Presenters: Wei-Chih Hsu Professor : Ming-Puu Chen Date : 03/21/2009

1. A computer game as a context for non-routine mathematical problem solving: The effects of type of question prompt and level of prior knowledge Presenters: Wei-Chih Hsu Professor : Ming-Puu Chen Date : 03/21/2009 Chun-Yi Lee & Ming-Puu Chen (2009). A computer game as a context for non-routine mathematical problem solving: The effects of type of question prompt and level of prior knowledge. Computers & Education, 52(3), 530-542.

2. Introduction(1/3) • The purpose of this study • Investigate the effects of type of question promptand level of prior knowledgeon non-routine mathematical problem solving. • The Frog Leaping Online Game as a context to present a challenge and to trigger learners’ curiosity. • Many researchers(Chi & Glaser, 1985; Ge & Land, 2003; Schoenfeld, 1985; Voss & Post, 1988) suggested that students should be encouraged to work on non-routine mathematical problem-solving tasks • Help students see the meaning and relevance of what they learned. • Facilitate the transfer of contextual knowledge in authentic situations. • Computer games were regarded as powerful mathematical learning tools(Betz,1995; Malone, 1981; Moreno, 2002; Quinn, 1994). • Great motivational appeals. • Multiple representations of learning materials.

3. Introduction(2/3) • Successful use of computer games in the classroom depended on the quality of the teaching, including the teacher’s skill(Sanford, Ulicsak, Facer, & Rudd, 2006) • Diagnosing pupils’ abilities. • Identifying learning objectives. • Developing games in appropriate ways to meet these objectives. • Question prompts • Proved effective in scaffolding students’ high-order thinking in various domains and contexts (Scardamalia, Bereiter, McLean, Swallow, & Woodruff,1989; Scardamalia, Bereiter, & Steinbach, 1984) • Help learners to elaborate thinking, make inferences, and most important, monitor and evaluate their own learning processes(Lin, Hmelo, Kinzer, &Secules, 1999).

4. Introduction(3/3) Different types of question prompts may serve different needs and purposes for students. Procedural prompts Help learners complete specific tasks, such as writing (Scardamalia et al., 1984) or problem-solving (King, 1991). Help students learn cognitive strategies in specific content areas (Rosenshine, Meister, & Chapman, 1996). Elaboration prompts Prompt learners to articulate thoughts and elicit explanations. Reflection prompts Encourage reflection on a meta-level that students do not generally con-sider (Davis & Linn, 2000). The prior knowledge under investigation was defined as the following four pattern reasoning styles (1) recursive relationship, (2) functional relationship, (3) algebraic expression, (4) knowledge about arithmetic sequences. 4

5. Literature review (1/5) Non-routine mathematical problem solving • Students always solve these routine problems by copying standard solution methods provided by the textbook or teachers(Harskamp & Suhre, 2007). • Have a great difficulty in solving problems that require the application of domain knowledge and routines in novel situations. • Schoenfeld (1985) suggested that students should be encouraged to work on non-routine problems in their learning of mathematics because these problems were generally considered to be effective in • Engaging students’ intellect. • Capturing their interest and curiosity. • Developing their mathematical understanding and reasoning processes. • Allowing for different solution strategies, solutions, and representational forms(English & Halford, 1995; NCTM, 1991; Stein, Grover, & Henningsen, 1996). • Many researchers indicated that students’ problem-solving failures do not always result from lack of mathematical knowledge but from the ineffective use of their knowledge.(Garo-falo & Lester, 1985; Schoenfeld, 1987; Van Streum, 2000)

6. Literature review (2/5) Non-routine mathematical problem solving Schoenfeld (1992) suggested that students need to learn to define goals and to self-regulate their problem-solving behavior in order to improve solving of non-routine mathematics problems. Game and learning Computer games Make learning meaningful to students and that they created a learning culture which was more in correspondence with students’ interests(Papert, 1980; Provenzo, 1992). Enhanced learning through visualization, experimentation, and creativity of play(Betz, 1995). Visualized and anchored abstract mathematical concepts in a meaningful real-life context (Gredler, 1996) Reduce the extraneous cognitive load Allow students to use their precious working memory for higher-order tasks. 6

7. Literature review (3/5) Game and learning • Incorporating computer games into instruction is an appealing way to solving mathematical non-routine problems. • Many current games used for facilitating learning lacked connection to curricula in schools (Ke, 2008). • This study emphasized the application of computer games in school education by exploring the potentials of games in facilitating the learning of math concepts and reasoning skills that are required by core curriculum content standards. 7

8. Literature review (4/5) Game and learning • It is important for players to be aware of reflection and how it can be facilitated. • Problem-solving gaming model with reflection support (PSG) is proposed for addressing this issue (see Fig. 1). • PSG describes learning as a cyclic problem-solving process through idea generating, goal identifying, outcome assessing, and schemata constructing in the game world. 8

9. Literature review (5/5) Problem-solving gaming model (PSG) • Refection should be enhanced by game-based learning environment support in the four problem-solving phases: (Clark & Mayer, 2002; Garofalo & Lester, 1985; Kiili, 2007). • Orientation: Build the context of realistic problem-solving situation • Help students become interested in the problem and understand the problem. • Organization: Provide suggestions of expert problem-solving actions and thinking • Help learners identify goals and sub-goals, make and implement a global plan, and draw diagrams and organize data into other formats. • Execution:Provide gaming activities on executing local goals and global goals to the learners • Promote their awareness of and reflection on their problem-solving process by making learners document their plans. • Verification: Expert model • Offer the power of computation, construction, and visual representation to help students evaluate decisions and check computations by showing maps of student and expert problem-solving paths. 9

10. Methodology(1/5) Research design The independent variables : Question prompts (specific and general),Prior knowledge(high and low). The dependent variables: Students’ non-routine problem-solving performances Reasoning for one variable, Reasoning for two variables. Participants Seventy-eight 9th graders. They participated in the 6-week experimental instruction. Participants were randomly assigned to The specific-prompt group: procedural prompts were offered. The general-prompt group: the supplementary elaboration and reflection prompts strongly related with the major tasks were provided. 10

11. Methodology(2/5) • Materials • The frog leaping problem • Two major tasks • Task 1: If there are n frogs in the left group and n frogs in the right group, how many times do you move the frogs to finish the game? • Task 2: If there are n frogs in the left group and m frogs in the right group, how many times do you move the frogs to finish the game? • Computer tools were provided to help students explore the two tasks fluently. 11

12. Methodology(3/5) 12

13. Methodology(4/5) Instruments Pattern reasoning test (PRT) The aim of PRT was to evaluate students’ prior knowledge in pattern reasoning skills before the experimental program. PRT was developed based on task analysis of the frog leaping problem. Including ‘recursive relationship’, ‘functional relationship’, ‘algebraic expression’, and ‘arithmetic sequences’. The reliability coefficient of PRT was 0.91. Mathematical attitude scale (MAS) The purpose of MAS was to examine students’ mathematical attitude before the experimental program. This five-point Likert scale contained 24 items to investigate students’ enjoyment of mathematics, motivation of mathematics, importance of mathematics, and freedom from fear of mathematics. The reliability of MAS was 0.93. 13

14. Methodology(5/5) • Instruments • Teaching website • The teaching website was designed based on PSG and provided many related computer tools, question prompts and expert modeling to help students finish the given tasks. • Worksheets • Worksheets were complements of this website with which students needed to complete the two tasks through the teacher’s guiding. • Participants’ problem-solving performances could be measured based on the evaluation of these worksheets. 14

15. Students got higher scores in reasoning for one variable than reasoning for two variables. Students with high prior knowledge got higher scores in the two problem-solving performances than those with low prior knowledge did. Students receiving specific prompts got higher scores in the problem-solving performances of reasoning for two variables than those receiving general prompts did. Results (1/2)

16. Results (2/2) • Table 4 presents the final regression models in predicting students’ two problem-solving performances. • The regression results indicated that prior knowledge and comprehensive mathematical ability were important predictors on the two problem-solving performances: reasoning for one variable and reasoning for two variables. • Question prompts and mathematics attitude were the significant predictors on predicting only one problem-solving performance: reasoning for two variables. 16

17. Conclusions (1/2) In this study, four findings were concluded. Firstly, regardless of the simple task or difficult task, students with high prior knowledge outperformed those with low prior knowledge in problem-solving performances. Complex problem solving should emphasize the importance of prior knowledge combined with learning (Meijer & Riemersma, 2002). Secondly, specific prompts did not have a more positive effect than general prompts in solving the simple task. Thirdly, students receiving specific prompts outperformed those receiving general prompts in performing the difficult task. Finally, question prompts and mathematics attitude were not the significant predictors on predicting the simple task but they couldpredict the difficult task. 17

18. Conclusions (2/2) Ways to incorporate students’ mathematics attitude or beliefs in the instructional design was the key to promoting their performance of solving difficult problems. Several questions have been raised as a result of our investigations. Future studies should increase the number of sample size to examine the effective use of type of question prompt on students’ prior knowledge (King, 1992). It seemed important to add more scaffolding techniques to help students with low prior knowledge enhance their performance of non-routine problem-solving processes. How effectively students internalize self-questioning when prompts are removed was also an important issue. Investigate the effects of various affective scaffolds in motivating students in solving complex, non-routine problems (Ge & Land, 2004). 18