The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms PapersThe 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" ExperienceConstructivist
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. 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. "
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
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 1: "The Harmful Effects of Algorithms in Grades 1-4,"
paper isn't available on the internet, but another paper, "The Harmful
Effects of Carrying and Borrowing" also covers this research and it's available at Constance Kamii's website.
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
are considered "standard." When forced to get specific, they say
"conventional U. S. algorithms," implying that the standard algorithms
are not standard elsewhere.
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
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,"
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
algorithms" students. But Kamii
and Dominick are leading advocates for constructivist computational methods,
so it's highly likely that the "No algorithms" students
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
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.
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."
- '"No Algorithms 'Test' Takers: "Most of
in the 'No algorithms' classes typically began by saying, 'A
hundred eighty and fifty is two hundred thirty.' This is why even if
they made errors, the answers of the 'No algorithms' classes were
mostly between 230 and 260."
- There's ample time in "No algorithms"
classes for mental math methods. This is a major focus for
constructivist math educators. We can be sure that the "No algorithms"
students had frequently seen problems similar to Kamii and
Dominick's 7 + 52 + 186 'test' problem.
- Algorithms 'Test' Takers: "Typically,
they started by saying “7 + 2 + 6 = 15. Put down the 5 and carry the
one. One and 5 and 8 is 14. Oh! I forgot what I put down here. I’ll
- The use of "Typically" suggests that several students were correctly
using the paper-and-pencil standard addition algorithm, but got lost because the rules didn't allow
them to say or write down the answer one digit at a time [first 5 for the ones
digit, then 4 for the tens digit, etc.].
- What do they mean by "give up their own thinking?"
The standard algorithms for addition, subtraction, and
require that the student work from right to left.
Kamii and Dominick say that children naturally think left to
What do they mean by "unteach" what children know about place value?
The standard addition algorithm
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.
Kamii and Dominick didn't go on to say that children should use "their
own thinking" to "invent" their own algorithms. Kamii
has promoted the idea of child-invented algorithms elsewhere, but
this is really a sham. In programs such as TERC's Investigations,
the same methods are always "invented" by the children.
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.
- Example 1:
7 + 52 + 186 = 180 + 50 + 7 + 2 + 6 = 230 + 7 + 2 + 6
= 230 + 15 = 245 Kamii and Dominick wrote: "Most of
the children in the “No algorithms” classes typically began by
“A hundred eighty and fifty is two hundred thirty.”
- Example 2: 7 + 52 + 186 = 8 + 52 + 185 = 60 + 185 = 245. This is Everyday Math's "opposite-change rule. In called called "changing the numbers." in TERC's Investigations.
- Example 3: 7 + 52 + 186 = 10 + 50 + 180 + 5 = 245. This is TERC's "changing the numbers" to "landmark numbers."
- Example 4:
Everyday Math's partial sums method compared to the standard algorithm for addition
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
- Example 5: Everyday Math's partial products method compared to the standard algorithm for multiplication
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
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
specific place value associated with each column.
be pleased, but they're seriously misleading students. The partial
sums and partial products methods can't be carried out
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
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:
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
- "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
- Strategy 1: Claim that the standard algorithms are obsolete due to the power of calculators. Consider these quotes:
calculator renders obsolete much of the complex paper-and-pencil
proficiency traditionally emphasized in mathematics courses." -
- "There is no need for students to
understand and be able to apply paper-and-pencil computations with
complicated numbers, since such computations can be performed more
quickly and accurately with a calculator." - Everyday Mathematics
- Strategy 2: Claim that learning the standard algorithms has a harmful effect on conceptual understanding.
- Strategy 3: Fill the curriculum with constructivist versions of more advanced mathematics.
percent of K-6 math should be devoted to the mastery of standard
arithmetic for whole numbers, fractions, and decimals.
Constructivist K-6 programs replace most of this with advanced
topics that can't really be covered without prior mastery of standard
- For example, TERC's Investigations
fifth grade program offers probability without multiplying fractions,
statistics without the arithmetic mean, 3-D geometry without formulas
for volume, and number theory without prime numbers.
- Strategy 4: Fill the curriculum with 'hands-on" activities.
example, TERC insists on the ongoing use of hands-on tools, or
manipulatives. They say concrete tools must always be available
and regularly used. TERC strongly rejects the idea that children
must eventually migrate from hands-on to abstract thinking.
- Strategy 5: Schools of education don't cover how to teach standard paper-and-pencil arithmetic.
- There is only one elementary education math course available at The University of Alabama Birmingham [where
Constance Kamii teaches]. Just one course can't even begin to
cover how to teach elementary mathematics. Here's the vague
- ELE 412 Elementary School Curriculum: Mathematics - Materials
and methods of teaching mathematics. Scope, sequence, and content of
mathematics program. Computational skills, problem solving, and
discovery learning. Includes field experiences.
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.
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
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 ARG.
- Computational proficiency with whole number operations
is dependent on sufficient and appropriate practice to develop automatic recall of addition and
related subtraction facts, and of multiplication and related division
facts . It also requires fluency with the standard algorithms for
addition, subtraction, multiplication, and division. [PDF page 19]
the term proficiency, the Panel means that students should understand key
concepts, achieve automaticity as appropriate, develop flexible, accurate, and automatic execution of the standard
algorithms, and use these competencies to solve problems.
- [PDF pages 17 and 50]
- The Panel cautions that to the
degree that calculators impede the development of automaticity, fluency in
computation will be adversely affected. [PDF pages 24 and
Kenneth 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
- An important feature of algorithms is that they are automatic and do
not require thought once mastered. Thus learning algorithms frees up the brain
to struggle with higher level tasks. On the other hand, algorithms frequently
embody significant ideas, and understanding of these ideas is a source of
mathematical power. - Page 273
- The virtue of standard algorithms—that they are
guaranteed to work for all problems of the type they
deal with—deserves emphasis. - Page 275
- We would like to emphasize that the standard algorithms of arithmetic are
more than just "ways to get the right answer"—that is, they have theoretical as
well as practical significance. For one thing, all the algorithms of arithmetic
are preparatory for algebra, since there are (again, not be accident, but by
virtue of construction of the decimal system) strong analogies between
arithmetic of ordinary numbers and arithmetic of polynomials. The division
algorithm is also significant for later understanding of real numbers. -
- In constructivistic terms, individuals may well understand and
visualize the concepts in their own private ways, but we all still have to learn
to communicate our thoughts in a commonly acceptable language. - Page 1
- Success in mathematics needs to be grounded in well-learned
algorithms as well as understanding of the concepts. None of us advocates
"mindless drills." But drills of important algorithms that enable students to
master a topic, while at the same time learning the mathematical reasoning
behind them, can be used to great advantage by a knowledgeable teacher. Creative
exercises that probe students' understanding are difficult to develop but are
essential. - Page 1-2
- The challenge, as always, is balance. "Mindless algorithms" are powerful
tools that allow us to operate at a higher level. The genius of algebra and
calculus is that they allow us to perform complex calculations in a mechanical
way without having to do much thinking. One of the most important roles of a
mathematics teacher is to help students develop the flexibility to move back and
forth between the abstract and the mechanical. Students need to realize that,
even though part of what they are doing is mechanical, much of mathematics is
challenging and requires reasoning and thought. - Page 2
Copyright 2013 William G. Quirk, Ph.D.