Liu Hui's Cube Puzzle has been successfully used with more than 12 classes of prospective elementary teachers in the teacher education program at Southern Illinois University Carbondale as well as several groups of inservice teachers at in-school professional development workshops in southern Illinois, from fall 2013 to spring 2016. Initially, paper folding was used as the primary pedagogical tool. GeoGebra modeling and 3-D design and printing were later incorporated when the technologies became accessible to the targeted audience. In teaching practice, either one or all of these activities can be used, depending on student need and the accessibility of technologies. The hands-on nature of these activities, their mathematical and aesthetic appeal, and their pedagogical flexibility resonated well with both prospective and classroom teachers. In spring 2016, a group of 18 prospective elementary teachers responded to a short questionnaire on their feelings about the instructional approach. The results are presented in Table 1, where one can see that almost all the prospective teachers reported a positive experience with the cube puzzle sequence, including a field trip to the university 3-D printing service.
Question |
Frequency |
The cube puzzle has worthwhile math ideas. | 18 |
The cube puzzle provides rich learning experiences. | 18 |
The paper-cutting activity is useful. | 18 |
The cube puzzle allows meaningful group work. | 15 |
The cube puzzle has an artistic appeal. | 17 |
I will consider using the cube puzzle with K-8 students. | 17 |
The cube puzzle is just confusing. | 1 |
There are many other questions I wanted to explore using the cube puzzle. | 3 |
3D modeling or printing is a useful teaching tool for teachers to know. | 17 |
The cube puzzle provides diverse learning experiences. | 17 |
I am interested in finding more information about similar puzzle activities for my future teaching. | 16 |
The cube puzzle offers useful problem solving opportunities. | 16 |
I learned useful things during the 3D trip to the library. | 15 |
Field trips are a waste of our time for learning. | 0 |
Table 1. Prospective teachers' perception of the cube puzzle activities (n = 18, AG = Agree, SAG = Strongly Agree).
Thirteen of the 18 prospective teachers also added their own comments about the cube puzzle and their field trip to the university library 3-D printing service. They were all very excited to learn about the process of 3-D printing and its intrinsic connections to mathematics. About the whole cube puzzle experience, their comments are essentially consistent with the survey statistics. One prospective teacher wrote:
I thought the cube puzzle was fun and a great way to get my brain cranking and to develop ideas on how to form the cube. It was not very difficult for me but I could see it being a challenge for children and a great learning experience when learning about 3D objects and geometry. I also loved the 3D printer field trip to the library. I had no idea we even could use the printer or how to use it. It will be super useful in the future as I continue to work with young children. I am very happy I had that experience (A prospective teacher, February 18, 2016).
Another prospective teacher expressed a similar view and further addressed the pedagogical benefits of the cube puzzle:
I have never seen a 3D printer or even thought that it would be available to me as a teacher, but now that I know I can think of many useful ways to incorporate it that will engage students. I also loved the cube puzzle because it was unlike any math I have done before. Not only was I problem solving, but I also got to be hands on which was very engaging. When multiple people worked together it also brought new ideas and really encouraged collaboration. This is something I will definitely use in my future classroom (A prospective teacher, February 16, 2016).
One of the prospective teachers, Rita (a pseudonym), was doing her practicum at a local elementary school in spring 2016. She decided to take the cube lesson to a first grade class and explore children's responses. She won the support of her cooperating teacher and eventually invited the university librarian in charge of 3-D printing to demonstrate 3-D printing to the children. After her lesson, Rita made a presentation to her college class about the children's work and her reflection on it. In her summary, Rita wrote about the children,
The students were very engaged from the start. They never lost focus and I never had any behavioral issues. Also some of the more advanced students were able to help the less advanced students.
About her cooperating teacher, Rita said,
She thought it was a great lesson and that it went very well. She was surprised to see every student so involved and engaged.
Reflecting on her own role as a prospective teacher, Rita reflected,
[I]n our practicums we have the choice to teach what we want. I do not think many people choose math lessons unless it is their area of interest. I was pretty insecure about my math abilities, but this definitely helped boost my confidence and teaching abilities.
In summary, there is consistent evidence that the cube puzzle and related activities provided an inviting experience for the prospective teachers to think deeply about mathematics, mathematic teaching, and their emerging ideas of technology use and student engagement. The cube puzzle served multiple purposes for the instructor to reach out to these novice teaching professionals, many of whom, unfortunately, did not experience school mathematics as relevant, meaningful, and beautiful.