In her book Knowing and Teaching Elementary Mathematics, Dr. Liping Ma proposed that teachers should possess a "profound understanding" of the mathematics they teach. Profound understanding requires a "connected, curricularly structured and longitudinally coherent knowledge of core mathematics" (Ball, et al. 2005). Furthermore, the American Mathematical Society (AMS) asserts that "effective teaching requires an understanding of the underlying meaning and justification for the ideas and procedures to be taught and the ability to make connections among topics." However, understanding of this kind is seldom achieved by teachers in schools anywhere in the world—in China, Singapore, Russia, France, Finland, or the United States, as the Programme for International Student Assessment (PISA) results clearly indicate.

Other adjectives better capture the aspirations of most American teachers. One alternative portrayal is "comprehensive" understanding for teachers and "practical" understanding for their students. The dictionary defines "comprehensive" as "dealing with all or many details," while "practical" means "capable of being used." These definitions seem consistent with American parents' expectations for their children and their teachers.

Decades of work with children and their teachers have led to the identification of ten steps that can assist experienced teachers, as well as novices, in achieving comprehensive understanding of the mathematics they teach and practical understanding for their students. These steps are sequential, active, and reiterative. They can be applied in the same manner and sequence again and again throughout a teacher's career. With each application, teachers' comprehension of the topics they teach will expand to include more understanding of the topics' roles in the curriculum, in mathematics, and in life. Ma and the AMS propose that students' practical understanding will be enhanced as teachers' comprehension broadens. Teachers who follow these ten steps will experience a new appreciation for and enjoyment of mathematics, as well as increased confidence in their own mathematical abilities. Likewise, their students will become more confident and competent.

The first three steps of the ten-step sequence are intended to precede the teaching of the topic. They provide the preparation needed to plan and conduct an effective classroom experience. Steps 4, 5, and 6 guide the lesson activities themselves, while steps 7 through 10 are completed following the lesson.

Step 1: Read as much about each topic as time permits.
Although there are many excellent professional publications to assist teachers, none is more readily available than the textbook's teacher's guide in which mathematical and pedagogical information is clustered together topic by topic. A good teacher's guide will refer teachers to pertinent primary sources that can be consulted if time permits. Mathematics is the most sequential subject in the elementary school curriculum. A good teacher's guide will allow the sequential nature of mathematics to enhance both comprehensive and practical understanding. Teachers should also look for connections with previous and forthcoming topics so the topic of a specific lesson need not stand isolated from other lessons.

Step 2: Review research that is relevant to the topic.
Thousands of research studies have been published on the teaching of mathematics, and hundreds more appear each year. High-quality, relevant research is key for building teachers' comprehensive understanding of the math they teach. Research-based commentary, like the following highlight of algebraic thinking, provides teachers with succinct summaries of topic-specific research results:

• Martinez (2002) suggests the following for teachers: algebraic expression should be taught starting with "familiar ideas and expressions from arithmetic" and working "through repeated patterns to introduce incremental changes in ideas and procedures." This plan will help students in two ways. First, it will give them a better conceptual understanding of the algebraic concepts, and second, it helps to alleviate some of the intimidation that students sometimes feel toward algebra.
• Nathan and Koedinger (2000) offer an alternative approach to the more traditional ways of teaching of algebraic concepts in sixth grade. They argue that informal methods for solving problems such as guessing and checking "have proven beneficial for students who are developing their understanding of a balanced equation." They go on to state that these informal methods help students to conceptualize, thereby making it easier to learn the more traditional or formal strategies that are needed in algebra.

Step 3: Read the textbook lesson from the student’s perspective.
Reading the lesson, not as a teacher but as a student, allows teachers to identify where students may go wrong or become confused. The ability to anticipate "trouble spots" within a lesson is one of the best indicators of a teacher's comprehensive understanding. For example, if teachers recognize that some students may need help understanding instructions accurately, they can plan activities to assist them.

Step 4: Keep the textbook closed at the beginning of a lesson.
In several movies about schools, teachers hold their students' attention by throwing textbooks out the window or by ripping out "offending" chapters. Teachers do not need to employ quite so much theater to make the point that learning can occur in the absence of a textbook. Since the teacher has already read the lesson from students' viewpoint, the class can begin without the customary preamble, "Open your books to page…." The teacher should introduce the lesson as a learner rather than as an "all-knowing teacher," thus participating as a learner—asking questions, making (intentional) errors, and encouraging students to challenge and correct the teacher's work. Most importantly, teachers should observe the errors their students make. As their comprehensive understanding grows, teachers will find their insights for these errors improving rapidly.

Here are some examples of possible student difficulties:

• By the first grade, children are faced with a number of difficulties when dealing with mathematics. First, they have only recently been acquainted with formal terms—for example, written expression and basic operations of addition and subtraction (Baroody and Wolks 1999; Fuson 1988; Hughes 1986). Also, when they begin to count above 12, they often make mistakes because they commonly are unable to identify patterns and relationships. Some studies have argued that the English language is not always conducive to pattern detection (Miller and Parades 1996). For example, decade transition is not consistent: after "ten," we do not have "ten-one," and instead, we have a unique-sounding number called "eleven." "Ten-two" is also unique—we call it "twelve." Further, the "teen" numbers are spoken as if the ones come before the tens—unlike the "twenties," "thirties," and above, where the first spoken number is the tens digit followed by the ones digit. Having children identify the tens and ones in numbers helps them better understand place value.

Teachers with comprehensive understanding find that interactions like these are especially helpful in meeting students' individual needs.

Step 5: Experiment with alternate approaches.
The advantage teachers gain by reading background materials in step 1 and step 2 and withholding the textbook in step 4 is flexibility. Increased comprehension encourages teachers to be flexible and to employ alternate approaches. If the textbook's approach does not meet students' needs, teachers should experiment with alternative activities of their own design or activities suggested by teacher's guides, handbooks, or other sources. As their comprehensive understanding increases, teachers will become more experimental. After all, teachers know their students better than any textbook author does, and thus should use their knowledge of the topic to vary the textbook's presentation.

Cooperative group activities like those described below may be welcome departures from more structured lessons for some students:

• A study by Irwin (2001) investigated the role of students' everyday knowledge of decimals in supporting the development of their further knowledge of decimals. In this study, students worked in pairs. Half of the pairs worked on problems in familiar contexts and half worked on problems presented without context. A comparison of pretest and posttest results revealed that students who worked on contextual problems made significantly more progress in their knowledge of decimals than did those who worked on noncontextual problems. Dialogues between pairs of students during problem solving were analyzed with respect to the arguments used. For those problems, the less able students more commonly took advantage of their everyday knowledge of decimals. It was postulated that the students who solved contextualized problems were able to build understanding of decimals by reflecting on their everyday knowledge as it pertained to the meaning of decimal numbers and the results of decimal calculations. These results must be considered when we select problems for our students to engage in.

While textbooks serve as curricular guides, teachers should feel free to depart from the lessons of the text when appropriate.

Step 6: Be guided by students' responses and questions.
In The Child's Conception of Number, Piaget was the first to place more importance on students' incorrect responses than on their correct ones. Comprehensive understanding permits the teacher to observe, analyze, and diagnose student responses. Teachers who "role play" as students rapidly develop understanding of the origins of errors and become more effective in correcting them. Role-playing activities that simulate real-world situations are especially useful, as shown in the following example for third-grade teachers:

• There are many in-class activities that build on the learning of money concepts. Teachers can set up a store center in their classrooms. Students can either bring in materials to "sell" or create fictitious items. In either case, discussion can and should ensue regarding the selling prices of items. As an extension, students can make use of newspapers, magazines, or the Internet to help determine prices. Once prices have been set, students can begin to buy and sell items. The key activity here is the process of making change. To reinforce the understanding of place value, use only pennies and dimes, or one-dollar, ten-dollar, and hundred-dollar bills. When considering patterns and skip counting, other coins or bills can be introduced.

Step 7: Connect the topic to past and future lessons.
The National Council of Teachers of Mathematics Curriculum and Evaluation Standards for School Mathematics (1989) stresses the importance of forming connections among various mathematical topics. A "knowledge package" is a collection of related ideas that promotes breadth of understanding. More closely linked ideas form "concept knots." Both knowledge packages and concept knots are essential for truly comprehensive understanding. Here is one example of how teachers can assist sixth-graders with deductive reasoning skills:

• In East Asian countries, children learn that just as numbers can be composed and decomposed as sets and subsets (Ma 1999), geometric figures can be composed and decomposed as well. Consistency and application of mathematical language across topics and strands builds coherence of learning and reinforces understanding.
• Many formulas for area and volume are based on the fact that geometric figures can be decomposed. This method of presentation helps students develop their deductive skills in relation to polygons.
• Students learn to find the area of a parallelogram by decomposing it into a triangle and a trapezoid. The two figures can then be composed into a rectangle. In the same manner, a parallelogram can be decomposed to find the area of a triangle.

Comprehensive understanding of the mathematics to be taught requires a network of links from past to future topics. Links and connections should point vertically to more advanced topics, as well as horizontally across the curriculum of the grade and even adjacent grades.

Step 8: Relate mathematical topics to the real world.
Connections, links, and knowledge packages involve not only mathematical ideas but real-world events as well. Connecting mathematical topics to the real world is critical in achieving practical understanding. Often, textbook problems are caricatures of real-world events. Resources highlighting "real" and "everyday" mathematics frequently do not include genuine applications as intended. Also, applications that connect mathematics with other curricular areas are especially useful.

Step 9: Determine what has and has not been learned.
"High-stakes tests" often attempt to determine "what the student knows." In reality, evaluation should focus upon "what the student has learned" under the teacher's leadership. In many cases, these things are quite different. A conscientious teacher can be discouraged by a cumulative measure of the knowledge possessed by a student who made very substantial improvement in the teacher's class despite low entering scores. Teachers with comprehensive understanding know students' entering characteristics and apply their knowledge of mathematics to diagnose difficulties.

Step 10: Share methods, outcomes, and insights with others.
Japanese teachers are given time during the school day to engage in "lesson study." Lesson study involves discussing specific lessons with other teachers to plan effectively and analyze lessons already taught. American teachers do not have the luxury of released time for informal lesson study; however, informal discussions are helpful especially for novice teachers. Lesson study has been used successfully as a central theme in professional development activities (West 2005).

Conclusion
During past decades many teachers have applied these ten steps to improve their understanding of the mathematics they teach. Invariably, they enhanced their understanding substantially. As teachers' comprehensive understanding increases, so does their students' practical understanding, which, after all, is the objective all educators strive to achieve.

References
Ball, D. L. et al. 2005. Knowing Mathematics for Teaching. American Educator, Fall.

Ma, L. 1999. Knowing and Teaching Elementary Mathematics. Mahwah, NJ: Lawrence Erlbaum Associates.

Martinez, J.G.R. Building Conceptual Bridges From Arithmetic to Algebra. Mathematics Teaching in the Middle School 7: 326–331.

Nathan, J.J. and K.R. Koedinger. Teachers' and Researchers' Beliefs About the Development of Algebraic Reasoning. Journal for Research in Mathematics Education 31: 168–190.

National Council of Teachers of Mathematics. 1989. Curriculum and Evaluation Standards for School Mathematics. Reston, VA: National Council of Teachers of Mathematics.

Piaget, J. 1965. The Child's Conception of Number. New York: W.W. Norton & Co.

Programme for International Student Assessment (PISA). 2004. Executive Summary. Paris: Organisation for Economic Co-operation and Development.

West, K. 2005. Teacher Education in Japan. New York: Columbia University.