The following section represents the work of Drs. King and Bright in reviewing
the mathematical soundness of the top four high school curricular materials for
Algebra 1 and 2, Geometry and Integrated Mathematics 1, 2 and 3. The team
selected key standards that represent important development of mathematical
concepts that allow students to be well-prepared to continue in mathematics
study. The selection of these standards does not imply that these are more
valuable than others; it simply provided a method for deep analysis on central
themes.
Review of Mathematical Soundness of High School Curriculum Materials
James R. King, Ph.D. and George W. Bright, Ph.D.
The OSPI alignment study of high school curriculum materials was organized in
three categories: Algebra 1/Algebra 2 materials, Geometry materials, and
Integrated Mathematics materials. This review of mathematical soundness is
organized in the same way. For each category, the Performance Expectations that
drove the review are listed first. However, we did not replicate the alignment
study that OSPI has already completed. Rather, we looked for evidence of
mathematical soundness; that is, mathematical correctness and coherent
development of ideas. Only the best-aligned materials (based on preliminary
analysis of the OSPI alignment study) were reviewed; the order of these reviews
reflects the order of these materials in the preliminary data analysis. A
summary/synthesis of the reviews is provided at the end of each section. Any
review of mathematical soundness of necessity reflects reviewers’ views about
mathematics itself about how an idea is, or should be, explained. Different
mathematicians will potentially have different views on the “best way” to
present an idea
so that it is clear. Geometers and topologists, for example, “see” mathematical
ideas differently, even though they study some of the same mathematical objects.
No review is likely to represent all possible views. We were looking for
evidence that materials provided opportunities for students to develop
mathematical understanding that would be rich and deep, as opposed to
compartmentalized.
In general, the materials we reviewed were found to be mathematically sound.
However, we found differences among the materials related to the development of
rich, deep mathematical understanding. These differences might be important to
districts as they consider choosing materials for instructional use.
4.1 Algebra 1/Algebra 2
One of the major organizing ideas in algebra is functions. Students in Algebra
1/Algebra 2 are expected to become very familiar with linear, quadratic, and
exponential functions and to gain some experience with other kinds of functions.
There are many ways that the mathematics ideas related to functions might be
examined. We have chosen two categories of ideas.
First, we chose to examine the development of one class of functions. The class
of functions that seems most extensively developed in the high school PEs is
quadratics; this is an important class of functions for high school students,
both for developing mathematical maturity and in terms of application to
science.
The relevant PEs are listed below.
A1.1.D (M2.1.B)
Solve problems that can be represented by quadratic functions
and
equations.
A1.5.A (M2.2.A)
Represent a quadratic function with a formula /convert-decimal-to-symbolic.html">symbolic expression , as a
graph,
in a table, and with a description, and make connections among the
representations.
A1.5.B (M2.2.B)
Sketch the graph of a quadratic function, describe the effects
that
changes in the parameters have on the graph, and interpret the x-intercepts as
solutions to a quadratic equation.
A1.5.C (M2.2.D)
Solve quadratic equations that can be factored as (ax + b)(cx +
d)
where a, b, c, and d are integers.
A1.5.D (M2.2.F)
Solve quadratic equations that have real roots by completing the
square
and by using the quadratic formula.
A2.3.A (M2.2.C)
Translate between the standard form of a quadratic function, the
vertex
form, and the factored form; graph and interpret the meaning of each form.
A2.3.B (M2.2.E)
Determine the number and nature of the roots of a quadratic
function.
A2.3.C (M2.2.G)
Solve quadratic equations and inequalities, including equations
with
complex roots .
To a lesser extent, we also examined how some general ideas related to function
were
developed. Understanding domain/range, developing skill at moving among
representations of functions, and identifying the role that parameters play are
all important ideas. The Performance Expectations (PEs) below provide focus for
these ideas.
A1.3.A (M1.2.A)
Determine whether a relationship is a function and identify the
domain,
range, roots, and independent and dependent variables.
A1.3.B (M1.2.B)
Represent a function with a symbolic expression, as a graph, in
a table,
and using words, and make connections among these representations.
A1.4.E (M1.3.B)
Describe how changes in the parameters of linear functions and
functions containing an absolute value of a linear expression affect their
graphs
and the relationships they represent.
A1.5.B (M2.2.B)
Sketch the graph of a quadratic function, describe the effects
that
changes in the parameters have on the graph, and interpret the x-intercepts as
solutions to a quadratic equation.
A1.7.A (M1.7.A)
Sketch the graph for an exponential function of the form y = abn
where n
is an integer, describe the effects that changes in the parameters a and b have
on
the graph, and answer questions that arise in situations modeled by exponential
functions.
4.1.1 Discovering Algebra/Discovering Advanced Algebra
In Discovering Algebra, significant groundwork for the study of functions is
laid in Chapter 7. It is significant that the ideas are developed here for
functions in general; this creates a coherent mathematical sequence that is
critical for helping students “see” the mathematical big picture. Domain and
range for relations and functions are introduced in Lesson 7.1 and reinforced
throughout the chapter. The vertical line test is introduced in Lesson 7.2, with
application to the graphs of a wide range of functions/relations. Lessons 7.3
and 7.4 develop critical understanding of how functions can be used to represent
different contexts; this helps motivate the need to study special kinds of
functions, beginning in Lesson 7.5 (absolute value function) and Lesson 7.6
( parabolas ).
Chapter 8 (Transformations of Functions) provides general background on how
different function rules (e.g., y = |x| and y = |x| + 3 or y = x^2 and y = x^2 +
3) generate graphs that look the same but are in different positions through
translation, reflection, and scaling. Dealing with these issues in general
prevents the need to deal with a collection of special cases when quadratic
functions are studied (Chapter 9). This approach provides coherence to the
mathematics ideas and would seem to make the
mathematics more easily learned. For example, when students encounter Chapter 9,
they will already know the effect of changing the value of a in the equation, y
= ax + b.
Chapter 9 deals with quadratic functions. The introduction is through the
modeling of real-world situations, but more standard ideas are addressed almost
immediately: roots and vertex (Lesson 9.2), vertex and general form (Lesson
9.3), factoring (Lesson 9.4), completing the square (Lesson 9.6), and quadratic
formula (Lesson 9.7). The extension to cubic equations (Lesson 9.8) provides a
“non-example” that helps cement understanding of properties of quadratic
functions. The development of critical ideas earlier in the context of many
different functions should help students develop rich cognitive understanding
that can be retained permanently.
In Discovering Advanced Algebra functions and transformations of functions are
addressed in Chapter 4; again, the ideas are applied to a range of functions as
a means of illustrating the power of these ideas. Lesson 4.4 specifically
addresses transformations of quadratic functions. Chapter 7 (Quadratic and Other
Polynomial Functions ) provides specific review and extension of the study of
quadratic functions. Topics include finite differences (Lesson 7.1), equivalent
forms/rules (Lesson 7.2), completing the square (Lesson 7.3), quadratic formula
(Lesson 7.4), and complex numbers (Lesson 7.5) which allows factoring of
previously “unfactorable” quadratic expressions. Extension to higherorder
polynomials provides a contrast quadratic functions; having examples and
nonexamples of the relevant ideas is important for helping students generalize
accurately.
In general, the “Discovering” series strikes a very good balance between
teaching general concepts/skills (e.g., transformations of functions) and
specific concepts/skills related to quadratic functions (e.g., equation of the
line of symmetry of a parabola). The mathematics is developed coherently (and
soundly). By the end of the Advanced Algebra course, students should be quite
ready to move on to pre-calculus.
4.1.2 Holt Algebra 1/Algebra 2
In Algebra 1, functions as rules are introduced in Chapter 1, but the ideas are
not developed until Chapter 4. Operations on polynomials, factoring, and
quadratic functions are addressed in Chapters 7, 8, and 9.
In Chapter 4, graphs are used to represent situations. Then the standard
characteristics of functions are discussed: relations and functions (Lesson
4-2), vertical line test (Lab Lesson 4-2), function rules (Lesson 4-3), graphing
(Lesson 4-4), and multiple representations of functions (Technology Lab Lesson
4-4). These ideas are treated somewhat compartmentally, however.
The second half of Chapter 7 addresses addition, subtraction, and multiplication
of polynomials, including special products of binomials (i.e., squares of
binomials and product of sum and difference of two quantities). Algebra tiles
are used to model the ideas, but symbolic manipulation (including FOIL) is the
technique used in the worked out examples in the lessons.
Chapter 8 addresses factoring, first for monomials and then of general
trinomials (i.e., x^2 +bx + c and ax^2 +bx + c), with special products (e.g.,
difference of two squares) following. In worked-out examples, factoring is
completed by identifying combinations of the factors of c and a to generate b.
The modeling with algebra tiles in the introductory Lab Lesson is not extended
into the “regular”
lessons. Lesson 8-6 brings all of the techniques together by discussing
“choosing a factoring method;” this is a nice way to help students reflect on
what they have learned in the chapter.
Chapter 9 deals with quadratic functions. In Lesson 9-1 the idea of constant
second differences is introduced and related to constant first differences
already developed for linear functions. Lab Lesson 9-2 provides an opportunity
for explorations leading to the equation for the axis of symmetry. Additional
worked-out examples highlight relationships among the zeros , the axis of
symmetry, and the vertex; graphing of parabolas (Lesson 9-3) is centered around
these relationships. Families of quadratic functions (Lab Lesson 9-4) and
transformations (Lesson 9-4) build on the ideas developed about graphing. The
second half of the chapter deals with solving quadratic equations, completing
the square, and the quadratic formula.
In Algebra 2 functions are reviewed and extended in Chapter 1; this includes
attention to transformations of functions and an emphasis on “parent” functions.
Chapter 5 (Quadratic Functions) begins from this orientation of parent functions
and leads to the vertex form of the quadratic equation. This is a very nice way
to provide conceptual grounding for the entire chapter. Lab Lesson 5-3 connects
the graph of a quadratic and the graphs of the factors of the quadratic
expression; this, too, provides very good conceptual underpinning for
understanding characteristics of quadratic functions. The primary extension for
the remainder of this chapter is complex numbers, with applications to solving
quadratic equations with no real roots.
Although the sequence of ideas in this series is fairly traditional, opportunity
is provided for students to make connections among the ideas. It seems likely
that students will exit with a rich understanding of the mathematics ideas
underlying quadratic functions. Mathematical soundness, thus, is clearly
evident.
