Maker Math: Exploring Mathematics through Digitally Fabricated Tools with K–12 In-Service Teachers
Abstract
:1. Introduction
2. Literature Review
2.1. Making and Digital Fabrication in K–12 Education
2.2. The Role of Technology in Teaching and Learning Mathematics
2.3. The TPACK Framework
- CK: Knowledge about the subject matter;
- PK: Knowledge about the methods and processes of teaching;
- TK: Knowledge about various technologies that can be applied to education;
- PCK: Knowledge of the pedagogical approaches appropriate for teaching a given content;
- TCK: Knowledge of how technology can create new representations for specific content;
- TPK: Knowledge of how various technologies can be used in teaching and understanding that using technology may change the way teachers teach;
- TPACK: Knowledge required by teachers for integrating technology into their teaching in any content area;
- XK: Knowledge required by teachers about the local and far-reaching affordances and constraints of teaching with technology.
2.4. The Impact of PD on Mathematics Teachers’ TPACK
“Strategic use of technology in the teaching and learning of mathematics is the use of digital and physical tools by students and teachers in thoughtfully designed ways and at carefully determined times so that the capabilities of the technology enhance how students and educators learn, experience, communicate, and do mathematics. Technology must be used in this way in all classrooms to support all students’ learning of mathematical concepts and procedures, including those that students eventually employ without the aid of technology. Strategic uses support effective teaching practices and are consistent with research in teaching and learning.”[65] (p. 1)
- In what ways do digital fabrication tools support teachers’ perceptions of mathematics teaching, learning, and curriculum?
- What challenges do in-service teachers face when applying digital fabrication tools to mathematics in terms of teaching, learning, and curriculum?
- In what ways do the workshops influence teachers’ conceptualization and practices of broadening participation in learning mathematics?
3. Methods
3.1. Research Context and Workshop Design
3.2. Participants
3.3. Data Collection
3.4. Data Analysis
4. Results
4.1. RQ1: In What Ways Do Digital Fabrication Tools Support Teachers’ Perceptions of Mathematics Teaching, Learning, and Curriculum?
“You can use slope and quadratics in the real world, but simplifying a rational expression is hard to explain to kids since you don’t know where you will need it in the real world. But things like statistics you are going to use in the real world. Sometimes it’s applicable and sometimes it’s not worth talking about. Just here is the standard and let’s move on.”
4.2. RQ2: What Challenges Do In-Service Teachers Face When Applying Digital Fabrication Tools to Mathematics in Terms of Teaching, Learning, and Curriculum?
4.3. RQ3: In What Ways Do the Workshops Influence Teachers’ Conceptualization and Practices of Broadening Participation in Learning Mathematics?
4.3.1. Visualizing Mathematics at the Intersection of Nature and Art
4.3.2. Broadening Participation in Mathematics
4.3.3. How Teachers Shared and Used the Tools
4.3.4. Recommendations for Future Workshops
5. Discussion
5.1. Misalignment between Standards and Broadening the Purposes of Learning Mathematics
5.2. Professional Development for Math Teachers to Integrate Technology
5.3. Added Value of the Present Work
5.4. Implications for Future Research and Practice
6. Limitations
7. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
Workshops | Activities | Examples |
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Workshop 1 |
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Workshop 1 (cont.) |
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Workshop 2 |
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Workshop 2 (cont.) |
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Workshop 3 |
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Workshop 3 (cont.) |
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Appendix B
- 1.
- What are some affordances and constraints of the laser-cut math tools in your perspective?
- 2.
- What are some of your ideas about how these laser-cut math tools and activities could be implemented in your classrooms?
- 3.
- What challenges do you have if you want to implement these laser-cut math tools in your math curricula? What viable solutions do you see for these challenges?
- 4.
- What connections between nature and mathematics came up for you during the workshops? Between the workshops?
- 5.
- What are some of your suggestions and recommendations for future workshops? How can we improve and better serve your needs?
Appendix C
Maker Tools | Students | Colleagues | Family | Non-Education Friends |
Decahedron tiles | ||||
Laser-cut rectangular puzzle | ||||
Voronoi flipbook | ||||
Voronoi stained glass | ||||
Fibonacci transparent plexiglass | ||||
Golden ratio calipers | ||||
Origami chompers | ||||
Icosahedron (rubber band shape) | ||||
String art | ||||
Nautilus gears | ||||
Mercader projections | ||||
Objects of constant width (the spheroids) |
Appendix D
Activity | Technological Knowledge | Pedagogical Knowledge | Content Knowledge | Contextual Knowledge | Aesthetics | Participant Examples | Misconceptions | Additional Notes |
---|---|---|---|---|---|---|---|---|
Decahedron (triangular pieces) | 3 mm plywood cutting speed/power, how to color plywood with water-based bingo dabbers. | Discovery learning/free play. | Teachers mention their observation of the triangular pieces are similar to Tangrams. | The Instructor offered the opportunity to get the instructional sheet that has five designs on it for the activity. Four teachers decided to take the sheet and explored the patterns, and the other four teachers decided to continue to try themselves. | After successfully coming up with new patterns, teachers immediately engage in conversations about how the shapes look like in the patterns and how colors play into the design. “Does that shape remind you of anything?” Teachers also want to make the patterns visually pleasing by coordinating the colored pieces. | Stop sign, decagon, paper airplane, star. | Connection to Pythagorean theorem (Triangular pieces have no right-angle, 36-36-108 degrees). | Participants were asked to use all 20 pieces to make a “round” shape after a period of free play. Participants were informed to not use the hexagonal piece from their baggie of pieces. |
The instructor asked the teachers the degrees of the triangles and asked them to think about their degrees. “How many sides are there in the pattern?” “10 sides, 1440-degrees (sum of the interior angles, (n-2)*180. (NOTE: Participants identified the interior with 36*10 = 360 degrees). | Instructor asked the kindergarten teacher what she thinks about this activity for kindergarteners, participants thought the activity was good for all ages. | Teachers commented that they began playing by grouping the pieces based on colors, and then, it came together into the full pattern. Most teachers agreed that colored pieces are more appealing than plain wood pieces, and the colors help with the play and design. | NOTE: Not mentioned during workshop, this is the Phi triangle, with phi as the short legs, and hypotenuse of phi + 1. Also called the Divine Triangle. See video: https://www.youtube.com/watch?v=z4hCcI_Ates(accessed on 19 August 2022) | |||||
Rectangular Puzzle | Activity packs were distributed to teachers, and teachers peeled off the stickers (blue painter’s tape) on the pieces. Teachers mentioned they smelled like a campfire. | Challenge-based learning. | Elementary teachers, similar to manipulatives. Middle school teachers, surface areas, and columns. High school teacher, very different from high school math teaching, rational expression, logs, imaging numbers. Everyone agreed that we were doing geometry. | Challenge: Find a rectangle with 5, 6, 7, and 8 pieces. | Some people liked the colors; one said color did not make any difference, and some preferred plain wood color. | (Real-world example) Similar to buildings and lands, this can be used by people doing architecture and lands. | 6 vs. 9 for upside-down piece of puzzle due to symmetrical piece. | “The challenge is to try to figure out how to make large rectangles and how to make the rectangles with 5, 6, 7, and 8 pieces”. One group figured it out immediately and said, “Let’s just do the eight pieces really quick”. Teachers celebrated once they completed a rectangle. Everyone was very engaged, and there was a lot of laughter. |
One teacher asked about the 3D shapes. Whether we can print the 3D shapes for these activities? The instructor showcased the 3D printed Pi vase to teachers, 28 h of printing. | How this activity will be easier or harder if we don’t have the numbers on it?” One teacher mentioned that it might be easier. | One teacher talked about city planning, figuring out the roads, parallels, different building shapes, and so on. | The instructor showcased the book, Earnest Irving Freese’s Geometric Transformations: The Man, the Manuscript, the Magnificent Dissections! written by Greg N. Frederickson and talked about 15 ways to do pentagons and talked about tiles and quilting. |
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Teachers | Gender | Age Range | Ethnicity | Years of Teaching | Grade Level | Content Area(s) | Certification(s) |
---|---|---|---|---|---|---|---|
Scarlett | Female | 31–40 | Other | 7 | K–2 | All | Early Childhood Education |
Mia | Female | 51–60 | White | 10 | 2nd | All | PK–5th |
Ava | Female | 31–40 | White | 8 | 3rd | All | PK–5th |
Riley | Female | 31–40 | African American | 8 | 6th–8th | Science | Math and Science (4–8) |
Lucas | Male | 41–50 | African American | 8 | 6th–8th | Special Education, STEM | Special Education, Social Sciences |
James | Male | 41–50 | White | 5 | 7th | Math | Math and Social Studies |
William | Male | 21–30 | Asian | 3 | 8th | Math, Science | Math (6–12), Science (6–12), Engineering and Technology, ESOL, Gifted |
Chloe | Female | 41–50 | Asian | 14 | 9th | Math (Geometry) | Secondary Math |
Olivia | Female | 31–40 | Hispanic/Latina | 9 | 11th | Math | K–5 Elementary, 6–12 Math |
Research Questions | Data Sources Used | Data Triangulation |
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Laser-Cut Math Tool | Students | Colleagues | Family | Non-Ed Friends | ||||
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n | % | n | % | n | % | n | % | |
Decahedron Tiles | 4 | 44 | 4 | 44 | 5 | 56 | 2 | 22 |
Rectangular Puzzle | 4 | 44 | 5 | 56 | 5 | 56 | 2 | 22 |
Voronoi Flipbook | 1 | 11 | 3 | 33 | 3 | 33 | 0 | 0 |
Voronoi Stained Glass | 1 | 11 | 3 | 33 | 6 | 67 | 3 | 33 |
Fibonacci Transparent Plexiglass | 0 | 0 | 5 | 56 | 5 | 56 | 2 | 22 |
Golden Ratio Calipers | 2 | 22 | 6 | 67 | 5 | 56 | 3 | 33 |
Origami Chompers | 0 | 0 | 2 | 22 | 4 | 44 | 2 | 22 |
Icosahedron | 1 | 11 | 2 | 22 | 4 | 44 | 0 | 0 |
String Art | 2 | 22 | 2 | 22 | 4 | 44 | 3 | 33 |
Conic Sections | 3 | 33 | 5 | 56 | 3 | 33 | 3 | 33 |
Nautilus Gears | 3 | 33 | 3 | 33 | 3 | 33 | 1 | 11 |
Mercator Projections | 0 | 0 | 3 | 33 | 1 | 11 | 0 | 0 |
Objects of Constant Width | 1 | 11 | 2 | 22 | 2 | 22 | 0 | 0 |
Grand Total (n = 117) | 22 | 19 | 45 | 38 | 53 | 45 | 21 | 18 |
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Harron, J.R.; Jin, Y.; Hillen, A.; Mason, L.; Siegel, L. Maker Math: Exploring Mathematics through Digitally Fabricated Tools with K–12 In-Service Teachers. Mathematics 2022, 10, 3069. https://doi.org/10.3390/math10173069
Harron JR, Jin Y, Hillen A, Mason L, Siegel L. Maker Math: Exploring Mathematics through Digitally Fabricated Tools with K–12 In-Service Teachers. Mathematics. 2022; 10(17):3069. https://doi.org/10.3390/math10173069
Chicago/Turabian StyleHarron, Jason R., Yi Jin, Amy Hillen, Lindsey Mason, and Lauren Siegel. 2022. "Maker Math: Exploring Mathematics through Digitally Fabricated Tools with K–12 In-Service Teachers" Mathematics 10, no. 17: 3069. https://doi.org/10.3390/math10173069