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There is a lively and vigorous debate currently underway among teachers, parents, and educators on how to educate our children, particularly in mathematics, so they can take advantage of the many opportunities offered by a scientifically advanced economy such as our own. I have created this website in order to contribute to this debate with specific suggestions of a pedagogical nature based on my many years of teaching mathematics at both the undergraduate and graduate levels. One particular suggestion I am promoting is Calcul Mental, which is an instructional method used in France (5th grade level through middle school) for developing computational skills in the context of a mathematically sound theory of arithmetic. The goal is for students to acquire, at each grade level, not only the age-appropriate computational skills, but the mathematical reasoning embedded in them. In particular, the computational algorithm must be consistent with, and indeed illustrate the underlying theory. In short, Calcul Mental prepares the middle school student for the more abstract thinking required in high school algebra and geometry. Students in the 5th grade level (age 10 years), for example, are taught how to simplify computing a sum by rearranging and regrouping the summands as in the following example: Consider the problem of adding the numbers 4+7+11+9+13 in the given order, versus simplifying the calculations by first rearranging the order and then regrouping as follows: (7+13)+(11+9)+4=20+20+4=44. Rearranging and regrouping the summands in this fashion are just simple applications of the commutative and associative laws of addition, although it would be too pedantic to explicitly point this out to a ten-year old child. The importance of this point is stressed repeatedly by Liping Ma in her penetrating book comparing teachers' understanding of fundamental mathematics in China and the United States . ``Elementary mathematics", writes Professor Ma, ``contains the rudiments of many important concepts in more advanced branches of the discipline. For instance, algebra is a way of arranging knowns and unknowns in equations so that the unknowns can be made knowable [that is solved for]. ….. the three basic laws by which these equations are solved-commutative, distributive, and associative-are naturally rooted in arithmetic." [Liping Ma (1999), Knowing and Teaching Mathematics, Lawrence Erlbaum Associates]

In our country, unfortunately, ``Many students enter High School not knowing how to add or subtract without a calculator; and even the better students do not understand fractions." [Washington Post, Section B, Sunday, March 21, 2010.] One of the consequences is that many manufacturing jobs, even today, go unfilled, despite the current high level of unemployment, ``because the automated factories demand workers who can operate, program and maintain the new computerized equipment. Many of those who have been laid off can operate only the old-fashioned manual machines.'' [Peter Whoriskey, Washington Post, page A1 February 20, 2012.] What is the difference between the new and old machines? Mr. Whoriskey explains: ``The old lathes and mills were operated by hand and turned out pieces one by one. The new ones, as big as minivans and arrayed with screens and buttons, must be programmed with codes that sometimes look arcane. The computer numerically controlled, or CNC, machines can cost hundreds of thousands of dollars. Once they are programmed, they churn out piece after piece unattended.

One day last week, for example, Greg Rowles, 27, a former tow truck driver who is now a CNC programmer, was working on a machine to shape a metal part at the Vickers Engineering plant in New Troy. He took some classes at the local community college. The codes he typed in looked like this: G54G90G0B0 M7; G4X3.; G81Z-.829R.1F28.;

The last line, he said, tells the machine to take a tool and drill 0.829 inches deep at a certain rate of speed. "It takes a while to learn," he said. The leap in technology means that many of the workers who once toiled on the old machines, and had become proficient on them, can no longer find jobs."

Here is another example, based on my own personal experience with a sales clerk at an Eastern Mountain Sports store located near Amherst, Massachusetts--where I was a professor of mathematics at the University of Massachusetts—that suggests the innumeracy problem is much more serious than most people can imagine. Needing a good winter parka and responding to an advertised ``20% storewide discount sale" at EMS, I went there to buy one. It didn't take me long to find just the parka I was looking for, but I couldn't find the price tag. So, I asked the sales clerk, ``What is the price of this parka after the 20% discount?" I expected her, naively perhaps, to first look for the price tag, subtract the 20% discount, and compute the final sale price. None of this, needless to say, required even a calculator. Without even attempting to look for the price tag, she replied,

``Oh, I can't do that kind of math in my head. But," noting the bewildered expression on my face, she quickly added with a smile, ``I can compute a 10% discount."

While I was pondering how one could compute a 10% discount but not a 20% one, she found the price tag, which was $180. Unable to restrain myself I gave a brief mathematical lecture.

``Look," I said, `` you can do a 10% discount, right?"

``Yes", she replied.

``Well, here's how to compute a 20% discount: 10% of $180 is $18, is it not?"

``Yes", she replied.

``So, 20% of $180 is two times $18, which equals $36, right?"

``Yes."

``So the final sales price is $180-$36=$144, correct?"

After briefly reflecting on my computations, she smiled and exclaimed, ``How neat!"

The author, Walter A. Rosenkrantz is a mathematics professor with many years of experience teaching in a wide variety of public and private universities, including Dartmouth College, New York University, University of Massachusetts (Amherst), and Purdue University; see the curriculum vita for more details. Upon his retirement from the University of Massachusetts (Amherst) in December 2004 he and his wife moved to Washington, DC to be close to their children. Shortly afterwards, upon the recommendation of a former student of his, he ``unretired" and has been teaching at George Washington University in both the departments of mathematics and statistics with the title of Professorial Lecturer.

email: rosenkrantzw@calculmental-teacher.com

- 1957 University of Chicago, B.A., B.S. (Math, with honors)
- 1959, University of Illinois, M.S. (Math)
- 1963, University of Illinois, Ph.D. (Math)

- Fall 2011-2012, George Washington University, Professorial Lecturer, Department of Statistics
- Fall 2005- Spring 2011, George Washington University, Professorial Lecturer, Department of Mathematics
- Fall 1971- Dec. 31, 2004, University of Massachusetts (Amherst), Professor of Mathematics
- 1965-1971, New York University, Washington Sq. College, Associate Professor of Mathematics
- 1962-1964, Dartmouth College, John Wesley Young Research Instructor

- Sept. 1985- August 1986, INRIA (Institut National de Recherche en Informatique et en Automatique, Rocquencourt, France), Directeur Scientifique. Note: This sabbatical leave was supported in part by the Air Force Office of Scientific Research (AFOSR), INRIA, and the University of Massachusetts.
- 1980 (June 1-30), Bell Labs, Holmdel, N.J., Consultant
- 1976-1977, Purdue University, West Lafayette, Indiana, Visiting Associate Professor
- 1964-1965, Visiting Member (Post Doctoral Appointment), Courant Institute of Mathematical Sciences, New York University

- Introduction to Probability and Statistics for Scientists and Engineers, McGraw-Hill Series in Probability and Statistics, 592 pages, 1997
- Introduction to Probability and Statistics for Science, Engineering, and Finance, Chapman& Hall, 667 pages, 2008

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