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Place value is a fundamental concept when solving mathematics as it forms the foundation of subtraction, addition, multiplication, and division. Lack of understanding of this concept will lead to a future problem in mathematics (Tucker, Singleton & Weaver, 2013). Place value is the basis of early counting, pinpointing cardinal numbers and knowing the value of each number in a digit, therefore, enhance the child to understand the base-10 structure.
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Case Study A– Counting Chips
The student is confusing the tens and ones when counting. The student misunderstands the procedure of counting and treats tens and ones as a single digit (Fraivillig, 2017). The digit 16 has one of the tens and six as the ones. However, the place value of 6 as the ones is six multiplied by one, which will give 6. The student was correct. Likewise, one on the tens side is one multiplied by ten, which provides a place value of ten, making the student’s answer of one wrong. This concept can prove to be challenging for the students to understand. Therefore, it is essential to implement various methods of helping the student understand the concept.
For the modified method, I will introduce the sixteen chips on the student’s desk. I would ask him to count the total number of chips and write the number down. Then, I will point to the six on the digit sixteen and ask him to count the number of chips six represent. After counting six chips, I will ask him to count the remaining chips on the desk, which will be ten. Then I will point to the one on the tens value of sixteen to determine whether he has understood the one has a place value of ten.
Alternatively, I can ask the student to draw a two-column table on the book and write one on the first column and six on the second column. Then follow the procedure of pointing and counting and write six on the second row under the second column and so on. I would carefully inform the student to put a zero under the other digits of the number to come up with on six or one depending on the digit we are looking for its place value. However, the second method may not prove to be effective when it comes to hundreds and thousands. Therefore, it can easily confuse the student unless carefully interpreted to them to understand the concept.
When using the secret code card, I would introduce two secret code cards with one card having the ones and the other the tens. I would ask the student to create a two-digit number using the secret code cards you have provided, which in this case is 16. On the book, let the student present the place value of each digit he has combined using the secret code cards. When using penny strips, the child should pull six penny strips to represent the place value of six in sixteen. He should have ten penny strips to represent the place value of one of the digit sixteen.
Case Study B – A Misguided Comparison
The student is using the value of the digits rather than the place value to determine the greater value (Cady, Hopkins & Price, 2014). Math talk can help the children understand her misconception. There are three stages to go about math talk (BartoliniBussi, 2011). The first stage is asking and modeling an open-minded question. Some of the questions to consider at this stage are: what makes you say that? Explain your thinking to me? What is another way to solve the question? The second stage involves providing visual cues for references. It will give answers to some of the questions the student answered during the first stage. The last stage is engaging all the students in the math talk. By following this procedure, the student will understand his or her misconception.
When using the secret code card, I will produce six secret code cards. Each of the two digits will be represented by three secret code cards with the various place values. For 379, I will have one with 300, 70 and 9. For 512, I will have 500, 10 and 2. Using the place values, I will explain to the student that 379 has 300 hundred while 512 has 500 hundred. Therefore, 512 is greater than 379. The secret code cards do not offer a direct solution of the value of the digits versus the place value. For example, in this case, the student might still be convinced that we used her misconception since five is greater than 3 in 500 and 300. The advantage is that the secret code cards offer a physical illustration, which is easier for children to understand.
Comparison and other strategies
These two misconceptions are similar since both students have difficulties in understanding the concept of place value. The students also lack the understanding of values of various digits in a number. For the differences, misunderstanding place values differently affect students. In the first case study, the student does not understand the place value of one in sixteen is ten, while in the second case study, the student cannot figure out which one is greater between 379 and 512. Another difference is the method that proves effective to use while helping the student realize their misconceptions. In the first case study, counting chips prove to be effective than other methods, while in the second case study, secret code card is the most effective method.
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Other approaches that can be used to help the students include: helping them using math drawings. It is an effective approach since it is easy to remember a drawing than something that is written down on the board. As well, doing math papers regularly to help the student understand the concept is important. Lastly, helping the student track their progress is beneficial. By doing so, the student will be motivated. They will post positive results.
In conclusion, it is always important to consider different methods of helping a child out of a misconception. The explanation is that each child has a different way of understanding and there are minimal chances that two children have the same way of understanding things. In class, it is important to move with the pace of the slowest student as every student will benefit from such a strategy.
- BartoliniBussi, M. (2011). Artefacts and utilization schemes in mathematics teacher education: place value in early childhood education. Journal Of Mathematics Teacher Education, 14(2), 93-112. http://dx.doi.org/10.1007/s10857-011-9171-2
- Cady, J., Hopkins, T., & Price, J. (2014). Impacting Early Childhood Teachers’ Understanding of the Complexities of Place Value. Journal of Early Childhood Teacher Education, 35(1), 79-97. http://dx.doi.org/10.1080/10901027.2013.874382
- Fraivillig, J. (2017). Enhancing Established Counting Routines to Promote Place-Value Understanding: An Empirical Study in Early Elementary Classrooms. Early Childhood Education Journal. http://dx.doi.org/10.1007/s10643-016-0835-5
- Tucker, B., Singleton, A., & Weaver, T. (2013). Teaching mathematics in diverse classrooms for grades K-4. Boston: Pearson.