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.

Note 2: 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.

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.

- '"No Algorithms 'Test' Takers: "Most of the children 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 start over."
- 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 multiplication require that the student work from right to left. Kamii and Dominick say that children naturally think left to right.
- Note: 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.
- 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.

- 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 saying, “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

53

+ 6 11 << Carry

100 (Add the hundreds) 185

130 (Add the tens) 53

14 (Add the ones) + 6

244 (Add the partial sums) 244

- 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

2400 395082

280

2400

360

42

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:

- "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
- "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

- Strategy 1: Claim that the standard algorithms are obsolete due to the power of calculators. Consider these quotes:
- "The calculator renders obsolete much of the complex paper-and-pencil proficiency traditionally emphasized in mathematics courses." - NCTM Standards
- "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.
- Constructivists cite Kamii and Dominick's papers as major evidence supporting this claim. Mathematician, H. Wu, directly addressed Kamii and Dominick in "Basic Skills Versus Conceptual Understanding - A Bogus Dichotomy in Mathematics Education." Knowing that the only possible problem was a teaching failure, Professor Wu explained how to correctly teach standard arithmetic.
- Strategy 3: Fill the curriculum with constructivist versions of more advanced mathematics.
- Eighty 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 arithmetic.
- 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.
- For
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 course description
**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.

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:

- 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]
- By 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 78]

- 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. - Page 275

- 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.