Error Patterns in Computation. Whole Numbers: Addition and Subtraction - Book Report/Review Example

?Running head: CHAPTER REFLECTIONS Chapter Reflections: Error Patterns in Computation CHAPTER REFLECTIONS 2 Chapter Reflections: Error Patterns in Computation Chapter 4: Whole Numbers: Addition and Subtraction Every teacher has encountered a couple of error patterns in computation among their students. This book's chapter discussed several of these error patterns that I find very familiar, and provides valuable intervention processes as well. I find these data valuable especially in teaching middle school mathematics. One of the error patterns I have encountered is that students forget basic addition, especially for additions involving whole number 1 to 9. For example, when one of the addends is 0, I have seen some students put 0 as the sum, which is obviously a rule for multiplication. Some consider adding 0 as similar to adding 1 to the other addend, thereby increasing the number by 1. Another common error I have encountered is when students add a multi-digit addend to a single-digit addend, some calculate for the sum of the individual digits together. Some add the single-digit to each of the digits of the multi-digit addend. Some get confused with place-values, thus adding the ones to the tens, and so on. Some even make up a new rule in addition by adding all the numbers in the ones position then placing the sum under the tens. This is followed by calculating the sum of all the numbers in the tens position, and then placing the sum under the ones. There are also ones who seem to think that adding from the left to the right is the correct way. While this may not be obvious in additions that do not involve renaming, this error pattern could be carried over in other mathematical calculations, so teachers need to watch out for such student slips as well. I would say then that in teaching middle school addition, teachers need to help students “never forget” the basic principles of addition, and focus on place-values especially of multi-digit numbers. CHAPTER REFLECTIONS 3 Middle school subtraction, on the other hand, has its own set of error patterns apparent in some students. Just like in addition, some students tend to forget that subtraction is just a reversed addition. Sometimes, although not very often, a student makes a mistake in subtraction that could have been easily corrected had the student remembered the connection between subtraction and addition. Some students also do subtraction by forgetting proper renaming and borrowing. Place-values are also sometimes not used properly. This is common especially in multi-digit minuends and single-digit subtrahends. Forgetting to put proper marks when renaming is also common. This causes several mistakes in the subtraction processes. Sometimes, students tend to subtract from left to right. However, this is not easy to observe since, just like in addition, this is only apparent when renaming is required in the equation. Another very common error pattern I have observed in the students is renaming involving one or a series of 0s. It is interesting to note that students who demonstrate a certain error pattern in subtraction is more likely to have the same error pattern in addition. This gets me to thinking again of the forgotten relationship between addition and subtraction. Furthermore, although in higher grade levels, use of calculators is very much allowed, sometimes even encouraged, this could cause a further setback in students' number sense. This could cause them to practice paper and pencil and mental computations less often, which would make them forget the basics even more. I have reflected, based on the planning instruction in this chapter and personal experiences with my students, that number sense could play a big part in helping the students realize their wrong answers. Students sometimes give answers to subtraction equations that are glaringly incorrect. If a student has a strong number sense, merely looking at the answer would tell them if they need to rethink and check their mathematical process. CHAPTER REFLECTIONS 4 Chapter 5: Whole Numbers: Multiplication and Division This chapter provides interesting data on the error patterns in multiplying and dividing whole numbers. In middle school mathematics, I have observed several of the discussed error patterns from among my students. The simplest is the knowledge of the multiplication table. Although it is expected that everyone in middle should know it by heart already, a few students can still be seen using their fingers in counting. This fact alone provides a wide range of possibilities in error patterns in multiplication. I believe that a lot of students have been pampered by calculators. Gone were the days when even double-digit multiplicands and multipliers can even be calculated mentally. However, this is not to say that students today have lesser mathematical skills than those of the previous generations. It is just that, with my experiences in error patterns in addition, subtraction, multiplication, and division among students, what is common is the lack of knowledge of the basics and number sense. For example, students know what multiplication is --- technically. Yet it is sometimes distressing to see products of double-digit equations that are, logically off even at a glance. Let us see this as an example, in the equation [30 x 30 = ?], I have seen students disregard the zero and multiply both 3s before bringing down the 0, obtaining a product of 90. A student with a strong number sense would easily realize that 30 30s can never be only 90. Even a $30 USD wage in 30 days is impossible to imagine to just come up to 90. With these, we already see several errors at play: the concept of 0, no partial products (both are double-digits), disregarding the 0 in the equations, and multiplying vertically. As a teacher, I would like to focus on correcting these errors, along with wrong place-values and incorrect carrying of numbers. I think students will benefit from exercises without the use of calculators. CHAPTER REFLECTIONS 5 Error problems in division is very much similar to that of multiplication. This is probably because of the relationship between multiplication and division. This is just like the relationship between addition and subtraction. One common error pattern I have encountered among students here is the concept of regrouping. Also, there is much difficulty in place-values particularly for multi-digit dividends and divisors. Knowledge of basic division processes seems to have been forgotten too. Students seem to be confused on how to treat 0 in both multiplication and division. In division, for example, a student is given this equation: [3500 / 35 = ?], a few would answer 10, while a few would answer 1000. Of course there will always be those who get the correct answer. But the incorrect answers portray the existing lack of understanding of the magnitude and order or numbers. Again, this is basic number sense. It just means that in their brain processes, they are not able to automatically link the variant of this equation in multiplication: [100 x 35 = 3500]. Another example of this is when I previously asked for the the quotient of: [5600/70 = ?]. I received several answers of 800. It appears that students tend to disregard the 0s when computing, then add them incorrectly after. Estimates to check the accuracy, or even probability, of the results would have been possible with strong number sense among students. If students deal with 0s incorrectly, how will they be able to calculate accurately especially when they need to round-off already --- a thing that happens in daily life. As a teacher, mathematics for me is not something that should be merely answered correctly. It is something that should be understood. Mathematics is a perfect science and no amount of estimates would do, that is right. But relying on estimates (due to strong number sense) is different from understanding the logic and reason behind the calculation of numbers. CHAPTER REFLECTIONS 6 References Ashlock, R. B. (2002). Error patterns in computation: Using error patterns to improve instruction. Upper Saddle River, N.J: Merrill Prentice Hall. Read More