long multiplication

  123
x 645

(Liping Ma example)

or,

 
  26
x 31

(M. J. McDermott example, see video below)

In her book, Liping Ma demonstrates that some teachers use a procedural method (long multiplication is like a recipe that you follow to obtain the correct answer) and some teachers use conceptual methods (explain how the procedure works by explaining how place value is obtained) in teaching long multiplication.

Liping Ma interviews the teachers involved. What emerges clearly is that nearly all the teachers who use procedural (recipe) methods don't actually understand the role of place value in long multiplication!

First up, should you place a zero after the 78, making it 780 in this example?

 
  26       or,       26
x 31              x  31
  26                 26
 78                 780
 806                806

Either way is acceptable provided the teacher can explain what is happening. If you don't place a zero after the 78, then the teacher needs to understand and explain that we are dealing with 78 tens or 78 lots of ten. If you do place a zero after the 78 then the teacher needs to understand and explain that in the second step we are multiplying 26 x 30, rather than 26 x 3

Explanations can vary but the teacher needs to understand what they are doing

Here is what procedural explanation teachers do and think (from Liping Ma's interviews):

They talk about place value as just a label of different columns, they don't explain the significance of that label. When they talk about the "tens column" and the "hundreds column" they use these terms as labels, not as values that require explanation.

They treat the gap after the 78 as just a placeholder which can be filled by other symbols apart from zero, such as an asterick or even something more memorable like a picture of an apple or animal

Some of them think that to add a zero after the 78 is actually incorrect

If students make a mistake, such as putting the 8 in the 78 in the units column, then they correct the error by restating the rule ("the 8 belongs in the tens column") or by using a placeholder in the units column. They do not explain why the 8 belongs in the tens column.

Here is what conceptual explanation teachers do:

They explain what place value actually means, ie. the 3 in 31 means 30

Sometimes they separate the problem into subproblems:
26 x 31
= 26 x 30 + 26 x 1
etc.

More knowledgable conceptual teachers explain this in terms of the distributive rule, which they may have taught to students earlier:
26 x 31
= 26 x(30 +1)
= 26 x 30 + 26 x 1
etc.

Some of them will deliberately put an error on the board and then engage the class in discussion about where is the error and how to correct it

Notice how M. J. McDermott fudges the place value explanation in the following video in the course of pronouncing the "correct" way of teaching long multiplication. She doesn't explain why the 8 in 78 goes in the tens column. Then after a moments hesitation she places a zero after the 8, with the words, "Sometimes we put a zero here", but does not explain why.



Liping Ma demonstrates through her interviews that the important issue is whether or not the teacher understands and can explain the maths. She also finds that many American teachers do not and that the Chinese teachers in her sample have superior understanding.

Reference:
Liping Ma, Knowing and Teaching Elementary Mathematics (1999)