Taking Action is a form of curricular organization that I have worked with for the past six years as part of a collaboration between Arcadia University and Philadelphia public schools. This article describes this way of teaching and learning mathematics, analyzing it as a form of culturally responsive, and culturally specific, pedagogy (Averill et al., 2009; Leonard 2008). Future teachers are trained in the use of this form of organizing the teaching and learning of mathematics. They begin in a semester course and, later, lead groups of elementary school students (Grades 1–8, approximately ages 6–14) through six-week units of study.
Taking Action is part of a larger curriculum-design structure that promotes children taking responsibility for their own learning. In this structure, the teacher introduces a set of topics and potential investigations and leads the children through the processes of investigation design and implementation. During investigations, the teacher assesses student progress on specific mathematics objectives while holding individual and group conferences with students, teaching mini-lessons on specific skills and concepts, and facilitating dialogue on the progress of the mathematical investigations. After the investigations are well underway, the teacher helps students to develop “taking action” projects. After the action projects, the unit culminates in an archaeology period, during which students consolidate and extend the skills and concepts that have been learned.
Although a careful study of experiences like Taking Action might reveal important insights into teaching techniques, my own reflections point more to characteristics of the curriculum design. As students develop mathematical understandings, they often recognize the importance of acting on their beliefs and the connections they see to issues in their lives and communities. Even if they do not see these at first, the process of figuring how to make an impact and following through with an action often helps them to see that they have been doing something that matters. Mathematics is in this sense not just stuff that has to be done to get through it, but the stuff of meaning and action.
The most important element of what we have been doing,
| Opening | Investigations | Archaeology | ||
|---|---|---|---|---|
| Develop Projects | Do Projects | Take Action | ||
Figure 1: The five-part structure of Taking Action
I believe, is the focus on this action work. At first it seems like it is not even part of the mathematics curriculum; it is in some sense an add-on to the “real” content. Yet, for many students, the action phase turns out to be a place where the mathematical understanding is developed or extended. As Gutstein and Peterson (2005) point out, “once learners are engaged in a project, like finding the concentration of liquor stores in their neighbourhood and comparing it to the concentration of liquor stores in a different community, they recognize the necessity and value of understanding concepts of area, density, and ratio” (p. 4). Indeed, we have found that action work often prompts demands for advanced instruction in skills and concepts. While there are many reasons to pursue meaningful and engaging pedagogical techniques and practices, taking action might make it possible for almost any form of pedagogy to be highly effective.
The hardest part of innovation, envisioning what it might look like, is especially challenging for mathematics. It seems we have a narrower set of images for math classrooms than for other subjects. How can we create when we do not know what we are creating? This work highlights the idea that curriculum design structures might be an interesting research direction for mathematics education. Rather than social theory, cognitive psychology, subject sequencing, or other areas that have dominated mathematics education in the past, there may be some ways of organizing activity independent of content or pedagogy that can lead to effective mathematics teaching and learning. Taking Action is, in this sense, a case study. This reflective cycle of creative curriculum design followed by analysis of its characteristics based on field experiences, then followed by theoretical connections with other forms of pedagogy that share its characteristics, is an example of a type of research in mathematics education that may serve our work more richly.
The unique aspect of Taking Action is that, in the middle of their investigation work, students are required to reconsider what they have done and identify key aspects of their experience. Based on this reflection, students are helped to design a way of interacting with people outside of their class in order to make an impact on the world using the mathematics they have been studying. This “impact” can be one of two types: either they must identify a person or group of people upon whom they believe they can make a worthy impact, or they must identify a person or group of people who they believe can make a considerable impact on their own mathematical work. This is done either individually or in small groups, based on investigations that have been carried out so far within the unit.
For example, in a first grade group working on a number of investigations involving a zoo – sizes of habitats, amounts of food, costs per year, and so on – the class came together around the issue of transporting animals. They wondered, for example, how zoo architects design the sizes of doorways between rooms and cages. So they undertook a group investigation around whether a mature, female giraffe could fit through their classroom doorway. Their action project entailed eliciting the help of a docent at the city zoo. They sent the docent descriptions of their research along with questions about giraffes that could not be answered through library and Internet research, showing that they had “done their homework” and “knew their stuff.” Within one week, they received a response. In this case, students felt like they were on the right track with their mathematics, but they needed confirmation and further support. The action work was designed to enlist an outsider who could make an impact. Their attempts to understand and apply the information in the letter became the initial experience of their archaeology work that followed. The docent suggested several questions that the zookeepers were considering related to transport of animals, having to do with minimal space requirements in transit. The students were thrilled that their work was relevant and not just a silly school exercise.
Many students do not feel ready to move on to the next segment. In the midst of mathematical investigations, they may feel like they are only getting started. Nevertheless, most choose some aspect of what they have done as standing out. Either they have a result, or they have identified a way of working that has served them well as mathematicians or they have found a direction of research lacking in results, which in and of itself is an interesting result. For example, a fifth grade class working on a video to introduce to a new videogame based on their neighborhood did not think that their work with scale models of buildings was the basis for an action project. When they discussed what was critical, however, they kept coming back to disagreements about what standard units of measurement would be most worthwhile, both in their estimates of the heights of the buildings and in the creation of their set backgrounds.
Once students have identified key aspects of their work, they need to consider an audience – people upon whom they can have an impact or who can make an impact on them. These might be other children, younger or older, that they want to share their results with. They may be professionals that use the mathematics who could serve as consultants. Or they may be people in the larger community who would be well chosen for such “action.” The first graders chose an outside expert. In the case of the fifth graders, the disagreements over units of measure became the main theme of a puppet show they performed for second graders who were studying measurement. The “real audience” for their work makes it important that the students prepare well. For this reason, it is not always during the investigations that the actual mathematical learning takes place. The crafting of the letter to the zoo docent demanded strong understanding of the mathematics of a giraffe walking through doorways. The students could estimate an average height of a mature giraffe, the range the space required given a giraffe’s limitations for bending down and folding their legs to move, and so on. They could also calculate the typical size of a zoo doorway, and so on, but in order to elicit from the workers at the zoo the details they needed about what one could reasonably demand of a giraffe, they also needed to describe their work in a way that would not simply be “cute”; they wanted a professional response, and this required careful reflection on what they already knew. In the case of the fifth graders, the need to design an action forced them to reconsider their disagreements as meaningful mathematical content themselves. Rather than being opinions, the choices of measurement unit highlighted for them the notion of a standard unit, as well as the role of the unit in the accuracy of calculations.
Note that this process and the larger five-part structure do not depend on any specific mathematical content. They can be used at any grade level for any set of mathematical skills and concepts. Critical to the taking action phase of the work is that students look back over what they have done and figure out a way to share what is important to them. Their plans must be serious and they must prepare well in order to take advantage of the opportunity that this action project presents. Since the students choose the specific aspects of the mathematics that form the basis of their actions, and since they identify the best members of the outside community for their actions, they have a greater stake in the success of these actions than in a typical classroom activity. While the teacher can help with arrangements and logistics, we find that it is best if the students themselves contact the community members and make arrangements, as much as possible. This is the case even for first grade students. If first graders contact an architect or zoologist or mathematician, or visit a class of sixth graders to invite them to visit with their class, it is more likely to be received well and more likely to be taken seriously than if the teacher asks. If the invitation is articulated in a way that demonstrates serious prior involvement with the mathematics, it is even more likely to get a strong, positive response.
Students must rehearse and prepare materials for the action, anticipating how their action will make its impact. If they want their audience to learn something or to make life choices that matter, their presentation of their work must be done in an entertaining, understandable and compelling way. It cannot be routine or boring. Thus, as part of the design of their project, the students themselves must identify and actualize what is compelling about the mathematics that they have been using. The curriculum unfolds under the assumption that pupils are capable of doing this.
Students must also consider how to communicate the mathematics. If the audience is less sophisticated mathematically, they need to select appropriate representations for the ideas, using them in ways that effectively communicate important points. If the audience is more sophisticated, students want to communicate in ways that are based on the importance of what they have learned. If they want their audience to help, they must communicate what they have ...
... for complete article, please go to: For the Learning of Mathematics 29, 2 (July, 2009) FLM Publishing Association, Edmonton, Alberta, Canada