The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms Papers

By Bill Quirk   []

The importance of repetition until automaticity cannot be overstated.  Repetition is the key to learning.  - John Wooden

Kamii and Dominick's Research Favored Those With "No Algorithms" Experience

Constructivist math educators regularly cite the "research" found in "The Harmful Effects of Algorithms in Grades 1-4," a paper by Constance Kamii and Ann Dominick.  Another paper, "The Harmful Effects of Carrying and Borrowing" also covers this "research" and it's available at Constance Kamii's website.  The authors described their research activities as follows:  "At the end of a school year, we asked 185 children in grades 2-4 to solve 7 + 52 + 186 (or 6 + 53 + 185) in individual interviews. Note that this problem had a one-digit number, a two-digit number, and a three-digit number. The children were allowed to look at the problem as long as they wanted to, but they were not permitted to write anything. "

Kamii and Dominick concluded that experience with the standard algorithm for addition was harmful, because children with "No algorithms" experience performed the best on the mental math 'test.'  But quotes in their paper shows that their mental math 'test' favored those with "No algorithms" experience in two ways: (1)   Most of the children with "No algorithms" experience had considerable experience with mental math methods.  (2) Several students with algorithms experience tried to solve the mental math 'test' using the paper-and-pencil standard algorithm for addition. They failed to get the right answer because they weren't allowed to write or even say the answer's digits one at a time, as they carried out the right to left computation.  

Note: When these authors use the phrase "No algorithms" they are referring to the standard algorithms of paper-and-pencil arithmetic.  As constructivists, they don't like to acknowledge that these algorithms are considered "standard."  When forced to get specific, they say "conventional U. S. algorithms," implying that the standard algorithms are not standard elsewhere.

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What Did Kamii and Dominick Actually Do?   And What Didn't They Do?

Kamii and Dominick reported that students who had no experience with standard algorithms performed the best on their mental math 'test'.  They claim that the 'test' results showed that the "No algorithms" students were more likely to get the right answer to the question, and they also claim that the "No algorithms" students wrong answers tended to be closer to the right answer.   But beyond "experience with algorithms," the authors didn't provide any details about the math education of any of the 185 children.  Most importantly, Kamii and Dominick failed to disclose anything about the math education of the "No algorithms" students.  But Kamii and Dominick are leading advocates for constructivist computational methods, so it's highly likely that the "No algorithms" students had considerable experience practicing mental math addition problems similar to the 'test' problems.  On the other hand, it's also likely that students working to master the standard algorithms had less time to practice mental math.

The following quotes
[on page 6 of "The Harmful Effects of Carrying and Borrowing"] show that the 'test' was a setup favoring the "No algorithms" students. Beyond concluding that experience with the standard algorithms led to poorer (mental math) 'test' results, Kamii and Dominick also claimed that "algorithms are harmful to most young children for two reasons: (1) They encourage children to give up their own thinking, and (2) they unteach what children know about place value, thereby preventing them from developing number sense."  
  1. What do they mean by "give up their own thinking?"  The standard algorithms  for addition, subtraction, and multiplication require that the student work from right to left.  Kamii and Dominick say that children naturally think left to right. 
  2. What do they mean by "unteach" what children know about place value?   The standard addition algorithm works with numbers in standard (compressed) place value form. The process works in the same way for each column: find the sum of the column's digits and carry to the next column to the left,  if the sum of the column's digits is greater than 9.  Because of this procedure's repetitive characteristic, students learn to carry out the steps automatically, without conscious thought about the specific place value associated with each column.   Constructivists object to the automaticity. 
It is a profoundly erroneous truism ... that we should cultivate the habit of thinking of what we are doing.  The precise opposite is the case.  Civilization advances by extending the number of important operations which we can perform without thinking about them.  - Alfred North Whitehead

Kamii and Dominick want us to believe 
that students are harmed if they carry out computations automatically, working right to left with numbers in standard place value form.  How do the authors defend or explain these claims?  They don't. They're just offering their reasons for the (claimed) poorer 'test' performance of those experienced with algorithms.  First consider the criticism that working right to left isn't natural. It's actually working (naturally) from smaller to larger.  Constructivists offer alternative algorithms [see below] where students carry out computations working from larger to smaller.  Is that more natural for children?  Next consider their criticism of automaticity. Constructivists believe that students can't achieve automatic use of the standard algorithms and also fully understand the concept of place value.  They should read Chapter 2, Addition and Subtraction, in Singapore Math [US Edition] Textbook 2A.  There they will find child-friendly pictorial to abstract methods for teaching "Addition with Renaming" and "Subtraction with Renaming."  Singapore Math never uses the terms carrying and borrowing, but Singapore Math students do go away fully understanding these place value concepts.

What are the Constructivist Alternatives to the Standard Algorithms?

Constructivists emphasize mental math methods.  The problems presented to students normally involve small numbers [such as those in the 'test' problem], because constructivists believe that problems with larger numbers should be handled with a calculator.  The first three examples below illustrate constructivist mental math methods for solving the 'test' problem.  Examples 4 and 5 demonstrate the constructivist's paper-and-pencil replacements for the standard addition and multiplication algorithms. 
                        185  Partial Sums Method       Standard Algorithm 
+ 6 11 << Carry
100 (Add the hundreds) 185
130 (Add the tens) 53
14 (Add the ones) + 6
 244 (Add the partial sums) 244

                                            Partial Products Method                                                                       Standard Algorithm
                         467       300000   [9 partial products                      467                                                              
x 846 70000 added using the partial x 846
320000 23000 sums method] 2802
48000 1900 1868
5600 180 [Note: Carry 1 3736
16000 2 to thousands column] 395082
2400 395082

Partial sums and partial products are the constructivist's replacements for the standard algorithms of addition and multiplication.  Computation proceeds left to right (really larger to smaller), and students must constantly think of the specific place value associated with each column.  Constructivists may be pleased, but they're seriously misleading students. The partial sums and partial products methods can't be carried out automatically, and they're not the algorithms students need for later mastery of algebra.  

The power to lead is the power to mislead, and the power to mislead is the power to destroy  -  Thomas S. Monson

There's more bad news.  Partial sums and partial products computations are increasingly inefficient as computations get more complex.  Students often lose their place.  Consider the difficulty of "keeping track" of  25 partial products for the partial products multiplication of two 5-digit numbers.  Also, as Example 5 above demonstrates, carrying isn't really avoided as computations get more complex.  Constructivists give the impression that their methods avoid carrying and borrowing, but this is only true for simple cases.  The following quotes show how constructivists avoid more complex cases.  Source: Everyday Mathematics Grades 4-6 Teacher's Reference Manual:
  1. "The partial products algorithm can get tedious for problems with very large numbers, but we recommend using a calculator for those, so this is not a serious drawback."  - Page 115
  2. "The authors of Everyday Mathematics do not believe it is worth the time and effort to fully develop highly efficient paper-and-pencil algorithms for all possible whole number, fraction, and decimal division problems.  Mastery of the intricacies of such algorithms is a huge undertaking, one that experience tells us is doomed to failure for many students.  It is simply counter-productive to invest many hours of precious class time on such algorithms.  The mathematical payoff is not worth the cost, particularly because quotients can be found quickly and accurately with a calculator."  - Page 120

Constructivist Strategies for Eliminating Standard Arithmetic from the K-6 Curriculum

Why are the Standard Algorithms Important? 

Constructivists promote computational methods that only work for simple case problems.  They recommend calculators for everything else.  They refuse to acknowledge the elegant design of the standard algorithms.  And they're blind to what is lost.  The standard algorithms are efficient, general methods.  For example, multi-digit multiplication only requires knowing single digit multiplication facts, single digit addition facts, and the idea of carrying.  And carrying works the same for both addition and multiplication.  Students can learn how to carry out this procedure automatically, freeing the conscious mind for higher level thought.
Mastery of standard arithmetic is critical foundational knowledge for later mastery of algebra, and algebra is the gateway to higher mathematics.  Consider the following quotes from the March 2008 Final Report of the National Mathematics Advisory Panel:
The NCTM asked leading mathematicians for their advice about the standard algorithms.  Roger Howe, Professor of Mathematics at Yale, responded for the American Mathematics Society.   See the following quotes in context following the first report at Reports of AMS ARGKenneth Ross, Professor of Mathematics at the University of Oregon responded for The Mathematical Association of America.  See the following quotes in context in Second Report from the Task Force.
Copyright 2013 William G. Quirk, Ph.D.