In grades 9-12, the mathematics curriculum should include the study of mathematical structure so that all students can--

• compare and contrast the real number system and its various subsystems with regard to their structural characteristics;
• understand the logic of algebraic procedures;
• appreciate that seemingly different mathematical systems may be essentially the same;

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

• develop the complex number system and demonstrate facility with its operations;
• prove elementary theorems within various mathematical structures, such as groups and fields;
• develop an understanding of the nature and purpose of axiomatic systems.

Focus

The structure of mathematics is like the steel framework of a modern building. Students should become aware of this structure, how it provides a strong foundation on which a variety of content strands are built, and how it simultaneously holds these different strands together. For example, one of the girders in this building is the associative property to which are attached objects and operations in such wide-ranging mathematical subjects as arithmetic, algebra, functions, and geometric transformations. An awareness of these broad structuring principles frees students to take a more constructive approach to new mathematical topics and provides them with a conceptual framework that facilitates long-term retention.

Mathematical structure in the form of lists of general properties is not a good starting point for instruction. Rather, students gain a sense of the structure of mathematics over an extended time period through the general accumulation of experience, as well as through more focused activities. It is neither necessary nor appropriate for them to hear constantly the word structure applied to their activities; occasional summary statements will serve them far better. It also is essential to recognize that mathematical structure and formalism are not synonymous. In mathematics, just as with a building, all students can develop an understanding and appreciation of its underlying structure independent of a knowledge of the corresponding technical vocabulary and symbolism. The degree of formalism must be consistent with the student's level of mathematical maturity.

Discussion

Student insight into the structure of mathematics should be fostered at both micro and macro levels. At the micro level, it is gained through the pervasive application of the notion of step (n + 1 ) following from step (n) because .....; for example, 3x = 7 follows from 3x - 5 = 2, since we know we can add 5 to each equation member without destroying the balance. Such arguments should be common to all mathematical study. At the macro level, insight is gained from the observation of quite different structures built on common features; the concept of an identity element, for example, plays a special role, not only in addition and multiplication within various number systems, but also in operations on functions, matrices, and geometric transformations.

Students come to understand the idea of structure through observation of the common properties in simple systems that seem on the surface to be quite dissimilar. For example, consider the structural commonality of binary multiplication, the logical operation "and," and a series electrical circuit (fig. 14.1).

Fig. 14.1. Systems with structural commonalities

The equivalent roles in these operation tables of 0, false, and "off," as well as of 1, true, and "on," are immediately evident.

The sets of symmetries of geometric figures, together with the operation of function composition, provide further concrete settings rich in opportunities for students to investigate mathematical structure. For example, the sets of symmetries for an isosceles triangle and a parallelogram each contain two elements, and their corresponding operation tables are structurally the same (fig. 14.2).

Fig. 14.2. Sets of symmetries that are "essentially the same" under function composition

Students work with a series of number systems as they move through school. It is important that they return to this topic in order to view the associated structural characteristics of these systems. They should come to appreciate what is gained, what is lost, and what is retained in the structural characteristics of each new system. As students move from the integers to the rationals, for example, they retain addition and multiplication, they extend division, and they gain the density property. When college-intending students extend the real-number system to the complex numbers, they retain the field properties of the reals, they extend roots, but they lose order. For college-intending students, these ideas of structure should be extended to a more formal level through the organizing properties of groups and fields.

Applications of matrices as representation tools lead naturally to the usual definitions of equality and addition. All students could revisit this topic from a structural viewpoint. In particular, they could consider the set of 2 x 2 matrices with integer entries under addition and explore whether the commutative, associative, identity, and inverse properties hold. They could compare the resulting structure with other systems with which they are familiar (such as the integers under addition). College-intending students could continue a similar exploration with 2 x 2 real matrices using the usual definitions of matrix addition and multiplication. This would provide an example of a system in which multiplication is not commutative. In addition, these students could show that the existence of a multiplicative inverse for a given 2 x 2 matrix is equivalent to the existence of a unique solution of a related system of two linear equations in two variables.

College-intending students should also see how their school mathematics fits into the larger picture of advanced mathematical studies. For example, an important concept and a fundamental connecting link of mathematics is isomorphism; a direct translation of its Latin roots (iso = same, morph = form) conveys the term's essential meaning. Exponents (or, equivalently, logarithms) provide an excellent first example of an isomorphism that can reasonably be conveyed to students. In the world of numbers, we have multiplications, b^m x bⁿ, but when this problem is translated into the world of exponents, the operation becomes addition, m + n. The result, translated back into the world of numbers, is b^(m+n). In this way, students can gain an insight--through the lens of the fundamental property of exponents "when multiplying powers with the same base, add the exponents" --into a deep mathematical concept, often expressed in abstract form as f(a * b) = f>(a) f(b). How this also applies to logarithms may be seen by direct substitution of log for f, x for *, and + for .

The concept of isomorphism also provides a means of clarifying some of the important connections between algebra and geometry. For example, college-intending students should recognize that the complex numbers under addition are isomorphic to two-dimensional translations under composition.

Consider the sum (3 + i) + (-5 + 2i) and the isomorphism indicated in figure 14.3.

Fig. 14.3. Application of an isomorphism

As an enrichment activity, some students could be encouraged to establish that the nonzero complex numbers under multiplication are isomorphic to the spiral similarities (composites of a rotation and dilation with the same center) under composition.

Although the curriculum standards for grades 9-12 recommend that Euclidean geometry no longer be developed as a complete axiomatic system, students (especially the college intending) should nevertheless develop an understanding of the nature and purpose of axiomatic systems. This objective can best be achieved through cumulative experiences across several contexts. Students should develop short sequences of Euclidean geometry theorems in which they explore the role of assumptions and the consequences of replacing them by other statements that also hold in Euclidean geometry. They should investigate some of the simpler consequences of deleting or contradicting the Euclidean parallel axiom. Finally, they should have experiences with reasoning from assumptions in connection with their study of other topics in the curriculum, such as the structure of number systems and algebra. This shift in emphasis provides students with a better perspective on, and appreciation for, the role and nature of both geometry and axiomatic systems in contemporary mathematics.

Throughout the Standards, there has been a consistent emphasis on organizing the curriculum and instruction so that the connectedness of mathematical ideas is established and capitalized on both in problem solving and in the learning of new content. The study of mathematics in grades 9-12 should also provide students with an awareness and appreciation of the broad underlying themes and logical consistency of mathematics.