As stated in the last reflection on the Folds & Fractals lesson plan, I recognized opportunities to strengthen the involvement of students with the use of appropriate technology. The affordance provided with carefully chosen tools “allows students to cross disciplinary boundaries and transfer ideas from one realm to another, deepening their insight into both domains” (Mishra & Koehler, 2009, p.18). This is one of the primary intents of this lesson, and TPACK can be useful in this by formalizing the relationships between content, pedagogical and technological knowledge within the particular contexts I work within as an outreach educator.
Recognizing those contexts allows for a greater understanding of the choices made in the development of the original lesson plan. We are often invited into art and science classrooms to present a period at time, the length of which varies between schools, but the lesson has to adapt to. We bring many of our own materials to each school, which must be setup in a reasonable amount of time and be easily transported. Each classroom setup and available of technology varies from school to school. We visit students that have been introduced to the concepts we cover already, but often many that have not, so there are very few assumptions we can make about student prior knowledge. We also reach many rural schools where poverty is prevalent. All of these factors play a significant role in determining the way we approach content, pedagogy, and technology in our lessons.
The lesson covers content that some teachers may consider to be extracurricular at first glance, but we work hard to make sure curricular connections are present with any program we present. The content of the lesson is drawn from two disciplines, mathematics and art. In math, we explicitly investigate repeating fractal patterns that draw on students’ knowledge of geometry, functions, variables, and operations, along with skills in measuring, estimating, and evaluating functions and apply those in the context of fractals, a topic that most students are not familiar with. On the art side, we investigate how paper, often a familiar material, can be used in original ways, and how folding patterns draw from geometric knowledge as well.
Both the subject matter and the constraints of the outreach format help us determine our pedagogy. In order to both draw and fold the Sierpinski Triangle pattern, direct instruction is used, since a well-defined algorithm is used in their construction and we can finish the steps quickly. It is during this time that students start to develop an understanding of what fractals are and how they are made without explicitly stating it, based on their repeated creation of fractals and noticing common features. If we had more time with students, I think there could be natural points where inquiry would be appropriate, such as once students have discovered what a fractal is and have created a couple examples, see if they could transfer that to the creation of their own fractal patterns.
Technology use is also constrained by the outreach format, as students will use inexpensive, easily transported rulers, pencils, scissors, and of course paper as their primary tools. I don’t know of any tools that will do the job better than those in this context, and TPACK is described as being appropriate for older technologies as well (Mishra, Koehler, & Henriksen, 2011). As an instructor, I make use of more recent technology to share ideas, such as my laptop, projector, and overhead camera. Since computers are so well-suited to the creation of fractals, we use a program called Xaos to demonstrate how a fractal can have infinite detail by zooming further and further into an example, but this is teacher driven as we only have the single laptop.
I believe the connection between content and pedagogy within this context is clear. The steps for creating and understanding the nature of a fractal would be difficult for students to discover on their own or even with guidance. But once that hurdle is cleared with direct instruction, the pedagogy options open up considerably. Allowing students to wander outside and document examples of natural fractals could result, as could taking knowledge about fractals in ethnomathematics and exploring it within their own culture. The technology used to support that pedagogy is also appropriate, as it allows for students to engage in the act of creation while learning, but these are also tools that learners from any district would be familiar with. We did in fact find that some middle school students struggled with using rulers correctly, so some further practice was appreciated by their teachers.
It is the relationship between content and technology that merits the most investigation. While the traditional tools we provide to students work well, there could be greater understandings gained if students had access to computers as well. When Benoit Mandelbrot was first investigating fractals, he had only crude representations to consider until computers reached the point where he was able to visualize these fractals in detail, which opened up a whole new world for him. We will have a set of laptops by this Fall, and I would like to revisit this program to see how having students explore a fractal of their own choosing in Xaos and the resulting discussion on what they observed could make the learning more student driven. I also see computers and spreadsheets in aiding with student understanding of how fractals are constructed, as the calculations needed are tedious, but well suited for a computer. As I continue to explore computer programming as a means of applying mathematical knowledge, having high school students program a visualization of a fractal might be a possible option.
In all of this, I would need to consider the constraint of varying comfort levels that students have with using computers and when it may not always be appropriate. I would also want to consider when using laptops may be more of a distraction rather than focusing learning, as I’ve heard some teachers who teach in a 1:1 student-to-laptop school complain. As Mishra and Koehler (2009) point out, any change to one element of TPACK requires modifications of the other two, so this would require constant reevaluation. I believe this need to be flexible and react at a moment’s notice to change is critical for success. In outreach, there is the need for a particular type of flexibility as circumstances and audiences can change dramatically, but I hope with further experience to be adaptable and begin realizing the many opportunities that technology affords.
Mishra, P. & Koehler. M. J. (2009). Too cool for school? No way! Using the TPACK framework: You can have your hot tools and teach with them, too. Learning & Leading with Technology, 36(7), 14-18.
Mishra, P., Koehler, M.J., & Henriksen, D. (2011). The seven trans-disciplinary habits of mind: Extending the TPACK framework towards 21st Century learning. Educational Technology, 51(2) 22-28.