The place of algorithms and proofs in school mathematics

Doing and proving: The place of algorithms and proofs in school mathematics

Ross, Kenneth A

In the fall of 1996, the NCTM’s Commission on the Future of the Standards asked several mathematics organizations to have groups respond to sets of questions about the Standards posed by the Commission. These groups are called ARGs (Association Review Groups for the NCTM Standards). The ARG for the MAA is the President’s Task Force on the NCTM Standards. This article is truly a joint article with all fifteen members of the Task Force as co-authors: Henry L. Alder, University of California-Davis; Thomas R. Berger, Colby College; Susanna S. Epp, DePaul University; Ladnor Geissinger, University of North Carolina; Deborah Tepper Haimo, University of California-San Diego; David E. Kullman, Miami University (Oxford, Ohio); Kathy Layton, Beverly Hills High School; James R. C. Leitzel, University of New Hampshire; Mercedes A. McGowen, William Raney Harper College; Robert E. Megginson, University of Michigan; Henry O. Pollak, AT& T Bell; Stephen B. Rodi, Austin Community College; Kenneth A. Ross (chair), University of Oregon; Alan C. Tucker, SUNY at Stony Brook; Hung-Hsi Wu, University of California-Berkeley.

The second set of questions posed by the NCTM Commission involved the role of algorithms and proofs in school mathematics. In this article, we state the questions and give the essence of our response. We hope this will give readers a feel for the sorts of issues with which we are dealing. For the full response, including illustrative examples, we refer the reader to our website: http://www.maa.org/past/maa nctm.html.

The first part of the second set of questions concerned algorithms and algorithmic thinking. (a) What is meant by “algorithmic thinking”? (b) How should the Standards address the nature of algorithms in their more general mathematical context? (c) How should the Standards address the matter of invented and standard algorithms for arithmetic computation? (d) What is it about the nature of algorithms that might be important for children to learn? Our response follows.

An algorithm is a procedure involving prescribed steps that lead to a specific outcome, which is often the calculation of something. At the elementary level, this frequently refers to procedures for doing basic operations of arithmetic. At a more advanced level, this refers to procedures designed to solve specific problems, often arising in computer science. Rather than discuss parts (a)-(d) individually, we interpret the main issue as how the teaching of algorithms should be handled in the schools. If this issue is clear, then questions of how the Standards should address aspects of algorithms will be relatively easy.

The NCTM Standards emphasize that children should be encouraged to create their own algorithms, since more learning results from “doing” rather than “listening” and children will “own” the material if they create it themselves. We feel that this point of view has been over-emphasized in reaction to “mindless drills.” It should be pointed out that in other activities in which many children are willing to work hard and excel, such as sports and music, they do not need to create their own sports rules or write their own music in order to “own” the material or to learn it well. In all these areas, it is essential for there to be a common language and understanding. Standard mathematical definitions and algorithms serve as a vehicle of human communication. In constructivistic terms, individuals may well understand and visualize the concepts in their own private ways, but we all still have to learn to communicate our thoughts in a commonly accepted language.

The starting point for the development of children’s creativity and skills should be established concepts and algorithms. As part of the natural encouragement of exploration and curiosity, children should certainly be allowed to investigate alternative approaches to the task of an algorithm. However, such investigation should be viewed as motivating, enriching, and supplementing standard approaches. Success in mathematics needs to be grounded in well-learned algorithms as well as understanding of the concepts. None of us advocates “mindless drills.” But drills of important algorithms that enable students to master a topic, while at the same time learning the mathematical reasoning behind them, can be used to great advantage by a knowledgeable teacher. Creative exercises that probe students’ understanding are difficult to develop but are essential.

Algorithms are a very important part of mathematics, but classroom teachers should watch out for their abuse as an instrument of mindless drills. They should not be over-emphasized just because they are easy to teach and test. We are greatly concerned that the Standards use careful language to convey this message. In general, we believe that it is far better to suggest that some aspect of mathematics, such as algorithms, not be over-emphasized rather than to say that it should be de-emphasized or receive reduced emphasis. These latter phrasings are too often interpreted to mean “eliminate,” which in turn may lead teachers to believe that they are somehow in violation of the Standards whenever they teach standard algorithms.

The challenge, as always, is balance. “Mindless algorithms” are powerful tools that allow us to operate at a higher level. The genius of algebra and calculus is that they allow us to perform complex calculations in a mechanical way without having to do much thinking. One of the most important roles of a mathematics teacher is to help students develop the flexibility to move back and forth between the abstract and the mechanical. Students need to realize that, even though part of what they are doing is mechanical, much of mathematics is challenging and requires reasoning and thought.

The second part of the second set of questions concerned proof and mathematical reasoning. (a) What mathematical reasoning skills should be emphasized across the grades? (b) How should the Standards address mathematical proof? Why? (c) How should the Standards address topics within mathematical structure? Here is our response.

One of the most important goals of mathematics courses is to teach students logical reasoning. This is a fundamental skill, not just a mathematical one. To accomplish this, teachers need to recognize mathematics as a lively, exciting, vibrant field of study that must have a primary role in every child’s education throughout the school years. They should recognize its theoretical nature, which idealizes every situation, as well as the utilitarian interpretations of the abstract concepts. This dual role of mathematics allows for applications to an incredibly broad range of seemingly unrelated areas, where answers to practical problems can be found with remarkable accuracy. However, the utilitarian aspects of mathematics should not be adopted to the virtual exclusion of its theoretical nature.

It should be emphasized that the foundation of mathematics is reasoning. While science verifies through observation, mathematics verifies through logical reasoning. Thus the essence of mathematics lies in proofs, and the distinction among illustrations, conjectures, and proofs should be emphasized. It should be stressed that mathematical results become valid only after they have been carefully proved. Results may be shown to hold in a small number of cases directly, but students must recognize that all they have in that case is evidence of a conjecture until the result has been firmly established. Construction of valid arguments or proofs and criticizing arguments are integral parts of doing mathematics. If reasoning ability is not developed in the student, then mathematics simply becomes a matter of following a set of procedures and mimicking examples without thought as to why they make sense.

Teachers of mathematics should, therefore, make it their aim to explain everything in mathematics to the extent that this is reasonable and effective at the student’s level of mathematical knowledge. The important thing is to be honest; if only illustrations and a plausibility argument are supplied, the students should be reminded that a logical reason or proof is needed. This point should not get lost now that technology provides a means for exploring mathematical ideas and testing conjectures. Of course, the emphasis on proofs should be more on their educational value than on formal correctness. Time need not be wasted on the technical details of proofs, or even entire proofs, that do not lead to understanding or insight.

It should also be emphasized that results in mathematics follow from hypotheses, which may be implicit or explicit. Although there may be many routes to a solution, based on the hypotheses, there is but one correct answer in mathematics. It may have many components, or it may be nonexistent if the assumptions are inconsistent, but the answer does not change unless the hypotheses change.

Starting no later than the 8th grade, the word “proof” and the phrase “this is not a proof” should be used consistently wherever appropriate. At this stage, students’ mathematical sensitivity should be sharpened. They need to start picking up on logical subtleties and appreciate the need for airtight arguments before making conclusions. Furthermore, they will soon be called upon to make distinctions between truths and pseudo-truths in the much more difficult context of human and social issues.

In their work with mathematical reasoning, students, in particular, beginning with eighth grade should (1) distinguish between inductive and deductive reasoning and explain when each is appropriate; (2) understand the meaning of logical implication, in particular, be able to identify the hypothesis and conclusion in a deduction; (3) test an assertion with examples; (4) realize that one counterexample is enough to show that an assertion is false; (5) clearly recognize that the truth of an assertion in a few cases does not allow one to conclude that it is true in all cases; (6) recognize whether something is being proved or is merely being given a plausibility argument; (7) identify logical errors in chains of reasoning involving more than one step.

As indicated above, the appropriate inclusion of proofs and mathematical reasoning in the mathematics curriculum is extremely important. However, we want to acknowledge how very difficult this is and address some concerns about the Standards’ section on Mathematics as Reasoning.

First, the way in which the sections on reasoning are separated from the subject-matter sections may have had the opposite effect from what the Standards’ authors intended. While we completely agree that “mathematical reasoning” should be a thread running through the entire school curriculum, the problem is that it is much harder for teachers to incorporate a thread than it is for them to add a topic. For instance, each example in the “Mathematics as Reasoning” standard occurs within some context, but since the standards dealing with those contexts are some distance away from the reasoning standard, these examples appear to be “extras” that an overburdened teacher might not think to consult when engaged in course preparation and that, even if consulted, might not fit into an already crammed syllabus.

A second concern is that, while the Standards establish lofty expectations for students’ reasoning and communication performance, they offer teachers little assistance about how to help students develop their abilities in these areas. Significantly, traditional teacher training in mathematics has not included preparation for dealing with these issues. But these are major and difficult issues.

We recommend that (A) The new Standards explicitly specify inclusion of material on logic and mathematical language in age-appropriate ways throughout the K-12 curriculum. We are referring primarily to what it means for sentences involving “and,” “or,” “not,” “if-then,” “some,” and “all” to be true and what it means for them to be false, as well as some strategies for determining the truth or falsity of such sentences. (B) Each subject matter standard give much greater attention to the issue of how to combine teaching the facts and applications of the subject with helping students develop their ability to reason and express their reasoning with clarity and precision.

We also recommend that materials and programs be developed for teachers to help them develop their own sensitivity and sophistication about teaching reasoning and mathematical writing. This recommendation is addressed to the wider community concerned with teacher preparation, which includes the MAA and the NCTM.

KENNETH A. ROSS served as President of the MAA during 1995 and 1996. One of his most important actions as President was the creation of the MAA Task Force on the NCTM Standards, which he subsequently agreed to chair. He received his B.S. from the University of Utah in 1956 and his Ph.D. from the University of Washington in 1960. He has been teaching at the University of Oregon since 1965, and he is spending the academic year 1997-1998 on sabbatical at Cornell University. University of Oregon, Eugene, OR 97403-1222

ross@math. uoregon. edu

Copyright Mathematical Association Of America Mar 1998

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