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

Abstract

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.Cross-national studies provide unique opportunities for us to understand issues of the teaching and learning of mathematics and then to provide diagnostic and deci- MATHEMATICAL THINKING AND LEARNING, 7(2),

Hong Kong and U.S. Teachers’ Perceptions of Mathematical Disagreements and their Resolution Processes

“…Since this seminal work, several studies have further explored mathematics teacher knowledge in Mainland China and the U.S. (e.g., An, Kulm, & Wu, 2004;Cai, 2005;Cai & Wang, 2006;Zhou, Peverly, & Xin, 2006). Here, we present a brief discussion of these studies.…”

“…Similarly, Cai and his colleagues (Cai, 2000(Cai, , 2005Cai & Wang, 2006) conducted a series of comparative studies between Mainland China and the U.S. on teachers' construction of representations. Both groups of teachers used concrete representations for developing the concepts of ratio and average.…”

Ortaokul Matematik Öğretmenlerinin Temsil Kullanımına İlişkin Öğrenci ve Öğretim Stratejileri Bilgileri

“…Buna karşın öğrenme-öğretme sürecini doğrudan etkileyen öğretmenler üzerine sadece uluslararası alanyazında sınırlı sayıda araştırmaya rastlanmıştır (Cai, 2005;Cai & Lester, 2005;Cai & Wang, 2006;Monoyiou, Papageorgiou, & Gagatsis, 2007;Patterson & Norwood, 2004). Bu araştırmalar da, öğretmenlerin temsil tercihinin öğrenci tercihine etkisi veya öğretmen inancının temsil kullanımını nasıl etkilediği üzerine yapılmıştır.…”

“…Bu sonuç Ball'un (1993) çalışmasındaki öğretmen adaylarının seçtikleri temsillerin matematiği kendi anlamalarından etkilendiği sonucu ile paralellik göstermektedir. Bunun yanı sıra, bu sonuç uluslar arası karşılaştırma araştırmalarındaki Çinli öğretmenlerin temsil kullanımı ile ilgili değerlendirmelerinde ortaya çıkan sonuç ile benzerlik göstermektedir (Cai, 2005;Huang & Cai, 2007). Çinli öğretmenlerde öğrencilerin problemleri cebirsel yollarla çözmelerini beklerken, temsil kullanarak çözülen problemlere cebirsel yolla çözülenlere göre daha az puan vermişlerdir (Cai, 2005).…”

Symbolic and Verbal Representation Process of Student in Solving Mathematics Problem Based Polya’s Stages

“…Through representation students can make mathematical ideas to be more concrete and facilitate to do reflection (NCTM, 2000). Even the representation is seen as an important part of mathematical activity and means to capture mathematical concepts (Cobb et al, 2002;Cobb, 2003;Gravemeijer et al, 2003;Meira, 2003;Kaput et al, 2008;Cai, 2005;Anwar et al, 2016).…”