U.S. and Chinese Teachers' Constructing,Knowing, and Evaluating Representations to Teach Mathematics

This study examined U.S. and Chinese teachers' constructing, knowing, and evaluating representations to teach mathematics. All Chinese lesson plans are very similar, because they are all based on the Chinese national unified curriculum in mathematics. However, the U.S. lesson plans are extremely varied, even for those teachers from the same school. The Chinese teachers' lessons are very detailed; the U.S. teachers' lesson plans have exclusively adopted the "outline and worksheet" format. In the Chinese lesson plans, concrete representations are used exclusively to mediate students' understanding of the concept of average. In U.S. lessons, concrete representations are not only used to model the averaging processes to foster students' understanding of the concept, but they are also used to generate data. The U.S. teachers are much more likely than the Chinese teachers to predict drawing and guess-and-check strategies. For some problems, the Chinese teachers are much more likely than are the U.S. teachers to predict algebraic approaches. For the responses using conventional strategies, both the U.S. and Chinese teachers gave them high and almost identical scores. If a response involved a drawing or an estimate of an answer, the Chinese teachers usually gave a relatively lower score, even though the strategy is appropriate for the correct answer, because it is less generalizable. This study contributed to our understanding of the cross-national differences between U.S. and Chinese students' mathematical thinking. It also contributed to our understanding about teachers' beliefs from a cross-cultural perspective.     

Assessing and understanding U.S. and Chinese students' mathematical thinking

If the main goal of educational research and refinement of instructional program is to improve students' learning, it is necessary to assess students' emerging understandings and to see how they arise. The purpose of this paper is to address issues related to assessments of students' mathematical thinking in cross-national studies and then to discuss the lessons we may learn from these studies to assess and improve students' learning. In particular, the issues related to assessing U.S. and Chinese students' mathematical thinking were discussed. Then, this paper discussed the findings from two studies examining the impact of early algebra learning and teachers' beliefs on U.S. and Chinese students' mathematical thinking. Lastly, the issues related to interpreting and understanding the differences between U.S. and Chinese students' thinking were discussed.     

Why do U.S. and Chinese students think differently in mathematical problem solving?: Impact of early algebra learning and teachers’ beliefs

This paper reports two studies that examined the impact of early algebra learning and teachers’ beliefs on U.S. and Chinese students’ thinking. The first study examined the extent to which U.S. and Chinese students’ selection of solution strategies and representations is related to their opportunity to learn algebra. The second study examined the impact of teachers’ beliefs on their students’ thinking through analyzing U.S. and Chinese teachers’ scoring of student responses. The results of the first study showed that, for the U.S. sample, students who have formally learned algebraic concepts are as likely to use visual representations as those who have not formally learned algebraic concepts in their problem solving. For the Chinese sample, students rarely used visual representations whether or not they had formally learned algebraic concepts. The findings of the second study clearly showed that U.S. and Chinese teachers view students’ responses involving concrete strategies and visual representations differently. Moreover, although both U.S. and Chinese teachers value responses involving more generalized strategies and symbolic representations equally high, Chinese teachers expect 6th graders to use the generalized strategies to solve problems while U.S. teachers do not. The research reported in this paper contributed to our understanding of the differences between U.S. and Chinese students’ mathematical thinking. This research also established the feasibility of using teachers’ scoring of student responses as an alternative and effective way of examining teachers’ beliefs.     

The same tasks,different learning opportunities: An analysis of two exemplary lessons in China and the U.S. from a perspective of variation

This study examined the learning opportunities afforded in two exemplary lessons based on a theory of variation. Implemented in China and the U.S., the two lessons focused on the same topic of patterns in a calendar and were carefully developed through a lesson study approach. Both lessons set similar learning goals but enacted these goals differently. When compared with the U.S. lesson, the Chinese lesson provided more learning opportunities through high cognitively demanding tasks focusing on different identities within patterns. However, the U.S. lesson, which featured fewer tasks and focused on a single pattern identity, may have better supported students in discerning the critical features within the objects of learning. The implications for task design and implementation for effective mathematics teaching are discussed.     

A Cross‐Cultural Lesson Comparison on Teaching the Connection Between Multiplication and Division

This study compared one lesson across four U.S. “traditional” textbook series, two U.S. reform‐based textbook series, and one Chinese mathematics textbook series in teaching the connection between multiplication and division. The results showed the differences across U.S. and Chinese lessons in both the teaching and the practice parts of the lesson across three dimensions (i.e., problem schemata, response requirement, and algebra readiness). In particular, the Chinese lesson's penetrating analysis or explanation of the topic is reflected in its deliberately constructed examples and wide range of problems (pertaining to problem types and difficulty levels) present in the teaching and practice sections of the lesson. None of analyzed U.S. lessons are comparable with the Chinese lesson with respect to the breadth and depth in teaching the topic. A deliberate emphasis, both arithmetically and algebraically, on problem schema acquisition as found in the Chinese lesson represents a promotion of symbolic or higher order of conceptual understanding. The findings are discussed within the context of teaching big ideas through problem schemata acquisition and the importance of symbolic level of conceptual understanding.     

Pedagogical representations to teach linear relations in Chinese and U.S. classrooms: Parallel or hierarchical?

This study investigates Chinese and U.S. teachers’ construction and use of pedagogical representations surrounding implementation of mathematical tasks. It does this by analyzing video-taped lessons from the Learner's Perspective Study, involving 15 Chinese and 10 U.S. consecutive lessons on the topic of linear equations/linear relations. We examined patterns of pedagogical representations that Chinese and U.S. teachers construct over a set of consecutive lessons, but also investigated the strategies of using representations to solve mathematical problems by Chinese and U.S. teachers. It was found that multiple representations were constructed simultaneously to develop the connection of relevant concepts in the U.S. classrooms while selective representations were constructed to develop relevant concepts in the Chinese classrooms. This study is significant because it contributes to our understanding of the cultural differences involving Chinese and U.S. students’ mathematical thinking and has practical implications for constructing pedagogical representations to maximize students’ learning.     

Preservice teachers’ use of mathematics and science learning processes

Mathematics and science have similar learning processes (SLPs) and it has been proposed that courses focused on these and other similarities promote transfer across disciplines. However, it is not known how the use of these processes in lessons taught to children change throughout a preservice teacher education course or which are most likely to transfer within and between disciplines. Three hundred and ninety lesson plans written by 113 preservice teachers (PSTs) from 10 sections of an elementary mathematics/science methods course were analyzed. PSTs taught an eight‐lesson sequence to children: five science lessons followed by three mathematics lessons. The findings suggested that: (a) PSTs needed to only teach three mathematics lessons, after five science lessons, to reach the same number of SLPs used in the five science lessons; (b) some SLPs are highly correlated processes (HCPs) and are more likely to transfer within and between science and mathematics lessons; and (c) PSTs needed to teach no mathematics lessons, after four science lessons, to reach the same number of HCPs used in the four science lessons. Implications include centering courses on multiple and varied representations of learning processes within problem‐solving, and HCPs may be essential similarities of problem‐solving which promote transfer.     

Chinese and English Mathematics Language: The Relation Between Linguistic Clarity and Mathematics Performance

In Study 1, 48 judges rated the clarity of Chinese, English, and “Chinglish” (Chinese words translated into English) mathematical words-for example, the Chinglish version of the Chinese word for quadrilateral is “four-side-shape.” Native Chinese-speaking judges achieve greater agreement on the relative clarity of Chinese words than do native English-speaking judges on the relative clarity of English words. More Chinese words are rated clear than are English. Chinglish mathematical words tend to be rated more clear than English. The inherent compound word structure of the Chinese language seems well suited to portray mathematical ideas.

In Study 2, we examined the relations among the clarity of Chinese mathematical terms, U.S. urban junior high school students' Chinese reading ability, and their mathematics performance. There is a strong correlation between Chinese reading ability and performance on test items with mathematics words rated clear by Chinese judges. The relative clarity of mathematical terms in the Chinese language may contribute to Chinese-speaking students' understanding of mathematics and to superior mathematics performance.     

Mathematical Thinking Involved in U.S. and Chinese Students' Solving of Process-Constrained and Process-Open Problems

This study examined U.S. and Chinese 6th-grade students' mathematical thinking and reasoning involved in solving 6 process-constrained and 6 process-open problems. The Chinese sample (from Guiyang, Guizhou) had a significantly higher mean score than the U.S. sample (from Milwaukee, Wisconsin) on the process-constrained tasks, but the sample of U.S. students had a significantly higher mean score than the sample of the Chinese students on the process-open tasks. A qualitative analysis of students' responses was conducted to understand the mathematical thinking and reasoning involved in solving these problems. The qualitative results indicate that the Chinese sample preferred to use routine algorithms and symbolic representations, whereas the U.S. sample preferred to use concrete visual representations. Such a qualitative analysis of students' responses provided insights into U.S. and Chinese students' mathematical thinking, thereby facilitating interpretation of the cross-national differences in solving the process-constrained and process-open problems.     

Middle School Mathematics Teachers' Knowledge of Students' Understanding of Core Algebraic Concepts: Equal Sign and Variable

This article reports results from a study focused on teachers' knowledge of students' understanding of core algebraic concepts. In particular, the study examined middle school mathematics teachers' knowledge of students' understanding of the equal sign and variable, and students' success applying their understanding of these concepts. Interview data were collected from 20 middle school teachers regarding their predictions of student responses to written assessment items focusing on the equal sign and variable. Teachers' predictions of students' understanding of variable aligned to a large extent with students' actual responses to corresponding items. In contrast, teachers' predictions of students' understanding of the equal sign did not correspond with actual student responses. Further, teachers rarely identified misconceptions about either variable or the equal sign as an obstacle to solving problems that required application of these concepts. Implications for teacher professional development are discussed.     

Preservice teachers learning to teach proof through classroom implementation: Successes and challenges

Proof and reasoning are central to learning mathematics with understanding. Yet proof is seen as challenging to teach and to learn. In a capstone course for preservice teachers, we developed instructional modules that guided prospective secondary mathematics teachers (PSTs) through a cycle of learning about the logical aspects of proof, then planning and implementing lessons in secondary classrooms that integrate these aspects with traditional mathematics curriculum in the United States. In this paper we highlight our framework on mathematical knowledge for teaching proof and focus on some of the logical aspects of proof that are seen as particularly challenging (four proof themes). We analyze 60 lesson plans, video recordings of a subset of 13 enacted lessons, and the PSTs’ self- reported data to shed light on how the PSTs planned and enacted lessons that integrate these proof themes. The results provide insights into successes and challenges the PSTs encountered in this process and illustrate potential pathways for preparing PSTs to enact reasoning and proof in secondary classrooms. We also highlight the design principles for supporting the development of PSTs’ mathematical knowledge for teaching proof.     

Teachers' Considerations of Students' Thinking During Mathematics Lesson Design

Teachers' abilities to design mathematics lessons are related to their capability to mobilize resources to meeting intended learning goals based on their noticing. In this process, knowing how teachers consider Students' thinking is important for understanding how they are making decisions to promote student learning. While teaching, what teachers notice influences their decision‐making process. This article explores the mathematics lesson planning practices of four 4th‐grade teachers at the same school to understand how their consideration of Students' learning influences planning decisions. Case study methodology was used to gain an in‐depth perspective of the mathematics planning practices of the teachers. Results indicate the teachers took varying approaches in how they considered students. One teacher adapted instruction based on Students' conceptual understanding, two teachers aimed at producing skill‐efficient students, and the final teacher regulated learning with a strict adherence to daily lessons in curriculum materials, with little emphasis on student understanding. These findings highlight the importance of providing professional development support to teachers focused on their noticing and considerations of Students' mathematical understandings as related to learning outcomes. These findings are distinguished from other studies because of the focus on how teachers consider Students' thinking during lesson planning. This article features a Research to Practice Companion Article . Please click on the supporting information link below to access.     

Hong Kong teachers’ views of effective mathematics teaching and learning

Twelve experienced mathematics teachers in Hong Kong were invited to face-to-face semi-structured interviews to express their views about mathematics, about mathematics learning and about the teacher and teaching. Mathematics was generally regarded as a subject that is practical, logical, useful and involves thinking. In view of the abstract nature of the subject, the teachers took abstract thinking as the goal of mathematics learning. They reflected that it is not just a matter of “how” and “when”, but one should build a path so that students can proceed from the concrete to the abstract. Their conceptions of mathematics understanding were tapped. Furthermore, the roles of memorisation, practices and concrete experiences were discussed, in relation with understanding. Teaching for understanding is unanimously supported and along this line, the characteristics of an effective mathematics lesson and of an effective mathematics teacher were discussed. Though many of the participants realize that there is no fixed rule for good practices, some of the indicators were put forth. To arrive at an effective mathematics lesson, good preparation, basic teaching skills and good relationship with the students are prerequisite.     

Teachers’ gestures and speech in mathematics lessons: forging common ground by resolving trouble spots

This research focused on how teachers establish and maintain shared understanding with students during classroom mathematics instruction. We studied the micro-level interventions that teachers implement spontaneously as a lesson unfolds, which we call micro-interventions. In particular, we focused on teachers’ micro-interventions around trouble spots, defined as points during the lesson when students display lack of understanding. We investigated how teachers use gestures along with speech in responding to such trouble spots in a corpus of six middle-school mathematics lessons. Trouble spots were a regular occurrence in the lessons (M = 10.2 per lesson). We hypothesized that, in the face of trouble spots, teachers might increase their use of gestures in an effort to re-establish shared understanding with students. Thus, we predicted that teachers would gesture more in turns immediately following trouble spots than in turns immediately preceding trouble spots. This hypothesis was supported with quantitative analyses of teachers’ gesture frequency and gesture rates, and with qualitative analyses of representative cases. Thus, teachers use gestures adaptively in micro-interventions in order to foster common ground when instructional communication breaks down.     

Exploring Probability and Statistics with Preservice and Inservice Teachers

NCTM's mathematics curriculum and evaluation standards (1989) have provided educators with the challenge of revamping high school mathematics curricula as well as pedagogies by which content is taught. This article presents a lesson designed for preservice and inservice teachers that permits participants to: (a) strengthen their conceptual understanding, and (b) experience learning in a cooperative environment that encourages communication. The lesson engages participants in the collection and representation of probabilistic data using dice with 4, 6, 8, 10, 12, and 20 faces. Opportunities are provided for participants to discover patterns and construct mathematical knowledge concerning theoretical probability. Teacher educators can facilitate reform of mathematics education by developing and delivering such lessons.     

Student teachers’ types of probing questions in inquiry-based mathematics teaching with and without GeoGebra

Previous studies have produced several typologies of teacher questions in mathematics. Probing questions that ask students to explain are often included in the types of questions. However, only rare studies have created subtypes for probing questions or investigated how questioning differs depending on whether technology is used or not. The aims of this study are to elaborate on different ways of asking students to give explanations in inquiry-based mathematics teaching and to investigate whether questioning in GeoGebra lessons differs from questioning in other lessons. Data was collected by video recording 29 Finnish mathematics student teachers’ lessons in secondary and upper secondary schools. The lesson videos were coded for the student teachers’ probing questions. After this, categories for the types of probing questions were created, which is elaborated in this paper. It was found that the student teachers who used GeoGebra emphasized conceptual probing questions during the explore phase of a lesson slightly more than the other student teachers.