Mathematical Analysis of Top-Ranked Programs

4.2.4 Prentice-Hall Geometry
Chapter 1 lays significant groundwork for the study of geometry. Topics include informal geometry, important definitions (e.g., parallel and skew lines, parallel planes, perpendicular lines), compass and straightedge constructions, the coordinate plane (e.g., formula for the midpoint of the segment), and the distance formula (based on the Pythagorean Theorem). The text carefully distinguishes the use of the word “segment” from the word “line.” Some exercises contrast circular definitions with the use of undefined terms in mathematics, and the discussion addresses the tension between the logical development of geometry as an axiomatic system and the fact that students will have already studied informal geometry in earlier grades. It attempts to make clear what is proved and what is not yet proved.

The development of logical tools for proof is taken up systematically in Chapter 2.
Section 2.1 introduces conditional (if-then) statements right away, with many examples, including rewording of statements not in if-then form into if-then form. Counterexamples and converses (and the truth value of the converse ) are introduced and illustrated. The chapter also includes Venn diagrams and standard arrow symbols. Section 2.2 contains a careful introduction to biconditional statements and definitions. Section 2.3 is about deduction, including the Law of Detachment and the Law of Syllogism. Examples and problems focus on the effective and correct use of these tools. Section 2.5 centers on the use of equations and algebra for solving questions in geometry. Section 2.6 uses these algebraic tools to make angle computations, including proving that vertical angles are equal. The chapter does a good job of presenting the important tools of logic and proof and addressing possible points of confusion. It is efficient in that it does not digress into a study of logic or algebra beyond what is needed for geometry.

Chapter 3 addresses parallel/perpendicular lines. Section 3.1 defines three pairs of angles formed by a transversal of any pair of lines and then moves to the case of parallel lines with the postulate that corresponding angles formed by a transversal intersecting a pair of parallel lines are congruent. The other angle relations formed by parallels and a transversal are proved. The teacher notes correctly point out that the Corresponding Angle Postulate is a variation of the Euclidean Parallel Postulate. This section is distinguished in that it moves briskly from definitions to the geometrical content of angles and parallels. Section 3.2 contains a postulate and then theorems stating the usual conditions that congruence of one pair of angles (corresponding, or alternate interior, etc.) formed by a transversal and two lines implies that the two lines are parallel. The theorems are correctly labeled as converses of the theorems in the previous section. Section 3.3 is about parallel and perpendicular lines. Perpendicular transversals are used to give a correct proof that two lines parallel to the same line are parallel. Section 3.4 proves that the sum of the angles of a triangle is 180 degrees. By proving this theorem in the chapter on parallels, the text provides an interesting and powerful application of the theory of angles and parallels. After this theorem, the exterior angle theorem is proved and classifications of triangles by angle are introduced. Section 3.5 proves angle sum theorems (both interior and exterior) for convex polygons. Sections 3.6 and 3.7 deal with the slopes of parallel and perpendicular lines. These relations are correctly presented as concepts that will be proved later rather than as postulates. Section 3.8 presents step-by2008 step straightedge and compass constructions of parallel and perpendicular lines. The treatment of parallels in this chapter presents the theorems about angles and parallels concisely but effectively. Distance does not appear in the section (thus avoiding some logical sequence problems). The mathematics is correct, including the appropriate distinction between logically necessary postulates and facts that are really theorems than can be proved later. Also, the understanding of the parallel postulate is correct.

Chapter 6 is about quadrilaterals, including application of the angle sum theorem for convex polygons, which was proved in Chapter 3. Section 6.1 begins with the definitions of special quadrilaterals, along with a diagram relating the logical relationships among the various kinds of quadrilaterals. Exercises develop examples and consequences of the definitions, including examples in the coordinate plane. Section 6.2 presents the standard properties of parallelograms. The equality of opposite sides is proved in a detailed proof. Included is one useful theorem that is often not stated: if three parallel lines cut off two congruent segments on one transversal, then they cut off two congruent segments on any transversal (a situation that occurs multiple times with notebook paper or street grids). Section 6.3 contains the sufficient conditions to prove that a quadrilateral is a parallelogram. Careful proofs are given of two of the theorems. Examples and investigations are included. The topic of Section 6.4 is special parallelograms, namely rhombuses and rectangles. Theorems about the diagonals are proved (i.e., necessary and sufficient conditions). Numerous exercises are included, some about problem solving and some asking for proofs. This development of
the theory of parallelograms is complete and clear. The extra theorem about transversals and congruent segments is an interesting and useful application. The examples of proofs do a good job of making clear how proofs are written.

The selected topics from the Washington Standards are covered fully in Prentice-Hall Geometry. Some things that distinguish this text are the unusual placement of the angle sum theorems and the inclusion of an additional theorem about parallels. More importantly, the text shows good mathematical judgment. The relationship between postulates about parallels and angles and the Euclidean parallel postulate is understood correctly. The text refrains from labeling every unproved fact as a postulate, instead stating them as “principles” that are merely as-yet unproved theorems. Also, the text avoids some tricky points making hidden and unproved assumptions about distance and parallelism. There is a generous supply of exercises and activities.

4.2.5 Conclusions: Geometry
The Mathematics Standards state that students should know, prove, and apply theorems about angles that arise from parallel lines intersected by a transversal. The development adopted by the reviewed texts is to assume as a postulate that for any two parallel lines intersected by a transversal corresponding angles are congruent. It is immediate to prove that a number of pairs of angles are either congruent or supplementary (for example, alternating interior angles are congruent). Then, as a second postulate, the converse of the first postulate is assumed. After this, it is proved that the necessary conditions in the earlier theorems are in fact sufficient conditions.

One important “backstory” for this development is that these postulates imply the Euclidean Parallel Postulate (EPP). To be precise, the second postulate can be proved as a theorem in Euclidean geometry and the first postulate is equivalent to the EPP. Some of the textbooks try to include some of this background, more or less successfully as the reviews note. It is not strictly necessary for students to know this background for their study of geometry, but if the choice is to introduce the EPP, it would be better to tell the story correctly.

The texts differ in the accuracy and completeness with which they present the relevant mathematics. Holt Geometry and Prentice-Hall Geometry seem to be the most successful in this regard. Teachers might have to be more careful in explicating the mathematics of the other two texts.

4.3 Integrated Mathematics
All of the integrated mathematics materials were three-book series. The same threads were examined here as were examined in the Algebra 1/Algebra 2 and Geometry materials.

One characteristic that distinguishes integrated mathematics materials from more traditional materials is the extensive use of contexts and applications as the focus of attention. Mathematics ideas are typically not presented as “naked” mathematics, but rather as ways to solve problems. This does not mean that the mathematics is less important or less well developed, but it does make a review of mathematical soundness somewhat more complex .

4.3.1 Core-Plus Mathematics
Functions. In Course 1, quadratic functions (Unit 7) are introduced through specific examples (e.g., projectile paths). This specific approach has the potential to create “stereotypical images” in students’ minds that may be difficult to overcome to create a general understanding of quadratic functions. It appears, however, that by the time students work through Investigation 3 a general
understanding should have developed. The teacher’s role in debriefing students’ work is probably critical so that students understand how the parameters for the general quadratic function influence the shape and position of the graph.

In Course 2, quadratic functions are treated as one kind of nonlinear function (Unit 5). This is a strength mathematically, since it helps reinforce the similarities and difference among different kinds of nonlinear functions. It is in this unit that domain and range are emphasized (Lesson 1, Investigation 2) and factoring is developed (Investigation 3). The area model (i.e., algebra tiles) is used to motivate techniques for factoring. Solving of quadratic equations is developed, and the quadratic formula is presented, but it appears to be developed only in the “On Your Own” section of problems/exercises. Lesson 2 focuses on Nonlinear Systems; this provides an immediate application of what was dealt with in Lesson 1.

In Course 3, quadratic functions reappear in Unit 5, Lesson 2: Quadratic Polynomials . Completing the square is the focus of Investigation 1; by this point, all students should be intellectually prepared to understand the mathematics of this idea at a deep level. The vertex form of the equation is addressed here, and complex numbers are introduced with the obvious extension to quadratic equations with no real solutions can be examined .

Geometry. In Course 1, the study of properties of figures begins in Unit 6. “The focus here is on careful visual reasoning, not on formal proof.” (Formal proof is addressed extensively in Course 3.) Unit 6 is “developed and sequenced in a manner consistent with the van Hiele levels of geometric thinking.” Senk’s data (1986) suggest strongly that students who attempt to study proof before the development of Level 2 thinking (e.g., Fuys, Geddes, & Tischler, 1988; Van Hiele, 1986) are unlikely to be successful. Unit 6 is organized to help students develop Level 2 thinking. Because the study of formal proof is delayed another year, there are additional opportunities for this kind of thinking to develop.

Unit 6, Lesson 1, deals with a variety of topics at an informal level, including conditions that determine triangles or quadrilaterals (e.g., triangle inequality ), angle sums for polygons, SSS/SAS/ASA properties of triangles, reasoning about shapes, and the Pythagorean Theorem. Some constructions are included as an extension of this work. Lesson 2 addresses symmetries of figures, angle sums of polygons, and tessellations. The tasks here emphasize relationships among different shapes; these help students internalize Level 2 van Hiele thinking. Specific attention is paid to interior and exterior angles of polygons. Lesson 3 deals with three-dimensional shapes. This work, too, is informal. It is much more exploratory, since students are likely to have less well-developed understanding of three-dimensional shapes.

The primary attention to geometry in Course 2 is coordinate geometry . This is important but does not relate directly to the threads being reviewed here.

In Course 3, Unit 1 addresses proof. The unit begins with an introduction to logical reasoning set in many different contexts, not just geometry. This is an obvious strength for the study of proof. Lessons 2, 3, and 4 address proof in geometry (mainly study of angles when parallel lines are cut by a transversal), algebra, and statistics. Both in this Unit and in Unit 3, the teacher notes are extensive, with considerable detail provided for each of the proofs. These notes would support teachers well in leading discussions that were effective at helping students internalize the critical mathematics ideas.
Unit 3 addressed triangle similarity (Lesson 1) and congruence (Lesson 2). In Lesson 1, students explore a variety of conjectures, for example, all isosceles right triangles are similar. There are numerous applications of similarity which provide a rationale and motivation for proofs. As one would expect in a “proof unit,” there are numerous classic mathematics relationships established and proved. In Lesson 2 congruence is studied as a special case of similarity. Included are the classic triangle congruence theorems, with attention also paid to perpendicular bisectors of sides, angle bisectors, and medians. This is followed by an equally extensive study of the properties of quadrilaterals, with particular attention to parallelograms.

In summary, the mathematics in Core Plus is mathematically sound and very well sequenced to support student learning at a deep level.

4.3.2 SIMMS Integrated Mathematics
Functions. In Level 1, quadratic functions are addressed in Module 10. Distance/time graphs are used as a context to support comparison of these graphs to determine average velocity over a time interval, leading to linear modeling for objects moving at constant speed. Quadratic functions are introduced in Activity 3; topics include coordinates of the vertex, vertex form of quadratic function rule, families of functions (based on y = x2), and translation of parabola graphs. The Chapter ends with an exploration of the quadratic modeling of data.

One difficulty in analyzing the Teacher’s Guide is that there is very little discussion of the mathematics; detailed answers are provided for each task, but there is no rationale provided for the sequencing of these tasks. It might be difficult for some teachers to lead appropriate debriefing of the exercises so that students truly internalize mathematical understanding. Merely solving the tasks correctly does guarantee depth of understanding.

In Level 2, quadratics are addressed in Module 6 as part of the study of polynomials, with parabolas highlighted in Activity 2. Topics addressed include fitting a parabola to three non-collinear points, roots and factors of polynomials, and effects of changing the parameter, a, in the general form of a quadratic function. Embedding quadratic functions in a more general context is a strength for supporting students’ understanding.

In Level 3, Module 11, transformations of functions are addressed. This is a general treatment, though some examples are quadratic functions. There does not appear to be a significant development of quadratic functions, per se, in Level 3.

Geometry. In Level 1, Module 1, simple ideas about angles are used to introduce techniques for studying mathematics. There is little development here. The Activities in Module 4 address surface area of three-dimensional figures, tessellations, and area of regular polygons. These ideas “feel” disconnected, with little obvious attempt to highlight common features of the ideas.

In Level 2, Modules 3 and 7 each address geometric ideas, but again the connections among them are not immediately obvious. Module 3 addresses area of regular polygons and surface area and volume of three-dimensional shapes. Module 7 addresses angles formed by a transversal of parallel lines, tangents and secants to circles , and dilations. Many teachers might need help in communicating to students what key mathematics ideas underlie the tasks. Module 12 is a more traditional treatment of proof. Three areas are addressed: Pythagorean Theorem, triangles, and quadrilaterals. However, there may not be enough tasks to support deep understanding by students of the nature of proof.

In Level 3, Module 6 is a more general treatment of proof. It is strange that this Module is after the Module in Level 2 on proof of triangle and quadrilateral theorems. Certainly students by Level 3 should be ready to learn this material, but it might also have been useful prior to the work with congruent triangles in Level 2.

In summary, the development of mathematical ideas is difficult to follow in SIMMS. This observation seems reinforced by examination of the alignment grid provided by the publisher. Many of the Performance Expectations are addressed in parts of problems scattered across a wide range of pages. It seems likely that some teachers might have difficult in helping students internalize the mathematical ideas based on the tasks they have completed. Also, the Modules seem too short to support in-depth development of mathematical ideas.

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