Place Value Papers

Place value is an important foundational area of Mathematics that underpins the development of more complex skills, including addition and subtraction, decimals and multiplicative thinking (Dawson, 2015). Students do not always have the place value knowledge and skills to apply learning to more abstract contexts.

Understanding base ten numbers is one of the most important mathematics topics taught in primary school, and yet it is also one of the most difficult to teach and learn. Base ten blocks are used to teach place value concepts, but in a lot of cases, children often do not perceive the links between numbers, symbols, and models. Research shows that many children have inaccurate or faulty number conceptions, and use rote-learned procedures

with little regard for quantities represented by mathematical symbols (Price, 2001). Children need to learn from the early primary school grades and how number are written in the base ten numeration system, and to construct accurate mental models for numbers, in offer to develop a proficiency with mathematics that will equip them to solve problems in later life (Price, 2001).

Place value is the idea that each digit in a number represents a certain amount, depending on the position that it occupies.

Therefore, a number like 356 has a 3 in the hundreds place, a 5 in the tens place, and a 6 in the ones place. The digit 3, in the hundreds place, does not represent 3 as it represents 300. This idea is generally introduced in the lower elementary grades in order to help students manipulate numbers and solve problems. If a child understands that number 356 is actually 300+50+6, the student can play around with this number more easily. Understanding this concept might make it easier to add or subtract numbers, as well as use multiplication in the future grades – 365 x 3 = (300 x 3) + (50 x 3) + (6 x 3). The concept that numbers can be broken apart and put back together gives the student a more solid understanding of how different operations work. Not only that, but the student can also figure out how to solve problems independently by playing with the numbers (Rumack,

2011).

Typical exposure to numbers occurs via a variety of media, through songs, play, interaction with adults and other children, picture storybooks and television programs. When a small child walks down the steps he/she may count them. It is important that students build an understanding of the numbers up to ten before they progress further. In the beginning of first grade, children will strengthen their skills in counting to 100, addition facts and subtraction facts, and skip counting. Around the middle of first grade, children will learn to separate numbers and objects into tens and ones (Smartfirstgraders.com, 2015). At the end of first grade, children will begin using regrouping when adding and subtracting 2-digit numbers whilst in second grade, children will work with regrouping and place value concepts for 3-digit numbers (Smartfirstgraders.com, 2015).

Ideas about place value develop from earlier ideas such as those related to counting and subtilizing. The ability to identify quickly the number of objects in a small set is known as subtilizing. This is different to counting and requires a rapid response. Before their first birthday babies can make judgements about the relative sizes of sets of objects (Amsi.org.au, 2015). They react differently when two sets of the same size are presented compared to when two sets with different amounts are presented (Amsi.org.au, 2015). It is not necessary to name the numbers to tell if one set has more or less than another. Using one-to-one correspondence the child matches up the elements of the two sets, pairing each item in one set with exactly one element in the other set (Amsi.org.au, 2015). The child can now tell if the sets are ‘the same’ or ‘different’ in number and which is larger or smaller.

Children need opportunities to see amounts in numbers. For instance, use base ten blocks or strips to ensure that students see the value of numbers. The place value blocks are most effective at showing the value of numbers. There are cubes to represent one, strips to represent ten, flats to represent 100 and blocks to represent 1,000 (Kliman, 2000). Students can easily see that 10 cubes fit into a 10 strip, 10 strips fit into the 100 flat and 10 100 flats fit into the 1000 block. Students should show numbers with the blocks and then writing the numbers. Students that practice building with numbers, with the place value blocks or strips will get a better grasp on the concept. As time progresses, a chart should be provided, like the image with this article and ask questions about the placement of specific numbers (Russell & Russell, 2015).
Understanding the process of knowledge change is a central goal in the study of development and education. Two essential types of knowledge that children acquire are conceptual understanding and procedural skill. We define procedural knowledge as the ability to execute action sequences to solve problems. This type of knowledge is tied to specific problem types and therefore is not widely generalizable. In contrast to procedural knowledge, we define conceptual knowledge as implicit or explicit understanding of the principles that govern a domain and of the interrelations between units of knowledge in a domain (Rittle- Johnson, Siegler & Alibali, 2000). As children do not already know a procedure for solving a task, they must reply on their knowledge of relevant concepts to generate methods for solving problems.

After interviewing ‘Georgia’ to assess her understanding of a place value and other mathematics concept, one clear pattern in the data from both interviews and teaching sessions was that there were a large number of misconceptions, errors, and limited conceptions evident in her response.

Georgia’s response revealed that she had not developed the idea that the “tens” digit represents a collection of ones, but believed that it represented only it face value. She has also thought the point is the decimal fraction in decimals. She requires the explaining of the value of each of the digits for numbers when there is a decimal notation. Using models such as ten frames became a vital tool to highlight the correct language of place value. When it is implemented into the lesson, it will provide opportunities to demonstrate, explain and justify the

learning.

Reference

Amsi.org.au,. (2015). Counting and place valueK-4. Retrieved 22 September 2015, from http://www.amsi.org.au/teacher_modules/Counting_and_place_valueK-4.html

Dawson, E. (2015). Lesson Study: Improving Teachers' conceptions of students' Understanding in Place value. Retrieved 23 September 2015, from http://www.alphinps.vic.edu.au/wp-content/uploads/2013/10/Improving-Teachers-Conceptions-of-Students-Understanding-of-Place-Value.pdf

Kliman, M. (2000). How do students build an understanding of place value in Investigations? | Home | Investigations in Number, Data, and Space®. Investigations.terc.edu. Retrieved 23 September 2015, from http://investigations.terc.edu/library/curric-math/qa-1ed/place_value.cfm

Price, P. (2001). http://eprints.qut.edu.au/15783/1/Peter_Price_Thesis.pdf. Retrieved 22 September 2015, from http://eprints.qut.edu.au/15783/1/Peter_Price_Thesis.pdf

Reys, R., Lindquist, M., Lambdin, D., Smith, N., Rogers, A., & Falle, J. et al. (2012). Helping Children Learn Mathematics. Brisbane: John Wiley and Sons Australia.

Rumack, R. (2011). The Importance of Place Value. Ruth Rumack's Learning Space Blog. Retrieved from https://ruthrumack.wordpress.com/2011/06/02/the-importance-of-place-value/

Russell, & Russell, D. (2015). 5 Crucial Principles of Counting. About.com Education. Retrieved 23 September 2015, from http://math.about.com/od/counting/a/The-Principles-Of-Counting.htm

Smartfirstgraders.com,. (2015). Place Value: An Essential First Grade Skill - Smart First Graders. Retrieved 22 September 2015, from http://www.smartfirstgraders.com/place-value.html