Description and Background to the Problem
This chapter will provide an overview of previous observations and interactions that I had encountered when teaching a group of grade two students ranging from ages 7 to 8 years old at an urban school. This chapter will introduce the problem that prevailed consistently for the majority of the class when students were given an assessment task. This chapter will also state and explain a strategy that was used to eliminate students’ failure to exhibit the mathematical skill of stable order counting in relation to one to one correspondence.
This problem was manifested while I was on my Fourth Year Teaching Practicum. An assessment task was given in my first week of observation which geared towards assessing, evaluating and retrieving evidence of the issue above. The results from the test allowed me (the researcher) to identify the misconceptions, challenges and problems the students faced in relation to mastering the skill of stable order counting. As students were engaged into the lesson by copying from the board, some students were not able to count within an orderly manner, the amount of objects that were placed in a given set. In addition, when I walked around the classroom to see if students understood the instructions given on the test, it was evident that some students were just writing and guessing the amount of objects that were in each set. It was also evident that students were not able to read or understand the written instructions that was provided for each section of the test from Section A, B and C. One question was done for students and they answered in a chorus to provide the answer. However, what I then noticed as I made some checks by walking around and observing what was being done by students, there were six persons out of the twenty-five (25) students present who had some serious problems with counting and one to one correspondence. Due this, I have realized that as an aspiring teacher, I need to do further studies into this problem to see what strategies I can implement to improve these Grade Two students to reach mastery level of understanding stable order counting. Therefore, my research problem is to find out why Grade Two students have not mastered the basic mathematical skill of stable order counting in relation to one to one correspondence. According to Kearns (2010), one to one correspondence is the ability to match an object to the corresponding number and recognize that numbers are symbols to represent a quantity. Young children often learn to count without having an understanding of one to one correspondence. This means that while they can count they may not understand the meaning or value of the numbers. Numbers and symbols are abstract for children so one to one correspondence is the connection between the symbol, the language and the quantity (Kearns, 2010). Therefore, I strongly believe at the Grade Two level, this skill should have been mastered by students but now I see where I had to intervene to facilitate these students for them to be on par with the other class population.
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Purpose of the study
The purpose of this study was to find the underlying factors that causes Grade Two students to not master the mathematical skill of stable order counting. The purpose of this study was to also identify and examine how the use of manipulatives can help to improve students’ understanding of stable order counting skills.
Research questions
This research was guided using questions that provide answers to the problem of my topic. Two (2) key questions were used as the main focus points which justified the causes of the above problem. These questions are as follows:
- What are the factors that cause Grade Two students to have not mastered the mathematical skill of stable order counting?
- How can the use of manipulatives improve stable order counting for a set of grade two students?
Significance of the study
The greater demand for graduates with good scores in mathematics from the primary level onwards justifies the need for more effective life-changing teaching approaches. The importance of hands-on activities cannot be overemphasized at this stage. These activities provide students an avenue to make abstract ideas concrete, allowing them to get their hands on mathematical ideas and concepts as useful tools for solving problem. As students use the materials, they acquire experiences that help lay the foundation for more advanced mathematical thinking (Burns et al, 2000, p. 55). Therefore, the findings of this study will showcase and outline the factors that causes students to not to master the skill of stable order counting which poses a difficulty when identifying sets. The integration of the use of manipulatives in lessons providing hands – on learning and experiences is yet beneficial to both teachers and students.
The findings of this research has shown teachers including myself that the use of manipulatives provide hands-on learning allowing students to retain content and understand concepts better. In addition, the findings of the research has educated teachers on the effectiveness of game-based assessment to test student’s knowledge rather than the use of giving worksheets to improve mathematical challenges. This findings of this research has also allowed classroom teachers an improved way to track student’s mathematical progress bearing in mind that there are different methods or ways to enhance teaching/learning activities for numeracy.
Secondly, early elementary students have benefited from the use of manipulatives, as it has helped to develop their number senses. Therefore, the integration of manipulatives into one’s lesson allowed students to create their own understanding and creativity when interacting with manipulatives in mathematics while learning. From this, students understood the conceptual knowledge of stable order counting with the use of different manipulatives; as they counted and identified the numerals, they were able to group sets of counters. The use of manipulatives facilitated a chance for students to visualize mathematical thinking and concepts (Todd, 2018). As a result, students have felt more confident and comfortable about doing Mathematics.
Operational Definition of key terms
Throughout this chapter, there are some words that are be consistently repeated. Therefore, using credible sources, these words were defined:
Stable Order Counting; Stable Order Counting is the counting and quantity principle stating that the list of words used to count must be in a repeatable order For example, it is always 1, 2, 3, 4, 5, 6, etc. Not 1, 2, 5, 3, 6, 8 (Pearce, 2017).
One to One Correspondence; One to One Correspondence is the ability to match an object to the corresponding number and recognize that numbers are symbols to represent a quantity (Kearns, 2010).
Mathematical concept; A mathematical concept is the 'why' or 'big idea' of math. Knowing a math concept means you know the workings behind the answer. You know why you got the answer you got and you don't have to memorize answers or formulas to figure them out (Study.com, n.d.)
Manipulatives; Manipulatives are physical objects that students and teachers can use to illustrate and discover mathematical concepts, whether made specifically for mathematics (e.g., connecting cubes) or for other purposes (e.g., buttons)” (John van de Walle et al 2013)
Literature Review
Introductory Paragraph
This chapter will provide an overview of what literature sources suggest about using counters to teach the mathematical skill: stable order counting, in relation to one to one correspondence. In this chapter, the researcher will then compare literature text to explain the factors of the research problem, suggestions from a theoretical perspective, impacts of why students’ misunderstanding the mathematical concept of and the benefits of the use of counters when teaching the skill: stable order counting.
Theoretical Perspectives
Jean Piaget is a theorist who is predominantly heard across the Early Childhood spectrum. Jean Piaget (1896-1980) is the first psychologist to make a systematic study of cognitive development by engaging in ingenious tests to reveal student’s cognitive development (McLeod, 2018). Piaget’s theory emphasizes the importance of play and the use of manipulatives to teach a concept. Researchers over the past forty years have generally found that manipulatives are a powerful addition to mathematics instruction. Meta-analyses by Parham (1983) found that achievement in mathematics could be increased by the long-term use of manipulatives. Therefore, the use of integrating manipulatives to facilitate learning is the best way to improve student’s understanding of stable order counting with relation to one-to-one correspondence (McLeod, 2018).
The well-known theorist who has been a great impact and contribution, in relation to how children develop cognitively stressed the need for integration by using manipulatives. Piaget strongly believes the inclusion of manipulatives a part of our lessons creates hands-on experiences for children and a channel for exploration of learning. The theory of Cognitive Development proposed by Jean Piaget consists of four stages of how a child develops intellectually. However, the stage of focus for this research is the pre-operational stage: students’ ages range from 2 years to 7 years old. At this stage, children learn through symbolic representation and discovery through play (McLeod, 2018). Therefore, the rote memorization of different concepts such as numbers in the mathematical field will not be an effective strategy to improve stable order counting for a group of grade 2 students. The integration of manipulatives plays a crucial role for students to critically understand the conceptual knowledge of stable order counting when learning.
Piaget cited on Linguasphereus.com (2012) made emphasis that children are active leaners. Hence, to lessen the old traditional: drill and practice method in school and to promote play in one’s lesson in particular Mathematics, the integration of using manipulative is indeed important.
Factors that may influence Grade Two students’ inability to not master Stable Order Counting.
Factors that may contribute to students’ inability to not master the principle of stable order counting may influence their attitude towards understanding other future Mathematics topics. According to Huttenlocher et al (1994), one such factor is that children may not been introduced to the early mathematical skill in their early years of development. As a result of this, students are unable to match the number of items to their correct numeral. There are instances where students are unable to perceive and understand quantities through counting. While stable order counting and one to one correspondence are indeed inter-related; there are still some students who are unable to make the connection between numeral and their quantity. Some may even skip an item or do not include it in the counting sequence and also assist more than one number word to a single item.
Another factor that may cause Grade 2 students to not master the mathematical skill of stable order counting is the lack of parental involvement. The relationship of a successful education for students depend greatly on the support given and shown from their parents or guardians. Due to the lack of parental involvement in some students’ educational life, it negatively contributes to student’s inadequacy of grasping the basic skills such as stable order counting. This is can be so because parent are illiterate (poor achievement) and from a low socio-economic status to provide the necessary instruments to facilitate learning (Kohl et al, 2000). Topor et al (2010) also argued that parental involvement is significantly related to academic performance and children’s perception of cognitive competence. Whereby, the parents just drill students with the order of how to count but has little to no time to allow and teach their children to understand the concept behind mathematical structures.
The development of Stable Order Counting may have on Grade Two students
Uttal et al (1997) noted that the literature is somewhat ambivalent about the use of mathematics manipulatives. They explained that, research on the effectiveness of manipulatives has failed to demonstrate a clear consistent advantage for manipulatives over more traditional methods of instruction’ (p. 38). According to the advent of Interactive Whiteboards, Sowell (1989) performed a meta-analysis of much of the literature on the use of mathematics manipulatives to that time. The results indicated that short term use of mathematics manipulatives was not effective and that long-term use was more effective. A critical component is the teacher. Teachers who lack conviction, in relation to the efficacy of the use of mathematics manipulatives will be less likely to persevere with their use, and implement systems for their distribution and collection.
According to Punkoney (2017), 10% of children enter kindergarten level who has proficient counting skills. The reason explained for this was that parents often think that it is the teachers’ responsibility to teach the basic counting skills; while teachers pass that responsibility to the parents. Therefore, children may have little chance of being successful in mathematics without the knowledge of basic counting skills. Stable Order Counting refers to rote counting in the correct order using number names (Punkoney, 2015). It is the basic 1, 2, 3, 4 and so on. Therefore, when a child is developing stable order counting skills, he or she may skip numbers. Therefore, to move children from rote counting, teachers should use various manipulatives to facilitate one to one correspondence.
Possible strategies to solve the problem
In carrying out this research using manipulatives to improve students’ stable order counting skill. There are other strategies that can also be used to develop students’ stable order counting skill. Other strategies include to teach and sing songs, use of pattern blocks and by storytelling. Adding songs to teach concepts, in particular mathematical concepts in this case can also help students to retain content that was learnt. A song can be used so that children can understand the conceptual knowledge of each or a number. Children at the preschool level are more interested to learn when familiar or catchy songs are used. The culture of our Jamaican society plays a major role in music so linking musical rhythm or songs can assist in the reinforcement and retention of the content taught.
Secondly, the use of pattern blocks will also help children to establish one to one correspondence and reduce counting errors like counting an object more than once or missing an object while counting. The use of pattern blocks can help children to keep track of the pattern involved by moving objects from one pile to another to limit confusion.
Thirdly, another strategy that can help students to understand Stable Order Counting is by using a number-sense story to facilitate conceptual understanding of a number. Often times, young children struggle with abstract concepts in mathematics. Using storytelling as a catalyst to mathematical instruction and learning is one that is enjoyable and versatile (Hyde, 1998). Storytelling appeals to children’s imaginations and emotions which help to make learning more meaningful and visual. The use of storytelling is another pedagogical tool to help children connect with mathematical concepts which they need to learn and understand. For instance, by using imagination to realize one apple plus another apple equal two apples.
However, among all other possible strategies that can be used to teach children how to count; my strategy of using manipulatives such as counters to aid in improving students’ understanding of stable order counting can help students to create a more realistic and visual image as they interact when learning Mathematics. Perhaps, this strategy of using manipulatives to improve stable order counting for a group of Grade Two students will significantly allow students to better understand the conceptual knowledge of why 3 is more than 2 rather to resort to learning by rote memorizations.
Benefits of using counters to teach mathematical concepts
My strategy of using manipulatives to improve student’s stable order counting skill is a feasible, accessible and affordable tool that can be used to enhance and promote learning through a wide range of game based activities which can enhance students’ conceptual knowledge to identify sets. This strategy also allow students to identify quantities and match it with the appropriate numeral or vice-versa. The concept of stable order counting with the use of manipulatives has allowed children to see sequences in pattern arrangement whenever counting. Using manipulatives is an effective strategy to improve children’s stable order counting skills because it facilitates children exploration, as they discover other ways for them to better retain information.
Secondly, the introduction of the use of manipulatives will allow children to count items within a set which will aid in life-long skills by moving from rote memorization to children counting independently. According to Potter & Levy (1968), when practicing one-to-one correspondence with the use of manipulatives, it is easier for students to keep track of items in a row or items that have been tagged and partitioned. To count using the stable order counting skill, students must first know number names and understand the relationship between each count. The concept of “one-to-one correspondence” is the understanding that each object being counted represents “one more” (Edu.gov.mb.ca, 2020). Before a child understands one-to-one correspondence, he will learn to count by rote memorization. When asked to count a small group of objects, students are likely to count quickly through the numbers they have memorized and randomly touch the objects being counted. For example, a child who has five beads may automatically count aloud from 1 to 5 when asked to count the beads, pointing to random beads as he proudly shows how well he can “count.”
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