Liping Ma's Knowing and Teaching Elementary Mathematics: Teachers’ Understanding of Fundamental Mathematics in China & the United States
From MathWiki
In the following, I wish to share some of the work of Liping Ma – specifically, her findings and her ideas regarding the education and skills of elementary mathematics teachers. If you are interested, I would encourage you to read her book (relatively short and available in the York library).
Table of contents 

The Study: Introduction & Motivation
As a graduate student at Michigan State University, Liping Ma served as a research assistant for the Teacher Education and Learning to Teach (TELT) study. In her classroom observations and review of TELT interview transcripts on the teaching of mathematics, Ma invariably compared these observations to her own experiences of being educated in China. Ma noticed that many of the U.S. teachers (ranging from student to experienced teachers) lacked a deep conceptual understanding of many areas covered in elementary mathematics. In addition, international studies of achievement showed that students in the United States are consistently “outperformed” by students in certain Asian countries, including China. While researchers have suggested potential reasons for this apparent gap (such as differences in number word systems – in Chinese, Ma explains, 20 means “two tens”, 30 means “three tens” etc), Ma hypothesized that teachers’ mathematical knowledge and training was a key factor in this equation. Thus, Ma, who was educated in China and familiar with the education system in the United States, decided to engage in comparative research into the knowledge of elementary teachers in China and the U.S.
As Ma explains, there is a clear difference between the training of teachers in China and the U.S. While most U.S. teachers complete at least a bachelor’s degree, Chinese teachers complete only between two and three years of formal training following the ninth grade. How, then, could it be possible that Chinese teachers have a better understanding of elementary mathematics? Ma hypothesizes that “elementary teachers in the two countries possess differently structured bodies of mathematical knowledge,” where pedagogical content knowledge (i.e. knowing how to represent the content in a comprehensible way) is central. The question of teachers’ mathematics subject matter knowledge – what does a teacher need to know to be well equipped to teach mathematics – has been a focus of mathematics education researchers since the late 1980’s.
Participants & Interview Questions
For the U.S. representation in her study, Ma interviewed twentythree “above average” (school inservice training leaders, or near the completion of a Master’s degree) U.S. elementary teachers. As well, Ma interviewed seventytwo Chinese teachers from five urban/rural elementary schools “ranging from very high to very low quality” educational status (Ma, 1999, p. xxiii). All these teachers were interviewed with four questions from the TELT study. The following topics were covered (questions paraphrased)
1. Subtraction with regrouping – how would you teach and explain this topic to a grade two class?
2. Multidigit number multiplication – how would you respond to student mistakes when dealing with questions from this topic?
3. Division by fractions – how would you represent (“realworld” situation, story, model, etc) this concept? [Margaret Sinclair brought this question up during her seminar in our 4100 class.]
4. The relationship between area and perimeter: This area is particularly relevant to our work this summer, since it is the only direct geometry question in the lot. Interestingly, this question is intended to determine how teachers may “explore new knowledge”. As Ma suggests (and most of us can attest to this based on personal experiences), students suggest novel ideas in math classrooms all the time. The focus of this question was on how teachers would respond to seemingly novel student claims. I am including this question in its entirety, and would love to hear any related feedback from the group. Based on Ma’s analysis (teacher reactions/levels of understanding), I will post a followup to this question after you have had a chance to think about it. (Prof. Whiteley has informed me that this question is so well known that it can no longer be used on exit exams for teachers!)
“Imagine that one of your students comes to class very excited. She tells you that she has figured out a theory that you never told the class. She explains that she has discovered that as the perimeter of a closed figure increases, the area also increases. She shows you this picture to prove what she is doing: [a 4cm by 4 cm square and a 4cm by 8cm rectangle, with respective perimeters and areas calculated correctly, and “in support” of her claim]. How would you respond to this student?”
The PerimeterArea Question: Results & Analysis
Of the U.S. teachers, 2 simply accepted the claim, 18 did not pursue a mathematical investigation, and 3 investigated the claim mathematically. Of the group, only one teacher achieved a correct solution (via counterexample). Approaches included consulting a book, calling for more examples, and mathematical approaches (such as proof by counterexample). Of the Chinese teachers, about 8% simply accepted the claim to be true (similar to U.S. sample) and 92% explored the problem mathematically. Of this 92%, 22% reached an incorrect solution due to problematic strategies, but 70% of these teachers reached the correct solution. Based on the Chinese teachers’ solutions, Ma presents four levels of understanding in relation to this problem:
1. Disproving the claim  finding a viable counterexample (14 teachers).
2. Identifying the possibilities – finding examples that display the possibilities of relationships between the area and perimeter of two closed figures (8 teachers).
3. Clarifying the conditions – determining the conditions under which the possible relationships between area and perimeter held true (26 teachers).
4. Explaining the conditions – elaborating on why the area and perimeter relate as they do under particular conditions, so as to support or refute the student’s claim (6 teachers).
Why were some teachers more successful than others?
In examining the interview results of the teachers, Ma explains that strategy and intention were key factors in the nature of teachers’ exploration of the student’s claim. Strategy appears to be an obvious ingredient, which includes knowledge of appropriate formulae, their underlying rationales, and modes of mathematical thinking such as the use of examples and counterexamples. I think that Ma’s idea of strategy is certainly connected the idea of mathematical maturity that we spoke about in our 4100 course. In contrast to mathematical maturity, Ma expresses her concern regarding the laypersonlike attitude that some of the teachers expressed in their investigations. For instance, the idea that a mathematical statement can be proved through the use of a single example was evident in the responses of several of the U.S. and Chinese teachers.
Intention, on the other hand, is dependent on a teacher’s interest in exploring a mathematical idea and their selfconfidence in tackling a new problem. The teachers in the study who thoroughly explored (or unpacked, as we would put it) the areaperimeter problem showed a genuine interest in the problem, which fuelled their desire to reach a plausible conclusion. Interestingly, Ma suggests that teachers’ confidence is a function of teacher attitudes towards the possibility of solving a novel problem. While some of the U.S. teachers showed difficulty with the strategy component of the problem, Ma suggests that their intention was their pitfall – many of the U.S. teachers knew the appropriate formulae, but lacked the necessary interest and confidence to approach the new problem mathematically.
Acculturation of Math teachers to the Mathematics Discipline – Thinking Mathematically
Ma notes that the U.S. teachers did not necessarily have less to say than the Chinese teachers, but their answers were, generally speaking, “less mathematically relevant and mathematically organized” (Ma, 1999, p.104). Furthermore, she suggests that the Chinese teachers’ proficiency in communicating mathematics may stem from their chosen style of teaching, which involves a significant component of lecture presentation. Thus, for each new lesson, Chinese teachers spend some time preparing a lecturestyle introductory, yet complete, presentation on the topic. According to Ma, these lecturestyle lessons require the teachers to practise and train their mathematical communication skills. I have always felt that lecturestyle teaching has its benefits (such as the suggested opportunity to organize and gather ideas), but I also feel that these benefits are not exclusively tied to lecturestyle teaching, nor do I think that Ma is suggesting this. Specifically, acquiring welldeveloped communication skills in mathematics is possible through other teaching styles. However, since the lecture element was a salient feature of the Chinese teachers’ pedagogical repertoire (and not necessarily for the U.S. teachers), Ma suggests that this may be a significant factor for the differences in communication skills between the two groups.
Knowledge Packages
Throughout her investigation, Ma used the term “knowledge package” when speaking about the subject matter knowledge of teachers. Ma explains that when a teacher begins to teach a new topic, that teacher has an idea in her mind about where this idea is situated in the field of mathematics. Thus, “given a topic, a teacher tends to see other topics related to its learning,” and such topics comprise the knowledge package for the topic to be taught (Ma, 1999, p.118). In knowledge packages, there are “key pieces,” which are certain related topics that are viewed as being more important to the comprehension of the topic at hand. Knowledge packages for any topic can contain both procedural and conceptual elements, and Ma asserts that the two are interrelated. Ma found that teachers with a conceptual understanding of a topic viewed related procedural topics as being essential to student understanding. “In fact,” Ma emphasizes, these teachers felt that “a conceptual understanding is never separate from the corresponding procedures where the understanding ‘lives’” (Ma, 1999, p.114). Ma believes that knowledge packages are important because it is from this information that a teacher attempts to construct a cohesive and comprehensive picture of a mathematical topic. With underdeveloped knowledge packages, it can be very difficult for a teacher to plan and facilitate a course of study for their students.
Learning Paths – a linear progression?
Knowledge packages also contain implicit sequences of student learning, where a student is expected to know and understand key related pieces before they can grasp the topic at hand. Ma (1999) explains, “the teachers believe that these sequences are the main paths through which knowledge and skill about the… topic develop” (p.114). While this may seem to be a linear progression of learning (you need to know x, y, and z, before you can learn topic A), Ma clarifies that topics in a knowledge package are interdependent, and that “linear sequences, however, do not develop alone, but are supported by other topics” (p.114). Thus, the learning paths generated through teachers’ knowledge packages are similar to the hypothetical learning trajectories and learning landscapes, where Ma’s “key pieces” are mirrored by Fosnot’s “learning landmarks.”
Profound Understanding of Fundamental Mathematics (PUFM)
Ma builds her analysis of teacher training in mathematics around the idea of teachers’ acquiring a profound understanding of fundamental mathematics (PUFM). Early on, Ma (1999) explains that a teacher with PUFM “goes beyond being able to compute correctly and to give a rationale for computational algorithms” (p. xxiv). A grasp of both the procedural and conceptual elements of topics in elementary mathematics is necessary. A PUFM teacher is “not only aware of the conceptual structure and basic attitudes of mathematics inherent in elementary mathematics, but is able to teach them to students” (p. xxiv). Thus, Ma situates subject matter knowledge (concepts, procedures & attitudes) and pedagogical content knowledge (how to teach math) as both being essential to a successful elementary teacher.
Ma highlights the fact that PUFM is possible at the elementary level because elementary mathematics is a field rich with “depth, breadth, and thoroughness” (p.122). It is not a superficial discipline that is easily and commonly understood in its entirety by people.
PUFM Characteristics
Ma explains that a classroom led by a PUFM teacher has the following characteristics:
Connectedness – the teacher feels that it is necessary to emphasize and make explicit connections among concepts and procedures that students are learning. To prevent a fragmented experience of isolated topics in mathematics, these teachers seek to present a “unified body of knowledge” (Ma, 1999, p.122).
Multiple Perspectives – PUFM teachers will stress the idea that multiple solutions are possible, but also stress the advantages and disadvantages of using certain methods in certain situations. The aim is to give the students a flexible understanding of the content.
Basic Ideas – PUFM teachers stress basic ideas and attitudes in mathematics. For example, these include the idea of an equation, and the attitude that single examples cannot be used as proof.
Longitudinal Coherence – PUFM teachers are fundamentally aware of the entire elementary curriculum (and not just the grades that they are teaching or have taught). These teachers know where their students are coming from and where they are headed in the mathematics curriculum. Thus, they will take opportunities to review what they feel are “key pieces” in knowledge packages, or lay appropriate foundation for something that will be learned in the future.
China & The U.S.: Differences In Teacher Education & Training
Overall, the Chinese teachers in the study demonstrated a deeper understanding of both the procedural and conceptual elements of elementary mathematics. On the other hand, many of the U.S. teachers displayed an almost purely procedural approach to the topics of elementary mathematics, and struggled when asked to communicate the rationales behind algorithms. Ma attributes these differences in ability and performance to the striking differences in teacher training programs in the two countries. Roger Howe (who wrote a review on Ma's book for the AMS, linked below) summarizes these differences as follows:
 "Chinese teachers receive better early training— good training produces good trainers, in a virtuous cycle." This can be traced back to the elementary education of the teachers themselves. In terms of early specialization, recall that teacher education in China begins after Grade 9.
 "Chinese mathematics teachers are specialists." Elementary math teachers in China, only teach math (and sometimes science). This contributes to the subject matter knowledge of teachers, as well as their interest in the topic that can serve as selfmotivation.
 "Chinese teachers have working conditions which favor maturation of understanding." While American (and Canadian!) teachers spend almost their entire day teaching in front of a classroom, Chinese teachers have time set out in their day for studying their teaching material, consulting with other teachers in the form of lesson studies, and reflecting on their own teaching practices. As well, with a lesser teaching load, elementary teachers have the energy  and dedicated time  to study their topic of specialization.
Current Trends
After reading this summary of Ma's work, it would appear that the U.S. education system should adopt some key ideas from their Chinese counterparts. While there has been a push in this direction, the adoption of ideas is currently flowing in the opposite direction. As Sean Cavanagh (EdWeek, Asian Equation (http://www.edweek.org/ew/articles/2007/06/06/39china.h26.html)) reports, the Chinese government is embracing some of the trends in the U.S. and encouraging its teachers to "move away from lectures, drills, and memorization in class, and to invite more discussion and studentled activity." As well, "schools are adding more elective courses and independent research projects." This is interesting in light of Ma's observation that lecturestyle teaching can help develop critical communication and understanding skills in elementary mathematics teachers (But at what cost to the students?).
As well, it is my understanding that China is adopting a teacher education model where prospective teachers will need to complete a bachelor's degree in order to teach. Interestingly, some prospective teachers even find it necessary to obtain graduate degrees in order to be competative in the Chinese job market (See Teaching Viewed as Stable and Respectable Profession (http://www.edweek.org/ew/articles/2007/06/13/41chinateach.h26.html) for more information). Nevertheless, these university degrees still produce specialist teachers for the elementary levels  a point which Ma identified as being key to the understanding of the teachers in her study.
Reference Information
Ma, L. (1999). Knowing and teaching elementary mathematics : teachers' understanding of fundamental mathematics in China and the United States. Mahwah, N.J.: Lawrence Erlbaum Associates.
Relevant Links
What Does Liping Ma REALLY say? (http://rationalmathed.blogspot.com/2007/06/whatdoeslipingmareallysay.html)  Math educator and blogger, Michael Paul Goldenberg, writes about his experience of hearing Ma speak in 2000, shortly after the publishing of her book. This blog entry contains some interesting elements, including a recorded Q&A session with Liping Ma. (Thank you to Prof. Whiteley (& Jerry Becker) for the link!)
American Mathematical Society (AMS) Book Review (http://www.ams.org/notices/199908/revhowe.pdf)  Yale math professor, Roger Howe, reviews Ma's book and comments on some of her views, looking towards math reform in the U.S. I will extract Howe's main points and include it here when the opportunity comes up.
Response to Mordy's post on Liping Ma's research study  this discussion page shows Tanya's initial thoughts about the perimeterarea question, and an ongoing dialogue on this topic.
Asian Equation (http://www.edweek.org/ew/articles/2007/06/06/39china.h26.html)  In this EdWeek article, Sean Cavanagh reports on recent changes to math and science instruction in China. The move is to adopt some of the U.S. strategies for inspiring "creative and analytical" thinking, versus mechanical and memorization skills, in their students. Towards the end, the article also includes a discussion of teacher education in China.
Teaching Viewed as Stable and Respectable Profession (http://www.edweek.org/ew/articles/2007/06/13/41chinateach.h26.html)  This EdWeek article describes the societal role of a teacher in China. Among other interesting points, author Sean Cavanagh explains that teaching positions are hard to find in China, since many people want to be teachers, and some prospective teachers must pursue graduate work in mathematics to distinguish themselves.
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