Subtraction refers to an arithmetic operation that represents the process of removing objects from a given collection. In mathematics, subtraction is normally signified by the minus (-) sign. It is important to realize that subtraction follows several important patterns. Firstly, it is anti-commutative. This implies that changing the order changes the sign of the answer. Secondly, subtraction is not associative. This means that when one subtracts more than two numbers, the order in which the operation is done matters. Thirdly, subtraction of zero does not actually change a given number. Moreover, subtraction obeys predictable rules regarding related operations such as addition and multiplication. Generally, performing subtraction is one of the simplest numerical tasks. Subtraction of small numbers is always accessible to young children. Gretchen’s video showing her performing a simple subtraction using three different strategies provides insight regarding the undesirable consequences arising from a student being taught an algorithm prior to being conceptually ready to use it correctly.
During her interview, Gretchen uses three different strategies to solve the problem 70-23. The first strategy that she employs is the standard algorithm, base ten blocks and hundreds chart. Performing subtraction using the standard algorithm requires knowledge about the place value of the numbers involved in the operation (Philipp & Cabral, 2005). While discussing place value with young students, emphasis should be placed on exchanging 10 ones for 1 ten, as well as 10 tens for 1 hundred as you work from right to left. While using standard algorithm to solve the problem, 70-23, Gretchen fails to get the right answer because she explains, “Zero take away three is 3.” This simple misunderstanding impedes her ability to accurately use the standard algorithm. Essentially, it is important for students to be taught all the essential rules involving subtraction. Some of the students may not be familiar with the basic concepts behind subtraction problems. Using the standard algorithm, Gretchen came up with the following answer;
Delegate your assignment to our experts and they will do the rest.
70
- 23
53
Simply put, Gretchen was very confident that her answer was correct. This was as a result of her belief that subtracting a number from zero equals to that number. In this case, Gretchen believed that subtracting three from zero equals to three. Gretchen used the standard algorithm to perform the subtraction because it is the most commonly used method. The standard algorithm relies on the way a child understands a given arithmetic problem. The concepts underlying a given arithmetic operation determine whether the answer obtained is right or wrong.
The second strategy employed by Gretchen is base ten blocks. The base ten blocks offer a spatial model of base ten number system. The smallest blocks, cubes measuring 1 centimeter, are called units. The long narrow blocks, measuring 10 centimeter, are called the rods, while the flat square blocks measuring 10cm by 10cm by 1cm are called the flats. Finally, the largest blocks available are the cubes measuring 10cm and are called cubes. While working with base ten place value experiences, the units is usually used to represent ones, the rod tens, the flat hundreds and the cube thousands. It is important to realize that providing names based on the shape rather the size allows for the pieces to be renamed when necessary. The size relationships among the various blocks make them ideal for the investigation of number concepts. Generally, the base ten blocks are useful in providing students with means to physically represent the concepts of subtraction of whole numbers. Gretchen, using base ten blocks was able to arrive at the correct answer, 47. Gretchen was able to come up with the correct answer because she could conceptualize better using physical objects rather theorizing.
Gretchen was able to make sense to solve the arithmetic problem before her using the base ten blocks because the blocks are especially useful in providing students with means to physically represent the concepts of place value and subtraction of whole numbers. Gretchen eased into the concept of trading and was able to see the logical development of each operation. It is also important to note that the blocks are very effective because they provide a visual foundation and understanding of the algorithms that children use when doing pencil and paper computations. Using the base ten blocks to arrive at her answer, Gretchen took 7 rods and then removed 2 rods and 3 units from the set. By so doing, he counted the remaining rods and units. This strategy is very simple and interesting. Using manipulative strategies make learning subtraction fun. Additionally, students can easily memorize concepts using based ten blocks.
The third strategy used by Gretchen was the hundreds chart. The hundreds chart refers to a chart with numbers one to hundred. The chart helps students to build number sense and effectively understand composition of 2-digit numbers. Therefore, the hundreds chart is an effective manipulative for teaching two digit subtraction. Basically, the hundreds chart is most powerful when presented as one of the various representations of numbers and operations. This particular approach to mathematics promotes critical thinking, as well as conceptual understanding. Gretchen was able to arrive at the correct answer, 47, using the hundreds chart. She was able to do that by counting down 23 steps from 70, finally arriving at 47. The strategy had the potential of promoting her conceptual understanding.
Of the three strategies used by Gretchen to solve the arithmetic problem, she failed to get the correct answer using one. Gretchen did not obtain the correct answer using the standard algorithm because she did not understand the concept of borrowing using the standard algorithm. As for the base ten blocks and hundreds chart, students can easily perform subtractions as they can physically count the units and boxes respectively. She was able to get the correct answer using the base ten blocks and the hundred charts because they promote effective conceptual understanding and reasoning.
There are various instructional implications for Gretchen for next year. The fact that she failed to get the correct answer using the standard algorithm does not necessarily mean that she is not sharp. It is just that there are discrepancies that exist in her expressed thoughts (Philipp, 2008). Gretchen is a good learner because she is able to reason very well with the manipulatives. As such, Gretchen’s teacher should focus on using manipulatives to enhance her conceptual understanding of standard algorithms. This is because her thinking can impede her ability to move ahead, particularly when instruction is not well sequenced. In order to ensure that Gretchen performs better next year, focus should be on ensuring that she understand the basic concepts underlying 2-digit subtractions.
Subtraction with trading is normally challenging for students. Subtraction with trading can be done using standard algorithm, base ten blocks and hundreds chart. Using the standard algorithm, 70-23 can be performed by borrowing 1 ten from 7 to make 0 ten in 70 as follows;
6
7 10
-2 3
4 7
While using base ten blocks, subtraction with trading involves borrowing some of the blocks of equal value so that the taking away can be possible. For example, to subtract 23 from 70, a student would be required to trade one of the rods representing 70 (7 rods) for ten units to form an equivalent representation of 70 (7 rods). The student would then take away 23 (2 rods and 3 units) and be left with a difference of 47 units (4 rods and 7 units).
Performing subtraction with trading using the hundreds table is very easy. Subtracting 23 from 70 requires the student to mark 70 on the hundred chart using a crayon. The student would then proceed to count twenty-three boxes downwards. By doing so, he will arrive at the correct answer, 47.
Gretchen’s reasoning using the three different strategies highlights the importance of identifying the mathematics embedded in each of the strategies. Using the original concept map as a foundation, I will begin to make essential connections. I will then conduct class discussions regarding the key elements of subtraction and subtraction solution strategies. Most importantly, Gretchen’s reasoning using the three strategies has made me realize that it is always important to ensure that students are conceptually ready to use a subtraction standard algorithm before teaching them such an algorithm (English, 2004). This will avoid the undesirable consequences of poor conceptual understanding. In order to ensure that students understand subtraction effectively, I will employ the use of manipulatives in order to develop the critical thinking and conceptual understanding of the students.
References
English, L. D. (Ed.). (2004). Mathematical and analogical reasoning of young learners . Routledge.
Gretchen’s interview: https://www.youtube.com/watch?v=CLApo35_FTM&t=7s .
Jacobs, V. R., Lamb, L. L., & Philipp, R. A. (2010). Professional noticing of children's mathematical thinking. Journal for research in mathematics education , 169-202.
Philipp, R. & Cabral, C. (2005). Gretchen Clip Three. IMAP: Integrating Mathematics and Pedagogy to Illustrate Children's Reasoning [Computer software]. San Diego, CA: San Diego University Foundation. Retrieved September 28, 2010. Available from http://www.sci.sdsu.edu/CRMSE/IMAP/main.html
Philipp, R. A. (2008). Motivating Prospective Elementary School Teachers To Learn Mathematics by Focusing upon Children's Mathematical Thinking1. Issues in Teacher Education , 17 (2), 7.
Philipp, R. A., Thanheiser, E., & Clement, L. (2002). The role of a children's mathematical thinking experience in the preparation of prospective elementary school teachers. International Journal of Educational Research , 37 (2), 195-210.
