Concept and Computation: The teacher's role
John T. Baldwin
The `concept-compute' debate may be impeded by misunderstandings of what is meant by
`teaching algorithms'. The diverse meanings of this phrase is illustrated in Liping Ma's book:
Knowing and Teaching Elementary Mathematics:Teachers' Understanding of Fundamental
Mathematics in China and the United} States. She compares the reactions of 72 `representative'
Chinese elementary school teachers with that of 23 `better than average U.S. teachers to four
mathematics problems. The 4 problems involved subtraction with regrouping, a multidigit
multiplication problem, division of fractions, and the relationship between perimeter and area.
The first two problems `just' require carrying out an algorithm. But the reaction of the two
groups of teachers reveal that this `just' is hiding a lot.
Perhaps, the multiplication problem was most revealing. The teachers were shown a problem where in multiplying by a three digit number, the student had placed the right digits of the three partial products directly under one another and added. All teachers correctly identified the error and could do the problem correctly themselves. But in explaining how they would remediate this problem a wide divergence emerged between the two groups. 70% of the American teachers and only 8% of the Chinese teachers had what Dr. Ma identifies as a `procedural' rather than a `conceptual' understanding of the problem. That is, most of the American teachers identified the difficulty as failure to memorize the technique of moving the products to the left. In contrast, the bulk of the Chinese teachers identified the difficulty as a failure to understand place value and the concept of multiple digit multiplication. This distinction is emphasized by the fact that 61% of the American teachers thought that putting in the `hidden zeros' at the end of each partial product was essentially alien to the computation. Many thought it simply wrong; several proposed using asterisks as placeholders.
In contrast, the Chinese teachers proposed understanding the algorithm in terms of the
distributive law and emphasized that multiplying by 4 in the 10's place is really multiplying by
40. (Most interesting, interviews of some of the American teachers elicit a superficially correct
reference to the 10's column, which further questioning showed to be simply a memorized name
for the second digit from the right that did not reflect an understanding of place value.) Th other
examples show similar emphasis on procedure over understanding. (Strikingly, only 52% of the
U.S. teachers could calculate $1\ 3/4 \div 1/2$; only 1 of the 23 U.S. teachers proposed a word
problem which required the calculation intended. Most modeled $1\ 3/4 \times 1/2$.)
Dr. Ma traces most of the difficulty to inadequate elementary school mathematics; 9th graders in
China demonstrated more competence and understanding than the American teachers though less
understanding than the Chinese teachers. Thus the difficulty is partly just the weak mathematical
background of U.S. elementary teachers. But, more incisively, the book grounds some of that
weak background in the American curriculum. E.g., teaching only the procedure for multidigit
multiplication rather than grounding it in an understanding of place value. Dr. Ma's remedy is
to address both teacher knowledge and student learning in our teacher preparation courses. Her
book surely provides a beginning. These 4 problems should be embedded in the mandatory
course for all elementary teachers in this country. But they are just samples. She also points out
that curriculum reform can play a major role. For example, much of the difficulty in the
subtraction problem came from the American teacher's use of a `borrowing' metaphor which
focuses attention on the relation between adjacent digits as opposed to a `regrouping' metaphor
which focuses attention on alternative representations of the same number. The borrowing
metaphor had been introduced to Chinese schools from the West about 100 years ago and was
removed in the 1970's.
Dr. Ma identifies 10% of the Chinese teachers in the study and none of the Americans as having `a profound understanding of fundamental mathematics': that is, a broad understanding of the connections and rationales for the various pieces of mathematics in the elementary curriculum. She notes that this broad understanding arises after teachers have been in service for many years. Such a development requires the careful study of both the mathematics and the techniques for teaching it. Several important aspects of the Chinese system further this development: teachers `rotate through' to teach students of all ages; elementary teachers have about four 45 minute classes per day; in between they have and take time to deepen their understanding of the mathematics and how to teach it.
[The comparison of Chinese and American teachers in Ma's book is perhaps a bit unfair to the individuals. The Chinese teachers were spread among grades K-8; the American appeared concentrated in the lower grades. 80% of the Chinese were mathematics specialists; I could not find similar information on the American group. But while this may not be a fair comparison of the individuals it was an appropriate comparison of the systems. The assertion that `above average' American teachers would react in such a procedural fashion is not a big surprise. But Dr. Ma performs a great service by making the deficiencies specific.]
These episodes provide several examples (borrowing and multiplication) where the unthinking
memorization of mathematically correct procedures may in fact impede understanding. Teachers
may introduce mnemonics which are helpful to the memorization (e.g. putting `barrels' in the
blank spaces in the multiplication problem) which confound true understanding. In assessing
current teaching we should examine what is actually taking place in the classroom rather than
relying on our recollections of our own childhood. I hope hat both sides of the
concept/procedure debate would agree on the approaches to teaching conceptually correct
algorithms suggested by the majority of the Chinese teachers.
Liping Ma: Knowing and Teaching Elementary Mathematics Teachers' Understanding of
Fundamental Mathematics in China and the United States, (1999) Lawrence Erlbaum
Associates,Mahwah, N.J., www.erlbaum.com available May 1999 \$45 hardcover