467

x 846

2802

1868

3736

395082

- 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].
- 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).
- For PP 2802: carry the 4 for 6 x 7 = 42 and carry the 4 for (6 x 6) +4 = 40
- For PP 18680: carry the 2 for 4 x 7 = 28 and carry the 2 for (4 x 6) + 2 = 26
- For PP 373600: carry the 5 for 8 x 7 = 56 and carry the carry the 5 for (8 x 6) + 5 = 53
- The standard algorithm of addition is used to add the 3 partial products. This algorithm uses the following single digit addition facts:
- 0 + 2 = 2, 8 + 0 = 8, 6 + 6 + 8 = 20, 2 + 3 + 8 + 2 = 15, and 1 + 7 + 1 = 9, 3 + 0 = 3
- Carrying is used 2 times (the 2 from the 20 and the 1 from the 15).

- Multi-digit multiplication is reduced to a problem in multi-digit addition.
- Multi-digit computations (for multiplication and addition) are reduced to single digit number facts.
- Carrying works the same way for every place value column. There's no need to think of the specific place value for any column.
- Carrying works the same way for addition and multiplication.
- With sufficient practice, the standard multiplication algorithm can be carried out automatically.
- This is an efficient, general method that works for decimals in exactly the same way it works for whole numbers.
- This is the multiplication algorithm that students need to master in order to prepare for algebra.

- Constructivists say that working right to left is not natural for children.
- 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.
- Constructivists say that the concepts of carrying and borrowing are too difficult for most children to understand.
- 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.

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

- 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.
- 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.
- For example, for the second column from the left, 10000 + 40000 + 20000 yields 70000.
- Next, working left to right, the 6 partial sums are added by adding column digits (this is the constructivist "partial sums method" discussed below).
- Note that this works fine for the first 2 columns (left to right), but the partial sums method (for this example) breaks down for the 3rd column (from the left) because adding digits for this column yields 4, but we need a 5 there because the sum for the 4th column (from the left) is 10 and that means that a 1 needs to be carried one column to the left [from the hundreds 3rd column to the thousands 4th column].
- More generally, the partial sums method doesn't work if the sum of the digits in any column is greater than 9.

- The student must consciously think of the specific place value associated with each column.
- 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.
- "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." - Everyday Mathematics Grades 4-6 Teacher's Reference Manual, Page 115
- Note: Partial products is the Everyday Math "focus algorithm" for multiplication.
- 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.

- The student must consciously think of the specific place value associated with each column.
- 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.
- 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.
- Constructivists favor the partial sums method because:
- The process proceeds left to right.
- Ongoing conscious thought about place value is necessary.
- Carrying is avoided when adding just 2 numbers and for other carefully chosen problems.
- Note: Partial sums is the Everyday Math "focus algorithm" for addition.

- We
do not think it wise for students to be left with untested private
algorithms for arithmetic operations—such algorithms may only be valid
for some subclass of problems. The virtue of standard
algorithms—that they are
**guaranteed**to work forproblems of the type they deal with—deserves emphasis. - Roger Howe. Professor of Mathematics, Yale University.**all** - 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. - Roger Howe. Professor of Mathematics, Yale University.
- 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. - Kenneth Ross, Emeritus Professor of Mathematics, University of Oregon
- 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. - Roger Howe. Professor of Mathematics, Yale University.
- Students must know basic arithmetic operations as applied to polynomials. That is, they must be able to add, subtract, multiply, factor, and divide polynomials and their fractions. They must be able to do this quickly (and accurately) without having to think too much about it. Thus it is important to learn to use the standard algorithms for these operations, the algorithms which have been developed by society over hundreds of years (keep in mind that we in the ``western world'' have only been using Arabic numerals for about 400 years). These algorithms generalize nicely to polynomials and the student with good facility with numbers will most likely be able to adapt the same skills to polynomials. This facility is gained in the lower grades. - W. Stephen Wilson, Professor of Mathematics, Johns Hopkins
- 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
- The importance of repetition until automaticity cannot be overstated. Repetition is the key to learning. - John Wooden

Related Essays by Bill Quirk

- The Bogus Research in Kamii and Dominick's Harmful Effects of Algorithms Papers
- The Parrot Math Attack on Memorization

Copyright 2014 William G. Quirk, Ph.D.