of various texts applied in mathematics. Others have also been focused on how the student’s reading competencies influence or manipulate their comprehension and adequate performance in mathematics as a subject (NMPR, 2007).

It is critical to understand that despite the subject or main focus of the investigations, the fact remains that mathematics is a unique language that stands purely on solitary grounds and its application has various implications and impacts on the learners (Chard, 2003). Therefore, it is imperative to understand that knowledge of this distinct language, thus, dictates, at least to some extent, the success of any classroom discourse as well as the understanding of all learners as they engage in the reading of these written texts with mathematical derivations or implications. The interest in investigating the association between the learner’s vocabulary comprehension and student performance has considerably developed from the motivation of past notable studies within the subject.

### Purpose of the Proposed Study

The purpose of this proposed study is to investigate or analyze the impact of using mathematical vocabulary strategies on students’ academic achievements. In this study, there will be a specific focus on those students without English as their first language. The researcher intends to examine the basic features of the various mathematical languages applicable in the study and tests based on the mathematical class and subject (Lake, 2009). The student’s convictions and belief on the existence of the notable association will also be investigated to effectively understand the basic underlying factors. There will be a critical analysis of a majority of diverse factors in the examination of features underlying this issue. Among some of these important issues to be addressed include the students’ accuracy and concise application and understanding of the vocabulary related to mathematics within written solutions. This is collectively termed as writing about math initiatives.

The learners’ convictions on the significance as well as the importance of applying concise mathematical vocabulary will also be assessed in this context. In addition, the researcher also recognizes that it will be important to investigate or study the number of precise and accurate assessment answers for problems depicting precise mathematical vocabulary. The investigator will be interested in the learning of the particular instruction or teaching language applicable in mathematics and how this would improve the learners’ comprehension of the basic concepts (Marzano, Pickering & Pollock, 2001). More objectively, the investigator would also link the basic observations and processes to how they can enhance the students clearly and concisely communicate mathematical issues as this creates the potential basis for excellent comprehension and understanding of the students.

The proposed research would apply the use of basic research questions. For instance, there would be an investigation into the likely occurrences ensuing students’ application of concise and accurate mathematical vocabulary (Rubestein & Thompson, 2002). This will be after these students receive particular, distinct instructions or directives within the mathematics vocabulary. The student’s perceptions on the application of concise and accurate mathematical vocabulary in written mathematical solutions as well as on other assessments will also be necessarily analyzed.

Generally, the dependence on the investigation of the most effective strategies would be applicable in enhancing the critical research outcomes of the study (Anderson & Little, 2004). The appearance and perception of the teaching or instruction method when in the application of the concise or accurate mathematical vocabulary within students’ written or outlined solutions would also be critical. The study will also analyze the effects of the application of diverse proposed strategies in teaching mathematical vocabulary. There will be consideration of the most affected classroom grades within the education systems, most probably, the high school as well as the eighth grades. The proposed study remains of great importance to most other instructors, even those within other subjects. There will be the provision of adequate information to these teachers, regarding the effect of basic languages on particular subjects by the end of the proposed study.

### Research Questions and/or Hypotheses

The definite research questions that will guide the proposed study include:

- What is the impact of applying mathematical vocabulary on students’ achievement? Is there a difference in the application of this vocabulary in the context of students with English as their second language?
- Is there a notable disparity between the scores observed in vocabulary examinations for students taught mathematics vocabulary through direct teaching minus keyword mnemonics relative to the others taught through a direct method involving keyword mnemonics?
- What is likely to occur to the learners’ mathematical comprehension after they get instructed in vocabulary?
- How will the learners’ questioning as well as performance in homework and examinations undertakings transform after receiving basic instructions using the mathematical vocabulary?
- How will the diverse teaching methodologies challenge, support, transform or enhance the student’s competencies and performance in mathematics as a general subject?

### Significance of the Proposed Study

The knowledge of vocabulary is crucial in learning mathematics. This is because it involves mechanisms of reading and comprehension of mathematical concepts. In the context of mathematical concepts, words are vital since they potentially give an implicit and explicit description of the activities as well as associations that are a dearth of visual counterparts. The significance of the proposed study cannot, therefore, be underscored in the context of education and student learning. This study is yet to provide critical information about the diverse impacts of the application of mathematical vocabulary strategies on the students’ achievement and performance. The information provided would be particularly pertinent to cases where or situations in which the students under context do not have English as their first language. This information will be pertinent not only to the teachers or students but also to the general education system distinct within particular areas. The information and results eminent from this study will enhance critical knowledge and open doors for additional analysis and further investigation into the subject.

### Definitions of Terms

The basic terms that will be applicable in this proposed study are outlined and defined as follows:

**Comprehension:**Refers to deliberate thinking out which specific derivation of meaning becomes constructed. This normally occurs with the processes of interactions notable between the learner as well as the actual content.**Achievement:**stands for the accomplishment of certain target objectives, or measurement of the level of a persons’ (in this context, student) performance.**Mathematics Vocabulary Instruction:**The application of mathematical language in the academic facilitation and guidance of students or learners.**Poor performing learner:**A learner with minimal or dearth experience of success within the outlined or contextual grade level or who lacks the competency to adequately meet the required or stipulated performance standards.**Students with disabilities:**A learner recognized with a limitation that is able of inhibiting his or her learning or education capacities.**Vocabulary understanding:**familiarity with the meaning of a specific word as well as its precise use or application.

### Limitations

The investigator forecasts possible limitations due to both external as well as internal validity for the proposed study. In terms of external validity factors of limitation, lack of adequate resources will constrict the expansiveness or extent of the intended research. This is because of the possible need for finance to execute some initiatives to complete the research within an extensive sample cluster (Stone, 2007). Teachers might not get recompense for taking part in the investigation. This might consequently result in considerably low rates of attrition. The proposed research or study is also likely to lack the element of inclusiveness of the entire population. Therefore, the sample size might not be purely representative of the entire population. Sampling students from the entire general population would probably provide considerable different and generalized results and study implications. The fact that all the instructional materials will be improvised principally by the investigator might cause a potential internal validity threat. The inclusion, as well as exclusion criteria for various sampling sites as well as study population, might also pose a potential limitation for the study. This is because mathematics is taught and examined within all cadres or grade levels within both elementary as well as high school institutions.

## Literature Review

Most studies have indicated that children seize the concept of quantity and other relational ideas when still young. Their progression for mathematical comprehension, language and other associated vocabulary remain critical cognitive association between the learners’ native sense of numbers and conceptual education. Yet investigations indicate that the students learn mathematics appropriately through its application as well comprehension of the language that mathematics use (Bandura, 1977). The mathematical language offers the students the appropriate skills that are applicable to imagine, talk about, and consequently incorporate new mathematical constructs as they get introduced. For instance, it is observable that students develop their restrictive knowledge, getting aware of how to label or define the objects (Zaslavsky & Shir, 2005). These objects might include the notable disparities between triangles, rectangles, or even polygons. Such capacities remain eminently applicable and critical for manipulating these objects by the students. Therefore, the language of mathematics, within such contexts, remains vital in assisting students to advance and sharpen their means or methods of acquiring novel concepts (Perie, Grigg & Dion, 2005).

Imperatively, while students get to familiarize themselves with the concepts of doing mathematics, they must also learn articulate concepts. In such scenarios, they must have the competency to identify and answer the questions within mathematical vocabulary to advance their critical problem resolution skills and capacities. In 1932, Pressey and Elam’s studies of this subject indicated that knowledge of particular subject content and vocabulary mastery is pertinent with one another (Pugaalee, 2001). During the conduct of their study, the two principals developed a list that contained around 117 critical words for mathematics as a basic subject. Consequently, they tested high school students based on their knowledge as well as familiarity with these vocabularies, not just regarding or giving important considerations to the definitions alone.

Critically, eminent challenges concerning vocabularies applied in math are evident globally (Thompson, 2004). This mathematical language has become harder to read, relative to any other subjects within our education systems. The associated materials are also difficult to comprehend “with additional concepts for each word, sentence, as well as paragraph relative to any other area” (Anderson & Little, 2004). Particularly, it is vital to stress vocabulary instruction about this specific content area. Monroe’s investigation or research examined the efficacy of graphic organizers during the instruction of the mathematics vocabulary. The outcomes he discovered were basically that the graphic organizers have the potential to assist in the development of comprehension. It is critical to observe that research and investigations on mathematical vocabulary have been done within all levels as well as social contexts.

In another instance, Stahl’s and Fairbanks’ (1986) investigation examined the issues with the effects and implications of various strategies for vocabulary instruction on the students’ capacities in the learning of word implications and their consequent comprehension (Scruggs & Mastropieri, 2000). In this study, the two investigators categorically considered the curriculum as well as the instructors or teachers. The study found two basic elements that had an impact on the concern or issue of vocabulary learning as well as comprehension. These major indicators included whether instruction remains potentially personal or conducted at least partially within-group environment. The second basic element discovered from the study was the quantity of time that was billed to instruction. Other principal investigators have like Schwarz’s mathematics curriculum requires to adequately and strictly focus intensively on the content vocabulary as well as its significance to mathematical communication through either written or oral means.

In her study, Schwarz considered the fifth-grade class in a K-12 school district (Rubestein & Thompson, 2002). This targeted student population had potentially depicted a low comprehension of the mathematical vocabulary. In this investigation, certain strategies that enabled the researcher’s target population to augment their mathematical vocabulary familiarity included vocabulary journals as well as mathematical journals. Additionally, word walls also played a critical stake in the increase of the learner’s or students’ competencies in mathematical vocabulary (Scruggs & Mastropieri, 2000). This implies the diverse strategies that can be applied within different populations in enhancing the performance of students in mathematics. According to investigators like Bromley, proper instruction of vocabulary remains an important aspect. Additionally, the scholar reiterates that vocabulary emanates as a solitary contributor to complete comprehension, fluency, as well as accomplishment. Following the analysis of various study articles as well as projects, the researcher discovered that vocabulary has potentially remained as an area with significant attention from different individuals for a long period (Perie, Grigg & Dion, 2005).

In 1932, Pressey and Elam developed an array of “important” words in the subject of mathematics. These words have significantly influenced the study and learning of this subject over a long period. Varied scholars and researchers within the mathematical education field have noted that although there has been the particular discovery of elements that can significantly improve reading, certain strategies can still be applied within other contents (NMPR, 2007). From these studies, critical instances and lessons can be concluded. For instance, it is notable that knowing the specific meaning of a particular vocabulary, not specifically its definition, can improve the general knowledge of mathematical concepts. Apart from just applying the crucial strategies that were discovered and proposed by these previously extensive investigations, additional research must be conducted (Marzano, Pickering & Pollock, 2001). This will enhance the process of addressing other critical emerging issues presently notable within the learning of mathematics.

‘Physicalists’ may adopt existent mathematical instrumentalism. Formalists, as would be invariably applied in this investigation, may assume the math expressions within theories or models to be potentially meaningless but still important syntactic tools. The proposed study is to adopt the application of instrumentalist theory as a mode of explanation of the basis for discrepancies in different mathematical performances amongst the students in general (Rubestein & Thompson, 2002). Relative to the definition-only approach that results in basic comprehension only, certain direct teaching strategies might be effective in assisting the learners to assign distinct and appropriate meanings to certain words. The graphic organizer that remains closely aligned with the present theory on the process, through which the human brain organizes information, can be a promising methodology (Fogelberg, 2008).

Visually, a graphic organizer indicates crucial concepts and other associations. Analytically, this graphic organizer is also perceived to elucidate and present the background knowledge concerning critical concepts to the student mind. This is believed to transpire in a more instantaneous and comprehensively more than given abstract prose (Pugaalee, 2001). There have also been important indications that the graphic organizers provide a system and structure for the fact or data to be learned by the student. Most investigators have, therefore, concluded the graphic organizer as an appropriate tool or theoretical approach towards teaching the technical mathematical vocabulary. Therefore, based on this assumption, the research will largely borrow from such studies and base its principles on the graphic organizer (Burton & Morgan, 2000). Additionally, other previous investigations have indicated that the students’ constructed graphic organizers enable and promote the free recall propensity of the primary students. This research will, therefore, analyze and utilize the basic theories underlying the graphic organizer approach.

## Methodology

### Sampling Procedure

The proposed research will purposively choose all the freshman students within the chosen school’s science department. After this purposive selection, there will be an application of random sampling procedures with equal chances and opportunities given to all the possible partakers in the proposed research. Application of various statistical formulas and inferences including the use of Fischer’s formula in determining the sample size requirement will be applicable. This is because the proposed study population is set to be less than ten thousand in total numbers. The exclusion and inclusion criteria will therefore depend on whether the student is a freshman within the school’s science department or not. The freshman within the science department will be chosen due to their intensive involvement with mathematical problems and solutions within a relatively new environment. Due to this factor, they are reportedly bound to experience notable challenges within the system and particularly, on the topic of effective comprehension of mathematical vocabulary and jargon.

### Materials, Equipment, Instruments used in data collection

The researcher will divide the sampled freshman students within the science department into basic groups that they will not be initially familiar with. In this approach, the investigator is supposed to apply diverse methodologies for instructing the mathematical vocabulary. Consequently, the concept papers shall be introduced in the late times to avoid confusion and fallouts from the already set groups or study sub-sets. Each student shall receive a two-part questionnaire (Mills, 2007). Notably, the first part of the questionnaire shall contain the mathematical calculations for the students to attempt and complete. Consequently, the second portion or part two shall have the section for the explanations of the procedures applied for these students to fill up.

It can be noted that structured mathematical questionnaires will be applicable as the main tools or instruments in the proposed investigation. The first portion will be applicable in determining the level of comprehension of these students in the arithmetic areas as well as the general individual comprehension of mathematical concepts (Leikin, 2010). On the other hand, the second portion of the instrument used shall be applied in the determination of the extent of comprehension of the students’ mathematical vocabulary. In this last section, each participant student shall be required to explain the process that they applied in arriving at their answers that are presented within the first portion. Notably, all these two sections shall bear a score.

### Research Design

Both the qualitative and quantitative approaches of research will be applicable in the proposed study. This mixed methodology will be critical in enhancing the triangulation objective and determining the best applicable strategies for teaching mathematical vocabulary based on the individual participating students’ qualitative assessments and interviews (Mills, 2007). The quantitative approaches shall considerably involve the application of such mechanisms like the structured questionnaires involving the basic mathematical calculations and general questions. The combined approach of both the qualitative and quantitative methodologies is generally widely applicable for most studies across different investigative fields. This is mainly because the approach has the competency to offer the unique and powerful depths of the distinct outcomes and results. The need to offer specific attention to the outlined aims as well as objectives and critical research questions largely emanates from the investigation. The alternative for the mixed method of research will therefore be based on these important study goals and objectives. Structured equation modeling will be applicable for the quantitative approaches within the proposed investigation.

The use of structured and unstructured surveys will also be crucial. The use of focus group discussion methods will help in the investigation and discovery of the students’ discourse capacities in handling sophisticated mathematical vocabulary (Leikin, 2010). This will be done within groups to enhance positive contributions and be able to identify the various existent disparities amongst these students. Sampling procedures are important for undertaking scientific investigations. The proposed investigation recognizes the significance of the application of the purposive sampling procedure to enhance the process of obtaining the necessary data. The proposed study is to be a longitudinal case examination based on the purposefully sampled student population. Additionally, the cross-sectional analysis will be used in strengthening the processes.

### Measurement Approach

There will be various measurement approaches applicable in the proposed investigation. For instance, vocabulary assessment responses will be used. Additionally, there will be an application of Curriculum-Based Measurement questions or Focal interviews specifically meant for the students. The application of pre-, as well as post- investigative student surveys, will also be eminent in the study (Marzano, Pickering & Pollock, 2001). The sampled student population will be divided into different groups. This is because there will be treatment groups as well as contrast groups. There will be consideration of basic factors during these classifications including the individual student performances in categorizing these groups and students. The treatment groups will then be observed versus the contrast group. A mixed approach of multi-strand investigative design will be executed to determine the effectiveness of various proposed instructional strategies.

### Procedures

There will be consideration of internal validity factors in all procedures. Closer observation and monitoring of the contrast and treatment groups will be critical in the identification of the notable variances. The analysis of both quantitative and qualitative data will follow the two matched and compared pairs within the research. The administration of the questionnaires and vocabulary assessment tools will follow an informed consent procedure to ensure all the ethical and legal measures are observed.

## Results

### Plan for Analyzing the Data

This research will employ ANOVA analytical test methods to scrutinize the results. This will employ 4 groups. Nonetheless, data analysis will occur at both quantitative as well as qualitative levels. In explanation, quantitative analysis will consider numerical aspects of research. In this context, it will handle issues related to volume, numbers, capacity, and other related quantitative provisions relevant to this research. The number/quantity of mathematical vocabularies will be discerned in this context. Conversely, qualitative analysis will consider particular/distinct research provisions demanded in this study. Statistical analysis techniques including the application of SPSS will be used for quantitative data management. On the other hand, the qualitative data will be appropriately discussed to assist in the reinforcement of the already presented quantitative information.

### Presentation Format

In explanation, the use of graphical presentation, pie charts, bar graphs, and distribution tables will be applied due to their precision, comprehensibility, and appropriateness. The presentation will follow a report format, with the inclusion of necessary figurative and tabulated dispersions and observations. On numbers, 5 tables will be drawn from the findings indicating the different abilities of the students coached with varied strategies in mathematical language. 3 pie charts will be used to represent the disparities in the level of performance amongst the students coached in different languages. The 5 bar graphs will represent the most preferred vocabulary for teaching mathematics by the students.

## References

Anderson, M. & Little, D. (2004). On the right path: Improving communication in an elementary mathematics classroom. *Teaching Children Mathematics*, 10 (9), 468-472.

Bandura, A. (1977). *Social Learning Theory*. New York, NY: General Learning Press.

Burton, L. & Morgan, C. (2000). Mathematicians writing. *Journal for Research in Mathematics Education*, 31(4), 429-453.

Chard, D. (2003). *Vocabulary strategies for the mathematics classroom*. Web.

Fogelberg, E. (2008). *Integrating literacy and math: Strategies for K-6 teachers*. New York: Guilford Press.

Lake, J. (2009). *Math memories you can count on: A literature-based approach to teaching mathematics in the primary classrooms*. Markham, Ontario: Pembroke Publishers.

Leikin, R. (2010). *Learning through teaching mathematics: Development of teachers’ knowledge and expertise in practice*. Dordrecht: Springer.

Marzano, R., Pickering, M., & Pollock, J. (2001). *Classroom instruction that works: Research-based strategies for increasing student achievement*. Alexandria: Association for Supervision and Curriculum Development.

Mills, Geoffrey E. (2007). *Action Research: A Guide for the Teacher Researcher*; London: Merrill-Prentice-Hall.

NMPR. (2007). *National Math Panel Report*. Web.

Perie, M., Grigg, W., & Dion, G. (2005). *The Nation’s Report Card: Mathematics 2005 (NCES 2006-453). U.S. Department of Education, National Center for Education Statistics*, Washington, DC: US Government Printing Office.

Pugaalee, D. (2001). Using communication to develop student’s mathematical literacy. *Mathematics Teaching in the Middle School,* 6 (5), 296-299.

Rubestein, R. & Thompson, D. (2002). Understanding and supporting children’s mathematical vocabulary development. *Teaching Children Mathematics*, 9 (2), 107-112.

Scruggs, T. & Mastropieri, M. (2000). The effectiveness of mnemonic instruction for students with learning and behavior problems: An update and research synthesis. *Journal of Behavioral Education*, 1(10), 163-17.

Scruggs, T. & Mastropieri, M. (2000). *The Inclusive Classroom: Strategies for Effective Instruction*. Upper Saddle River, NJ: Prentice-Hall, Inc.

Stone, R. (2007). *Best practices for teaching mathematics: What award-winning classroom teachers do*. Thousand Oaks, CA: Corwin Press.

Thompson, I. (2004). *Enhancing primary mathematics teaching*. Maidenhead: Open Univ. Press.

Zaslavsky, O. & Shir, K. (2005). Students’ conception of a mathematical definition. *Journal for Research in Mathematics Education*, 36 (4), 317-346.