In April, the MAA Task Force on the NCTM Standards was sent another set of questions for which the NCTM Commission on the Future of the Standards requested input. There were two sets of questions, one on algorithms and algorithmic thinking and one on proof and mathematical reasoning. The questions and the responses of the MAA Task Force are given below.
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 acceptable 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, like 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.
Algorithms may be viewed as special types of theorems; as with other theorems, they need verification. In most cases, students should know why the algorithms work, which leads us to the closely related topic of proofs and mathematical reasoning. However, it should be admitted that it is not always a bad thing to know how to apply an algorithm without knowing the reasoning behind it. There is considerable value in understanding what the algorithm says and when it can be applied. On the other hand, advanced algorithms (enciphering and deciphering a Huffman code, finding an Euler circuit for a graph, finding a critical-path using a scheduling algorithm, etc.) should be accompanied by discussions of why they work, since we do not want to reinforce the notion that mathematics is all about "mindless algorithms."
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; see Examples in the Appendix. If it is not possible to do so at that level, it should be made clear what justifications have been omitted. The important thing is to tell no lies; 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 its 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
Some examples are given in the Appendix.
APPENDIX ON IMPLEMENTATION
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 and 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" which an overburdened teacher might not think to consult when engaged in course preparation and which, 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
APPENDIX OF EXAMPLES
Example A. Students can be sensitized to mathematical language, especially
with truth and falsity of quantified statements, at a very young age. For
example, they should realize that the negation of "All teachers wear
glasses" is not "No teacher wears glasses."
Symbols should be introduced as efficient and effective means of communication, once the students acquire some understanding of the notions and processes under study. The notion of variable also should be introduced early and nurtured to ever greater sophistication over the years. It should not be a new and intimidating concept in the eighth grade.
Example B. Once variables have been introduced, questions requiring more and more logical sophistication can be posed in grades 7-12. One can start with ones like "For what values of x is 10x = x, or xx = 1 or 6x = x6?" which only require imagining what happens when x takes different values. Next, because it's easier for students to grasp the idea of counterexample than proof, one can go on to ask things like "True or false? For all real numbers a and b, (a+b)2 = a2+b2. Justify your answer." Then one can ask questions like "Is the equation x(x+5) = x2+5x true for all real numbers x? Justify your answer" which students can answer merely by citing a known property, such as the distributive law. Finally, one can ask students questions that require them to prove simple general statements. For instance: "Indicate whether the equation a2 + b2 = -1 is true for all values of a and b, for some values of a and b but not for others, or for no values of a and b. Justify your answer." Besides those found in geometry courses, other appropriate statements high school students can reasonably be asked to prove involve divisibility properties of integers, properties of rational numbers, properties of logarithms, one-to-one and onto functions, increasing and decreasing functions, and properties of trigonometric functions. We also advocate devoting some class time to logical puzzles starting perhaps at age 8 and continuing through high school.
Example C. The proof of the fundamental theorem of arithmetic usually is not given in high school, but the theorem can be studied in the 7th grade. Students are easily convinced of its correctness when the integer is small. To convince them of the need of a proof in general, give them two five digit numbers, preferably a prime and a composite. This will keep them busy and highlight the nontriviality of just getting a prime decomposition, let alone checking uniqueness. At this point, prove that if a natural number n has a factor, it has one in the interval [2,sqrt{n}]. Next try a number such as 12371237, and then 123777771237. The last one will probably discourage all the students. If someone suggests that, regardless of the size of the number, given enough time any number can be decomposed as a product of primes, then give them something like 12377777777....(300 7's).....77777771237. Point out that, in the foreseeable future, all the computers in the world together could not hope to get the prime decomposition of a number this large. At this point, students should be convinced that a proof is needed, one that works regardless of the size of the number. Emphasize that sophisticated reasoning is needed to prove the uniqueness part of the fundamental theorem of arithmetic. At the appropriate level, this theorem can be used to show that the square root of a natural number which is not a perfect square is irrational.
Example D. A combination of mathematical theorems and algorithmic thinking may be more powerful than either one alone. Consider, for example, the problem of finding the least common multiple (lcm) of two numbers a and b. One approach is to decompose both numbers into their prime factors, and then create the smallest pool that contains all factors of both numbers. This is very educational, but may be hard to carry out because factoring is itself hard to carry out. Since the product ab is certainly a multiple of both a and b, one could devise algorithms that test all numbers between 1 and ab to see which are common multiples. One could begin with 1 and work up, or begin with ab and work down. A better algorithm would only test multiples of a or b, probably the bigger one. One can continue this sort of thinking in search of efficient algorithms.
The related problem of finding the greatest common divisor (gcd) is a lot less work because one can use the efficient Euclidean algorithm. Of course, this doesn't solve the original lcm problem directly, but we can get the solution indirectly using the theorem: The product of the gcd of a and b and the lcm of a and b is the product ab itself. So one can use an algorithm for finding the gcd, and then divide that into the product to get the lcm.
Example E. A discussion of pi can lead to a meaningful discussion of circular reasoning. Ask a class what the number pi is. A typical response will be that pi is the circumference of a circle of diameter 1. "Good. So what is the circumference of a circle, of say diameter 1?" With luck, the answer "pi" will draw some laughter and lead to the realization that these definitions are circular. If the class manages to avoid this circular definition, then it will be led to an analysis of the geometry of circles. For a calculus class, this would lead naturally to a discussion of arc length and hence to the idea of limit. Since there is a variant of the above discussion involving the area of a disk, the discussion could also lead to a discussion of area.
Kenneth Ross
Department of Mathematics
University of Oregon
Eugene, OR 97403-1222
phone: 541-346-4721
fax: 541-346-0987
web: http://darkwing.uoregon.edu/~ross1/
ross@math.uoregon.edu
MAA Online is edited by Fernando Q. Gouvêa Last modified: Fri Sep 19 10:38:07 1997