How Children Complete Mathematics Tasks - Case Study Example

Assessing Place Value Introduction One of the key objectives of the Australian education system is to ensure that all students attain a solid foundation in literacy and numeracy. Numeracy can be described as “the ability to comprehend, critically respond to and employ mathematics in different work and social-cultural contexts” (The South Australian Curriculum, Standards and Accountability Framework, 2001, p. 225). The ability to comprehend and employ the number system is important in the development of numeracy and thus essential to mathematics in primary school (Shuard, 1982; Pengelly, 1991). Students’ understanding of the number system and how it works helps them to use the connections and relationships between numbers to interact with data, quantity and measurements. Nevertheless, one of the main difficulties in teaching numeracy is assessing what students understand or know about a particular aspects of mathematics. Assessing the understanding of students in a particular aspect of mathematics is important since it helps teachers to understand the strengths and weaknesses of students in a particular area and find ways of helping students to improve their weak point. In addition to this, assessing the understanding of students in a particular aspect of mathematics helps teachers to find ideas on where to start teaching new materials. This paper seeks to examine how children complete mathematics tasks and diagnose what they know about place value. This study will focus on two children in Year Four. Foremost, this paper will provide a brief background of the two children who will participate in this study. Secondly, it will define place value and discuss the position of place value in mathematics education. Subsequently, this paper will highlight the two tasks that the children will be asked to complete in order to assess their knowledge on place value. It will subsequently provide a discussion on how I completed the tasks. Moreover, this paper will provide a diagnostic assessment for each child. Furthermore, it will discuss what needs to be considered when teaching place value and some of the effective teaching strategies that can be used to help children understand place value. In addition, this paper will provide a lesson plan that will address an aspect of place value identified in one of the children’s diagnostic assessment. Background This study will involve two students in Year Four namely; Jane Doe 9years and John Doe 10 years. In order to establish their backgrounds an interview was conducted with their parents about their understanding in mathematics. Through the interviews conducted I was able to get a general idea about their backgrounds. Jane Doe is a 9 year old girl in Year Four. Prior to joining school at the age of 4, she had already mastered the number system and knew how to count from 1 to 50. Her quick progress in learning the number system can be attributed to her interactions with her elder siblings. Furthermore, much of her knowledge on the number system was mainly gained through play. Before joining school Jane actively engaged in various play activities that involved counting, thus much of her knowledge on the number system gained through play. When she joined school at the age of 4, her learning progress in ordering, writing, reading and interpreting numbers has been significantly faster than that of other children in her class. Over the years, Jane’s performance in different aspects of mathematics has been remarkable. Her general understanding of the number system at a young age has enabled her to easily understand basic mathematics concepts such as addition, multiplication, division, fraction and place value. As a result, her performance in mathematics has gradually shown consistent improvements as she progresses in primary schooling. In Year Four, Jane is one of the top scoring students in mathematics. John Doe is a 10 year old boy in Year Four. He did not attend pre-school rather he was home-schooled due to medical reasons. While being home-schooled John’s learning progress was slow but he showed considerable understanding on the number system. Being an only child, John did not engage in much play and as a result he became somewhat introverted. At the age of seven, he joined primary school. His learning progress in ordering, writing, reading and interpreting numbers has been significantly slower than that of other children in his class. However, with time John has managed to show improvements in different aspects of mathematics. His general performance in mathematics has not been consistent but he has shown improvements in his understanding on additions, multiplication, division, fractions and place value. Place Value This study will focus on assessing how John and Jane complete mathematics tasks in place value and diagnose what they know about place value. In mathematics, place value can be regarded as the value of a digit that is determined by the position of the digit in a number. Place value can also be considered as the name of the location or place of a digit in a number. For example in 230, “2” refers to the hundreds place, “3” refers to the tens place whereas “0” refers to the ones place. The ability of children to understand place value is an essential mathematics skill since it builds on the broader aspect of the number system. Some of these ideas can be difficult and complex for some children to learn (Ross 1989; Mersky, Harris & Turkington, 2000). Place value is one of the basic concepts imbedded in the middle and elementary school curriculum. Solving problems that involve rational and whole numbers is mainly dependent on expressing and understanding digit values and quantities. It is therefore important that students should develop a strong understanding on the place value concepts. Students may require many instructional experiences in order to develop their understanding on place value. Liping (2010) observes that it is difficult for students to attain a thorough understanding of place value in day but step by step they can be able to know about place value and complete tasks on place value. Learning place value is important especially for children beginning their primary school mainly because, it provides a strong basis in mathematics that children can use so as to realise a greater understanding for the application of numbers in their day to day activities and experiences. Understanding place value helps children to build skills in mathematics that are necessary to succeed not only in mathematics but also other subject areas (Mersky, Harris & Turkington, 2000). Rumack (2011), notes that, many students often find themselves struggling in mathematics as they progress in their studies. The struggles they experience in mathematics can be attributed to assumptions that in order to successfully master mathematics one has to memorise the formula. Nevertheless, Rumack recommends that in order for one to succeed in mathematics, it is important to understand basic concepts taught in elementary school such as place value. Basically, the concept of place value in mathematics is embedded on the premise that numbers can be broken apart and then put back together. This premise gives children a more solid understanding of how various operations work. Furthermore, this notion helps children to figure out how they can solve mathematical problems independently by playing with numbers. Once children have a good understanding of place value, they are able to have an easy time doing addition, division, multiplication subtraction and expanded notation among many other mathematical operations. It is thus evident that place value is one of the basic concepts of mathematics that goes beyond repetition and memorization (Rumack, 2011). Completing the tasks The assessment will involve two tasks. In the first task I will ask the children to enter 1 into a calculator and then progressively enter 0. Subsequently, I will ask them to say which place value 1 is in. I will ask them to progressively add 0 and then say the place value of one. I will also ask them how many places across the number they have entered in the calculator does 1 need to move so as to say 10000 and what is the pattern in the name of the places? When completing this task I entered 1 then 0, in this case, I identified 1 to be in the tens category. Then 1 entered another 0 and got 100, in this case, I identified 1 to be in the hundreds category. I entered another 0 and got 1000 and identified 1 to be in the thousands category. In the second task, I will ask each child to classify numbers into their suitable groups. I will give each child a card with numbers ranging from 1000 to 1000000 and then I will ask them to organise the numbers into their suitable groups. I will then ask them what the numbers they have organised in the groups have in common. When completing the tasks, I organised the numbers 1000 to 9000 in the thousands groups. I organised 10,000 to 90,000 in the tens of thousands group. I then organised 100, 000 to 900, 000 in the hundreds of thousands group (See appendix on the samples of tasks). Diagnostic assessment The students This assessment will involve two students in Year Four namely; Jane Doe 9years and John Doe 10 years. Both of them are native English speakers and have undergone lesson lessons on place since they were in Year Two. Selection of tasks This assessment will involve two tasks on place value that will be used to gauge the understanding of each child on place value. I chose these tasks mainly because they are simple and they can efficiently assess what children know about place value. For instance, the first task requires the children to enter digits in the calculator and then identify the place value of one number. This task is very simple and will be used to gauge whether the children can identify the place value of one number in different positions. The second task is also very simple and can be used to assess whether children can group numbers based on their place value. Planning for the tasks The diagnostic assessment with both Jane and John will take approximately 30 minutes for each child. Some of the resources that will be used in the course of this assessment include; pencils, rubbers, papers, card numbers and calculators. Before the assessment, I will ensure that I carefully review the tasks and present them in a manner that the students will be able to easily understand and complete. I will also ensure that I have all the required resources. During my session with each child, I will ensure that I carefully explain to them what they are supposed to do. For instance, I will let them know that the aim of this assessment is not to teach them about place value rather, it is to find out what they know about place value. The assessment will be interactive in nature. While the students complete each task, I will keenly observe what they are doing and ask questions in relations to what they think is the right way of completing the tasks. In case any of the children experiences difficulties, I will ask them to explain the difficulties that they are experience while completing the tasks. Subsequently, I will record the performance of each child. Results Jane Doe was able to correctly complete the first task without experiencing any difficulties. She keyed in one in the calculator and progressively entered zeros. As she progressively entered zeros in the calculator she was able to correctly state the place value of one until she reached hundreds of thousands 100,000. Nevertheless, she was not able to explain how she reached to her conclusion particularly with regards to the place value of one in various positions after she added the zeros. In the second task, Jane Doe was able to group the numbers in the number card according to their place value. However, just like in task one she was not able to explain how she arrived at her conclusion. When completing the tasks her pace was generally fast she did not experience any difficulties or have any questions. John Doe was also able to complete most of the tasks correctly. In the first task, he was able to correctly identify the place value of after adding about four zeros. After entering about five zeros into the overall number he had entered into the calculator he incorrectly identified the place value of one. Just like Jane, he was also not able to explain how he arrived at his conclusions. He was able to complete the second tasks correctly although he was very slow and experienced difficulties in grouping bigger number. Just like in task one, he was also not able to explain how he arrived at his conclusions. Discussion Based on the performance of Jane Doe in the two tasks, it is apparent that she has a solid knowledge on place value. She completed the tasks correctly, this shows that she has done a lot of practise in the past. However, the fact she was not able to explain how she arrived at her conclusions shows that her understanding on place value is not holistic and much of what she knows may be as a result of cramming and memorization rather than understanding. Rumack (2011), observes that most of the struggles experienced by students in mathematics can be attributed to a focus on memorisation rather than understanding. On the other hand, based on the performance of John Doe in the two tasks, it is apparent that he has basic knowledge on place value however as compared to Jane, he has done less practise involving tasks on place value. One of his areas of weakness involves dealing with big numbers. In order for him to improve his performance, it is essential that he does a lot of practise on place value tasks especially those involving big numbers. Teaching place value When teaching place value to children, it is important to use simple language and instruction that the children can understand. Moreover, it is important to use a student –centered approach to teach the children. A student centered approach of teaching focuses on the specific needs of each student. It is also important to ensure that the learning sessions are interactive such that the students can be able to ask questions and give necessary feedback. When teaching place value, it is important to start with the simple aspects and then gradually progress to the complex aspects. For instance, one can first start teaching about the number system and then progress towards teaching on the positioning of digits in the number system. One of the effective strategies of ensuring that children understand place value is the use of concrete representations such as card numbers, pop-sticks and Multi-base Arithmetic Blocks (MAB) among many others. Using these representations, children can easily interpret these concrete materials as representing the number system. A number of studies show that concrete representations can help to enhance children’s understanding (Boulton-Lewis, 1996). Another strategy that can help to enhance the understanding of children in learning value is by incorporating play activities in learning. Generally, the attention span of children is short and as a result they require interactive activities such as games so that they can be able to learn effectively. Play activities related to the subject can help children to think more critical and apply what they have learnt (Tipps, Johnson & Kennedy, 2010). When teaching place value, my lesson plan will first address the number system and the position of digits in the number system. Subsequently, the lesson plan will address how the position of digits can be identified in a number. Lesson Plan Year : 4 Subjects: Maths Topic: Place Value Materials Chalk card numbers Paper Pencil Objectives To help students understand the number system. To help students learn how to identify the position of digits in a number. Time Activity 5Minutes Introduction Explain to the class what the lesson will entail Describe the fun activities that will take place during the lesson. Describe the aim and importance of the lesson to the class Main Activity 20 Minutes -Organise the students into groups of two - Give each group four card numbers, a pencil and a piece of paper -Ask the members of each group to hold the card number while the other group member identifies the number and the position of all the digits in the number. -Ask the student to record what their group member have stated - Ask the students to alternate so that each student participate -Discuss the tasks with each group and address the specific needs of each group with regards to their performance in the task 10 Minutes Plenary -Ask the each student to explain what they have gained from the lesson -Provide time for the students to ask questions Conclusion This key aim of this paper was to examine how children complete mathematics tasks and diagnose what they know about place value. This study focused on two children in Year Four. Based on the performance of the students on the two assessment tasks that they were given, it is evident that they have general knowledge on place value.. However, the fact they was not able to explain how they arrived at her conclusions shows that their understanding on place value is not holistic and much of what they knows is based cramming and memorization rather than understanding. When teaching place value to children, it is therefore important to use a student –centered approach to teach the children. It is also important to ensure that the learning sessions are interactive such that the students can be able to ask questions and give necessary feedback. Referencing Boulton-Lewis, G (1996). Representations of place value knowledge and implications for teaching addition and subtraction. In J. T. Mulligan & M. C. Mitchelmore (Eds.), Children’s number learning (pp. 75–88). Adelaide: Australian Association of Mathematics Teachers & MERGA. Department of Education, Training and Employment (2001). South Australian Curriculum, Standards and Accountability Framework. Retrieved on January 24, 2011 from Liping, M. (2010). Knowing and teaching elementary mathematics: teachers’ understanding of fundamental mathematics in China and the United State. New York: Taylor & Francis. Mersky, K., Harris, J. & Turkington, C. (2000). Get ready! For standardized Tests: Grade 3. New York: McGraw-Hill Professional. Pengelly, H. (1991). Understanding the structure of the number system. NSW: Ashton Scholastic. Shuard, H. (1982). Primary Mathematics Today and Tomorrow. London: Longman. Ross, S. (1989). Parts, wholes, and place value: A developmental view. Arithmetic Teacher NCTM, Vol. 36, No. 6. Rumack, R. (2011). Importance of place value. Retrieved on January 24, 2011 from Tipps, S., Johnson, A. & Kennedy, L. (2010). Guiding children’s learning of mathematics. London: Cengage Learning. Appendix 1 Task 1 Ask the children to enter 1 into a calculator Ask them to progressively add 0 and then state the place value of one Ask them how many places across the number they have entered in the calculator does 1 need to move so as to say 10000 Ask what is the pattern in the names of the places? Ask each child how they arrived at their conclusions Keenly observe and record the performance of each child. Appendix 2 Task 2 Provide each child with number cards ranging from 1000 to 1000000 Ask them to organise the cards into their suitable groups depending on the place value of the number in the card. Ask children to explain what the numbers they have organised in the groups have in common. Ask each child how they arrived at their conclusions Keenly observe and record the performance of each child. Read More