## Introduction

About 5 million school-going children in the United States do not have English as their first language. Thus, when teaching numeracy and mathematics in schools, particularly to young learners, effective skills and strategies must be put in place to ensure they understand mathematical concepts. In recent years, modules and techniques have been developed to aid in the effective teaching of mathematics, particularly to young children, not well conversant with the English language. The National Council of Teachers of Mathematics has developed techniques that have been incorporated into the curriculum to ensure effective teaching of mathematics. This paper focuses on effective ways of understanding mathematics for middle school-going children among them English Language Learners, where standards and techniques from the Mathematics Florida Standards and the NCTM have been analyzed for effective passing of mathematics concepts.

## Count with Me

The first step to understanding mathematics and a mathematical formula is from counting numbers. Rochel Gelman, an early childhood theorist, explains that counting exposes learners to fully apprehending mathematical concepts. Gelman developed principles of counting that assist educators and teachers today to introduce middle school and pre-learners to counting as the basics of understanding mathematics. There are five principles of counting that are vital in the introduction of mathematics to middle school learners and preschoolers.

The first principle is the one-to-one principle, which states that when counting, numerical values are assigned to the specific materials used. Teachers can apply this principle when introducing learners (ELLs) to mathematics, by relating verbal numbers to actual counting materials like sticks. For instance, when children are taught to use sticks then one stick represents a specific number i.e., one. The principles of Gelman are sequential, as understanding the first allows you to understand and appreciate the next. The second one is the stable order principle, which states that counting needs to follow an arrangement or order, (Ahn et al., 2011). The effective implementation of the second principle calls for use of linguistic skills to ensure numbers follow a rather ‘predictive’ order, which makes it easy for middle school learners to comprehend.

The third principle of cardinality aims at ensuring learners gain a mental understanding of the mathematical concepts; counting, before relating them to real experiments. The order irrelevance nonetheless states that it is getting the correct count of numerical value that counts and not necessarily the manner or order in which the value is obtained. The last principle of Piaget states that the principles mentioned and discussed above should apply to all mathematical concept understanding, counting for this case, (Ahn et al., 2011). For instance, the same process used to count the number of logs in a truck should be the same process to count the number of say books on the shelves.

## Learning Activities for Understanding Mathematics Concepts

The NCTM in conjunction with learning bodies in the United States developed learning activities that educators and teachers can employ in the classroom to assist middle school learners and preschool learners understand mathematics. The activities involve the effective application of theoretical principles and physical and psychological activities that the educators use for effective teaching, (Kaplan, 2019). Discussed below are some of the activities that may be implemented to assist the understanding of numeracy among ELLs.

The educators are to create an inclusive and accommodative learning environment, for all learners including the ELLs. The essence of the accommodative environment is to create a ‘common’ understanding of concepts that are appreciated by all the students. For instance, in the case of understanding a concept, the educator may use a value, limited to their learners (including ELLs) for them to understand the concept in question. For example, teachers may opt to say 1M, about one million or 1,000,000 for the general understanding of the concept. Similar to the initial principle of stable order discussed earlier, teachers and educators of mathematics can use procedural, systematic, and instructional teaching mechanisms to teach mathematical concepts. Mathematical procedures (often known as formulae) are globally accepted and applied to solve the mathematical problem, (McLeod et al., 2019). The formula ensures the same steps are followed to solve problems; the knowledge of the steps is beneficial to the ELLs.

More importantly, group discussions and informal classroom interaction between students and students, or teachers and students are vital in confidence-building. Finally, constant assessments (formal and informal) may be conducted by the teachers of mathematics, discriminately against ELLs, to track their understanding of the mathematical education that they receive. Assessment is often vital in determining strengths and weaknesses, educators may also learn areas to emphasize while delivering their services (Ahn et al., 2011). Assessments can be conducted in form of continuous assessment tests (CATs) or random assessment tests (RATs), open-ended questions and quizzes may be employed as assessment modules as well.

## Principles According to the MAFs and NCTM

The Mathematics Florida Standards (MAFS) and the NCTM have had a set of guidelines and principles that educators across the United States apply in their duty while teaching mathematics to learners, and ELLs as well. the MFS standards are categorized into arguable all mathematical branches; referred to as ‘Domains’ in the standards. The MAFS categorizes the standards into each learning category to ensure teachers give appropriate education, considering age, and understandability at each level, (Farfan & Murata, 2019). The standards are divided into categories with the prefix MAFS, attached to each level. The middle MAFs include:

- Counting and cardinality
- Operational and algebraic
- Numbers and operation
- Measurement and data
- Geometry

The NCTM, on the other hand, focuses on ways of creating a productive learning environment in the classroom that will foster a deeper and easier understanding of the mathematical concept. They are a psychological process that teachers apply when teaching their students. They consist of critical lesson planning activities that ensure eagerness among learners to want to know about mathematics (Naja, 2018). The NCTM psychological guidelines for teaching mathematics aim at creating a positive environment for learning and understanding.

Creating anxiety and desire to know; when this concept is applied, teachers get responses from their learners before solving the mathematical problem for them. A free interactive environment achieved by this technique helps learners develop a positive attitude towards learning. Developing a teaching plan is considered by the NCTM for productive mathematical learning. Categorized into three, ‘‘Three inquiry-based tasks highlight the planning, classroom discourse, positive results, and growth in one class’s journey’’ (McLeod et al., 2019). Finally, the NCTM suggested an assessment module known as ‘prove it to me. Teachers are advised to involve the learners in rather technical tasks whose main aims are to drive given concepts to learners when going about the problems.

To conclude, when on duty as a mathematic educator or teacher to middle school learners, I can employ the concept discussed in this paper to fulfill my practice for ELLs and English-speaking students. I would implement the procedural or instructional learning mechanism to give step-by-step guidance when teaching mathematical concepts. Besides, I would appreciate and implement the MAFs rubric of providing mathematical concepts to learners. I believe this rubric is vital in ensuring the right concept and mathematical topics are introduced to the learners at the appropriate time. For ELLs, my recommended module would be the interactive study sessions, which have proven to create a free learning environment that allows learners to share opinions and responses. The interactive lessons are in line with the MAFs and the NCTM standards and modules of teaching and understanding mathematics.

## References

Ahn, R., Yeong I, J., & T Wilson, R. (2011). *Teaching mathematics to English languagelearners using Moses’ Five-Step Approach*. Web.

Farfan, G., Murata, A., & Roehrig, A. (2019). Teachers’ learning of teaching with multiple strategies: Understanding challenges to the mathematics Florida standards during a lesson study cycle.

Kaplan, E. (2019). *6 Essential Strategies for Teaching English Language Learners*. Edutopia. Web.

McLeod, S., Harrison, L. J., & Wang, C. (2019). A longitudinal population study of literacy and numeracy outcomes for children identified with speech, language, and communication needs in early childhood. *Early Childhood Research Quarterly*, *47*, 507-517.

Naja, A. R. (2018). Analysis of students’ creative thinking level in problem solving based on national council of teachers of mathematics. In *Journal of Physics: Conference Series* (Vol. 1008, No. 1, p. 012065). IOP Publishing.