Matrix Calculations in Virginia


      It cannot be denied that the best way to invert a given matrix is to

enter the numbers in a calculator or computer and push the right buttons.

This is true even of a 1 X 1 matrix:  For example, the machine gives me

[.0268168] as the inverse of [37.29], correct to more digits than is

reasonable, in view of the apparent accuracy of the original.


     The probability that I will in a practical situation want to

compute the inverse of [0] is very low, for it is extremely likely that my

measurement (or other data source) in such a case will not have been [0]

at all, but something like [.00036]. A perfect zero is never found in a

laboratory experiment, after all, and for this particular (approximate)

zero, i.e. [.00036], the calculator will give me [2778] (more or less),

rather than the legend "MA ERROR" I get for the inverse of [0].


     Who then really needs to know that zero has no inverse? In other

words, my calculator is practically infallible, and I am glad someone (or

some printed manual) taught me how to use it, in this application as in

others more complicated. With real-life data and a calculator there is no

need to trouble myself with rote memorization of rules concerning division

and multiplication by zero.


     Now, the Standards of Learning for Virginia Public Schools

(published by the Virginia Board of Education, P.O. Box 2120, Richmond, VA

23216-2120 in June of 1995) is one of the better documents of its kind, at

least in its mathematics section, which occupies pages 3-29 of an

attractively printed double-columned 8.5 X 11 paperbound book.  Its

language is almost free from the educationist jargon that afflicts so many

others, and it seldom exhibits any downright mistaken conceptions of

mathematics. It is not a complete curriculum guide, nor does it intend to

be such, but it does outline fairly specifically the main things it

expects Virginia students to be taught, grade by grade and subject by

subject, with the unspoken assumption that such details as are missing,

but without which these main things could not be taught, will also

necessarily be part of the program. Nor does it get into matters of

pedagogy, for which Virginia teachers have other sources of information.


     As is common these days, the Virginia Standards urges the use of

calculators in all applications where their use is convenient, or

instructive, or in common use in the outside world.


     "A major goal of the mathematics program is to help students

become competent mathematical problem solvers," says page 3, where also is

written, "students must learn to use a variety of methods and tools to

compute, including paper and pencil, mental arithmetic, estimation, and

calculators. Graphing utilities, spreadsheets, calculators, computers, and

other forms of electronic information technology are now standard tools

for mathematical problem solving in science, engineering, business and

industry, government, and practical affairs.  Hence, the use of technology

must be an integral part of teaching and learning. However, facility in

the use of technology shall not be regarded as a substitute for a

student's understanding of quantitative concepts and relationships or for

proficiency in basic computations."


     Apart from the implication that "practical affairs" differ from

science, business and all the other preceding items on the list (education

writers are addicted to lengthening lists, I'm sorry to say), there is

nothing to object to in these statements.  Statements of this kind,

including the closing disclaimer about "not ... a substitute ...  for

proficiency in basic computations", are routinely offered by most State

Standards in answer to members of the public who complain that their

children in the "new"  mathematics programs are "not being taught to

multiply", or worse.


     Why should the entirely reasonable desire to prepare children for

the practical world, guarding the intellectual qualities of fundamental

understanding at the same time, give rise to such heated objections? Once

a student understands the meaning of 1/7, or even a/b more generally, and

has done a few simple calculations to get decimal equivalents, is there

any reason to deprive the child of the machines we all use anyway? To the

contrary, it is argued, not having to learn the algorithms of decimal

computation, or division of lumpy fractions to produce quotients with

remainders, frees time for use in other instructional purposes.


     I would here like to quote another entry in these very Virginia

Standards to support, by analogy, an opposing point of view. It is most

usual, when arguing by analogy, to take for examples things simpler than

that which is to be illuminated, but in this case I will choose something

more complicated. To explain what is missing in education at the fourth

grade level, when calculators are urged on children in place of tedious

hand computation, "long division", say, I shall proceed to the Virginia

Algebra II standards -- an advanced high school course -- where in the

context of matrices and linear systems an analogy will be found to

illustrate what is wrong with the urging of calculators on present-day

fourth-grade arithmetic students (even in replacing tedious computation

only, be it understood, not "mathematical understanding", or

"connections", or "learning to value mathematics").


     On page 21, then, under Algebra II, the last two entries read as



     (AII.11) "The student will use matrix multiplication to solve

practical problems. Graphing calculators or computer programs with matrix

capabilities will be used to find the product."


     (AII.12) "The student will represent problem situations with a

system of linear equations and solve the system using the inverse matrix

method. Graphing calculators or computer programs with matrix capability

will be used to perform computations."


     Problems of this sort abound; certainly one can imagine a teacher

asking students to find the point of intersection of three planes whose

Cartesian equations are given; and of course statistical questions

involving large data sets might require horribly lengthy computations of

this kind, computations which were in fact impossible for practical

purposes as little as fifty years ago.


   What has been left out of these two quoted requirements? Presumably the

students have been taught how to multiply two (small) square matrices, and

have been taught what the identity matrix is, and so understand the nature

of the problem they are now solving with machinery. They have also been

taught how certain problems (e.g. the three planes' intersection) are

modeled by systems of equations whose coefficients form matrices. The

Virginia students might, though this is not certain, have been taught why

(as well as how) the inverse matrix produces the desired result; one is

not certain here because the understanding of the process requires an

understanding of multiplication for non-square matrices, among other



     (My suspicion concerning the depth of understanding of linear

algebra, or matrices, among teachers of high school algebra was raised by

a sample problem printed in the Standards of another State, a document

produced by battalions of teachers and supervisors, and reviewed by the

highest officials of the State's mathematics education heierarchy. In this

problem, A was the name given to a certain 3 X 3 matrix and B a certain 3

x 2 matrix, and the student was to compute AB and BA, and tell what law

was illustrated by the result.)


     Now the Virginia Standards does not mean to suggest a full course

in linear algebra at this point, but if "understanding" of linear systems

and matrices, even on a primitive level, is the goal, Virginia is surely

concentrating on the wrong thing, by requiring calculator answers for

inverses of (nonsingular in all likelihood) square matrices. There are

occasions where singular matrices have some theoretical meaning, for even

if a zero determinant would be a miracle in a genuine laboratory

experiment, determinants close to zero have a meaning in terms of

approximate linear dependence, something of great importance to (say)

psychologists concerned with factor analysis.  There is not much a

calculator solution will tell the student about such a situation, unless

he knows something about rank to begin with.


     Again, a problem might well be modeled with fewer or more

equations than the number of variables; what then?  Suppose the teacher

asks the students to find "the point of intersection" of *two* planes in

space? The kid is helpless.  Where's the button?  Trick question, by no

means permissible for NAEP multiple-choice purposes.


     Any mathematician will agree that the way to begin the study of

linear systems is by eliminations in the equations, rather than immediate

analysis of associated matrices.  The steps may need calculation, of

course, and a calculator can certainly be used to find 5 X 9 when

necessary, but that sort of thing, like the inverse matrix button, will be

of little help.  The steps have to be explained (by the teacher, or by the

student) as one goes along, in the form, "if there are numbers x,y, and z

satisfying (1) and (2) then they will satisfy (3) as well", and so on. If

the end result is "5 = 17," a valuable lesson in the logic of algebraic

processes will be taught, or must be thought about; but the machine will

not do this.  If in the end the equation "0 = 0" appears, yet another

mathematical statement will need interpretation.  In both cases the

student will be impelled to learn something about the possibilities. With

more examples to play with, he will learn more, especially if the teacher

or the book begins to organize the possible results, gradually

abbreviating the notation until matrices appear.


     There is no need to go on with this particular lesson in the present

discussion, except to summarize: that the major intellectual value of

studying matrices in connection with linear systems is concealed rather

than elucidated by calculator exercises as suggested by the Virginia

prescriptions. Nor is it merely an intellectual value -- "merely", indeed!

Without that intellectual background, anything but the most trivial uses

of even the calculator will be incomprehensible to that student, for all

that he appears to be getting an up-to-date education in linear systems,

using "up-to-date methods".


     It is not calculators, in short, that can teach us linear algebra;

it is we who must teach the calculators linear algebra. All the

calculator can do is count.


     Yes, the calculations will be tedious, when students perform

eliminations to achieve superdiagonal forms, for example, but they are not

being done in school for the sake of numerical answers. The tedium is

necessary, since two equations in two unknowns will not suffice to

illustrate the phenomena in a memorable way. Actually, as was seen above,

even one equation in one unknown illustrates part of the lesson, the

singular case for the square matrix, but it is doubtful that anyone who

has not been through a higher dimensional workout will recognize this way

of looking at "division by zero". For some illustrations, four dimensions

are probably desirable. You and I and the hypothetical student will still

use calculators for what they are good for, in business or in practical

things (either one), but in teaching mathematics something else is called

for, and that something is obscured by the hasty use of calculators to

solve cut-and-dried problems, even if these are the kind that occur most

of the time.


     Good teachers have, by the way, contrived exercises of this sort

where the calculations are unreasonably simple, not at all the sort of

thing that turns up in a clinical experiment.  They are deliberately not

"real-life" numbers; otherwise the hand calculations in eliminating

variables would obscure the lesson in algebra.  The prescription of a

calculator makes multi-decimal entries as easy as single digits are to the

hand, but that sort of simplification of the problem has no didactic value

at all.  When this kid becomes a medical researcher he will of course have

to deal with awful-looking numbers; sufficient unto the day is the evil



     To return now to the elementary school, these same Virginia

Standards prescribe the use of calculators in and above the fourth grade

where multiplication of decimally expressed numbers involves factors of

more than two significant figures; and the same for divisions. I myself

bristle at the suggestion, and immediately liken it in my mind to the

matrix instruction I have just described. My question now is, how does the

linear algebra example help prove that children should learn the algorithm

for "long division"? To me it seems relevant, but I recognize that not

everyone will see it so.


     I can hear the opposition now: Shall they also be required to

learn how to take square roots by the 19th century algorithm that begins

with grouping the digits by twos and playing games with the pieces?  Shall

they later have to learn Horner's Method for solving polynomial equations?

Memorize Cardan's Formulas for cubics and quartics?


     One can become quite sarcastic about the apparent love that mathe-

maticians and old-fashioned parents have for "long division". It might be,

someone will suggest, that we want to teach them "mental discipline" for

some Victorian moral purpose scorned by the Deweyite educational

philosophers of sixty years ago. And to deprive them, too, of the

sweetness of childhood.


     But I say the line between a truly inessential drill, like

Horner's Method, or lessons in interpolation in seven-place logarithm

tables, and a tedious calculation that teaches something valuable, like

the elimination of terms in a set of linear equations, is a matter of

mathematical perspective. Only a person ignorant of mathematics and its

uses in (say) such places as statistics and physics would imagine that

Gaussian elimination and the search for linear dependence was a mindless

ritual, forced on bored children along with Horner's Method and a page or

two of practically identical trinomials to be factored.  Outdated, in this

Nintendo Generation! Pointless, in a world where there is an Internet to

explore! Mere memorization!


     It is not hard to remind a mathematician of what is missed by a

student who follows the Virginia prescriptions concerning matrices and

linear systems; and probably with a little time a college teacher of

algebra can teach a potential high school teacher much the same lessons.

But it seems not to be so easy to exhibit the difference between the waste

of time involved in learning Horner's Method and the "waste of time"

involved in learning "long division". There are even people with

Doctorates in mathematics education who believe, with the Virginia

Standards, that "long division" is quite useless now that we have

calculators, and that spending valuable fourth grade time on it alienates

students and does them no ultimate good.


     It is because of this difficulty of explanation that I brought up

the analogy with linear algebra just now. It seems to me that any

mathematician should instinctively know the difference between the fourth

grade lessons that can be taught via the long-division algorithm and the

lessons that were "taught" in 1920's school exercises in square roots or

Horner's Method. In my recent experience, my mathematician friends are

generally horrified at the thought that children should be driven to

calculators for operations of this sort from principle, as an

*educational* necessity. They deplore this tendency in the schools. On the

other hand, the instincts of the school mathematics supervisors, the

writers of Standards in the State Education Department, are just the

opposite: they feel themselves in the vanguard of a revolution in

mathematics teaching when they drop the hated algorithm in favor of the



      Demonstrably they haven't seen the point in the case of Virginia's

lessons in solving linear systems, either. But while in this case it is

not too hard to explain what is missing, and what is to be condemned, in

teaching nothing but a computer scheme for solving the "most common"

linear systems, in the fourth grade case what is missing when a child gets

insufficient exercise in decimal computations is a little harder to get

hold of, particularly as fourth grade students are taught by fourth grade

teachers, and not specialists in mathematics education, let alone

mathematics. Yes, it is essential that one appreciate the structure of our

decimal notation; it is an aid to mental approximations, for example. Yes,

long division has later generalizations of importance, in the treatment of

polynomial algebra, for example. These might be explained to the composer

of fourth-grade curricula, yet rejected on the grounds that most people do

not need such subtleties.  Sure, and "most" division problems do not have

a zero denominator


     To me it seems that certain exercises in calculation are probably

mainly of psychological advantage, a sort of wiring-in of the nature of

the decimal system (with possible later generalization made easier),

rather than something easily pointed to, like the meaning of rank in

linear transformations. But to point such things out is not as convincing

as the exhibition of a set of two linear equations in three variables, to

the person with the all-powerful graphing calculator.


     I trust the instinctive reaction of the mathematicians I know,

their disapproval of the current fashion of leaving arithmetic to machines

when teaching young children its "meaning" and "use", free of irrelevant

tedium. I therefore hope this mathematical community could come up with an

explanation concerning decimal arithmetic that would also convince those

in charge of our young children's mathematical education that the issues

lie deeper than that, and before it is too late. Analogy is seldom a

convincing argument, except to people who see the analogy, and these are

usually an audience that doesn't need it. Intuition is never considered a

trustworthy argument, though it is often persuasive to its holder, and

maybe his admirers. How shall we ever be able to get the point across to

strangers, and non-mathematicians?


   (There is of course the possibility that my own instincts are wrong

here, or not as generally shared as I imagine, in which case I'd like

to see the proof of that, too.)


     The usual arguments against exercise in the classical algorithms

of decimal arithmetic (and arithmetic of fractions, too!) tend to refer to

all those generations of children who in fact didn't learn anything of

intellectual or mathematical import from their grade-school lessons in

arithmetic. The usual counter-argument (from the mathematicians' side) is

that they weren't taught properly. I am willing to hear statistical

information on the results of various kinds of teaching, as well as

theoretical arguments tending to show what must be lacking if certain

sorts of lessons are not offered.


     Footnote: There are those who think mathematicians are definable

as people who are able to multiply -- very rapidly -- very large numbers

together. Such people can be made to believe that mathematicians oppose

the use of calculators in the schools because they fear for their own

honors and emoluments, once the public discovers that their services,

having been taken over by machines, will no longer be needed. Such people,

too, have to be persuaded that their children should learn multi-digit

arithmetic in the schools, if indeed they should be, even if they

themselves are no good at it, and suspect we have a vested interest in

offering arguments in favor of arithmetic calculation.  Even so, they are

not going to be as hard to convince as the educational theorists, whose

ignorance is also harder to forgive.


Ralph A. Raimi