Common Core Statement
The goals of teaching and learning mathematics must emphasize conceptual
understanding and problem solving.
“But what does mathematical understanding look like? One hallmark of
mathematical understanding is the ability to justify, in a way appropriate to the
student’s mathematical maturity, why a particular mathematical statement is true
or where a mathematical rule comes from.” (Common Core p. 4)
Specific characteristics of a problem-solving teaching approach include the following
(from Margaret Taplin - Institute of Sathya Sai Education, Hong Kong)
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interactions between students/students and teacher/students (Van Zoest et al.,
1994)
mathematical dialogue and consensus between students (Van Zoest et al., 1994)
teachers providing just enough information to establish background/intent of the
problem, and students clarifying, interpreting, and attempting to construct one or
more solution processes (Cobb et al., 1991)
teachers accepting right/wrong answers in a non-evaluative way (Cobb et al.,
1991)
teachers guiding, coaching, asking insightful questions and sharing in the process
of solving problems (Lester et al., 1994)
teachers knowing when it is appropriate to intervene, and when to step back and
let the pupils make their own way (Lester et al., 1994)
A further characteristic is that a problem-solving approach can be used to
encourage students to make generalizations about rules and concepts, a process,
which is central to mathematics (Evan and Lappin, 1994).
The Common Core Standards for Mathematical Practice are: (Common Core p. 7)
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
General Education Preparatory Mathematics (GEPM)
3-4 semester hours
Prerequisite: Basic Algebra with a “C” or better or appropriate placement
This course is designed to help students develop conceptual understanding and problem
solving ability. In particular this course must satisfy the Common Core Standards for
Mathematical Practice.
1 Make sense of problems and persevere in solving them.
2 Reason abstractly and quantitatively
3 Construct viable arguments and critique the reasoning of others.
4 Model with mathematics.
5 Use appropriate tools strategically.
6 Attend to precision.
7 Look for and make use of structure.
8 Look for and express regularity in repeated reasoning.
Purpose: This course is designed to be a second course in algebra and serves as a
prerequisite for General Education Statistics, General Education Mathematics,
Quantitative Literacy, or Elementary Mathematical Modeling. Students wishing to enroll
in courses other than these courses should take Intermediate Algebra. Students may also
take Intermediate Algebra upon completion of this course if they choose to pursue
courses beyond general education mathematics.
The primary goal of this course is to enable students to develop conceptual understanding
and problem solving competence at the intermediate algebra level. This course differs
from traditional courses in that it does not focus solely on procedures. However certain
procedures are essential to the study of algebra and they will be included.
There must be some flexibility in this course in order to meet the unique needs of
colleges. Colleges must determine how to best prepare their students for future study.
Modalities
1. One semester of Elementary Algebra followed by one semester of GEPM.
2. Eight weeks of Elementary Algebra (3 semester hours) followed by eight
weeks of GEPM (3 semester hours)
3. An integrated that model that covers both elementary algebra and GEPM in
one semester (6 semester hours)
4. An inverted model for GEPM that includes digital lectures for procedural
learning and classroom time for modeling and problem solving.
Course Description: This course focuses on developing mathematical maturity through
problem solving, critical thinking, data analysis, and the writing and communication of
mathematics. Students will develop conceptual and procedural tools that support the use
of key mathematical concepts in a variety of contexts. The instruction should emphasize
the connections between verbal, numerical, symbolic and graphical representation of the
concepts being taught whenever possible. Emphasis should be placed on modeling and
problem solving, with techniques and manipulations covered in context. The appropriate
use of technology, such as a graphing calculator, is strongly encouraged.
Note: The three strands of the course are Algebra, Functions, and Modeling. Each strand
must be covered but colleges are free to determine the amount of time spent on each
strand. The strands together with their descriptions are taken from the Core Standards.
Algebra Overview
Seeing Structure in Expressions
• Interpret the structure of expressions
• Write expressions in equivalent forms to solve problems
Arithmetic with Polynomials and Rational Expressions
• Perform arithmetic operations on polynomials
• Understand the relationship between zeros and factors of polynomials
• Use polynomial identities to solve problems
• Rewrite rational expressions
Creating Equations
• Create equations that describe numbers or relationships
Reasoning with Equations and Inequalities
• Understand solving equations as a process of reasoning and explain the reasoning
• Solve equations and inequalities in one variable
• Solve systems of equations
• Represent and solve equations and inequalities graphically
Functions Overview
Interpreting Functions
• Understand the concept of a function and use function notation
• Interpret functions that arise in applications in terms of the context
• Analyze functions using different representations
Building Functions
• Build a function that models a relationship between two quantities
• Build new functions from existing functions
Linear, Quadratic, and Exponential Models
• Construct and compare linear, quadratic, and exponential models and solve problems
• Interpret expressions for functions in terms of the situation they model
Modeling Overview
Modeling links classroom mathematics and statistics to everyday life, work, and
decision-making. Modeling is the process of choosing and using appropriate mathematics
and statistics to analyze empirical situations, to understand them better, and to improve
decisions. Quantities and their relationships in physical, economic, public policy, social,
and everyday situations can be modeled using mathematical and statistical methods.
When making mathematical models, technology is valuable for varying assumptions,
exploring consequences, and comparing predictions with data.
Course Objectives:
1. Demonstrate understanding of the characteristics of functions and apply this
knowledge in modeling and problem solving.
2. Perform operations on expressions and functions and make use of those
operations in modeling and problem solving.
3. Solve equations in the context of modeling and problem solving.
4. Represent mathematical information symbolically, visually, numerically, and
verbally.
5. Estimate and check answers to mathematical problems in order to determine
reasonableness, identify alternatives, and select optimal results.
6. Recognize the limitations of mathematical models.
7. Use mathematically correct vocabulary and symbolism to communicate orallyand in writing- problem statements, problem-solving methods, and interpretations
of the solutions to problems.
Topics must include the following:
1. Characteristics of functions including graphical analysis.
2. Operations on expressions and functions. (must include factoring)
3. Modeling with functions. These may include linear functions, but should
emphasize at least three types of nonlinear functions (polynomial, rational,
radical, exponential, logarithmic functions).
Topics must also include at least two of the following:
1.
2.
3.
4.
Modeling with systems of equations
Modeling using probability and statistics.
Modeling using geometry and right triangle trigonometry.
Modeling using proportional reasoning.