A Conversation with Dr. Liping Ma

Q: What do you believe is the most important factor in teaching elementary mathematics?

A: To fully promote mathematics learning, teachers must first have a profound understanding of fundamental mathematics. They must know well the mathematics they teach each day and feel both confident and comfortable talking it.

Q: What constitutes a profound understanding of fundamental mathematics?

A: It includes four key features. The first is Connectedness. A teacher's knowledge of mathematics should be fully connected. The teacher must be able to see how math concepts relate to one another so that they can build new knowledge upon what students already know.

The second feature is Multiple Perspectives, which means having the ability to approach mathematics in a variety of ways. Teachers who develop multiple perspectives for each basic math topic have complete knowledge of that topic.

Thirdly, teachers must understand that elementary math consists of Basic Ideas. These basic ideas that recur throughout math learning create a solid foundation on which to build future math learning.

The last feature is Longitudinal Coherence. What is taught today becomes the base for future knowledge, just as current mathematics teaching builds upon students' previous knowledge, however fragmentary that knowledge may be.

With this profound understanding of fundamental mathematics, we as teachers will be able to teach students more successfully.

Q: How does this profound understanding inform Knowing Mathematics?

A: Most people generally see elementary math as being simple math, but actually it demands a deep understanding of the basic ideas or topics such as addition and subtraction, multiplication and division, and place value. Realizing that elementary math requires such a deep understanding is part of the philosophy behind Knowing Mathematics.

Q: Can you tell me more about the program?

A: Knowing Mathematics is a tightly structured program that takes between 12 and 14 weeks to complete. Each lesson is carefully designed to ensure longitudinal coherence so that knowledge is continually building upon previous knowledge. It starts by repairing the fragmentary knowledge that struggling students have, and then uses it as a foundation on which future learning can occur.

Each grade level begins with a focus on place value, addition and subtraction concepts and facts, and multiplication and division concepts and facts. Once students master these critical understandings and develop a sound mathematics foundation, then geometry, statistics, and algebra are added. At the end of 12-14 weeks, students who were once struggling are now able to participate successfully and confidently in grade-level instruction.

Each lesson consists of two pages. The left-hand page highlights a mathematical conversation during which Ms. Park and her students, fictional characters within the lessons, discuss a specific math topic to build conceptual understanding. The right-hand page includes practice and word problem exercises, carefully designed to support each student's learning of the concept.

All of the lessons within the program are cohesive — taking one day to teach each lesson, taking one step to grade-level achievement each day.

Q: It is clear how the program helps students learn the basic ideas of math, but how does it support the teachers?

A: Knowing Mathematics supports teachers in two ways:

1. It clearly explains the rationale behind the lessons, giving the content and goal of each lesson, as well as the reasoning behind each goal. Additionally, it puts each lesson into the larger picture of mathematics learning.

2. It anticipates students' questions and provides possible answers, prompts, and other appropriate support for the teacher. And each mathematical conversation allows time for the teacher and students to reflect on the conversation and the mathematics within it before beginning the exercises. This time offers the teacher an opportunity to talk about the math with his or her students.

Q: What impact does this new way of learning math have on the students?

A: Historically, struggling math students are plagued by negative math experiences, feelings of failure and frustration, and low self-confidence. Knowing Mathematics, with its innovative approach to math learning, ultimately offers a positive math experience for students — giving them daily successes that lead to grade-level achievement in just 14 weeks. These successes build their confidence, provide them with motivation, and help change their self-image. Knowing Mathematics not only provides a challenging experience of working with math, but it is also a rewarding one.

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