And the Rest is Just Algebra

And the Rest is Just Algebra is a book that addresses the increasing weakness in algebra skills among college students. It not only considers the causes of the algebra deficiency but also offers potential solutions for how to improve students’ long-term success and understanding of mathematics. There are 15 contributing authors for the book who are all experts in many different fields. Each author presents key evidence using screen shots of students work. With the interest in students learning algebra, starting from the “Algebra for All” movement in math education in the 1990s, the book presents the difficulties students face, outlined as follows:

• Sources of students’ difficulties in algebra (e.g., impoverished understandings of fraction and proportion).
• The interplay between affect and mathematical understanding.
• The role of reflection in students’ successful algebra learning.
• Alternative conceptions of algebra (e.g., emphasis on functions and modeling).
• The centrality of generalizing and particularizing in algebraic thinking.
• Critical ways of thinking that curricula and instruction fail to foster (e.g., variables vary and expressions have numerical values).
• Uses of computer algebra systems to help students develop symbol sense and structure sense while, ironically, removing any need for them to engage directly in algebraic manipulations.
• New ways to think about linear algebra that both deepen and broaden students’ school algebra.

In short, this book presents to the reader new perspectives on how students think about algebra and their common misunderstandings about the concepts and the operations.

As pointed out in the introduction, every student goes through many different levels of difficulty in mathematics, particularly in algebra. There also comes a time, as seen in calculus, where most or all the computations involve orders of operations, knowledge of trigonometry, and algebraic manipulations. College professors are seeing students being able to set-up complex problems correctly but unable to carry out the computations.

The purpose of the book is not to offer a magical remedy or a set of best practices to solve this problem, but rather to reach out to the mathematics community and motivate us to fix things. Thus, the main question this book addresses is: What are our responsibilities in regard to college students’ poor algebra knowledge?

In Chapter 1, “Algebra Underperformances at College Level: What Are the Consequences?”, Stewart and Reeder provide us eye-opening examples of students’ solutions. They quote a Calculus I student’s comment: “Although I couldn’t really prove a lot on the exams, I did learn how to solve calculus problems, unfortunately what held me back was the algebra.” This sets the stage for this body of research, and for the entire book. On page 10–11, Stewart and Reeder discuss “Cancellation of Symbols.” Here are some examples:

Evaluate the limit if it exists. \[ \lim_{x\to\infty} \frac{6x^2+1}{2x^2-1}=\frac{6x^2+1}{2x^2-1} = 3+\frac{1}{-1}=3-1=2\]

Let \(f(x)\) be the function \[ f(x)=-\frac{1}{x+2}.\] Use the limit definition to find \(f'(x)\). \[ f'(x)=\frac{f(x+h)-f(x)}{h} = \frac{-\frac{1}{(x+h)+2}-\left(-\frac{1}{x+2}\right)}{h} =\frac{-\frac{1}{(x+h)+2}+\frac{1}{x+2}}{h}=\frac{h}{h}=1\]

Find \(\frac{dy}{dx}\) for \(y\cos x=1+\sin(xy)\). \[-\sin\left(\frac{dy}{dx}\right)(1)+\cos(x )(y)=\cos x\left(\frac{dy}{dx}\right)\] \[-\sin\left(\frac{dy}{dx}\right)+\cos(xy)-\cos x\left(\frac{dy}{dx}\right)=0\] \[-\sin\left(\frac{dy}{dx}\right)-\cos\left(\frac{dy}{dx}\right)=-\cos(xy)\] \[\left(\frac{dy}{dx}\right)=\frac{-\cos(xy)}{(-\sin)+(-\cos)}=\frac{xy}{-\sin}.\]

We teachers see these types of misunderstandings on a daily basis. Many questions can be posed as to why this is happening. Is it our responsibility at the college level to fix these problems? Stewart and Reeder offer the suggestion to give students assessment exams before registration for any college mathematics courses and to do remediation work on skills they are deficient in. Many schools in the United States are implementing assessment exams with modules to help students determine which courses they should start their college mathematical careers with. For example, at the Pennsylvania State University, the ALEKS assessment exam is given to all students. In addition, we should be working more closely with high school mathematics teachers to pin point why and how to better assist student’s algebra skills.

Chapter 9 is on “Cognitive Neuroscience and Algebra: Challenging Some Traditional Beliefs.” Two cognitive neuroscience studies are highlighted, from Lee et al. (2010) and Waisman et al. (2014). These two studies help raise our awareness of the cognitive limitations students have in algebra. The findings should serve as a means for us to reflect on this ever increasing handicap students have and how we can try to help. In Chapter 12, “School Algebra to Linear Algebra: Advancing Through the Worlds of Mathematical Thinking,” we learn that many students find the transition between the two subjects very difficult. The most relevant aspect are the proofs in linear algebra. On page 230, Figure 12.13 shows how three students start a proof by assuming what they are trying to prove, try to resort to a symbolic matrix, and make an attempt to find a determinant of an arbitrary matrix. Students soon discover that learning the concepts requires more than computing matrices.

The art of proofs does not come naturally to students. As Harel and Sowder (2005) point out, advanced thinking in mathematics can potentially start as early as elementary school and must not wait until students take courses such as linear algebra. Both elementary and high school mathematics are rich with problems that require deep thinking. There is room for self-discovery projects.

Instructors who are frustrated with students’ misunderstanding of algebra will find this book most helpful. Many of us who see these careless errors over and over again have the tendency to point the finger at high school teachers and even laugh at the results. The days of sitting on the sidelines are over, however, and we need to do something about this growing trend. Our responsibility and, more important, our duty to our students and their future, is to correct their mistakes and help them get over these inaccurate understandings in algebra. I highly recommend this book for those who want to help change this phenomenon.

Peter Olszewski is a Mathematics Lecturer at The Pennsylvania State University, The Behrend College. His research fields are in mathematics education, Cayley Color Graphs, Markov Chains, and mathematical textbooks. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar and bass, reading, gardening, traveling, and painting landscapes.