In *Prospective Mathematics Teachers’ Knowledge of Algebra,* Rongjin Huang brings to the forefront a comparative study of the knowledge of algebra 21^{st} century teachers have in China and in the United States. Huang further narrows his study to middle and high school teachers, which play an integral role in the development of more advanced mathematical study. The main result of the study is that the preparation of Chinese teachers is stronger than the United States teachers: “The data analysis showed overall the Chinese participants have a much greater knowledge of algebra for teaching than their U.S. counterparts.” However, as the book points out, there are some comparable and different results. As Huang points out:

- The U.S. participants showed a better understanding of introducing the concept of slope from multiple perspectives than the Chinese counterparts.
- Both the U.S. and Chinese participants revealed weaknesses in presenting numerical relations and algebraic equations using geometrical representations.
- The Chinese participants tended to make their judgments based on visual information and underlying conceptual understanding and logical reasoning while the U.S. participants tended to make their judgments mainly based on visual information without paying close attention to underlying concepts and logical reasoning.
- Chinese participants demonstrated strong knowledge and skills in algebraic manipulation and quadratic functions/equations/inequalities.

Huang begins his book with talking about KAT, which stands for the Knowledge for Algebra Teaching. As pointed out in the book, algebra became part of the civil rights movement, but today it is important to note that the learning of algebra has a different importance. In short, the knowledge of algebra 21^{st} century teachers have is quite different. Huang talks about the research team formed in Michigan State University and how the development of KAT came to be and how it could be measured. The four main research questions Huang addresses in the book are:

- What are the differences and similarities of KAT between China and the U.S. prospective teachers?
- What are the relationships among different components of KAT within and between China and the U.S.?
- What are the differences and similarities between Chinese and U.S. prospective teachers’
*Knowledge for Teaching Concept of Function (KTCF)?*
- What are the relationships between prospective teachers’ status of KAT and their course-taking?

On page 7, Huang references Shulman’s 1986 seminal work on teachers’ knowledge, which is still very relevant today, distinguishing *content knowledge, curriculum knowledge, *and *pedagogical knowledge.* With these fundamental ideas, one thing is true for teachers of any generation, “teachers’ knowledge of mathematics alone is insufficient to support their attempts to teach mathematics effectively.” This is even more so the case today, as many students want to know “why are we studying this” or “how will I use this in the future.” As we go through our education and higher education, we often learn the computations without making connections as to *how* the concepts are used in the real world. We often make these connections after we graduate and either begin teaching or working in industry.

Throughout the book, there are many studies done on classical mistakes students make and how the teachers’ from each country are either prepared for the questions, underprepared, and how they can address them. Some of the most interesting examples are on pages 17–18 and deal with functions, a fundamental part of preparing for Calculus. Page 19 talks about expressions and the mistake of applying the distributive property to \(\sin(\alpha + \beta)=\sin \alpha + \sin\beta\) or \(\displaystyle\frac{AX+BY}{X+Y}=A+B\), or how if \((X - A)(X - B) = K\), then either \( (X - A) = K\) or \( (X - B) = K\). The subtlety of solving \(3x^2+3x+3=0\) versus how one cannot divide all the coefficients by 3 in the function \( y = 3x^2 + 3x + 3\) is a very important example all mathematical algebra students should see.

Further in the book, Huang presents to the reader *Measuring Knowledge for Teaching the Concept of Function* through an exam taken by teachers. There are several types of these assessments presented for both the teachers and students. One of the most interesting is the problem \( (x+3)(x-4)>0\), discussed on pages 96–101, where students were asked to come up with two different solutions. This, I believe, is the clearest example of how the teaching and the knowledge teachers attain gets passed down differently in each country. Here, only one U.S. student provided two correct algebraic solutions, and none used a graphical approach. Interviewing the students afterward revealed that they “had no interconnected knowledge network.” In other words, the students had no ability to connect what they knew to the problem of solving the inequality. However, when motivated and “enlightened” (by being asked if \(ab>0\), what can you conclude about \(a\) and \(b\)?), they were able to make the connections to solve the problem. On the other hand, Chinese students were able to provide multiple ways to solve the inequality. In fact, as the book points out, 80% of the Chinese students provided two solutions, one algebraic and the other graphical.

Huang has presented and laid out some real pressing issues we, as teachers need to address. Blindly going through a mathematics major without stepping back and asking ourselves “how can this be used in the real world” is a fate many students face. With the ever-increasing push for graduates to attain working experience in their field even before graduation, it is not enough to teach the mathematics on a concept-by-concept level. Connections, applications, and logic to solve problems must be passed down to the next generation in order for our students to be globally competitive.

Peter Olszewski is a Mathematics Lecturer at Penn State Erie, The Behrend College, an editor for Larson Texts, Inc. in Erie, PA, and is the 362^{nd} Pennsylvania Alpha Beta Chapter Advisor of Pi Mu Epsilon. He can be reached at pto2@psu.edu. Outside of teaching and textbook editing, he enjoys playing golf, playing guitar, reading, gardening, traveling, and painting landscapes.