Introduction - About the Book
- Page ID
- 210149
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)The idea of creating this textbook is to provide a resource for a course that Lance Vobornik and I developed in 2022 when Dr. Jeff Thunder asked us to be creative in ways we can better support students' college mathematics learning experience. The result of this second edition is a product of collaboration between instructors who have taught this class and provided feedback. For example, Peter Sassman and Alexandra Hill provided insightful approaches to emphasize conceptual understanding, Mike Mutersbaugh and William O'Conner met weekly with me to discuss the course and provided meaningful feedback of approaches we use in this second edition. Other insights from Jeanne Padilla, Roy Quintero, Scott Rexford, and Allison O'Conner have also made this book possible.
The textbook's approach is to develop strong preparation for the precalculus and calculus sequence through college-level skills to complete mathematical work and to develop conceptual understanding, rather than focusing solely on memorization and repetition of processes. I like to think about the book in three parts: Logic and Critical Thinking, Number Sense, and Finances and Statistics.
Logic and Critical Thinking (An introduction to Chapters 1 and 2)
Doing mathematics requires more than just the traditional mathematical skills (e.g., adding fractions, multiplying natural numbers, evaluating variables, solving equations). In the first part of this textbook (Chapters 1 and 2), it is expected that students will learn the foundation of mathematical logic, thinking, and problem-solving. Students will learn different types of thinking, including creative, analytical, and critical thinking. In addition, we expect students to learn different strategies for solving problems and effective ways of explaining their work.
This part has the main objective of explicitly providing students with tools for solving problems and communicating their work to others. Specifically, these tools will help students explore different types of thinking and strategies for solving problems. In addition, the last section of this part provides students with a checklist on how to effectively communicate their work to their instructor when writing their homework. These communication techniques to explain their work will not only help them write clear homework and mathematical solutions, but also develop communication skills to solve problems with other students and future coworkers. Students can look back at these chapters when they are working on their homework or studying for exams, even if they are preparing for future math classes' exams.
As a supplement to this part, we encourage you to check the Huskie Academic Success Center: Student Success Tips and Tools. Especially, the “study smarter” section with the following resources: growth mindset, studying for math and physics, studying for STEM, and using office hours effectively.
Number Sense (An introduction to Chapters 3 and 4)
In college, there are two major concepts that every student is expected to master: fractions and algebra. In this part of our textbook, we provide students with tools to develop a deeper conceptual understanding of a fraction and the rationale behind the processes involving fraction operations. Similarly, we reflect on the foundations of algebra representations.
Regarding fractions, we expect that students develop a deeper understanding and their own answers to the following questions:
1. Is a fraction one number or two numbers? What can I represent with fractions?
2. When adding or subtracting fractions: Why do we need to have the same denominator? Why are we supposed to add or subtract the numerators?
3. When multiplying fractions: Why do we need to multiply the numerator by the numerator and the denominator by the denominator? Why does this process work?
4. When dividing fractions: Why do we need to “invert and multiply”? Why does this process work?
The authors of these chapters will facilitate tools for understanding these questions by presenting the concepts in several ways. It is expected that by understanding these questions, students can master fraction concepts and processes and avoid common fraction mistakes.
Regarding algebra, we expect to develop the concept of variables through understanding patterns. The authors of this chapter facilitate learning algebra by generalizing patterns in figures and using tools to explore unknown values. In addition, we finalize the algebra ideas by connecting to linear and exponential growth. It is expected that at the end of this chapter, students can recognize linear and exponential behaviors in graphs and equations. This knowledge is foundational in finance and calculus courses.
Finance and Statistics (An introduction to Chapters 5 and 6)
In the final part of this textbook (Chapters 5 and 6), students connect their developing mathematical skills to real-life contexts involving personal finance and foundational statistics. These chapters are designed to help students understand how mathematics informs everyday decisions—whether managing a savings account, navigating credit card interest, or interpreting data in school, work, and daily life.
In the finance chapter, we provide students with mathematical models rooted in algebra to analyze common financial situations. Students will explore how savings grow over time, and how interest rates affect loans and credit cards. Rather than memorizing formulas, students will learn why these financial models work and how algebraic relationships help them make informed decisions. Our goal is to empower students to understand the long-term consequences of financial choices and to apply mathematical reasoning to their personal financial planning.
In the statistics chapter, students are introduced to the fundamental concepts of data analysis. They will learn measures of central tendency, including the mean, median, and mode, and explore how each measure helps describe a data set. In addition, students will examine key ideas such as variability, distribution shapes, and the importance of context when interpreting data. Through real-world examples, this chapter emphasizes how statistics allow us to make sense of information, identify meaningful patterns, and avoid common misinterpretations.