Standard Algorithm for Multiplication vs. Constructivist Partial Products

By Bill Quirk   [wgquirk@wgquirk.com] 

The Standard Algorithm for Multiplication

                   467                                                              
x 846
2802
1868
3736
395082

Details of the Standard Algorithm Solution for 846 x 467

  1. This example uses 9 single digit multiplication facts [6 x 7, 6 x 6, 6 x 4, 4 x 7, 4 x 6, 4 x 4, 8 x 7, 8 x 6, and 8 x 4] to compute the 3 partial products (PP)  [2802, 18680, and 373600].  
  2. Working right to left, carrying is used a total of 6 times. It used 2 times for computing each of the 3 partial products (PP).
  3. The standard algorithm of addition is used to add the 3 partial products.  This algorithm uses the following single digit addition facts: 

Design Features for the Standard Algorithm for Multiplication

Constructivists Math Educators Find Fault With the Standard Algorithms

  1. Constructivists say that working right to left is not natural for children.  
  2. The standard algorithms work exclusively with numbers in standard (compressed) place value form. Constructivists believe that children need to continually work with numbers in expanded place value form.
  3. Constructivists say that the concepts of carrying and borrowing are too difficult for most children to understand.
  4. Constructivists strongly object to the fact that the standard algorithms can be carried out automatically. They fail to understand that it's often desirable to move to automatic use of knowledge.  This frees the mind to think at higher levels of complexity, without consciously revisiting underlying details.

Constructivist Partial Products Method for Multiplication                                                                   

               467            300000   [9 partial products                                                                       
x 846 70000 added using the partial
320000 23000 sums method]
48000 1900
5600 180 [Note: Need to Carry 1
16000 2 to thousands column]
2400 395082
280
2400
360
42


Details of the Partial Products Solution for 846 x 467
  1. Working left to right, 9 partial products are found by multiplying 800 x 400, 800 x 60, 800 x 7, 40 x 400, 40 x 60, 40 x 7, 6 x 400, 6 x 60, and 6 x 7.
  2. Next, 6 partial sums [300000, 70000, 23000, 1900, 180, and 2]  are found by working left to right (in the column of 9 partial products) to separately total each place value column. 
  3. Next, working left to right, the 6 partial sums are added by adding column digits (this is the constructivist "partial sums method" discussed below).  

Design Features for the Partial Products Method for Multiplication

  1. The student must consciously think of the specific place value associated with each column.  
  2. This method is inefficient.  For example, there are 9 partial products for this example vs. 3 partial products for the standard algorithm.  Consider the difficulty of "keeping track" of 25 partial products for the partial products multiplication of two 5-digit numbers.  
  3. This method does not qualify as an algorithm, because the partial sums method for adding the partial products breaks down, if the sum of the digits in any partial sums column is greater than 9.

Design Features for the Partial Sums Method for Addition

  1. The student must consciously think of the specific place value associated with each column.
  2. This method avoids carrying for simple problems, such as adding just 2 numbers, but it doesn't work more generally, if the sum of the digits in any partial sums column is greater than 9.
  3. Since the partial sums method does not always work, the partial sums method is not a general computational method and does not qualify as an algorithm.
  4. Constructivists favor the partial sums method because:
  5.   Note: Partial sums is the Everyday Math "focus algorithm" for addition.

Why Are the Standard Algorithms So Important? 

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Copyright 2014 William G. Quirk, Ph.D.