I recently read Knowing and Teaching Elementary Mathematics by Liping Ma. I really enjoyed this book and I wanted share my thoughts on some of its key messages. But first, a brief description of the book. The book compares the responses to a series of interview questions given to both US and Chinese elementary teachers. These quesions are part of the Teacher Education and Learning to Teach (TELT) Study. The four questions from the book deal with subtraction with regrouping, multidigit multiplication, division with fractions and the relationship between area and perimeter. Liping Ma analyses teachers' responses to discern how their understanding of mathematics influence how they instruct and respond to their students. Richard Askey wrote ane extensive review of Liping Ma's book for the American Educator, a journal of the American Federation of Teachers. The key message that I took from this book is the importance for teachers to develop a profound understanding of fundamental mathematics. Liping Ma describes this as having, "an understanding of the terrain of fundamental mathematics that is deep, broad, and thorough." (p. 120) It's through this conceptual understanding of elementary mathematics that teachers are able to think flexibly about mathematics, appreciate and understand a variety of methods for solving problems and respond to their students' novel ideas and questions. As Tracy Johnston Zager noted in her TMC16 keynote address, K2 mathematics is not "the basics" rather it is, "student's introduction to what it means to do mathematics, what is math as a discipline" as well teaching kids "what it means to have a mathematical discussion, a mathematical idea, a mathematical question". Liping Ma says that "elementary mathematics is not a simple collection of disconnected number facts and calculational algorithms. Rather, it is an intellectually demanding, challenging, and exciting field  a foundation on which much can be built." (p. 116) Indeed, no matter what level of mathematics you teach, you should know it deeply and how it connects to related areas of mathematics. Liping Ma states, "Being able to calculate in multiple ways means that one has transcended the formality of an algorithm and reached the essence of the numerical operations  the underlying mathematical ideas and principles. The reason that one problem can be solved in multiple ways is that mathematics does not consist of isolated rules, but connected ideas. Being able to and tending to solve a problem in more than one way, therefore, reveals the ability and the predilection to make connections between and among mathematical areas and topics." (p. 112) One question that remains for me after reading this book is how teachers can best gain this profound understanding of mathematics. Liping Ma writes that, in China, teachers spend a significant amount of time studying teaching materials intensively. They closely exam both their curriculum guides and their textbooks. They study how the textbook has interpreted and presented the concepts in the curriculum and why the textbook's authors might have chosen this. Teachers review the examples and exercises in their textbook for each unit and reflect on the purpose of each exercise in relation to the curriculum outcomes. Teachers often spend significant time interacting with their teaching colleagues, "to share their ideas and reflections on teaching." (p. 136). The author notes that, "Time is an issue here. If teachers have to find out what to teach by themselves in their very limited time outside the classroom and decide how to teach it, then where is the time for them to study carefully what they are to teach? U.S. teachers have less working time outside the classroom than Chinese teachers, but they need to do much more in this limited time. What U.S. teachers are expected to accomplish, then is impossible. It is clear that they do not have enough time and appropriate support to think through thoroughly what they are to teach. And without a clear idea of what to teach, how can one determine how to teach it thoughtfully?" (p. 149). This is a current area of discussion in Nova Scotia, with working conditions being a significant issue in teachers' contract negotiations with the provincial government. The chapter in this book about division by fractions was especially timely. In the book, teachers were asked to how they determine the following quotient: 1 and 3/4 divided by 1/2. I recently had a conversation about the strategy of dividing fractions with a common denominator with a pair of junior high teachers (see Christopher Danielson's blog Overthinking my Teaching for a nice summary of this strategy). We discussed how the examples in the textbook were presented, what the curriculum guide had to say about this strategy, and how it could be used to enhance students' understanding of division. The discussion led me to investigate division of fractions more deeply (there is some great material on Ontario's EduGains website). This type of conversation is what Liping Ma suggests helps to promote a deeper understanding of mathematical content, and in my experience, has been a benefit of teachers' professional learning communities. If you are interesting in other teachers relfections on this book, below are two excellent blog posts to read: EL
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