With a traditional culture that has generally emphasized standardized testing and the Confucian ‘sage on a stage’ model of instruction, there is a lot of controversy regarding Taiwanese education reform. In fact, “the fundamental purpose of education has long been debated in Taiwan. This ongoing debate has led to a learning system that over-emphasizes academic performance and neglects other dimensions of learning. But recently, the Taiwanese government adopted the use of a constructivist approach to teaching mathematics. This new approach to teaching and learning focuses on the whole child” (Eisenhart, 2011).

Contemporary education reform in Taiwan started during the late 1980’s, when a team led by Dr. Fou-Lai Lin “gradually began to investigate mathematics teaching through research and literature studies instead of only through their own experience. As a product of these occurrences, mathematics teacher education in Taiwan moved towards a new realm, combining practical experience with mathematics education research” (Hsieh et. al., 2009). In 1996, “in-service and pre-service math teachers throughout Taiwan began to deeply consider the way students think, shifting the view towards teaching from teacher-centered to student- oriented” (Hsieh et. al., 2009). The following year, the Ministry of Education implemented a new national curriculum for junior high school students. Many of these changes “centered on students; the links between mathematics and life; the cultivation of students’ creativity, thinking, as well as reasoning abilities; and on an active attitude towards learning mathematics and appreciating mathematics (Hsieh, 1997).” The intent of these reforms “means that in mathematics education the emphasis will shift to problem-solving and process-monitoring and away from memorizing and plugging into formulas. Problem solving through which one can learn the methods of acquiring knowledge is one aspect of mathematics education that has been more or less neglected in Taiwan, but is now gaining attention alongside the emphasis on mathematics education for lifelong learning” (Hoyles, Morgan, & Woodhouse, 1999).

In modern-day Taiwan, the Ministry of Education is currently piloting a new national curriculum that will be rolled out during the 2019-2020 school year. One of the Ministry’s noted goals is to “progressively implement the 12-Year Basic Education program, incorporating development of adaptive learning and completely non-exam-based secondary school admission” (Ministry of Education, 2017). Policy makers are are planning to adapt the Taiwanese curricula to encourage creative problem solving (Hoyles, Morgan, & Woodhouse, 1999). The Ministry has also put forward that “teachers are required to pay closer attention to the learning process and children’s conceptualization of content and ideas rather than focusing on simply attaining the correct answer” (Eisenhart, 2011). These proposed reforms look to address the pitfalls of current educational practice and intends to inspire students to collaborate through project-based learning and standard-based grading. During one interview, a teacher noted how these changes will take the future generation of Taiwanese students onto a positive new path that will prepare them for the adaptive challenges of our increasingly globalized world.

As the vision of Taiwan’s new 12-year basic education program is developed, its ideas of “spontaneity, interaction, and common good” are synthesized with reference to the educational ideas of John Dewey (1938), postmodernism, and complex thought (Morin, 1999; 1993). These instructional shifts encourage Taiwanese teachers to let students drive their learning and take ownership of their thinking with an aim to inspire rather than to control (Fan, 2016). After all, “if we continue to ignore the power of students’ own ideas and conceptions, we will only perpetuate the notion that mathematics and science (among other subjects in our school curricula) are irrelevant, uninteresting, and difficult to learn” (Sahlberg, 2018).

These progressive changes are not unique to Taiwan, either: “China, the leading economic competitor of the United States, is decentralizing its curriculum, diversifying assessment, and encouraging local autonomy and innovation. Meanwhile… Singapore is promoting a creative environment characterized by ‘Teach Less, Learn More’” (Finnish Lessons 2.0). In other Asian countries, schools “are limiting direct instruction and mere recitation of facts and looking for more innovative pedagogies and encourage students to design and make things” (Wagner & Dintersmith, 2016). When observing classrooms throughout Taiwan, it is apparent that lesson structure plays an important role both during class and when a teacher is preparing for a lesson. This idea was featured prominently in Elizabeth Green’s critically-acclaimed book *Building a Better Teacher*:

“One striking example was the way teachers structured their lessons. American teachers rarely talked about lesson structure – the way class proceeds from a beginning to a middle to an end – and yet, watching each individual teacher at work, Stigler felt as though they’d all read the same recipe. ‘A cultural script,’ he called it… Some American teachers called their pattern ‘I, We, You.’ The Japanese teachers, meanwhile, turned ‘I, We, You’ inside out. You might call their version ‘You, Y’all, We.’ They began not with an introduction, but a single problem that students spent ten or twenty minutes working through. Next, the teacher brought them back to the whole group, asking students to present their different ideas for how to solve the problem on the chalkboard.Give the answer and the reason for the answer. Finally, a teacher led a discussion, guiding students to a shared conclusion –What did you learn from today’s problem, or what new questions do you have, if any?” (Green, 2015)

To fully capitalize on harnessing student’s own ideas and conceptions, many schools in Taiwan (and throughout the world) are recognizing the importance of teaching students how to work collaboratively, create viable arguments, and critique the reasoning of others. Student voice is featured prominently within many Taiwanese math classes, often for students to share their strategy on how to solve a complex problem. Unlike in the U.S., most Taiwanese high school math classes only complete a few rigorous problems during each period, as opposed to drilling a few dozen scaffolded problems over the trajectory of a lesson. This means that students spend more time thinking deeply about a few hard problems, which enables them to reflect critically about their solution strategy. When students are solving these problems, the types of questions that Taiwanese teachers ask their students are noticeably different than the types of questions often posed by American teachers:

“In comparisons of mathematics teaching in the United States and in high-achieving countries, U.S. mathematics instructions has been characterized as rarely asking students to think and reason with or about mathematical ideas. [American] teachers sometimes perceive student frustration of lack of immediate success as indicators that they have somehow failed their students. As a result, [American math teachers] jump in to ‘rescue’ students by breaking down the task and guiding students step by step through the difficulties. Although well intentioned, such ‘rescuing’ undermines the efforts of students, lowers the cognitive demand of the task, and deprives students of opportunities to engage fully in making sense of mathematics” (NCTM, 2014).

To this end, some Taiwanese teachers are moving away from rigid algebraic algorithms to flexible divergent thinking. For an algebraic example that highlights this phenomenon, consider the simplification of the following expression, which was recently given to an 8^{th }grade class at a junior high school in Taiwan. How would most American students go about simplifying such an expression?

Most American children would follow “PEMDAS” (the rigid algorithm commonly used for order of operations), and start by multiplying 6 times 14 times 21, and then dividing by 42 OR simplifying the 21 and the 42 to ½ first. Look instead what one Taiwanese 8^{th }grader wrote on the board:

Before jumping immediately into the problem, the student reflects for a second and sees that by re-grouping the six, she can attain 42, which allows for a more straight forward simplification. The student then had to only multiply 3 times 14 to get the correct answer.

Another example was seen during a 9^{th}grade geometry class. After deriving the ‘interior angle’ formula of a polygon, a student worked a problem down to the following expression:

Again, most American students would start by distributing the 180 to the parenthesis, or by simplifying 360 times five equals 1800. Instead, consider what one Taiwanese 9^{th }grader wrote:

Because Taiwanese students were encouraged to think divergently about the algebra at hand instead of rigidly following an algorithm, the students could regroup certain terms to make the complex expression simpler. In many classroom observations, different students were solving algebra using a multitude of different strategies, allowing them to think more concretely about the algebra and open up the world of mathematics.

Another exemplar aspect of Taiwanese math pedagogy is how teachers prominently feature multiple modalities in their pedagogy, as well. The National Council of Teachers of Mathematics has put forward that “effective mathematics teaching includes a strong focus on using varied mathematical representations” (NCTM, 2014). In fact, multiple studies have found that “when students learn to represent, discuss, and make connections among mathematical ideas in multiple forms, they demonstrate deeper mathematical understanding and enhanced problem-solving abilities” (Fuson, Kalchman, and Bransford, 2005). Taiwanese teachers in particular focus heavily on different visual representations of abstract mathematics, which help students “advance their understanding of mathematical concepts and procedures” (Arcavi, 2003).

Creating arguments and critiquing the reasoning of others, on the other hand, is a pedagogical shift that Taiwanese teachers are struggling to implement. In one classroom observation, a teacher in Kinmen repeatedly told students that, “we cannot work independently anymore; we need to work with others and learn to cooperate more.” Although this teacher had strong messaging, they struggled to give students concrete strategies to help facilitate meaningful groupwork.

During another school visit, several educators in Kaohsiung have asked how teachers in the United States facilitate rigorous discussions and Socratic seminars with their students. In Newark, the Office of Mathematics argues that “mathematical discourse should be well-planned, intentional, and embedded in whole-class and small-group settings.” Classroom discussion is one of the most important levers in student success: when educators “decrease the teacher talk and increase the student talk by providing them with learning intentions and success criteria, and a deeper understanding of how to have a discussion with the class” (DeWitt, 2017). In fact, “students who learn to articulate and justify their own mathematical ideas, reason through their own and others’ mathematical explanations, and provide a rationale for their answers develop a deep understanding that is critical to their future success in mathematics and related field” (Michaels, O’Connor, and Resnick, 2007). These shifts are most profound when teachers view themselves as a facilitator of knowledge instead of a giver of knowledge, a shift that will be enduring for many teachers (NCTM, 2014). In a country with a strong culture that has many roots in Confucianism, this instructional shift will inevitably take time to fully implement.

While these are just some of the pedagogies that Taiwanese math teachers use throughout their practice, we still have a far way to go as a global math community until every school has implemented research-informed best practices that will help students learn better. Perhaps NCTM summated this global shifting landscape most succinctly: in math classes in 2018, “students must rethink what it means to be a successful learner of mathematics, and teachers must rethink what it means to be an effective teacher of mathematics” (2014). Let us now resolve to work relentlessly to achieve this end and share the innate beauty of mathematics with everyone.

__Works Cited__

Arcarvi, A. (2003) “The Role of Visual Representations in the Learning of Math” *Educational Studies in Mathematics, *52, no. 3 pg. 215-241

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Eisenhart, C. (2011). *Why do Taiwanese Children Excel at Math?. *The Phi Delta Kappan. Retrieved from http://www.academia.edu/987689/Why_do_Taiwanese_Children_Excel_at_Math

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Ministry of Education (2017). Ministry of Education Objectives for 2018 (January-December) released 7/19/2017. Taipei, Taiwan: Ministry of Education.

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