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.
