Middle-Grades Mathematics Classrooms

Middle-grades Mathematics Classrooms

Instruction in China: A Case Study

Su Liang

California State University, San Bernardino

Despite Chinese students outperform their western peers in mathematics in the international comparison studies; they have been characterized as rote learners (Au and Entwhistle, 1999; Biggs 1996, Chow, 1995; Liu, 2006; Martinsons and Martinsons, 1996). Do Chinese students really learn by rote? Some research revealed that it may be a wrong perception that Chinese students are only rote leaners who memorize the materials without actually understanding. According to Kontoulis &Williams (2000) and Watkins (2000), memorization is only a part of the learning process for Chinese learners. On (1996) summarized the four stages of Chinese systematic learning process: At first, get familiar with the context; second, understand it; third; reflect upon it; finally question it. Memorization is used by Chinese learners as a tool to understand the context learned (Au & Entwhistle, 1999). Marton et al called this method of learning “Memorization with understanding” (1996). According to Cooper (2004), this Chinese way of Leaning through repetition could lead to a deeper understanding and high performance. Nield’s research (2007) indicated that Chinese learners “adopt a more strategic approach to their learning” instead of learning by rote (Neild, 2007, p.39).

The learning environment in China appears to be not ideal for effective learning. It is common that each class has more than 60 students; in some places, each class has to hold 70 to 80 students. Classrooms are usually very crowded. However, the negative environment has not seemed to prevent students from achieving high performance. Being described as “paradox of the Chinese Learner” (Watkins and Biggs, 2001), Chinese students constantly outperform their American peers on international tests in mathematics (Biggs, 1996; Liang, 2010; Wang & Lin, 2005). Many studies have been conducted to look for the reasons causing the learning gap. Reviewing the literature of international comparison studies, four factors emerged to be blamed for US students to fall behind in learning mathematics: 1. Culture difference (Cai, 2003; Chang, 1985; Hess, Chang, & Mcdvitt, 1987); 2. Less focused and more repetitive curriculum (Askey, 1999; Jiang & Eggleton, 1995; Schmidt, Houang, & Cogan, 2002); 3. Teachers’ mathematics content knowledge (Ma, 1999); 4. Teachers’ effectiveness in instruction including clear explanations, efficiency in use of class time, and engaging students in inquiry-based learning (Perry, 2000; Steverson & Lee, 1995; Stigler & Hiebert, 1999). Although these research findings have helped us better understand mathematics teaching and learning in China, more extensive studies are needed to analyze how teaching and learning take place in Chinese mathematics classrooms. Researches have been agreed that teachers play central role in teaching and learning. It is reasonable to assume that what happens in Chinese classrooms may be essential for their students’ higher learning achievement.

Aiming to find some special figures of mathematics teaching in China that may help produce effective learning outcomes, this study attempted to identify the characteristics of typical Chinese mathematics classroom instructions by closely examining and analyzing six middle school mathematics classes in northeast urban area of China.

The Framework

What does an effective mathematics classroom instruction looks like? In other words, what characteristics of mathematics teaching are important in determining lesson quality? Many researchers have tried to answer this question. National Research Council (1999) stated that a high quality lesson should include activities providing students opportunities to explore, communicate, extend their thinking, and evaluate their learning progress. Selecting the nationally representative sample of schools in the United States, a research team observed 364 mathematics and science teachers’ teaching at those school and concluded that high quality lessons are characterized as engaging students with the subject content, creating an environment conductive learning, ensuring all students have access to the lesson, and helping students make sense of the subject content (Weiss, I. R., Pasley, J. D., Smith, P. S., Banilower, E. R., & Heck, D. J., 2003). Some study specifically brought up classroom discussions to help students develop mathematical strategic thinking (Artzt & Armour-Thomas, & Curcio, 2008).

Synthesizing the evidence of effective teaching from existing research, Anthony and Walshaw (2009) described the characteristics of effective teaching in the west countries from the following four aspects:

1.  classroom community: an effective teaching should start with a caring community that focus on developing students’ mathematics identities and proficiencies and provide opportunities for students to understand ideas both independently and collaboratively.

2.  classroom discourse: effective classroom discourse should facilitate effective communication focusing on mathematical argumentation and the teacher should model communicating mathematical ideas using appropriate mathematical terms and language.

3.  classroom tasks: effective mathematics tasks should be 1) worthwhile, facilitating students doing mathematics, and help students making sense of mathematics; 2) creating connections between mathematical topics, between mathematics and students’ life experience; and between various solutions for a problem; 3) selective to work with other tools and representations to strengthen students’ thinking,

4.  teacher knowledge: sound content knowledge make it possible for teachers to present mathematics coherently as a connected system and to be able to respond to students’ mathematical needs at different levels.

This study closely examined the six lessons of 6-9th grades at the schools located

at an urban area of northeast of China from the four aspects discussed above and found these lessons fit the description of effective mathematics classroom from western educators’ point of views. It was also worth to note that analysis of these lessons revealed some more special figures that were not included in the description of western standard’ effective classroom.

Additionally, the Bloom revised Taxonomy was also used to determine the types of questions asked by the teachers in these Chinese classrooms to facilitate students’ learning. Developed by Benjamin Bloom and his research group in 1956, Bloom’s Taxonomy classified six levels of the important learning behaviors that are knowledge, comprehension, application, analysis, synthesis and evaluation in the order from low to high. During the 1990’s, a former student of Bloom, Lorin Anderson and a group of cognitive psychologists revised the taxonomy and used nouns to replace verbs at each level. The new version of Bloom’s Taxonomy described the six levels of learning as remembering, understanding, applying, analyzing, evaluating, and creating (Anderson, L. W. & Sosniak, L. A., 1994; Barton, L., 2007)). Interpreted Bloom’s taxonomy for mathematics teaching, some mathematics educators adapted the taxonomy to develop effective strategies of mathematics questioning. For level one – remembering, questions at this level would ask students recall a definition, a theorem, a fact, a concept, a method, or a procedure that is relevant to what is learning; at the level two – understanding, questions should aim to help students construct meaning orally or in written through multiple representations and understand concepts, methods, facts, terms, or procedures; at the level three – applying, questions should facilitate students to apply what learned to new situations and solve problems; at level four – analyzing, questions should be designed to help students compare, organize, and explore relationship for understanding; at level five – evaluating, questions should allow students to hypothesize, verify, critique, experiment, and judge the value of ideas, materials, and methods; at the level 6 – creating; questions should provide opportunities for students to design, develop, or invent their own classification system , and make generalization and conclusion by applying and integrating multiple strategies. The first three levels are involving with lower-order thinking and the last three levels are facilitating higher-order thinking. In a single mathematics lesson, all six levels of questions should be used and designed purposefully to achieve the teaching goal of the lesson (“Bloom’s Revised Taxonomy: Mathematics”, n.d.; Shorser, L., 1999).

Methodology

Data Collection

The three visited schools (Grades 1-9) are located in a middle sized city with a population of 570,000, in northeast of China. The observed classes were one 6th grade class, two 7th grade classes; two 8th grade classes, one 9th grade class.

The data were collected through multiple ways, including:

·  Classroom observations. Field notes were taken during observations. Classes observed were also video-taped.

·  Lessons plans. Lesson plans of the observed classes were collected.

·  After-class discussions. Discussions with the teachers observed were conducted to learn background information and make some necessary clarifications. Notes were taken during the after-class discussions.

·  Observation of Collective lesson-planning sessions. Notes were taken while observing the sessions.

Data Analysis

Qualitative research methods were employed to analyze the collected data. Qualitative data analysis was described as “the process of bringing order, structure, and meaning to the mass of collected data” (Marshall Rossman, 1995, p. 111). The ultimate goal of the data analysis was to identify the characteristics of the classroom teaching in these observed lessons. The videos of the six lessons were repeatedly watched to strongly sense the data (Weber, K., Maher, C., Power, A. B., & Lee, H. S., 2008), capture most relevant information from the videos (Clement, J., 2000; Martin, L. C., 1999; Pirie, S. E. B., 1996), identify themes and patterns of class discursive interactions, as well as enhance data triangulation (Powell, A. B., Francisco, J. M., & Maher, C. A., 2003) . Field notes were compared while watching the videos. The process of analysis involved videotape data analysis (Powell, A. B., Francisco, J. M., & Maher, C. A., 2003), constant comparison (Corbin & Strass, 2008), inductive coding (Boyatzis, 1998; Miles & Huberman, 1994; Patton, 2002), categorizing (Miles & Huberman, 1994), and making conclusion.

Finding

The six observed lessons had covered different topics which were problem solving using fraction and percent, equations with one variable involving solving word problems, some basic concept of probability, problem solving using system equations; circle, and representing data. Although these classes were from different schools and covered different topics, all of them consisted of the same discourse components as follows:

•  Students were asked to recall a previously learned knowledge that is relevant to the knowledge currently learning;

•  Students were required to use knowledge learned to solve various problems from easy ones to sophisticated ones ;

•  Students explained mathematics ideas explicitly to other students;

•  Students were guided to interpret and compare different strategies to solve a problem;

•  Students had to justify their own work and evaluate other students’ work;

•  Students created a plan to solve a problem in various situations;

•  Students were required to generalize mathematical conclusions.

In the classrooms, students not only showed they remembered related mathematics concepts or properties but also understood these concepts and properties and smartly applied them into solving various new problems. Throughout the lessons, it had been the students who were solving the problems given and reasoning why they solved the problems the ways they did. The students were put in the center stage of learning and the teachers were like a director moving the class from one activity to another. Taking an example, in the probability lesson, the teacher designed a game of turning the wheel. By playing the game, the students experienced the learning process of exploring, observing patterns, recording their experiment, making their hypotheses, using their experiment data to verify the hypotheses, and making conclusions. Students learned from their own experimenting process.

It was also worth to note that in each of these classrooms, the teachers’ encouraging comments were often heard, such as “his explanation is very clear, let us clap for him”; “you did a good job, keep going”; “very smart idea, isn’t it?”, “what did you state is not bad, but can you say it more clearly?”… The teachers always tried to motivate students to think, compare, make connections, and correct their own mistakes, using the questions like “what do you think? any ideas about this problem? why do you think it works that way? are there any differences? Why don’t you agree with what he said? Why is this condition important in this situation? What have you discovered? ...” When a student made a mistake while demonstrating his/her work or answering a question, the teachers would ask the student questions until guiding him/her to realize the mistake and correct it. Sometimes other students were asked to identify and correct the mistake with clear explanation. Students did not seem to fear making mistakes when solving problems on the blackboard. Most of the students were volunteering to demonstrate their work on the blackboard. The students were not shy to explain their ideas in front of the whole class. They looked very comfortable with their learning environment.

All the lessons were content-rich, providing students various topic-related questions from simple problems to sophisticated ones. The questions were intentionally designed to situate in a variety of context for students to deepen their understanding and increase the ability of problem solving.

Analysis of the data in this study revealed that their classroom teaching not only fits the descriptions of effective teaching and learning summarized by Anthony and Walshaw (2009) (See the section of Framework on page 3) but also adds some their own characteristics. The classroom teaching in this study can be characterized as

•  Well-designed lesson plan

•  Teacher-guided, student-centered, and inquiry-based classroom instruction

•  Coherent

•  Immediate feedback to students’ answers and reasoning

•  Questioning to generate higher order mathematics thinking

•  Collaborating learning

•  Using modern technology to enhance teaching and learning

The details are explained as follows.

Well-designed lesson plan

All the lessons observed are based on a well-designed lesson plan. The teachers carefully prepared the lesson plans that 1. emphasized the particular math concepts or properties; 2. identified both the important points and the difficult parts in the lesson; 3. pointed out possible misconceptions or mistakes; and 4. provided a variety of problems at different levels starting from simple ones to sophisticatedly ones for students to solve.

Teacher-guided, student-centered, and inquiry-based classroom instruction

The lessons were teacher-guided, student-centered, and inquiry-based classroom instruction throughout the lessons, students took turns to get on the center stage demonstrating how they solved a problem verbally and in written. The teachers provided guiding questions to help students to correct their own mistakes or solve a problem when needed. There were a great deal of interactions between teacher and students and between students and students.