A review of Geometry: tools for a changing world

Geometry: tools for a changing world by Laurie E. Bass, Basia Rinesmith Hall, Art Johnson, and Dorothy F. Wood, with contributing author Simone W. Bess, published by Prentice-Hall, 1998.

This textbook is on the list of accepted books for the states of Texas and New Hampshire. It's a glitzy book filled with pictures to keep the attention of the students. That's fine. It's the content that bothers me, in particular, the lack of logical content.

Chapter 1 introduces postulates on page 14 as accepted statements of facts. The four postulates stated there involve points, lines, and planes. Unfortunately, the first two are redundant. Postulate 1-1 says 'through any two points there is exactly one line,' and postulate 1-2 says 'if two lines intersect, then they intersect in exactly one point.' The second one should not be a postulate, but a theorem, since it easily follows from the first. And what better time to introduce logic than at the beginning of the course. No statement should be taken as a postulate when it can be proved, especially when it can be easily proved.

A number of definitions are also given in the first chapter. Later postulates deal with distance on a line, lengths of line segments, and angles.

The book does not properly treat constructions. Constructions can be either postulates or theorems, depending on whether they're assumed or proved. For instance, postulate 1-1 above is actually a construction. On pages 40 through 42 four constructions are given: 1) to cut a line segment equal to a given line segment, 2) to construct an angle equal to a given angle, 3) to construct a perpendicular bisector of a line segment, and 4) to bisect an angle. Later in the book, these constructions are used to prove theorems, yet they are not proved here, nor are they proved later in the book. There is no indication whether they are to be taken as postulates (they should not, since they can be proved), or as theorems. At the very least, it should be stated that they are theorems which will be proved later.

Next, the concept of theorem is given: a statement with a proof, where a proof is a convincing argument that uses deductive reasoning. The theorem "vertical angles are congruent" is given with a proof. It is followed by a two more theorems either supplied with proofs or left as exercises.

Also in chapter 1 there is an introduction to plane coordinate geometry. Unfortunately, there is no connection made with plane synthetic geometry. Here in chapter 1, a distance formula is asserted with neither logical nor intuitive justification. Of course, the justification is the Pythagorean theorem, and that's not discussed until chapter 5. In that chapter there is an exercise to prove the distance formula from the Pythagorean theorem. The Pythagorean theorem itself gets proved in yet a later chapter.

In summary, the constructions should be postponed until they can be justified, and then they should be justified. The same for coordinate geometry.

Chapter 2 begins with theorem that the internal angles of a triangle sum to 180°. The proof is postponed until an exercise in chapter 7, and is based on two postulates on parallels. (Chapter 7 suffers from unnecessary postulates.) Other theorems that follow from the angle sum theorem are given as exercises to prove with outlines.

In a return to coordinate geometry it is implicitly assumed that a linear equation is the equation of a straight line. A proof would depend on the theory of similar triangles in chapter 10. At this point it is suggested that one can conclude that parallel lines have equal slope, and that the product the slopes of perpendicular lines is -1. The only justification given is by experiment. (A proof would require the theory of parallels.) What's worse is what comes next on the page 85:

Questions 10 and 11 do not demonstrate the following theorems. The students are not asked to prove 10 and 11, and even if they were, they would depend on the unproven statements about coordinate geometry slopes of lines. Thus, we have two purported theorems totally without justification. A proper justification can only be given after chapter 10.

(And this occurs in the section in which 'conjecture' is discussed. "Test your conjecture by graphing several equations of lines where the values of m are the same." What's the proper conclusion? That theorems may be justified by looking at a few examples?)

In summary, the material in chapter 2 should be postponed until after elementary geometry is developed.

Chapter 3 is about isometries of the plane. The entire chapter is entirely devoid of logic. How are the theorems proved? "The Work Together illustrates the two properties summarized in the theorems below. Theorem 3-1: A composition of reflections in two parallel lines is a translation. ..." Moving a bunch of paper figures around in a "work together" does not constitute a justification of a theorem.

The theorems can be proven once a little actual geometry is presented, but that's not done until the last half of the book. A little honesty is needed here. Why not tell them that the proofs will be postponed until a later chapter? Or that we just don't have time to do the proofs for this chapter. Even better: don't label statements as theorems (like many other unproved statements in the chapter). Like the theorems in chapter 2, those in chapter 3 cannot be proved until after elementary geometry is developed.

In summary, either this chapter should be inserted in the proper place in the course, or else tossed out entirely. Since there's a lot to learn in geometry, it would be best to toss it out.

Chapter 4 begins the study of triangles. The first theorem states that base angles of an isosceles triangle are equal. But the proof doesn't occur until chapter 8. It's getting a bit much. Almost every proof is being omitted or postponed until several chapters later. And when the proofs do come, they depend on constructions that were never justified. For instance, this first theorem is proved in chapter 8 using an angle bisector, but the construction for an angle bisector (chap.1) is given without justification.

The proofs of the next two theorems are postponed until chapter 8. The next four theorems which only involve addition and subtraction of angles appear with their proofs (which depend on the angle sum of a triangle whose proof doesn't occur until chapter 7).

Another theorem in this chapter states that the line joining the midpoints of two sides of a triangle is parallel to the third and half its length. A coordinate proof is given, but as the properties of coordinates are never proved, the proof is unsatisfactory. An actual proof can be given, but not until the basic properties of triangles and parallels are proven.

The text again shows contempt for logic in the section on triangle inequalities. In a silly "work together" students try to form triangles out of various length straws. Rather than try to figure out the relations between the sides of a triangle for themselves, they're led by the nose to "conjecture about the sum of the lengths of two sides of a triangle compared to the length of the third side. Think and discuss. Triangle Inequality Theorem. Your observations from the Work Together suggest the following theorem," and the statement of the theorem follows. Honesty out the window. Can any student armed with this book prove this theorem? Can a teacher? Can the authors? Is it possible to prove it without using the postulates of chapter eight? No. What is this theorem doing here?

The next two theorems depend on that one, and their proofs are either given or left as exercises, but the following four are not proved in any way. Theorem 4-12 says a point on a perpendicular bisector is equidistant from the ends, and the next theorem is its converse. What's the justification? Draw the figure and measure the lines. That's no justification. But it's easy to prove. All you need is a little basic geometry. Here we are on page 219 of a 633-page book (excluding the appendices and index), and, still, the students haven't seen the basics of geometry!

For the 19 theorems in chapter 4, three proofs are postponed until a later chapter; two proofs are given but depend on results in later chapters; two proofs are validly given or left as exercises; one is given with a coordinate proof (and therefore, ultimately unproved); one cannot be proved until a later chapter, and the next two depend on it; four more proofs are not given but would depend on results in later chapters, and the next two given proofs depend on them; and the last two proofs are not given (but would depend on later results).

In summary, chapter 4 is a dismal chapter. It's a prime example of what should never be done in a mathematics book. It shows no respect for logic, and unproved and proved theorems are not distinguished. ("Unproved theorem" is an oxymoron. If it's not proved, then it's not a theorem.) But the material is important and should be treated with respect. It should all be postponed (except the theorems on right angles) until after the basic theory of triangles and parallels is developed.

Chapter 5 is about areas, including the Pythagorean theorem. It begins with postulates about area: the area of a square is the square of the length of its side, congruent figures have equal area, and the area of a region is the sum of the areas of its nonoverlapping parts. A theorem follows: the area of a rectangle is the product of its base and height. There is no proof given, not even a "work together" piecing together squares to make the rectangle. An actual proof is difficult. It would depend either on limiting processes (which are inappropriate at this level), or the construction of a square equal to a rectangle (which could be done much later in the text). It would be just as well to make this theorem a postulate and drop the first postulate about a square.

The next two theorems about areas of parallelograms and triangles come with proofs. Then come the Pythagorean theorem and its converse. In a "work together" students try to piece together triangles and a square to come up with the ancient Chinese proof of the theorem. "The Work Together presents a justification of the well-known right triangle relationship called the Pythagorean Theorem." At this time, however, it's not a complete proof since the theory of parallels has not been done; the existence of any square requires the parallel postulate. Garfield's proof of the Pythagorean theorem is offered in chapter 5, and other proofs later, but all depend on the theory of parallels discussed in chapter 7. The converse is apparently not proved anywhere in the book.

Next 45°-45°-90° and 30°-60°-90° triangles are solved, and areas of trapezoids and regular polygons are found.

The tenth theorem in the chapter claims the circumference of a circle is pi times the diameter. It is apparent (but not explicit) that pi is defined in this theorem as the ratio of circumference of a circle to its diameter. So the content of the theorem is that all circles have the same ratio of circumference to diameter. This theorem is not proven. It would require the basic geometry that won't come for a couple of chapters yet, and it would require a definition of length of a curve and limiting processes. It would be nice if a statement were included that the proof the the theorem is beyond the scope of the course.

Theorem 5-12 states that the area of a circle is pi times the square of the radius. A "work together" has students cutting pie-shaped pieces from a circle and arranging them alternately to form a rough rectangle. That idea is the best justification that can be given without using advanced techniques.

In summary, chapter 5 could be fairly good, but it should be postponed until after the Pythagorean theorem can be proved.

Chapter 6 is on surface areas and volumes of solids. It is strange that surface areas and volumes are treated while the basics of solid geometry are ignored.

There are 11 theorems, the only ones that can be proved without advanced mathematics are the ones on the surface area of a right prism (box) and a regular pyramid. Most of the theorems are given with little or no justification. The area of a cylinder is justified by unrolling it; the area of a cone is unjustified; Cavalieri's principle is stated as a theorem but not proved (it can't be proved without advanced mathematics, better to make it a postulate); the volumes of prisms and cylinders are found using Cavalieri's principle; and the volumes of pyramids and cones are stated without justification. The only argument for the surface area of a sphere involves wrapping yarn around a ball, and that's unlikely to get within 10% of the formula. Finally, a limiting argument is given for the volume of a sphere, which is the best that can be done at this level.

In summary, there is little mathematics in chapter 6. Most of the results require more than what's possible in a first course in geometry. Surface areas and volumes should only be treated after the basics of solid geometry are covered. Alternatively, surface areas and volumes may be left as an application of calculus.

Chapter 7 is on the theory of parallel lines. It begins by postulating that corresponding angles made by a transversal cutting two parallel lines are equal. One postulate is enough, but for some reason two others are also given: the converse to the first postulate, and Euclid's parallel postulate (actually Playfair's postulate). A proliferation of unnecessary postulates is not a good thing. One postulate should be selected, and the others made into theorems.

Four theorems follow, each being proved or left as exercises. Then there are three constructions for parallel and perpendicular lines. Proofs of the constructions are given or left as exercises. But the constructions depend on earlier constructions which still have not been proved, and cannot be proved until the basic theory of triangles is developed in the next chapter.

In summary, postpone the presentation of parallel lines until after chapter 8, and select only one postulate for parallel lines.

Chapter 8 finally begins the basic theory of triangles at page 406, almost two-thirds of the way through the book. This chapter suffers from one of the same problems as the last, namely, too many postulates. The three congruence theorems for triangles, SSS, SAS, and ASA, are all taken as postulates. One is enough. The other two should be theorems.

There are only two theorems in this very important chapter. There's a trivial proof of AAS (by now the internal angle sum of a triangle has been demonstrated). Then the Hypotenuse-Leg congruence theorem for right triangles is proved. Some of the theorems of earlier chapters are finally proved, but the original constructions of chapter 1 aren't.

In summary, this should be chapter 1, not chapter 8. Results in all the earlier chapters depend on it. The book is backwards. How did geometry ever become taught in such a backward way? Done right, the material in chapters 8 and 7 and the theorems in the earlier chapters that depend on it, should form the bulk of the course. Much more emphasis should be placed here.

Chapter 9 is on parallelograms and other quadrilaterals. Nearly every theorem is proved or left as an exercise. By this time the students should be doing their own proofs with bare hints or none at all, but several of the exercises have almost complete outlines for proofs. Only one theorem has no proof (base angles of isosceles trapezoids, and one is given by way of coordinates.

Too much is included in this chapter. The sections on rhombuses, trapezoids, and kites are not important and should be omitted.

Chapter 10 is on similarity and similar figures. (Appropriately for this level, the difficulties of proportions are buried in the implicit assumptions of real numbers.) One postulate is taken: triangles with equal angles are similar (meaning proportional sides). The first five theorems are are accompanied by proofs or left as exercises. The proofs are omitted for the theorems which say similar plane figures have areas in duplicate ratios, and similar solid figures have areas in duplicate ratios and volumes in triplicate rations. At least there should be a proof that similar triangles have areas in duplicate ratios; that's easy since the areas of triangles are already known.

Chapter 11 covers right-triangle trigonometry. It's hard to see how there's any time left for trigonometry in a course on geometry, but at least it should be possible to prove the basic facts of trigonometry once the theory of similar triangles is done. The section of angles of elevation and depression need not appear, and the section of vectors omitted. (By the way, who ever calls the sum of two vectors the "resultant" of the two vectors?) The one theorem of the chapter (area of triangle = 1/2 bc sin A) is given for acute triangles.

As the trig functions for obtuse angles aren't covered, and applications of trig to non-right triangles aren't mentioned, it would probably be better to remove this chapter entirely.

Chapter 12 discusses some geometry of the circle, in particular, properties of radii, chords, secants, and tangents.

There are 16 theorems, some with proofs, some left to the students, some proofs omitted. This is one of the better chapters in the book.

Final conclusion. Much more emphasis should be placed on the logical structure of geometry. Postulates should be carefully selected, and clearly distinguished from theorems. Every theorem should be proved, or left as an exercise, or noted as having a proof beyond the scope of the course. Very few theorems, or none at all, should be stated with proofs forthcoming in future chapters.

It should be emphasized that "work togethers" do not substitute for proofs. They can lead to an understanding of the statement of the theorem, but few of them lead to proofs of the theorem. It must be emphasized that examples do not justify a theorem.

David E. Joyce
June, 1998