On Mathematical Reasoning in School Mathematics

Part 2 of A Mathematical Manifesto
by Ralph Raimi for NYC HOLD
October, 2002

2. On Mathematical Reasoning in School Mathematics

Civilized people have always recognized mathematics as an integral part of their cultural heritage. Mathematics is the oldest and most universal part of our culture, in fact, for we share it with all the world, and it has its roots in the most ancient of times and the most distant of lands.

The beauty and efficacy of mathematics both derive from a common factor, that distinguishes mathematics from the mere accretion of information, or application of practical skills and feats of memory. This distinguishing feature of mathematics is called mathematical reasoning, reasoning that makes use of the structural organization by which the parts of mathematics are connected to each other, and not just to the real world objects of our experience, as when we employ mathematics to calculate some practical result.

This essence of mathematics is its coherence, a quality found elsewhere, to be sure, but preeminently here. Knowledge of one part of a logical structure entails consequences which are inescapable, and can be found out by reason alone. It is the ability to deduce consequences, results which otherwise would require tedious observation and disconnected experiences to discover, that makes mathematics so valuable in practice. Only a confident command of the method by which such deductions are made can bring one the benefit of more than its most trivial results.

In 1997 the National Council of Teachers of Mathematics (NCTM) asked the Mathematical Association of America (MAA) for advice in revising the 1989 NCTM Standards, and the resulting report, written by Kenneth Ross, formerly President of MAA, argued for early emphasis on mathematical reasoning:

"... [Teachers] should recognize [the] theoretical nature [of mathematics], which idealizes every situation, as well as the utilitarian interpretations of the abstract concepts... If reasoning ability is not developed in the students, then mathematics simply becomes a matter of following a set of procedures and mimicking examples without thought as to why they make sense."

Even a young child can be shown how the memorization of tables of addition and multiplication for the small numbers (one through ten) necessarily produces all other information on sums and products of numbers of any size whatever, once the structural features of the decimal system of notation are fathomed and applied. The proper teaching of the standard algorithms of arithmetic is in fact the teaching of a particular example of mathematical reasoning, even though at the fifth grade level the procedures are not codified into "theorem-proof" format. At a more advanced level, the knowledge of a handful of facts of Euclidean geometry -- the famous Axioms and Postulates of Euclid, or an equivalent system -- necessarily imply (for example) the useful Pythagorean Theorem, the trigonometric Law of Cosines, and an inexhaustible tower of truths beyond.

Any program of mathematics teaching that slights these interconnections doesn't just deprive the student of the beauty of the subject, or his appreciation of its philosophic import in the universal culture of humanity, but even at the practical level it burdens that child with the apparent need for memorizing large numbers of disconnected facts.

People untaught in mathematical reasoning are not being saved from something difficult; they are, rather, being deprived of something easy.

It is unfortunate that so many of today's textbooks and programs, both the traditional and those called "reform" by their authors, conflate problem-solving and mathematical reasoning in the sense considered here. Both are important, but they are not the same. Mathematical reasoning is not found in the connection between mathematics and the real world, but in the logical interconnections within mathematics itself. The kind of reasoning by which some aspect of a real-world situation is translated into mathematical terms is certainly reasoning, and by no means easy to come by, and it is the basis of the uses of mathematics in science and daily life as a model for phenomena, thus naturally of extreme importance -- but it is not the subject of the present commentary.

In the American schools of the past century, however, the uses of mathematics have unfortunately overwhelmed the deductive reasoning which is the basis for our trust in the efficacy of mathematics itself, and the means by which we are able to organize and civilize, as it were, the innumerable truths of mathematics without ourselves becoming overwhelmed by their immensity.

Since children cannot be taught from the beginning "how to prove things" in general, they must begin with experience and facts, until, with time, the interconnections of facts manifest themselves and become a subject of discussion, with a vocabulary appropriate to the level of sophistication of the ideas at issue. Children then must learn how to prove certain particular things, memorable things, both as examples for reasoning and for the results obtained. The quadratic formula, the volume of a prism, and why the angles of a triangle add to a straight angle, for example. What does the distributive law have to do with "long multiplication"; why is the product of two numbers equal to the product of their negatives; why do the areas of similar figures vary as the squares of corresponding linear measurements?

It is neither necessary nor wise to attempt to construct an axiomatic basis for all mathematics and cause children to prove everything else before they can use it. It has not even been done satisfactorily by the mathematical profession itself, for the study of the foundations of mathematics continually throws up puzzles, contradictions, philosophical conundrums which there is reason to believe will never be fully resolved. Furthermore, even a mathematician uses facts that others have proved, and sometimes would be hard pressed to prove some of them if suddenly asked (though he would surely have done it at some point in his career). He uses these facts, drawn from memory, as if they were axiomatic, in obtaining the next fact, whether theorem or calculation, used in his work.

For a student much the same thing should be the case. Some of the most fundamental facts used in mathematics are beyond the power of a school child to deduce from a minimal set of "self-evident" principles. Even the very definitions of mathematical objects used every day are not easily understood without some prior education in deduction. "What is a real number?", for example, was answered not much more than a century ago, and the details even now are not for children; yet children must use real numbers and get used to knowing some of their properties. Still, this does not mean that mathematics for such children should be nothing but the following of rules given on authority alone. Despite the difficulty of the subject, a child in middle school can learn enough about real numbers to be able to reason further, whether about quadratic equations or geometric constructions. And it remains mathematical reasoning even if he cannot prove every earlier statement back to the self-evident truths of counting.

Two things are most important in this connection: First, that the student be aware, at every stage of his education, what he is entitled to believe and use; and second, that he know whether some things he believes are logically connected with others even if he himself cannot yet fully elucidate the connection. Whatever Authority might wish to say of two logically dependent truths, he should be taught, it cannot command us to believe both the first and the denial of the second. One hopes, and one hopes the student hopes, that the full logical connection will some day be discovered, or taught, but even short of this there is value in learning, via reason, as much of that connection as suits the mathematical maturity of the student.

For example, in Euclidean space, which is an idealization of the space we seem to live in, we know axiomatically such things as that it is filled with lines, planes and points, that two points determine a line, that in a plane one and only one line parallel to a given line exists through any point not on the given line, that the three angles of a triangle add up to 180 degrees, and that in a right triangle the square on the hypotenuse is equal in area to the sum of the areas of the squares on the two perpendicular sides. And we know a few other things, about ordering of points on a line and the like. Yet the facts just listed are not independent. Once we know about parallels it is not possible to declare that the three angles of a triangle, however artfully constructed, add up to 179 degrees, or that the square on a hypotenuse should be anything other than the sum of the other two squares.

Mathematical reasoning more than two thousand years ago established the dependence of these apparent facts, though not always easily. Students therefore may well be encouraged to use theorems such as the 180 degree theorem for triangles and the Pythagorean theorem without immediately knowing their proofs, getting used to their statements at an early age, but they must at the same time be taught that there are proofs, and that in due course they will get to learn them themselves. Otherwise we are committing an intellectual fraud on them, inducing the belief that mathematical knowledge is only a set of conventions. Furthermore, students must be exercised in drawing at least the simpler deductions from the facts already known, rather than presented with them all as if they were axioms. While it is pedagogically not wise to belabor the demonstration of what is apparently obvious it is pedagogically very important indeed to have students able to give a reasoned, deductive argument for some of the more important non-obvious facts of algebra and geometry. The question of which, and at what stage, is a pedagogical one of great importance, and cannot be decided in the abstract, or in a summary essay.

In many of today's commercial textbooks, and in much official doctrine within the teaching profession, however, mathematical reasoning is either garbled or missing. Children are taught that lines are perpendicular if they form right angles at the intersection. In another chapter, concerned with straight lines laid out on a coordinate plane, they are taught that straight lines are perpendicular if the product of their slopes is -1. There is nothing obvious about perpendicularity that dictates the value -1 for the product in question, yet innumerable books state both truths about straight lines without explanation, and then require the students to solve problems in indirect measurement by using the supposed equivalence of the criteria.

But they are related; only one of them can be the definition, with the other a truth provable by reasoning. If one were to define perpendicularity by stating that the product of the slopes is 5, we would find (given the other properties of Euclidean geometry) that the angles formed by such lines are not right angles. A textbook has no right to declare, blandly, the product-of-slopes rule for perpendicularity as if it were either a definition or axiomatic, a fact to be learned separately from the rightangle essence of perpendicularity, for despite their equivalence as facts such a claim plants in the young mind an entirely mistaken conception of the nature of mathematics. If mathematics can say whatever it wants to, it would have no use in describing the world, and it would constantly be stating one thing and its negation in the same breath. If a textbook fears a proof or even a mention of a proof would confuse the reader, it would be better not to mention (say) the rule concerning the product of the slopes at all, and save it for a time when the connection with perpendicularity can be made logically clear. Once the Pythagorean theorem is in hand, of course, the product-of-slopes rule becomes provable; but even before that time, if the rule is mentioned at all, the student should understand why a proof is needed -- needed -- by our very souls, and not just as an end-of-semester examination performance. Needed, whether we can at the moment provide it or not.

The major projects and textbooks of the "New Math" era of the 1960s, those that seriously came to grips with the problem of mathematical rigor and offered reasonable solutions, were difficult, often pedantic, and never widely used, though as experiments they were influential. The commercial textbooks of that time which attempted to follow their example either failed to sell and were withdrawn from the market, or achieved financial success by giving only lip-service to the professed ideals of the time. Both the correct (though tedious and difficult) books and programs, and the diluted and inconsistent ones, were rejected by the public after a time, though for varying reasons, and the textbooks of the "back to basics" era that followed were in most cases no better than the terrible texts that had generated the "new math" movement to begin with. From about 1970 to 1990 mathematical reasoning disappeared from school mathematics, and even Euclidean geometry gradually evaporated in favor of the treatment of geometry as a sort of empirical science.

Even so, the official dogma has never been other than an announced emphasis on "real understanding". In the "new math" era this gave a perfect excuse to the mediocre textbook-writer or teacher to avoid the teaching of the basic facts which were, after all, what mathematics is intended to produce for use. In the "back to basics" era it became the connection of mathematical computation with practical problems, rather than deductive reasoning, that was called "real understanding", and mathematical reasoning languished. Since the publication of the 1989 Standards of the NCTM an orthodoxy of "conceptual understanding", which consists mainly in reducing all mathematics to what can be seen and felt without the use of systematic deduction, has become the intellectual support of a host of nearly vacuous programs, with mathematical reasoning again, except for some trivialities, almost invisible.

In the 1960s the complaint arose that "Johnny can't add" because he was (supposedly) spending his time learning deductive reasoning, and "The New Math" died. But in truth, Johnny was at least no worse at arithmetic in 1970 than he had been in 1940, and -- also in truth -- he wasn't learning deductive reasoning either. But since both adding and reasoning are important, however, the pedagogical problem of how to include an optimal amount of both is still before us. An abstract answer to this problem being impossible I shall instead comment on a few examples of things taught in the schools, which should indeed be taught and taught better, to illustrate the question of how to present them within the framework of mathematical reasoning and to suggest a few particular answers. All this is by way of example, and a real program will of course require careful choices and a long series of textbooks. The typical questions in each case are: Should it be done entirely logically, i.e., should the theorems in question be proved by the student? If so, on the basis of what axioms? Is a half-proof better than none? Is it always even necessary to mention the fact that a given theorem is linked to something else? Should a practical application accompany every logical advance?

At the very beginning, for the first example, the positive integers are given as matters of experience. Children learn to count, 1,2,3,4,..., and they get the idea of "and so on." Certainly this is not the place to teach them Peano's axioms, or the principle of mathematical induction as used in the proof of certain theorems. Yet theorems turn up very early that do require all that if one is to be rigorous. Even definitions, multiplication for example. It is not hard to see the associative law for multiplication, however defined for young children, but while (ab)c = a(bc) can be made more or less evident, it is not so obvious from the three-factor associative property -- at least, it is not obvious to a professional -- that [(ab)c](de) = a{[(bc)d]e} for all numbers a, b, c, d and e. Yet we use such chains of multiplication all the time, even as children. The proof that all such combinations give the same answer requires "proof by induction". This is something best unmentioned in the early grades, and it was one unfortunate aspect of some versions of "The New Math" of the 1960s to attempt to incorporate it into a complete, airtight system for the K-8 level. If some genius of a child asks about it, however, the teacher should not claim it is obvious, but should quietly remark that this is something that child can look forward to learning about "when he grows up". It looks obvious, it is true, but while it is not independent of other things already taken to be true it would only be confusing to go on about it to children who do not see a problem there at all.

A second early theorem concerns the decomposition of a positive integer into primes. Every child who has learned to multiply should understand what primes are, and should be able to calculate the decomposition for fairly large numbers, say of three digits, by trial and error. So 260 = 2x2x5x13. What is subtle in this connection is that this is the only way such a decomposition can be made. Looking at this particular decomposition assures us that 7 does not divide 260, for example. At the fifth grade level such a fact should be known, and it should also be made known that there is a proof, but that the proof can wait. The difference between this fact and the generalized associative law in our first example above is that the uniqueness of prime decomposition is not obvious (and it is easy and fun to give an example of an arithmetic system in which it is not even true). In such a case, unlike the case of the associative law, the non-obviousness must be mentioned, even if the proof is beyond the student. And if some students don't follow this particular part of the lesson, there is no reason to spend a week making sure they do. Not all students will learn all there is to learn, and that which does not impede further progress can be dropped after a time, and considered an enrichment for the others.

A third example is that the angles of a triangle add to 180 degrees. Here is something that is both non-obvious and amenable to proof even by the 7th grader who is first learning it. It depends on axiomatic facts concerning the behavior of parallel lines, and every part of the development is important and beautifully illustrative of the power and the necessity of deductive reasoning.

Finally, an example that offers some possibility of controversy, or an ambiguous lesson concerning the reach of deduction in the curriculum:

3*4 may be construed as a rectangle partitioned into unit squares, in which case 3*4 = 4*3 is plain obvious; but 3*4 may also be construed on the number line, as the concatenation of four 3" lengths or of three 4" lengths. The commutativity in the concatenation interpretation is not so obvious (consider that 315 * 87.06 = 87.06 * 315, where the rectangle idea still shows commutativity as obvious); hence it must be that the two definitions are not obviously saying the same thing. Is a teacher, or textbook, entitled to take the two definitions without mention of this difficulty?

Any thoughtful educator will say yes: leave it alone. Do not try to prove the obvious, or bring up as a problem something not yet seen by the children as a problem; children will not be interested, nor understand what the fuss is about, not when they are eight years old. Anyone with experience of the New Math books, with their definition of addition as set unions, and the proof (e.g.) that any two orderings of a finite set yield the same ordinal number, will remember how futile the attempt at rigor is at such an age.

Yet even here I do not entirely agree, for an artful program can, albeit subliminally, convey the equivalence of the two most usual definitions of multiplication (concatenation of intervals, and squares within a rectangle), and it is here that American textbooks are generally inferior to certain foreign ones, especially those of Singapore which are today attracting so much attention. For children of an early age, in grades K-6 in particular, the lessons of reason can be best conveyed by indirection. Anyone who studies the K-6 Singapore texts will see how this is done:

For many explanations (in the Singapore series) a child is pictured trying to solve a problem, and a thought balloon shows an intermediate step one ought to think before putting the pieces back together. No formal proof is given that this intermediate result is logically apt, or that it is useful, but both are obvious once the thought is expressed in the balloon alongside the puzzle it elucidates. Yet the final answer is not quite given away, and the child goes on to use the thought to complete the problem. Only many pages later, or perhaps a year or two later, long after the child has forgotten quite how he came to see such things, the systematic algorithm that makes use of the earlier insight will become easily assimilated. "If..., then ..." statements can wait until then, but this doesn't mean that mathematical reasoning is absent from the earlier lessons.

Thus, for example, it becomes useful to revert to the question of the commutativity of multiplication, and especially the distributive laws, long after they have been used "thoughtlessly" by children for arithmetic, but this is a matter to be brought up only in an algebra course some years later, when examples have emerged that show the observation to be nontrivial, hence memorable.

Finally, it should be unnecessary to belabor the fact that formal reasoning should be taught explicitly, and not just subliminally, at the high school level, with patient attention to the way theorems are derived from axioms, definitions and prior knowledge, and that it is not only Euclidean geometry that is or should be subjected to such scrutiny, but that the structures of algebra admit proofs of important and non-trivial theorems as well. For students intending college mathematics, for example, the complex numbers should be defined as points in the plane with a certain arithmetic, and all properties including the placement of the real numbers within them provable with rigor; and mathematical induction and the properties of the binomial coefficients and the like should also be at their command. In other words, it is more important at the pre-collegiate level to use mathematical reasoning in mathematical contexts than to "appreciate" reasoning or study it abstractly, with truth tables and quantifiers best suited to a philosophical course in foundations of mathematics at a collegiate level. One does not need such abstraction in the schools, though of course the working language of logical discourse, including the correct use of words such as implication, contrapositive, negation, and, and or, should be part of a student's general mathematical equipment by high school's end.

2. We therefore ask school mathematics programs to include mathematical reasoning in some form at all levels of instruction, the completeness and rigor of reasoning being adjusted to the age level of the students. Whatever the level might be, it is essential that students appreciate and make use of the connections of one part of mathematics with another, and never come to believe that mathematical truths are merely conventional, to be learned singly. They must instead be taught how to simplify thought and reduce the burden upon memory by making full use of a gradual increase of deductive skill, in arithmetic, algebra, geometry, and their applications.

Ralph A. Raimi
Department of Mathematics
University of Rochester
Rochester, NY 14627

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