Hiebert, J., Carpenter T. P., Elizabeth F., Fuson K. C., Wearne D., Murray H., Olivier A., & Human P. (1997). Making Sense: Teaching and Learning Mathematics with Understanding. Reviewed by Aslıhan Osmanoğlu, Middle East Technical University

Hiebert, J., Carpenter T. P., Elizabeth F., Fuson K. C., Wearne D., Murray H., Olivier A., & Human P. (1997). Making Sense: Teaching and Learning Mathematics with Understanding. Portsmouth, NH: Heinemann xx + 184 pp.     $22.50     ISBN 0-435-07132-7 Reviewed by Aslıhan Osmanoğlu Middle East Technical University January 31, 2007 What is learning with understanding and how is it achieved? How does one design classrooms that foster such learning? Making Sense provides answers to these questions. The researchers leading four different projects come together in this book with the purpose to come to an agreement on the core features of classrooms that foster meaningful learning and increase students’ understanding of mathematics. The projects are called Cognitively Guided Instruction (CGI), Conceptually Based Instruction (CBI), Problem Centered Learning (PCL), and Supporting Ten-Structured Thinking (STST) directed by Carpenter, Fennema, and Franke; Hiebert and Wearne; Human, Murray, and Olivier; and Fuson, respectively. When such high-quality researchers who work on instruction in order to improve conditions of education come together and cooperate, it is predictable that a strong, well designed book will result. The book frames the core features of classrooms that support learning with understanding in five dimensions: The nature of classroom tasks, the role of the teacher, the social culture of the classroom, mathematical tools, and equity and accessibility. These dimensions are explained in detail in individual chapters. The chapters in the book are organized into four main parts: The introduction providing an overview-chapter 1; description of five dimensions of classrooms that foster meaningful learning-chapters 2, 3, 4, 5, 6; stories of classrooms from the four projects-chapters 7, 8, 9, 10; and the conclusion-chapter 11. As a foreword, Mary Montgomery Lindquist, past-president of NCTM, clearly describes why it is important to understand mathematics. She gives some real-life examples to draw a picture of how students see mathematics. One example is telling; a student says that no matter how meaningless a statement is, it might be assumed to be true in mathematics. Lindquist’s mentioning the history of perspectives in education provides a comprehensive understanding of understanding, and her explanations of the reasons for little progress in achieving learning with understanding help readers see the purpose of the book. In the first chapter, the authors explain why it is necessary to learn mathematics with understanding, and they go on to show how to design classrooms that foster such learning. The dimensions that they describe--tasks, teachers’ role, social culture, tools, and equity--provide the reader, especially teachers, a good chance of applying what they read to their own experiences. These features also seem closely connected to the U. S. National Council of Teachers of Mathematics (NCTM) Principles for School Mathematics. According to these principles, “effective mathematics teaching requires understanding what students know and need to learn and then challenging and supporting them to learn it well,” “students must learn mathematics with understanding, actively building new knowledge from experience and prior knowledge,” and “excellence in mathematics education requires equity—high expectations and strong support for all students.” One of the principles in NCTM is the curriculum principle, according towhich a curriculum should connect mathematical topics in order for students to build relationships between them, and understand mathematics. Actually, this is the aim of new curricula versus traditional curricula mentioned in several studies and articles. The new curricula are meant to create an environment for children to connect mathematical ideas and get a deep understanding of mathematics. What is lost sometimes is what those authors mean by learning with understanding when writing about the gains of such curricula. At this point the book gives a clear definition of what we need to understand when we talk about understanding. Thus, it is important that the authors have given a clear definition of understanding in their book especially with examples (p.4). Understanding is defined as connecting or relating what is learned to other things that are known. But the work does not end there. As they claim, not every connection is useful, and it is necessary to look into the processes of reflection and communication that play important roles in making useful connections. These two processes make a lot of sense when we think about how we come to an understanding. If understanding is making useful connections between what we learn and what we know, then the processes that we experience internally (reflection) and externally (communication) while building those connections play an important role in understanding. I believe these two processes constitute the most important basis of the book as the five dimensions that are explained in the book are mainly based on those processes as well. In the second chapter, the authors focus on the first dimension of classrooms that support learning with understanding, namely, the nature of classroom tasks. They claim that tasks play an important role in building understanding. The tasks that lead to building relationships between various ideas or between what is already known and what is learned help students with understanding. They are also important in shaping students’ perceptions of the subject. What is important is that the tasks should be problematic and interesting and that this problematic side of the tasks should come from mathematics. They also should be appropriate for students’ level of skills and knowledge in order for them to leave some mathematical value behind. According to the authors, what is important is for teachers to select tasks that encourage reflection and communication, that are suitable for the use of tools, and that leave some residues behind like mathematical relationships and strategies or methods that students understand. In the third chapter, the focus is on the second dimension, the role of the teacher. The most important role of the teacher is defined as creating an environment for children to be able to reflect on and communicate about mathematics. The responsibility of the teacher is to provide direction and help students with creating their own understanding without a great deal of interference. The teacher is responsible for selecting and sequencing appropriate tasks that permit reflection and communication. The authors suggest that teachers use their own learning goals for students, and set tasks and select sequences of tasks with those goals in mind. Another responsibility of teachers is that of being able to provide relevant information. How much information a teacher should share with her students is well explained by the statements taken from Dewey. It is important that teachers do not provide too much information, unless it is necessary, or tell the students the right answer directly without letting them find it on their own. This would reduce the interaction between the students, and also prevent them from having discussions which might open their minds (Leinhardt, 1988). The fourth chapter deals with the social culture of the classroom, the third dimension. The authors state that students’ working on problems together and sharing the methods that they develop is doing mathematics, and a social culture that fosters such interactions is needed. They underline the importance of such a social culture as it is only possible to bring it about when students have opportunities to build understanding through communicating and interacting with each other, to share different problem solving methods, and to approach problems from different points of view. Such an environment makes it possible for students to experience cognitive conflicts that bring about reevaluating and then reorganizing their thinking. It is also crucial that the social culture of the classroom should permit students learning from teaching others while explaining their solution methods; it should value mistakes as opportunities for learning, and give the responsibility of determining correctness to students through working on problems. The fifth chapter focuses on the mathematical tools. What is meant by tools is language, materials, and symbols. According to the authors, it is a necessary condition for students to spend time with using tools to be able to develop meaning for them. Construction of meaning requires that students interact with the tools, so active use of them is a necessary condition. Also the discussions held during the interaction with the tools are very important as construction of meaning for tools happens through communication. The tools also should be used to connect what is learned to what is already known in order to make sense of what is done. Their use also makes it easier for students to work on problems as they leave space for focusing on more creative aspects of the problem rather than trying merely to memorize things. The teacher’s role becomes critical at this point in that only the appropriate use of tools leads to opportunities for students to shape their thinking and develop understanding. The sixth chapter deals the fifth and last dimension of mathematics classrooms, equity and accessibility. This chapter explains why it is important to give equal chances to any child to learn with understanding. This dimension may be the hardest one to accomplish, but also the most critical and necessary one. It is well explained in the book that one can not expect students to learn to the same degree. Providing equal chances for learning with understanding to everyone can and will decrease the gap between different groups. According to the authors, again the responsibility of the teacher to create a learning environment in which equity and accessibility is achieved plays an important role. Not only does the teacher have an effect on building such an environment, but also the other dimensions mentioned so far have shared effects. In the third part of the book, the authors provide four different stories from the four projects that they lead. In my opinion, this part is the most useful as it makes what the authors try to explain throughout the book clear and concrete. Reading real stories from actual classroom settings is a best way to digest what is read as well as to reflect on our own experiences. In chapter 7, a story from a Cognitively Guided Instruction (CGI) classroom is shared with the readers. The CGI program focuses on teachers’ understanding of their students’ thinking. In the episode, the teacher deals with a diverse class in several ways. The choice of tasks, valuing different strategies, the interaction between the students, connecting mathematics with real life experiences and with other subjects, students’ spending sufficient time with tools, giving equal word to every student, and the variety of activities chosen are all related to what the authors provide in theory. The reader is expected to connect the theory to the given episode, and digest it through making connections between the idea of designing classrooms that foster learning with understanding and what they read in the episode. This is a good way of portraying what it means to talk about effective classrooms. All the details provided in the episode help the reader to visualize that classroom, and the conversations between the teacher and her students provide a deeper understanding of the structure of the lesson. The teacher’s role, use of multiple strategies, learning from mistakes, the use of tools, and the sharing atmosphere in the episode lead to an understanding of the design of a classroom in which understanding is supported and developed. In chapter 8, the authors mention the Conceptually Based Instruction (CBI) project first, and then give an example of a classroom. This project provides an alternative instruction, and focuses on students building relationships between their previous knowledge and new knowledge, on different ways of representations, and on multiple strategies. This focus fits with the definition of understanding that the authors present in the first part of the book, and gives clues to the design of the classroom in this episode. In this episode, it is possible to see how real-life problems are chosen, different representations are constructed and discussed by students, different approaches are valued, the teacher make use of any opportunities to create discourse, and misconceptions are valued as opportunities for learning. Chapter 9 provides background information on the third project, Problem-Centered Learning (PCL) first, and then follows with a story of a problem-solving session. In PCL, the focus is on the conceptual development of students, and it is achieved through working on problematic and meaningful computation problems. In this project also it is possible to see evidences of reflection and communication as well as the facilitator role of the teacher who doesn’t intervene much or quickly. Here, the main aim is to create a learning environment for children to personally construct meaning. Social interaction and communication are taken as the ways to increase reflection. The episode explains well enough how a classroom environment fosters children’s mathematical understanding through discussing a challenging problem with their peers and turning the ideas around in their minds in a way that results in personal construction of meaning at the end. This episode is especially useful for the reader to see what happens when teacher intervention is quite low and the students work on their own. In chapter 10, the authors share a story from a Supporting Ten-Structured Thinking (STST) classroom. The focus of the project at the beginning of the year is on student learning through decomposition of numbers and transforming their stories into word problems. These are very helpful for students to understand mathematics as they provide several opportunities for them to approach mathematics from different directions. The way that the students are asked to pose different questions about numbers given in a situation seems like a great opportunity for them to engage in mathematical thinking. Also the responsibility is placed on students for being active, sharing their thinking with others, and helping others foster learning. As in the other episodes, the reader sees the importance of task selection, the role of the teacher, the necessity of a classroom culture that fosters meaningful learning, the use of tools as supports for mathematical thinking, and finally how equity is maintained through giving the role of teaching others to every student and helping them to try their best to work on advanced methods that they are capable of. Although it is obvious that reading real life classroom stories provides easy understanding of what the authors talk about, the reader may think that to what extent the teachers in these episodes represent the teachers that we see in real classrooms. The teachers in these episodes sound like experts who can accomplish to design classrooms that foster meaningful learning. Selection of such teachers and episodes make sense since the authors try to draw a picture of how a classroom would look like if understanding is the target. The problem is that not every teacher can achieve to carry out all dimensions as these teachers do. How much is expected from teachers to get out of this book or how much teachers can make use of it is somewhat unclear. It would be naïve to expect every teacher to be able to fulfill the responsibilities laid out in the book. The last part of the book--chapter 11--is the conclusion and summary. In this chapter, the authors revisit the core features of classrooms that foster meaningful learning, and this time support them with examples from the stories of classrooms that they share with the reader. This helps the reader to see more clearly how the core features of classrooms explained throughout the book come to life in those episodes from actual classrooms. The authors conclude the book by putting the responsibility on teachers’ shoulders to design classrooms that provide students learning with understanding. They hope the teachers make use of what they read about the dimensions and core features of classrooms as well as the episodes from the project classrooms. In spite of the fact that much of the responsibility is placed on the teachers to design instruction, some of the duties need to be carried out by administrators, parents, and students as well. Teachers are already asked to do much, and the expectations are very high (Lortie, 1975). So, supportive administrators should make teachers’ work easier for them by providing them an environment that enables creating classrooms that foster understanding. Parents should be willing to send their children to schools where understanding rather than memorization and test-preparation is valued. Students themselves should be interested in learning with understanding, and willing to help to establish equity in their classrooms. Designing classrooms for understanding requires a team work. What is missing in the book is that the authors seem to neglect the importance of classroom conditions on being able to foster understanding. They seem to assume adequate classroom conditions. Although Cohen (1998) thinks that the classroom conditions are not sufficient to account for teachers’ resistance to engage in adventurous instruction, it would be naïve to think that it is easy to accomplish teaching for learning with understanding in classrooms with a large number of students. Delay, denial, interruption, and social distraction are all produced by the crowded conditions of the classroom (Jackson, 1990). What about time and the teacher’s responsibility to cover a lesson plan? In the first episode, the class spends 40 minutes of the lesson on solving two problems, and then the students share their solutions. In real life, the lesson plans ask teachers to cover more in a class period. This book is a powerful statement about the necessity of teaching for mathematical understanding, and it provides teachers a chance to examine their teaching and make changes in their instruction through reflecting on the explanations and episodes provided. I believe all teachers should read this book if they want to improve their teaching. Although I think that it is not possible that every teacher can make use of this book to the fullest degree, still I believe there are many lessons to be learned here, especially from the stories of actual classrooms. References Cohen, D.K. (1998). Teaching Practice: Plus Ca Change. The National Center Research on Teacher Education, 88-3. Jackson, P. W. (1990). Life in Classrooms (pp. ix-33). NY: Teachers College Press. Leinhardt, G. (1988). Expertise in Instructional Lessons: An Example from Fractions. In Grows, D. A., Cooney, T. J., Jones, D. (Eds.). Perspectives on Research on Effective Mathematics Teaching, 1, 47-66. Lawrance Erlbaum. Lortie, D. C. (1975). School Teacher: A Sociological Study (pp.134-161). Chicago Press. National Council of Teachers of Mathematics. Principles and Standards for School Mathematics. Retrieved November 23, 2006, from http://standards.nctm.org/document/chapter2/.