Description and evaluation of a large-scale project to facilitate student engagement in learning mathematics

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Description and evaluation of a large-scale project to facilitate student engagement in learning mathematics ⁎

Fou-Lai Lina, Ting-Ying Wangb, , Kai-Lin Yanga a b

Department of Mathematics, National Taiwan Normal University, Taiwan, ROC Department of Mathematics, National Taiwan Normal University, 88 Sec.4 Ting-Chou Rd., Taipei, Taiwan, ROC

A R T I C LE I N FO

A B S T R A C T

Keywords: Affective engagement Cognitive engagement Grounding activity module Teacher professional development

Responding to students’ performance in TIMSS and PISA in East Asian countries/regions – low interests, low confidence, not seeing the value of mathematics, and high-achieving performance in average but a substantial percentage of “lowest performers”– a project JUST DO MATH has been launched in Taiwan. The themes of the project included developing students’ fundamental prerequisite mathematical ideas before regular classes, employing concrete manipulative representations as a starter for learning, and embedding learning activities in games. The design and implementation of the project involved the development of grounding activity modules and facilitation of professional development of mathematics teachers to implement instruction according to modules to engage students in learning mathematics. The findings – confirming the project significantly facilitated students’ cognitive and affective engagement in learning mathematics – can inform the government regarding educational reform and can suggest teachers and educators feasible instructional approaches to facilitate student engagement in East Asian countries/regions.

1. Introduction East Asian countries/regions have outperformed their Western counterparts in the international comparative studies of mathematics achievements such as the Trends in International Mathematics and Science Study (TIMSS) and the Programme for International Student Assessment (PISA) since 1999 (e.g., Mullis et al., 2000; Mullis, Martin, & Foy, 2008; OECD, 2013). However, one severe problem of mathematics education in the high-achieving East Asian countries/regions was students’ low interests and confidence in mathematics as well as not seeing the value of mathematics (Mullis, Martin, Foy, & Arora, 2012). Furthermore, except Shanghai of Mainland China, the other six high-achieving East Asian countries/regions in PISA 2012 still had a substantial percentage of students categorized as the “lowest performers” whose mathematics achievements were under the baseline proficiency level; the situation is most severe in Taiwan where the percentage of lowest performers was as high as 12.9% (OECD, 2013). The findings have informed these countries/regions that it was urgent to deal with the problems regarding students’ affective and cognitive engagement in learning mathematics, especially for low achieving students. The literature has shown that it is more difficult to engage students in mathematics classes than in those of other subjects (Kong, Wong, & Lam, 2003; Plenty & Heubeck, 2011), while student

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engagement with learning mathematics is influential to their development of mathematical literacy which is crucial for everyone in the era of globalization, fast-changing economy, and information explosion (Kilpatrick, Swafford, & Findell, 2001; OECD, 2013; Steen, 1990). Thus, the challenges for these countries/regions are what and how policymakers, mathematics educators, and teachers can do to deal with the problems revealed by TIMSS and PISA results. A large-scale project launched in Taiwan to change the present situation and its evaluation of the continuing process was used to monitor and appropriately change the implement. The intervention and the evaluation of this project involved multiple levels systematically, including the levels of students’ learning, teachers’ professional development, and the design of teaching materials, which were rare in most projects (Chalmers & Gardiner, 2015; Hum, Amundsen, & Emmioglu, 2015). This article describes the project and discusses the evaluation of its effectiveness regarding facilitating student engagement in learning mathematics, which was the final goal of the project. The effectiveness of intervention on other levels – such as mathematics performance of students, qualities of instructional materials, or outcomes of teacher preparation programs – will be reported in other publications.

Corresponding author. E-mail addresses: linfl@math.ntnu.edu.tw (F.-L. Lin), [email protected] (T.-Y. Wang), [email protected] (K.-L. Yang).

https://doi.org/10.1016/j.stueduc.2018.03.001 Received 10 December 2016; Received in revised form 3 March 2018; Accepted 4 March 2018 0191-491X/ © 2018 Elsevier Ltd. All rights reserved.

Please cite this article as: Lin, F.-L., Studies in Educational Evaluation (2018), https://doi.org/10.1016/j.stueduc.2018.03.001

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liking, happiness, and confidence (Bodovski & Farkas, 2007; Connell & Wellborn, 1991; Guthrie et al., 1996; Miserandino, 1996; Skinner & Belmont, 1993). Some research takes students’ willingness and persistence of spending time on learning as the major aspects of affective engagement (Kong et al., 2003; Plenty & Heubeck, 2011; Steinberg, Brown, & Dornbush, 1996; Williams & Ivey, 2001). And some other research relates affective engagement to students’ appreciation and value of specific subjects (Attard, 2012; Eccles et al., 1983; Finn, 1989; Fredricks et al., 2004; Krapp, Hidi, & Renninger, 1992; Martin, 2007). Cognitive engagement usually relates mental effort invested in academic work, stressing inner psychological investment and the use of the mind in learning rather than simply participating and doing the work (Fredricks et al., 2004). Regarding what kind of mental activities students’ efforts are targeted at, the literature mentioned comprehending concepts, creating connections among ideas, thinking and solving problems, and mastering tasks (Blumenfeld et al., 1992; Bouta & Paraskeva, 2013; Connell & Wellborn, 1991; Helme & Clarke, 2001). The project JUST DO MATH was intended to facilitate Taiwanese students’ affective engagement in learning mathematics, that is, to help them increase interest, enjoyment, and confidence in learning mathematics, to promote their willingness and persistence of spending time on mathematics, and to allow them to see the value of mathematics. The other intention of the project was to increase low-achieving students’ cognitive engagement in mathematics. That is, to help these students stop being “guests” but to be “insiders” within their mathematics classes. The project was intended to provide students opportunities to think and to understand mathematics, as well as to solve problems in mathematics. The project approached these goals through holding mathematics camps (Fun-math Camps) in which grounding activity modules were used as instructional materials and the activity instructors taught students mathematics in the modules. These activity instructors were primary or lower secondary mathematics teachers who had been trained by the mathematics teacher professional development programs (Activity Instructor TPD Programs; Lin, 2013) conducted by the project.

Fig. 1. Design of the project JUST DO MATH.

2. Design of the project “JUST DO MATH” and rationale To deal with the problems revealed by TIMSS and PISA findings in Taiwan, authors’ institution with the support of the Ministry of Education, launched in 2014 a project – JUST DO MATH – which has since been implemented in many schools. Building on the idea that the learning environment involves a series of interactions among teachers, students, and instructional materials (Cohen & Ball, 1999), the project JUST DO MATH consisted of the levels in mathematics education as shown in Fig. 1 (Lin, 2013). To reach the goal of facilitating student engagement in learning mathematics, the project employed gamified activities which have been shown to be effective for helping students’ learning (Evans, Nino, Deater-Deckard, & Chang, 2015; Hsu, Tsai, & Wang, 2012). Different from some studies that focused on helping students obtain familiarity with the concepts or procedures they already learnt, JUST DO MATH aimed to help students develop fundamental prerequisite mathematical ideas in gamified activities before formal mathematics learning in classes. This idea accorded with Skemp’s (1989) rectangular numbers game which could help students engage in learning mathematics cognitively and affectively. Whilst, most students participated in JUST DO MATH had low interests, confidence, and performance in mathematics.

2.2. Grounding activity modules To effectively engage students in mathematics cognitively and affectively, the instructional materials were designed under careful consideration of mathematics content, representations, and learning activities. After several years of making efforts in conducting supplemental instruction in mathematics for low-achieving students without gaining satisfactory results, a new thought emerged to be adopted as the fundamental theme of the project JUST DO MATH. That is, the approach to help students engage in mathematics cognitively was to help them establish the fundamental prerequisite ideas before learning a mathematics topic in regular classes instead of assisting them with supplemental instruction after they have already failed to learn that topic. According to Piaget (1952), understanding is a progressive reorganization of mental structure to integrate what one already knows to what one newly discovers. However, the gap between what students already have in mind and what they need to learn in mathematics curriculum make the building of the connections between them not possible for students (Bransford, Brown, & Cocking, 1999). The approach to develop students on fundamental prerequisite ideas for mathematical topics was an attempt to bridge the gap so as to elicit students’ meaningful learning (Attard, 2012; Ausubel, 1961; Bennett & Desforges, 1988). In our study, instructional materials were designed in accordance with this thought which was also the reason why the instructional materials were named grounding activity modules. The fundamental prerequisite ideas of the topics in the four main fields – number and quantity, algebra, geometry, probability and statistics – in mathematics curriculum at the primary and lower secondary levels in Taiwan were identified through employing content analysis of the

2.1. Student engagement Engagement stands for active commitment, involvement, being occupied and attracted, rather than apathetically superficial participation (Fredricks, Blumenfeld, & Paris, 2004; Newmann, Wehlage, & Lamborn, 1992). Regarding student engagement in academic work, several researchers put forth similar definitions; they regarded it as a psychological process of expending attention, effort, investment, and interest in the work of learning (Guthrie et al., 1996; Marks, 2000; Newmann et al., 1992). From these definitions, the multifaceted nature of engagement is implicated. For example, the definition of Newmann et al. (1992) elaborates the promotion of intellectual involvement during learning activities, including using the mind, experiencing cognitive challenges, comprehending knowledge, and mastering skills. This definition emphasizes the cognitive facet of engagement. Newmann et al. (1992) also pointed out that interest, enthusiasm, and enjoyment are indispensable to the actual engagement, revealing the affective facet of engagement. Affective engagement plays a key role in activating and maintaining cognitive engagement (Blumenfeld, Puro, & Mergendolter, 1992; Goodenow & Grady, 1993; Sancho-Vinuesa, Escudero-Viladoms, & Masià, 2013). It is the research topic of numerous studies, but the constructs studied vary between studies. Most studies focus on students’ emotional reactions to academic work, including interest, enjoyment, 2

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quadrants, and some grid points on them; (2) abilities to recognize and use the symbolic representations of grid points. Since the purpose is to help students connect what they knew to what they have to learn in school rather than normally teaching them the concepts regarding Cartesian coordinate plane, the content is constrained within the integers between −10 to 10, and some technical mathematics language such as axes or quadrants will not be used. The module includes a series of games: Game 1 to Game 4. In each game, students have to sit at their desks in the classroom. In Game 1, one student will be assigned as the master of attacking monsters and one student will be assigned as the monster. To catch the monster, the master has to run to the monster to touch his/her desk. Before being caught, the monster can yell the name of another student who will become the new targeted monster. If the student being the monster is caught, he/she will be the new master to catch the monster. The rules for Game 2 are the same as Game 1, but the monster cannot call the names of other students directly. Instead, he/ she has to use one student as the base and say like “two units in front of Claire (the base).” The questions such as what should the master do if the monster say “two units in the front” or “two units from Claire” will be discussed after the game. In Game 3, the student to be the base (the origin) will be assigned by the teacher before the game starts. The monster only needs to say like “three units forward and two units to the left.” After the game, a discussion of simplifying the statements for identifying the new monster will be conducted. The goal of the discussion is to raise the symbolic representation of the points on a Cartesian coordinate plane – the ordered pair (a, b), which will be used in Game 4. In Game 3 and Game 4, the teacher can assign an appropriate student to be the base such that the whole classroom will be one or more quadrants combining with part of axes (see Fig. 2). In Game 1, only the schemas of students’ daily life are needed, which serve as meaning sources for students to learn mathematics in the following games. From Game 2 to Game 4, students can gradually develop the concepts regarding the origin, units, directions, axes, and quadrats (but not the technical terminologies), and the representations students use gradually change from the concrete to the abstract. Furthermore, the series of games provide students the feelings of excitement and fun, support students’ autonomy, and gradually increase the levels of mathematical challenges to engage students cognitively and affectively in the learning activities. In addition to invitational designs, the project also started to conduct teacher professional development programs (Module Designer TPD Program) to cultivate primary and lower secondary mathematics teachers for designing grounding activity modules (Lin, 2015).

mathematics curriculum guidelines and textbooks. Representations are essential elements in supporting students’ cognitive engagement in mathematics (Capraro, Capraro, Carter, & Harbaugh, 2010; National Council of Teachers of Mathematics, 2000) because students form meanings of mathematical ideas from them (Pape & Tchoshanov, 2001; Zazkis & Liljedahl, 2004). Bruner (1966) proposed three main types of representations: enactive, iconic, and symbolic representations in the order from the concrete to the abstract. Lesh, Post, and Behr (1987) expanded Bruner’s three types of representations to a five-type representation system. The five types were real scripts, manipulative models, static pictures, spoken language, and written symbols. Regardless of employing which representation system, a common belief shared by many researchers was the use of concrete and nonlinguistic representations, such as, manipulatives, physical movements, and pictographs at the start of teaching mathematical ideas to enhance student understanding (Hall, 1998; Marzano, Pickering, & Pollock, 2001; Thompson & Rubenstein, 2000). The design of grounding activity modules in our study was aligned with this perspective. Whether or not students can understand mathematical ideas which they are learning is a crucial factor in influencing their affective engagement (Wang & Hsieh, 2016). Thus, both the aforementioned considerations – regarding mathematics content and representations to increase students’ cognitive engagement – helped students’ affective engagement. Regarding learning activities which can cater for the need of enhancing student engagement, several perspectives in the literature enlightened us in our study. Appropriate challenging level and interesting level are two critical characteristics of tasks which could facilitate student engagement (Fredricks et al., 2004; Guthrie & Wigfield, 2000; Newmann et al., 1992). Researchers argued that interesting tasks could help enhance students’ motivations of learning mathematics, and that tasks which offered appropriate challenges could involve students in the process of deep mathematical thinking and conceptual understanding (Attard, 2012; Henningsen & Stein, 1997). Another concept commonly mentioned in the studies of student engagement is autonomy support, which is often characterized by control, choice, and power (Deci & Ryan, 1987; Deci, Vallerand, Pelletier, & Ryan, 1991). Providing choices to or sharing the power of making decisions with students have been shown to empower the enhancement of student affective engagement (Attard, 2012; Park, 2005; Rathunde & Csikszentmihalyi, 2005). Through implementing student activities, such as hands-on activities, small-group work, peer-cooperation activities, or competition games in mathematics class, students were provided with sufficient time and space – according to their own needs – to understand ideas and clarify their thinking in self-directed work (Cavanagh, 2011; Rathunde & Csikszentmihalyi, 2005). Inspired by Dienes’ (1973) proposition of a six-stage theory of learning mathematics which were embedded in games, the project JUST DO MATH employed games as the form of learning activities in grounding activity modules for its affordance of appropriate challenges, interesting tasks, and autonomy. The grounding activity modules were first developed by expert primary and lower secondary mathematics teachers who were members of compulsory education advisory groups. These groups were operated by Taiwanese national or city governments and were responsible for guiding other teachers in teaching mathematics. The expert teachers were regarded as competent and knowledgeable in mathematics teaching, and thus were invited by the project JUST DO MATH to design the grounding activity modules. The following is one grounding activity module named Attacking Monster designed by an expert teacher (Chiang, 2014), which exemplified how the theories and ideas of mathematics content, representations, and learning activities were put into practice in designing instructional materials in the project. The objective of the module is to develop students’ fundamental prerequisite ideas for learning Cartesian coordinate plane. That is to develop students’ (1) basic understanding of two axes and four

2.3. Competence and willingness of activity instructors Teachers’ quality has been considered as a powerful contributor to student engagement (e.g., Boaler & Greeno, 2000; Marks, 2000). Thus, the project JUST DO MATH considered that it was necessary to only enroll in-service primary and lower secondary mathematics teachers to be trained as activity instructors. Taiwanese future mathematics teachers at both primary and secondary levels have proved to be of high quality (Hsieh et al., 2011) and only a very small proportion of them can obtain teaching jobs in the competitive screening and selection process (Hsieh, Lin, & Wang, 2009). Based on the high quality of inservice teachers, the project stressed teachers’ competence to teach with the use of grounding activity modules and their willingness to hold Funmath Camps. To cultivate teachers’ competence of teaching according to grounding activity modules, the project provided Activity Instructor TPD Programs in which workshops were held. Primary and lower secondary mathematics teachers had to be trained in the workshops to become activity instructors and to have the certification to hold Funmath Camps. In each workshop, activity instructors studied 7 to 14 grounding activity modules. The modules were introduced by the educators who are expert teachers having rich experience of designing 3

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Fig. 2. Examples of a teacher’s assignment of a student to be the base in the module Attacking Monster.

instructors indicated that they realized and appreciated the educational ideas of instruction according to grounding activity modules (99.6%); they believed instruction according to the modules could facilitate lowachieving students’ interests in learning mathematics (99.0%) and could develop their mathematical confidence (98.5%). The other approach is to create easy access for activity instructors to hold Fun-math Camps. The grounding activity modules were uploaded to the website of Authors’ Institute. The teaching aids and learning materials were prepared by the project and then sent to the activity instructors who applied to hold a Fun-math Camp. After the workshops, 91.2% of activity instructors would like to hold Fun-math Camps, and a majority of them were willing to help introduce the modules to their colleagues (95.3%) and invite them to participate in workshops of Activity Instructor TPD Programs (99.5%). The activity instructors also showed their willingness to continue learning about instruction with the use of grounding activity modules by participating in workshops of Activity Instructor TPD Programs at advanced levels (97.6%).

modules and implementing instruction according to the modules as activity instructors. The educators were knowledgeable about the modules and aware of the situations and the difficulties which activity instructors might face when holding Fun-math Camps. In the workshops, the educators demonstrated how to teach according to the modules and the activity instructors actually played the games designed in the modules. Sometimes, the educators demonstrated how to teach by directly holding Fun-math Camps to teach students. The activity instructors had discussions with each other and with the educators regarding which fundamental prerequisite ideas were for students to learn, what kind of representations were used and how they were ordered, what functions of learning activities embedded in games were, and how and why modules worked to engage students in mathematics cognitively and affectively. The activity instructors were expected to develop their competence through seamlessly learning mathematics content knowledge (MCK), mathematics pedagogical content knowledge (MPCK), and teaching practice from the educators. Thus, the Activity Instructor TPD Program was named a transformative cascade model (Lin, Yang, & Wang, 2016). To understand whether or not the TPD Program helped the activity instructors increase their competence, the activity instructors were required to complete a reflective questionnaire which contained the questions regarding their studying of MCK, MPCK, and their appreciation of the ideas of the project after the workshops. Among 2256 activity instructors who participated in the workshops in 2015, as high as 97.8% of them confirmed their progress in learning MCK; they indicated that they concretely perceived the commonality and invariance of connotations of mathematics concepts.1 The majority of the activity instructors also confirmed their development in learning MPCK (98.9%); they indicated that their views about mathematics teaching had been broadened (99.7%) and that they sufficiently understood the necessity of manipulative activities and how the activities helped develop mathematics concepts (99.4%). To keep activity instructors’ willingness of holding Fun-math Camps to engage students in mathematics, the project employed two approaches. One approach was developing activity instructors’ appreciation of the ideas of the project. After the workshops, almost all activity

3. Scale of the project Since April in 2014, 3183 activity instructors have been trained in 115 workshops of Activity Instructor TPD Programs. Among them, 2559 were primary school teachers and 624 were lower secondary mathematics teachers. The number of grounding activity modules were 103, among which 69 were of primary school level and 34 were of lowersecondary school level. Fun-math Camps started since the end of July in 2014, and 721 camps have been held by activity instructors until now. A total of 21635 students have so far attended the camps; among them, 17284 were primary school students (third to sixth graders) and 4351 were lower secondary school students (seventh and eighth graders). The project first targeted at helping mathematics low-achieving students. Although the teaching aids and the learning materials were supported by the project to allow the students to attend camps for free, the activity instructors still had to employ various methods to persuade the students to attend camps in the beginning, for example, using external motivations by providing snacks or fast food.2 However, other students, including those who were not low-achievers, were also

1 The questions were in the form of a five-point Likert scale. The points 1 to 5 represented “strongly disagree,” “disagree,” “neutral,” “agree,” and “strongly agree.” The present study combined “agree” and “strongly agree” to calculate the percentage of the activity instructors who endorsed the statements of the questions.

2 The information was obtained from the activity instructors’ experience sharing in the annual conferences of final presentations.

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engagement in the Fun-math Camps. The internal consistency reliability was estimated by calculating Kuder-Richardson 20 (KR20) coefficients. The KR20 values obtained for the overall questionnaire, the cognitive engagement questions, and the affective engagement questions were 0.90, 0.73, and 0.86 respectively. All values were higher than 0.7, indicating the acceptable level of internal consistency. Qualitative data to deepen our understanding of the effectiveness of the project were collected through using open-ended questions and interviews with students by activity instructors. The data were collected after the Fun-math Camps from 35 cases (camps). Examples of the questions asked were “Regarding the activities we have played, please write down what impressed you the most” and “Which were the most interesting parts in the activities we have played?” A content analysis was conducted by the authors to identify and categorize the themes embedded in students’ responses regarded as pertinent to their cognitive and affective engagement. The categories of the students’ responses were then linked to the structure of the quantitative questionnaire which were developed based on the critical issues discussed in the literature. Students’ vivid descriptions of their perceptions would enrich our understanding of their endorsement of the statements in the questionnaire or enlighten us on some unexpected perspectives.

Table 1 Students’ agreement on the statements regarding cognitive engagement. Question

Percentage of agreement

1 2 3 4

90% 91% 93% 95%

5

I like to do thought experiments in Fun-math Camps. Mathematics instructed in Fun-math Camps is easy. I can understand the mathematics in Fun-math Camps. I think that it is likely for me to understand the mathematics taught in regular classes in the future if the mathematics has been mentioned in Fun-math Camps. I know what to do and how to do it in Fun-math Camps.

91%

attracted by the camps when they heard about the activities implemented and their peers’ experiences in Fun-math Camps. Thus, the project set a criterion that at least half of the participants in each camp should be low-achieving students. 4. Evaluation of the effectiveness of the project 4.1. Data sources and analysis

4.2. Students’ self-evaluation on their cognitive and affective engagement in mathematics

To evaluate the effectiveness of the project JUST DO MATH, in terms of students’ cognitive and affective engagement in mathematics, the quantitative and qualitative data collected were analyzed. Quantitative data were obtained from the students’ responses to the questionnaire which each student has to complete after attending the Fun-math Camps. The questionnaire included five questions involved students’ self-evaluation regarding their cognitive engagement and 10 involved their affective engagement. The questions were developed based on the critical issues discussed in the literature (see the aforementioned section pertinent to student engagement). The exemplified questions for cognitive engagement and affective engagement were “I like to do thought experiments in Fun-math Camps” (See Table 1 for more questions) and “I like mathematics in Fun-math Camps” (See Table 2 for more questions) respectively. All the questions employed the form of a six-point Likert scale, of which 1 to 6 represented “strongly disagree,” “disagree,” “slightly disagree,” “slightly agree,” “agree,” and “strongly agree.” In the present study, the data analyzed were from those students attending the Fun-math Camps in 2015, including 8959 students. Their responses were re-categorized by combining 1 to 3 as “disagree” and 4 to 6 as “agree” to calculate the percentages of their agreement on the statements regarding their

4.2.1. Cognitive engagement Table 1 shows the students’ self-evaluation on their cognitive engagement in Fun-math Camps. The percentage of the students who agreed on the statement (POA will be used hereafter to stand for Percentage of Agreement) of each question was higher than 90%. The students perceived that, in Fun-math Camps, they were capable to think, the mathematical ideas in the instruction were not too difficult for them to understand, and they knew what to do and how to do it in the activities (question 1, 2, 3, and 5). The students also believed that they could understand the concepts back to regular mathematics classes if the pertinent mathematical ideas have been taught in the camps (question 4). Qualitative data evidenced the findings from quantitative data aforementioned and provided rich connotations for them from three facets: mental efforts in thinking in Fun-math Camps, comprehending mathematical ideas in Fun-math Camps, and cognitive engagement in future regular classes. The students in the case studies confirmed their cognitive engagement in Fun-math Camps by illustrating their mental efforts in thinking. Some of their ideas were as follows: “… Some are very difficult. But we all have done brain storming [and] to figure [things] out.”

Table 2 Students’ agreement on the statements regarding affective engagement. Question

1 2 3 4 5 6 7 8 9 10

I find that Fun-math Camps are interesting. I feel joyful in Fun-math Camps. I like mathematics in Fun-math Camps. I like mathematics more than before because of attending Fun-math Camps. Fun-math Camps let me believe that I can learn mathematics well. I perform well when learning mathematics in Funmath Camps. I hope that I can learn every mathematics topic in Fun-math Camps before regular classes. I would like to participate in Fun-math Camps to learn topics other than those instructed today. I think that we can discover a lot of new things in the mathematics instructed in Fun-math Camps. I think that mathematics in Fun-math Camps can teach people how to analyze problems and think of their solutions.

Percentage of agreement

“My feelings are that [it is] super fun, super interesting, and [it] allows us to think.”

96% 95% 96% 92%

“We actually manipulate and discuss, not like in regular mathematics classes.” Numerous students mentioned their acquiring understanding of mathematical ideas in the camps:

91%

“I think that attending the mathematics camp is helpful to me. [I] still have [learned] some concepts though [they] are not very clear…”

91% 92%

“I think that I understood many mathematics expressions and several real examples which were not taught in [regular] classes but could allow me to connect [what I learned in the camps] to what I learned in [regular] classes. This reinforced my impression…”

94% 95% 93%

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“…we could learn knowledge at the same time when we played ‘Space Master3’…”

“I find that attending this activity makes me very happy and I learn a lot.”

“...I have made progress in learning mathematics since attending [Fun-math Camps]! I learned a lot of things, for example, how to read statements and write out expressions…trained my abilities to communicate and express.”

The students also mentioned several critical factors that were deemed as facilitators of motivation. For example, the following students’ descriptions involved the concept of challenge, curiosity, and fantasy,

“Every game we played was fun; it could have made my mathematics competence better and better. I hope my mathematics competence to become even better.”

“I think that Fun-math Camps are very interesting. The approaches in every instructional material are very challenging, for example, 72 Transformations of Triangles,5 building blocks, geo sticks...If there is a next time, I must attend the camps again.”

“I think that I could learn mathematics quickly [in the camp].”

“The first lesson today was very exciting.”

In addition, some students mentioned that their learning in Funmath Camps could raise their cognitive engagement in future regular mathematics classes and could even help their examinations:

“The game is very special. I thought they are normal cards in the beginning, but found them different afterwards. So much fun.”

“[It is] very helpful. Because attending mathematics camps allows me to have some concepts of what is taught. When the teacher teaches in [regular] mathematics classes, I can have better understanding. [The camp] also allows me to have better understanding about fractions and angles. I think that it is very helpful.”

“I think that the poker cards are very magical.” Both the students’ responses – to the quantitative questionnaire and the open-ended questions or interviews – revealed that students engaged in mathematics cognitively and affectively in the learning activities of the grounding activity modules conducted by the activity instructors. That is, from the evaluative perspective, there is evidence to support that JUST DO MATH project is effective in facilitating students’ engagement in mathematics.

“I find it useful. Because of this way, I will not be afraid of being incompetent in examinations.” 4.2.2. Affective engagement Table 2 shows the students’ self-evaluation on their affective engagement in Fun-math Camps. The percentage of the students who agreed on the statement (POA) of each question was higher than 90%. Firstly, the students’ emotional reactions to the camps were positive. The POAs of question 1 to 4 revealed that the students liked Fun-math Camps, found the camps interesting, felt joyful to learn in the camps, and believed that Fun-math Camps increased their affection of mathematics. Moreover, the students increased their confidence in learning mathematics through learning experiences in the camps (questions 5 and 6). Secondly, the students’ willingness and persistence of spending time on learning mathematics revealed. They would like to keep learning in Fun-math Camps according to the high POAs in questions 7 and 8. Thirdly, according to the POAs of question 9 and 10, the students approved that the activities in Fun-math Camps could provide them the actual experiences which cultivated their appreciation and value of mathematics so that they believed that new things can be discovered in mathematics by oneself and that mathematics can teach people how to analyze and solve problems. The majority of the students’ responses in open-ended questions or interviews evidenced findings from the quantitative data. They considered the activities in the camps interesting, they liked them, and they had fun in them. They illustrated these perceptions by the following:

5. Conclusion and discussion Approximately two decades ago, Taiwan launched a reform on mathematics curriculum. Aligning with the new curriculum, innovative mathematics instruction – emphasizing student-centered approaches rather than traditional teacher-centered approaches – has since been recommended (Lin & Li, 2009). In instruction characterized as studentcentered, students are expected to engage in mathematics cognitively and affectively. This is a common value shared in both Eastern and Western countries (Kaur, 2009; Kilpatrick et al., 2001; Martinez-Sierra, 2014; Murray, 2011; OECD, 2013; Pang, 2009; Steen, 1990), and is also clearly stated in the curriculum standards as a goal to pursue in the curriculum reform (Ministry of Education, 2009). However, Taiwanese TIMSS and PISA results, high percentages of the students with low achievements, low interests, or low confidence in mathematics, remind policymakers, mathematics educators, and teachers to think whether or not they have put the effort in the right place or in the right direction. The project JUST DO MATH was launched as a response to these results and has been evaluated by students, whose perspectives can be regarded as consumers’ viewpoints, as being effective to facilitate students’ cognitive and affective engagement in mathematics. Our wonder is that whether or not students’ engagement in mathematics enhanced in Fun-math Camps lasts when they get back to regular classes. The project took a step further to investigate students’ engagement in regular mathematics classes. The data were collected one month after the Fun-math Camps from 17 cases (camps) through interview. The exemplified questions were “Do you understand mathematics taught in classes nowadays? How was it before attending Fun-math Camps?” and “Do you feel confident of solving mathematics problems? Why?” The results revealed that the students maintained their engagement even though they sometimes encountered difficulties in solving some problems in class or in examinations. They mentioned “I can understand the majority of the contents in class. My major problem is on the examinations. I do not know how to solve the problems if they are different from those in textbooks or working sheets”, “I cannot

“Very much fun. Because the building blocks can make lots of interesting objects, such as vehicles and guns. Furthermore, I play tricks several times when playing Green Market4 It turns out that mathematics is so simple.” “We can learn mathematics in games. I find it very interesting.” “After taking part in the activities for a whole day, I found the activities today very interesting. Here, I learn a lot of knowledge related to mathematics. Very happy.” “I think that attending this camp allows me to learn mathematics with fun. I can play [there] happily.”

3 The teaching aids of a grounding activity module which involves the concepts pertinent to patterns of numbers and figures in the mathematics curriculum of sixth grade in Taiwan. Please check Appendix A for more information. 4 A grounding activity module involves the concepts of ratio which is in the mathematics curriculum of fifth grade in Taiwan.

5 A grounding activity module involves the concepts of various types of triangles, including equilateral and isosceles triangles, as well as acute, right and obtuse triangles. These topics are in the mathematics curriculum of fourth grade in Taiwan. Please check Appendix A for more information.

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training their skills of problem-solving in tests. Secondly, based on the findings in the project JUST DO MATH, the Minister of Education is initiating discussions about the possibility of including grounding activity modules in official mathematics curriculum as starters of topics. Taiwan is not the only East Asian countries/regions with the problems revealed by TIMSS and PISA aforementioned. For these countries, the findings in JUST DO MATH can suggest teachers and educators possible instructional methods to facilitate student engagement in learning mathematics. Furthermore, the design and implementation of the project, from developing of grounding activity modules, spreading the ideas by training teachers in Activity Instructor TPD Programs, practicing by holding Fun-math Camps, to evaluating the project with questionnaires and interviews, provide the government and the policymakers feasible ideas and approaches of execution to solve the common problems in East Asian countries/regions.

understand all in mathematics classes. I am not afraid or disappointed although I still have difficulties in understanding the solutions to some questions. Before attending the camps, I often give up and ignore the problem if I cannot solve it”, or “…there are still many problems that I cannot solve…the biggest difference between now and before attending the camps is that I dare to ask the mathematics teacher the problems I cannot solve. This is different from the way I dealt with those problems before – just leave them there.” The phenomenon revealed students’ engagement was different from that before their attending Fun-math Camps. The phenomenon also reflects Taiwanese present educational practice of employing difficult and complex problems in mathematics teaching and assessments. Though this is not what JUST DO MATH can directly change, two ideas pertinent to the newest curriculum reform (National Academy for Educational Research, 2016) provide opportunities to improve the undesirable situation. Firstly, the new curriculum focused on developing students’ mathematical literacy rather than Appendix A. Space master Space master

Space Master is the initial part of the module Mathematics Hidden in Toys (Hong, 2014a). It aims to develop students’ concepts pertinent to patterns of numbers and figures. The teaching aids are composed of 15 components which can be inset together to become a cube. Space Master includes the explorative learning activities of using cubic bricks to compose those five components of different sizes. Students are encouraged to figure out some pattern to compose the objects such that they can calculate the amounts of cubic bricks used easily and that they can extend the pattern to compose other objects which are of similar shapes but different sizes. Then the activity instructor will guide students to have discussion on the various patterns of composing the objects and the according approaches to calculate the amounts of the cubic bricks which students come up with by themselves. This is to initially develop students’ abilities to analyze, decode, and construct patterns of numbers and figures. In terms of the whole module of Mathematics Hidden in Toys, a relay game follows these explorative learning activities in Space Master. Eight students are grouped together as a team. Each team has a desk in the front of the classroom with eight tasks of composing objects on it. Four of the tasks are basic ones and four are advanced ones. Each student take turns to go to the front to work on a task. When he/she finishes one task, he/she can go back to the team and hand off the baton to the next one. The team which first finishes all eight takes wins.

72. transformations of triangles The module (Hong, 2014b) aims to develop students’ preliminary concepts involving various types of triangles, including equilateral and isosceles triangles, as well as acute, right and obtuse triangles. The module starts with the explorative learning activities of constructing triangles with the use of geometric buckle bars. The buckle bars are of six different lengths; those with the same length are of the same color which is different from the colors of other lengths. Students have to accomplish five tasks including constructing triangles with three buckle bars of the same length, of two different lengths, and of three different lengths, and constructing right triangles with three buckle bars of two different lengths and of three different lengths. Then the activity instructor will guide students to have discussion on the categories of the triangles they construct and to induct to the conclusion of acute, right, and obtuse triangles. Two games follow these explorative learning activities. The first game is to familiarize students with mathematical terminologies. The teaching aids are 26 cards (shape cards): eight cards with the terminology “acute triangle” on them, eight with “right triangle” on them, and eight with “obtuse triangle” on them, and the remained two cards are Angel Cards which can be used as any type of the triangles. The cards are dealt out to four students so that two of them obtain six cards and the other two of them obtain seven cards. The students cannot view their cards. Each of them has to place and reveal one of his/her cards in the middle of the table in turn in a clockwise manner, and speaks one of the terminologies simultaneously. The terminologies spoken are acute, right, and obtuse triangles in consecutive order. When the terminology on the card laid down matches the one spoken, every student has to slam the pile of cards in the middle. The last student picks all cards up. At last, the student with the most cards loses. 7

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There are another eight cards with the terminology “equilateral triangle” on them and another eight cards with “isosceles triangle” on them. These cards can also be included in the games. The second game further develops students’ concepts of acute, right, and obtuse triangles through their own manipulation of the geometric buckle bars to construct triangles. The teaching aids include the shape cards aforementioned and another 26 color cards which are eight cards with “one color” written on them, eight with “two colors” on them, and eight with “three colors” on them, and the remained two cards are Angel Cards which can be used as any color. At first, the pack of shape cards, the pack of color cards, and 72 buckle bars (12 of each length) are put in the middle of the table. In each round, each student takes one shape card and one color card, and then uses the buckle bars to construct a triangle which fits the conditions provided by the cards he/she gets. The student who first accomplishes the task wins the round.

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