It is apparent that formal education has the strongest influence on mathematical achievement (Marmasse, Bletsas, & Marti, 2000). Focus of the traditional arithmetic curricula on the basic numerical skills helps children develop their innate abilities and acquire new ones. The concept of numbers undergoes a progressive transformation with the development of a kid (Ojose, 2008). In the words of a Professor Stanislas Dehaene (1997) “every human brain is endowed with a primal number sense, an intuition about numerical relations. Whatever is different in adult brains is the result of successful education, strategies, and memorization” (p. 34).
It is important that a teacher understands the level at which each child in a class operates to plan stage-appropriate activities accordingly and thereby accelerate their cognitive development (Ojose, 2008). This paper will explore the scenario in which an eight-year-old second-grade primary student Olivia has completed an assessment item on the concepts in the mathematics strand called Number. The focus will be on the child’s answers that reveal her developmental level of understanding.
Addition and Subtraction
The student demonstrated the understanding of mathematical methods and described them using mathematical language (MA1-1WM). She clearly explained mathematical situations using such words as counting on, counting back, plus, minus, total and double. Her conclusions were supported by the use of objects and diagrams. The student was able to apply efficient mental strategies to solve simple addition and subtraction problems with one and two-digit numbers (MA1-5NA). Specifically, Olivia used doubles and near doubles for simple addition problems and without difficulty mentally combined pairs of numbers that add to ten when appropriate (BOSTES, 2016). However, Olivia struggled with counting back. It is important for the student to remember forward and backward counting sequences so she would be able to apply them to solving simple addition and subtraction problems (Department of Education and Training, 2002). However, even “children reflecting all the counting principles show confusion or uncertainty about the nested inclusion or previous numbers” (Reys, Linquist, Lambdin, & Smith, 2010, p.141). The student will be presented with the counting line and after it is clearly understood it will be replaced with the number line. She will also be encouraged to explore an activity that uses a calendar for practicing counting backwards (Reys et al., 2010).
Place Value
The student showed a clear understanding of mathematical terminology and was capable of representing mathematical situations in many different ways (MA3-1WM). Olivia was capable of using the concepts involved in place value along with properties of operations to add and subtract within hundred. Specifically, the student showed the ability to state the place value of digits in different numbers. She also had a clear understanding of the role of zero and applied it to reading and writing of numbers up to hundred. However, the student had problems with naming the number of tens in 100. She also struggled to recall how many tens in other whole numbers. The student must understand the concepts of sequences, quantity and positions of numbers. The focus must be on developing mental computational strategies related to place value techniques for later use in written methods for addition, subtraction, multiplication and subtraction (Department of Education and Early Childhood Development, 2009).The use of a place-value mat might prove very beneficial for teaching place value. The student will be able to see the connection between what she is building and how numbers relate to each other. Another strategy for developing an understanding of place value would be playing Race to 100 game, where a child has to trade up and down.
Counting by 10s and 100s
The student was not able to count backwards from various starting points using skip counting by 10s and 100s. Although Olivia showed a great proficiency in counting forwards, she could not successfully count backwards from 100 to zero skipping 10s. The child also had some difficulties bridging decades. She could not name the right number following 109 and when asked to make a guess she opted for 1000. The student must learn how to produce the number sequence forwards and backwards with and without skipping tens by using connections between number words. She needs to understand the patterns embedded in the number sequence so she would not have to produce them from memory. Olivia will be presented with a collection of items which she would have to group in tens or hundreds and count while a teacher reduces their number by ten. Her attention will be focused on the decrease of each new decade name with the decrease of items. It will help the child understand the relationship between the number words and will teach her to bridge 110. She will also learn how to produce a backward number word sequence.
Combining and Partitioning
Olivia demonstrated the great ability to combine, separate and partition a collection of objects and described her actions using everyday language (MAe-5NA). The child was able to mentally partition whole numbers into number parts. Olivia also had a clear, automated understanding of combinations with 10s and was able to partition 10 into 7 and 3. She also solved tree two-digit tasks without counting by ones. However, she lacked fluency with combinations and partitions in the range of 1 to 30. The child had difficulty with recognising that 19 can be partitioned as 10 and 9. The student needs to learn how to double numbers in the range of 1 to 30. Olivia must also acquire knowledge of combinations and partitions of numbers beyond 10. It will help her to develop strategies of counting which do not rely on counting by one. The child will use place value cards to have an additional practise of combining and partitioning two-digit numbers (Acara, 2014). She will also be presented with a place value chart to have a better understanding of two-digit numbers and how to properly partition them.
Numerals Identification
The student was able to identify all numerals in the range from 1 to 100 (MA1-4NA). She consistently demonstrates an ability to recognize any number in the number sequence. Olivia displayed the presence of prerequisite knowledge necessary for producing long number sequences. The child needs to increase the numeral identification skills in the range from 1 to 1000. It will help her to develop mathematical skills necessary for addition and subtractions of two, three and four-digit-numbers. The child also needs to be able to read and write any numbers up to 1000. Olivia will be presented with flash cards that would teach her different strategies to correctly represent four digit numbers. The child will engage in the activity called The Four-Digit Number Hunt where she will have to discover examples of four-digit numbers in her daily surroundings.
Multiplication and Division
The student used a number of strategies for multiplication and division (MA1-6NA). The child demonstrated the use of reasoning skills for dividing a collection of objects into equal groups and was able to distinguish between such concepts as the number of groups and the number in each group (BOSTES, 2016). Olivia had some difficulties with dividing items into equal groups when their number was not equal. She could not divide 22 items into 5 equal groups. Also, she was not able to use an area model for multiplication of one-digit numbers by two-digit numbers. Understanding of the area, or box, method will help the student to become experienced with multiplication concept and will serve as a basis for the switch to the paper and pencil methods of calculation. Olivia will be presented with Cuisenaire Rods along with simple sketches and examples. The teacher will explain the concept of the part left over using a number of small objects and sharing them into separate groups.
Conclusion
The student has demonstrated a clear understanding of basic mathematical methods and was consistently able to describe them using mathematical as well as everyday language (MA1-1WM). She supported her answers with the use of objects and diagrams. Olivia was able to identify all numerals in the range from 1 to 100 (MA1-4NA). The student experienced some difficulties when dividing items into equal groups if their number was not equal. Even though Olivia showed a great proficiency in counting forwards using skip counting, she could not successfully count backwards from 100 to zero skipping 10s. She also had difficulties with combining and partitioning in the range of 1 to 30.
The child should continue to work on the understanding of the relationship between the number words as well as expand her knowledge on doubling numbers in the range of 1 to 30.
References
Acara (2014). Work sample portfolio. Web.
BOSTES (2016) Mathematics K-10.Web.
Dehaene, S. (1997). The number sense. New York, NY: Oxford University Press.
Department of Education and Early Childhood Development (2009). Mental computation and estimation. Web.
Department of Education and Training. (2002). Developing efficient numeracy strategies. Sidney, Australia: Professional Support and Curriculum Directorate.
Marmasse, N., Bletsas, A., & Marti, S. (2000).Numerical mechanisms and children’s concept of numbers.Web.
Ojose, B. (2008). Applying Piaget’s theory of cognitive development to mathematics instruction. The Mathematics Educator, 18(1), 26-30.
Reys, R.E., Linquist, M. M., Lambdin, D.V., & Smith, N. L. (2010). Helping children learn mathematics. In R. E. Reys & M.M Lambdin (Eds.), Developing counting and number sense in early grades (pp. 130-153). New York, NY: Wiley.