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Composing Tangram Puzzles to Support Shape Transformation
Abstract
Different types of tangram puzzles can encourage students to make sense of problems and engage in the computational thinking practice of debugging.
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Thirty-seven third graders and thirty-two first graders engaged in solving a tangram puzzle in the shape of a fox. They had five minutes to solve the puzzle, and after this time, they received guidance on the particular piece they had difficulties with. Through the lenses of navigating flexible abstraction, reinterpretation, combinations, and borrowing structure to expand upon the existing 2D shape composition and decomposition learning trajectory, we examined ways in which students’ puzzle-solving processes and their challenges related to the fox puzzle’s features. Students’ initial shape placements suggest that parts of the fox puzzle primed the use of particular pieces, which reduced the abstraction of the puzzle. The most challenging part of the puzzle for students was navigating reinterpretation to place the square and two small triangles on the fox’s head in nonstandard orientations. Even though students faced challenges at different steps, they overcame them similarly by trying new combinations and by borrowing structure. Some students did not complete the puzzle even though they used flips and turns (reinterpretation) strategically. The results suggest potential modifications of the current learning trajectory to account for differences between tangram and pattern block puzzles and differences due to tangram puzzles’ features. Because the puzzle’s features played a role in students’ challenges, future work needs to focus on the interaction between students’ puzzle-solving processes and puzzles’ features for a variety of tangram puzzles.
This study blended the fields of debugging, spatial thinking, and change detection.
We explored 12 K-2 students’ debugging skills as they fixed a Lego structure by using multicolor or grayscale manuals. Students in both groups were most likely to debug the bugs on the top and bottom of the structure, while the multi-color group was more successful at debugging bugs in the middle of the structure than the grayscale group, possibly because they were also more likely to trace the steps in the manual and use an understanding the structure strategy. Overall, students used debugging strategies similar to those found in programming, providing initial evidence that young students can successfully engage in debugging during unplugged, non-programming activities such as Lego building.
Elementary students’ early development of embedding and disembedding is complex and paves the way for later STEM learning. The purpose of this study was to clarify the factors that support students’ embedding (i.e., overlapping shapes to form a new shape) and disembedding (i.e., identifying discrete shapes within another shape) through the use of filled shapes as opposed to shape frames. We recruited 26 Grade 1 students (~6–7 years old) and 23 Grade 3 students (~8–9 years old), asked them to work on two layered puzzle designs from the Color Code puzzle game, and interviewed them about their thinking processes. The first graders had higher success rates at fixing and embedding the tiles correctly, and students at both grade levels improved on the three-tile design when encountering it a second time about two months later. The four-tile design was more difficult, but students improved if they could identify a correct sub-structure of the design. Successful students used a combination of pictorial shape strategies and schematic location strategies, systematically testing tiles and checking how they could be embedded. The results suggest that helping students focus on sub-structures can promote their effective embedding.
Although learning to program can improve students’ computational thinking skills—specifically, creating and debugging algorithms—we need to determine instructional strategies that foster such skills in young students. We investigated the role of analyzing worked examples that focused on creating and debugging programming algorithms with 28 first and 27 third graders. Students played a tangible, block-based programming game, Coding Awbie, across six, 20-minute sessions and were randomly assigned to also analyze programming worked examples during sessions one to three (immediate group) or sessions four to six (delayed group). To measure changes in creating and debugging algorithms before and after their worked example intervention, students completed a pretest, midtest (after session three), and a posttest. By midtest, students who were in the immediate group wrote significantly more accurate programs, although both groups had similar accuracy when given the chance to debug their programs. By posttest, both groups made significant gains in accuracy of their programs. Overall, analyzing worked examples proved to be a powerful support for students’ programming skills in creating and debugging algorithms. However, students’ common bugs indicate that additional worked examples should focus on double-counting errors.
The purpose of this study was to explore preschool children's conceptual understanding of geometric shapes, the square in particular. There were a total of 115 children, 61 girls and 54 boys, from state preschool education programs, who participated in the study. The data were collected in two semesters through interviews in a one-on-one setting, where the researchers administered a paper-pencil test to the participants. The test included six questions. One question asked children to draw a square, one question asked to select the square among three other geometric shapes, three questions asked to differentiate the square among five to seven geometric shapes which were printed in rotated directions and in various fonts and sizes and one question asked to identify a picture of a square-like real life object among a selection of pictures. The findings showed that 65% of children were able to draw a square accurately, and 77% of children were able to identify a picture of a square-like object. Approximately 69% were able to differentiate three squares in different sizes among five geometric shapes, while 27% of the remaining were not able to identify the square in smaller sizes. Approximately 79% in one task and 56% in another task were unsuccessful in identifying squares in rotated directions. Moreover, there was no gender difference in the test between boys and girls. Findings were interpreted linking to Duval's theory, van Hiele's theory, Prototype theory and Simon's task design model.
The purpose of this research is to chart the mathematical actions-on-objects young children use to compose geometric shapes. The ultimate goal is the creation of a hypothetical learning trajectory based on previous research, as well as instrumentation to assess levels of learning along the developmental progression underlying the trajectory. We tested both the developmental progression and the instrument through a series of studies, including formative studies (including action research by 8 teachers) and a summative study involving 72 children ages 3 to 7 years. Results provide strong support for the validity of the developmental progression's levels and suggest that children move through these levels of thinking in developing the ability to compose 2-dimensional figures. From lack of competence in composing geometric shapes, they gain abilities to combine shapes-initially through trial and error and gradually by attributes-into pictures, and finally synthesize combinations of shapes into new shapes (composite shapes).
A qualitative analysis of debugging strategies of novice Java programmers is presented. The study involved 21 CS2 students from seven universities in the U.S. and U.K. Subjects "warmed up" by coding a solution to a typical introductory problem. This was followed by an exercise debugging a syntactically correct version with logic errors. Many novices found and fixed bugs using strategies such as tracing, commenting out code, diagnostic print statements and methodical testing. Some competently used online resources and debuggers. Students also used pattern matching to detect errors in code that "just didn't look right". However, some used few strategies, applied them ineffectively, or engaged in other unproductive behaviors. This led to poor performance, frustration for some, and occasionally the introduction of new bugs. Pedagogical implications and suggestions for future research are discussed.
Debugging is often difficult and frustrating for novices. Yet because students typically debug outside the classroom and often in isolation, instructors rarely have the opportunity to closely observe students while they debug. This paper describes the details of an exploratory study of the debugging skills and behaviors of contemporary novice Java programmers. Based on a modified replication of Katz and Anderson's study of novices, we sought to broadly survey the modern landscape of novice debugging abilities. As such, this study reports general quantitative results and fills in the picture with qualitative detail from a relatively small, but varied sample. Comprehensive interviews involving both a programming and a debugging task, followed by a semi-structured interview and a questionnaire, were conducted with 21 CS2 students at seven colleges and universities. While many subjects successfully debugged a representative set of typical CS1 bugs, there was a great deal of variation in their success at the programming and debugging tasks. Most of the students who were good debuggers were good novice programmers, although not all of the good programmers were successful at debugging. Students employed a variety of strategies to find 70% of all bugs and of the bugs they found they were able to fix 97% of them. They had the most difficulty with malformed statements, such as arithmetic errors and incorrect loop conditions. Our results confirm many findings from previous studies (some quite old) - most notably that once students find bugs, they can fix them. However, the results also suggest that some changes have occurred in the student population, particularly an increased use of debugging tools and online resources, as well as the use of pattern matching, which has not previously been reported.
Adding computer science as a separate school subject to the core K-6 curriculum is a complex issue with educational challenges. The authors herein address two of these challenges: (1) the design of the curriculum based on a generic computational thinking framework, and (2) the knowledge teachers need to teach the curriculum. The first issue is discussed within a perspective of designing an authentic computational thinking curriculum with a focus on real-world problems. The second issue is addressed within the framework of technological pedagogical content knowledge explicating in detail the body of knowledge that teachers need to have to be able to teach computational thinking in a K-6 environment. An example of how these ideas can be applied in practice is also given. While it is recognized there is a lack of adequate empirical evidence in terms of the effectiveness of the frameworks proposed herein, it is expected that our knowledge and research base will dramatically increase over the next several years, as more countries around the world add computer science as a separate school subject to their K-6 curriculum.
The overall aim of our research project is to explore the impact of dynamic geometry environments (DGEs) on children's geometrical thinking. The point of departure for the study presented in this paper is the analytically and empirically grounded assumption that as the geometric discourse develops, the direct visual identification of geometric shapes gives way to discursively mediated identification, that is to a process in which one needs to perform a discursive procedure, prescribed by a formal definition of the shape, in order to ascertain the name of the shape. Previous research, conducted in static geometry environments, has already shown that many children, even in the middle school grades, rely on static, visual prototypes when identifying geometric shapes and that formal definitions, even if known, play no role in this process. Our study aimed at testing the conjecture that DGEs, in which the shapes can be continuously transformed, may flex the routine of identification, allowing for greater diversity in the shapes recognized as deserving a given name (e.g. triangle). This, we believed, would be an important step toward the discursively mediated routine of identification. The study, conducted among 4-5 year-old children working with Sketchpad, furnished some supporting evidence. In this paper, the focus is on one 30-min lesson during which the children observed, described, created and transformed triangles of different sizes, proportions, and orientations. During this one meeting the children's thinking evolved, in that the diversity of three-sided polygons they were prepared to call 'triangle' grew substantially. Not surprisingly, however, this rapidly-induced change was local and object-level rather than meta-level: it changed the children's use of a specific word rather than causing a transition to a discourse-mediated routine of identification.
A constructivist instructional project on developing geometric problem-solving abilities using pattern blocks and tangrams with young children
- C Sales
Sales, C. (1994). A constructivist instructional project on developing geometric problem-solving abilities using pattern blocks and tangrams with young children. [Unpublished master's thesis]. University of Northern Iowa.
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