PLACE VALUE, ADDITION AND SUBTRACTION NOT YOUR MOTHER’S ARITHMETIC

PLACE VALUE, ADDITION AND SUBTRACTION

NOT YOUR MOTHER’S ARITHMETIC

Understanding Place Value

In Chapter 1, the authors discuss the necessity for a proper understanding of place value to perform the arithmetic operations of addition and subtraction.  The authors state that in order to properly understand the standard algorithm for adding multi-digit numbers, students need to have a precise understanding of each number’s place value.  Place value is an understanding of how numbers are read.  These values are what gives numbers meanings.  To ensure that a student has a complete understanding of place value, teachers need to be aware of their students’ prior instruction.  Kindergarten starts place value with 10 ones and some more.  This understanding is expanded through each year, until a student is aware that 3,425 is Three Thousand, Four Hundred Twenty-Five and that there are 3 thousands, 4 hundreds, 2 tens and 5 ones.  Further, a student needs to break down the numbers to understand that each higher place value represents ten units of the lower values.  In other words, 1 thousand is equal to 10 hundreds, 1 hundred is equal to 10 tens and so on.

Once students have a complete understanding of place value, it is much easier for them to perform the arithmetical operations necessary to reach the solutions to an equation.  In reaching the solutions to an equation, the authors emphasize that the “standard algorithm” (the usual way to solve a multi-digit equation) should be an offered strategy, not the only strategy offered to solve an equation.  This concept would be more difficult to include for a teacher that learned the standard algorithm by rote memorization, rather than exploring other ways to represent the problem.  As one of those students that learned to “carry and borrow the one,” it is somewhat difficult to distinguish alternate strategies from my usual method.  

I can see the benefits of allowing students to reach their answers with invented strategies, rather than insisting on the standard algorithm.  A student who discovers that 325 + 435 = 760 with a his/her own strategy allows the teacher to discover the processes that the student uses.  If the student’s strategy is rationally related to solving the equation, then the student is more likely to be able to grasp more complex strategies.

Each of the strategies offered in the chapter allows students to discover their own path to reaching a needed sum or difference.

Breaking Down Story Problems

The use of story/word problems becomes very important in mathematical instruction.  These types of problems allow a student to connect mathematical ideologies to real world settings.  Teachers need to be cognizant of word usage in these types of problems.  It is important for students to not only look for key words, but to read the problem and use their reading comprehension strategies to determine the operation needed to solve the problem.  The use of key words are helpful, but may not be the operation necessary.  There are many anchor charts and bulletin board sets that contain the key words for addition, subtraction, multiplication and division.  These charts can sometimes cause confusion, if the student believes the words are only for one operation.  As an example of the confusion for “altogether”, consider the following word problem.

John has 17 pencils.  How many pencils does John need to get to have 20 pencils altogether?

In this problem, the student is required to subtract 17 from 20.  However, students may confuse “altogether” to be an addition problem and add 17 to 20.  In order to solve the problem correctly, students need to use all of their strategies, including reading comprehension, to solve the problem.

Choosing Our Words Carefully

The authors emphasize that teachers need to choose their words carefully, when teaching a child to regroup.  Regrouping is what most adults would call carry (addition) and borrow (subtraction).  As many of us learned to add multi-digit numbers using the term carry, it is important to consider the possible confusion of the “carried” number’s place if you simply say, “carry the one.”  We understand, now, that carrying the one is actually regrouping 10 ones to 1 ten.  However in the following problem, these words may confuse the students.

        236

+      147

  However, our students may not grasp that concept.  In order to help students grasp the concept of regrouping, it is important to state, “6 + 7 is 13 ones.  We can separate these numbers into 1 ten and 3 ones.  Now, we can place the 1 ten into the tens column and leave the 3 ones in the ones column.”  This wording maintains the proper place value, rather than stating it as “carrying the one.”