In grades 9-12, the mathematics curriculum should include numerous and varied experiences that reinforce and extend logical reasoning skills so that all students can--

• make and test conjectures;
• formulate counterexamples;
• follow logical arguments;
• judge the validity of arguments;
• construct simple valid arguments;

and so that, in addition, college-intending students can--

• construct proofs for mathematical assertions, including indirect proofs and proofs by mathematical induction.

Focus

Inductive and deductive reasoning are required individually and in concert in all areas of mathematics. A mathematician or a student who is doing mathematics often makes a conjecture by generalizing from a pattern of observations made in particular cases (inductive reasoning) and then tests the conjecture by constructing either a logical verification or a counterexample (deductive reasoning). It is a goal of this standard that all students experience these activities so that they come to appreciate the role of both forms of reasoning in mathematics and in situations outside mathematics. Furthermore, all students, especially the college intending, should learn that deductive reasoning is the method by which the validity of a mathematical assertion is finally established.

A second goal of this standard is to expand the role of reasoning, now addressed primarily in geometry, so that it is emphasized in all mathematics courses for all students. In addition, this standard proposes that college-intending students should learn the more formal methods of proof required for college-level mathematics.

A third goal, also a departure from the existing curriculum for college-intending students, is to give increased attention to proof by mathematical induction, the most prominent proof technique in discrete mathematics. (The term mathematical induction refers to a formal technique used to prove statements defined for subsets of the integers. It should not be confused with inductive reasoning.)

Discussion

In grades 5-8, students will experience inductive reasoning and the evaluation and construction of simple deductive arguments in a variety of problem-solving settings. In grades 9-12, as the depth and complexity of content is increased, this emphasis on the interplay between conjecturing and inductive reasoning and the importance of deductive verification should be maintained.

All students, for example, should examine numerical patterns that result from algebraic manipulation, make conjectures about general algebraic properties based on their observations, and verify their conjectures with numerical substitutions. The properties of logarithms provide a context to illustrate our meaning. Using their knowledge of exponents and the fact that log ab = log a + log b, students could explore cases such as the following:

log 5² = log (5 · 5) = log 5 + log 5 = 2 log 5

log 5³ = log (5² · 5) = log 5² + log 5 = 2 log 5 + log 5 = 3 log 5

log 5⁴ = log (5³ · 5) = log 5³ + log 5 = 3 log 5 + log 5 = 4 log 5

Students could be asked to examine the emerging pattern and generalize to log 5ⁿ = n log 5 for each nonnegative integer n. Students could then use the [y^x] and [log] keys on a calculator to confirm this generalization with several numerical values such as n = 9, 14, and 0.

As in the setting decribed above, algebraic processing itself often suggests generalizations. Some students might observe, for example, that their algebraic manipulations did not depend on the particular number 5, thus leading to the more general assertion, log aⁿ =n log a for any a > 0.

Students can be introduced to the forms of deductive argument by examining everyday situations in which such forms arise naturally. Political claims and commercial advertisements are especially good sources for arguments that illustrate logical errors. For instance, a sign on the front of an ice-cream store declares, "If you want the best ice cream in the country, try Great Northern." The sign is meant to entice people into the Great Northern Ice Cream store, presumably to get the best ice cream in the country. However, the converse of the statement on the sign actually is needed to argue (by modus ponens) that Great Northern ice cream is the best in the country. This sign merely asserts that those who want the best ice cream in the country (i.e., almost anyone) should try Great Northern. It makes no claim about the quality of Great Northern ice cream.

Informal deductive arguments like that illustrated in the preceding example are applicable in mathematical settings. Students can begin to appreciate the power of deductive reasoning by providing simple valid arguments as justification for their solutions to specific problems and for algorithms constructed for various purposes. For example, all students should be able to compute the Euclidean distance between points with coordinates (2, 3) and (-4, 5) using the distance formula; additionally, all students should be able to provide a valid argument to justify why the computed distance is correct. This argument would be based on an appropriate figure, the Pythagorean theorem, and the knowledge of how to compute the distance between two points on a horizontal and on a vertical number line. However, it need not follow a particular format, and it may be presented orally or in writing in the student's own words. College-intending students, however, should be expected to derive and write a general proof of the distance formula. In contrast to the earlier argument, the proof would require arbitrary coordinates, as illustrated in figure 3.1. The proof itself would entail a careful sequence of steps with each step following logically from an assumed or previously proved statement and from previous steps. In addition, an argument should be made that the general formula applies when the points lie on the same horizontal or vertical line.

Fig. 3.1. Diagram with arbitrary coordinates

Although reporting proofs in two-column form may be a useful teaching tool as students first learn to write proofs, eventually they should be expected to write proofs in paragraph form. Only college-intending students should be expected to learn more specialized argument forms such as indirect proof and proof by mathematical induction. A proof of the conjecture that log aⁿ = n log a for a > 0 and each nonnegative integer n would require students to use the principle of mathematical induction as follows:

1. For n = 0, log a⁰ = log 1 = 0 = 0 log a for any a > 0. Thus, when n = 0, the equation is true.
2. Assume the equation is true for n = k where k is any positive integer, that is, log a^k = k log a. Our goal is to show that the equation is true for the case n =k + 1. This is established by the following chain of reasoning:

   log a^{(k + 1)} = log a^ka¹ = log a^k + log a = k log a + log a = (k + 1) log a

   Thus, if the equation is true for n = k, then it is true for n = k + 1. It follows by the principle of mathematical induction that log aⁿ = n log a for a > 0 and for every nonnegative integer n.

It is important that college-intending students have numerous and varied experiences with this proof technique in contexts beyond the familiar setting of series.

In grades 9-12, college-intending students will have their first experience of reasoning within an axiomatic system, the context that is so essential for work in mathematics. This higher-order thinking may not come easily, since the requirement to verify statements with a deductive proof by reasoning from axioms is unique to the discipline of mathematics; hence, it is a completely new way of thinking for high school students. Their previous experience both in and out of school has taught them to accept informal and empirical arguments as sufficient. Students should come to understand that although such arguments are useful, they do not constitute a proof.

It is also important that students recognize the difference between a statement that is verified by mathematical proof (i.e., a theorem) and one that is verified empirically using statistical arguments. Most mathematical theorems have a (sometimes hidden) quantifier, "for every." Thus, for example, the angle sum of every triangle in Euclidean geometry is 180 degrees, and sin²x + cos²x = 1 for every real number x. However, statements like "men are taller than women" or "women score higher than men on certain vocabulary tests" are true only in the sense that, in most empirial studies of these phenomena, the mean for a sample of men has differed from the mean for a sample of women at a level greater than some specified chance difference. In both cases, the men's and women's distributions actually overlap a great deal; that is, some women are taller than some men, and some men score higher on vocabulary tests than some women. It is not just incorrect logic to assume that statements arising from empirical and statistical arguments apply to every member of the groups in question; this error, when widely accepted in a society, contributes to inaccurate racial, ethnic, and gender sterotyping. A discussion of these reasoning issues can serve as a connection between mathematics and social studies classes in which the (usually negative) social consequences of such stereotyping are studied.

The potential for transfer between mathematical reasoning and the logic needed to resolve issues in everyday life can be enhanced by explicitly subjecting assertions about daily affairs to analysis in terms of the underlying principles of reasoning.