# Essay on Understanding The Arithmetic Of Arithmetic

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According to Ashlock, not only do students need to understand the computations of arithmetic, but also need to have computational fluency, in other words, conceptual understanding. As stated in the Principles and Standards for School Mathematics:
Knowing basic number combinations – the single-digit addition and multiplication pairs and their counterparts for subtraction and division – is essential. Equally essential is computational fluency – having and using efficient and accurate methods for computing.
The following examples demonstrate fictitious students and the common error patterns they exhibit along with the suggested instruction on how to teach the students to correct the errors. In the first example, Albert appears to comprehend his multiplication facts; however, he approaches the problems looking at each column as a separate multiplication while also applying a conventional addition algorithm. In addition, when the multiplicand has more digits than the multiplier, Albert continues to use the leftmost digit of the multiplier.
Albert

In order to teach Albert the correct procedure in multiplying, the following suggestions may help with his understanding. Start with the introduction of the distributive property, so that Albert understands multiplying a single term and two or more terms inside a set of parentheses, for example, the equation 2 × 93 represents the same as 2(3 + 90). Initially, ask Albert to rewrite the above problems as two problems with one of the problems with the multiplier as a multiple of ten and the other as the remainder (see below).

For the purpose of verifying that Albert multiplied correctly, teach him to use a calculator to compute each partial product, ensuring he applies multiplica...

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Paper 3 – The store had a sale of red and blue shirts. There were 46 red shirts left after the sale, and 28 blue shirts were left. How many shirts were left after the sale? 18
Introduce visuals to the student to aid in determining the operation needed in word problems. For the problem above, have the student draw a bar model showing 46 red shirts and 28 blue shirts, all in one bar, to help show how many total shirts remain after the sale.
Key strategies for students with error patterns and to avoid overgeneralization, involve intermixing addition and subtraction problems early, and intermixing two-digit and three-digit problems early and, of course, continuous practice is essential for conceptual fluency. According to Ashlock (2009), “we want to see the operations of arithmetic applied in real-world contexts where students observe and organize data” (p. 3).

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