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Teaching Prospective Teachers about Mathematics and Culture Page 1
Teaching Prospective Teachers about Mathematics and Culture:
An Example from a Teacher Education Program in Alaska
Anthony Rickard
Professor of Mathematics Education
University of Alaska Fairbanks
Teaching Prospective Teachers about Mathematics and Culture Page 2
Abstract
This paper examines standards for addressing culture in Alaska’s K-12 schools and the aligned mission of the School of Education at the University of Alaska Fairbanks to prepare culturally responsive, effective practitioners. Unpacking how mathematics and culture may be taught together, prior research is used to show how such an approach can serve the needs of Alaska Native students and how a supplementary K-7 mathematics curriculum can support teaching mathematics in the context of Alaska Native culture. An example and analysis is then provided of how prospective undergraduate K-8 teachers at UAF use the supplementary curriculum in a mathematics course to learn about teaching standard and nonstandard units using traditional Yup’ik body measures. Conclusions and implications for learning about teaching mathematics and culture are discussed.
Teaching Prospective Teachers about Mathematics and Culture Page 3
Introduction
The Alaska Standards for Culturally Responsive Schools were developed by the Alaska Native Knowledge Network (ANKN), have been adopted by the Assembly of Alaska Native Educators, and are endorsed by the Alaska State Board of Education and Early Development (ASBEED) (see ANKN, 1998). Included in the cultural standards are five major standards for Alaska educators that provide guidance on being culturally responsive and explain multiple facets of the role of a culturally-responsive educator:
• Culturally-responsive educators incorporate local ways of knowing and teaching in their work
• Culturally-responsive educators use the local environment and community resources on a regular basis to link what they are teaching to the everyday lives of students
• Culturally-responsive educators participate in community events and activities in an appropriate and supportive way
• Culturally-responsive educators work closely with parents to achieve a high level of complementary educational expectations between home and school
• Culturally-responsive educators recognize the full educational potential of each student and provide the challenges necessary for them to achieve that potential (see ANKN, 1998, pp. 9-12).
The above standards describe how culture, including that of Alaska Native students, should play an important role in Alaska’s schools. The University of Alaska Fairbanks (UAF) School of Education (SOE) is committed to supporting prospective teachers in developing the skills and dispositions needed to address both academic and cultural standards in classrooms. The UAF SOE mission is to prepare teachers who are culturally responsive, effective practitioners with particular commitment to Alaska Native K-12 students (SOE, 2009). Fulfillment of this mission, in the context of teaching and learning K-8 mathematics, requires the UAF SOE elementary education program to address the cultural standards above and K-8 mathematics standards for Alaska (ABEED, 2012).
Combining mathematics and Alaska Native culture in K-8 classrooms is supported from both mathematical and cultural perspectives. Regarding teaching and learning mathematics from a cultural perspective, Sleeter (1997) notes, “a rich body of theory and research documents how effective education, framed within high expectations for learning, builds on the cultural, experiential, and linguistic resources that children bring with them to the classroom” (p. 680). Moreover, Alaska Native elders and leaders advocate for school reforms that can increase achievement and success of Alaska Native students in all subjects, including mathematics, so that future generations are effective problem solvers and critical thinkers (e.g., Rickard, 2005). In other words, looking at mathematics through a cultural lens highlights that mathematics is a human endeavor that can be learned through cultural skills and practices. Similarly, viewing culture through a
Teaching Prospective Teachers about Mathematics and Culture Page 4
mathematics lens can help to unpack the mathematics embedded in cultural knowledge. For instance, consider a pattern for a ceremonial Yup’ik headband1:
Understanding the geometry (e.g., shapes, symmetry) of the above border pattern can help students, both Alaska Native and non-Native, learn important mathematics (e.g., what does symmetry mean?). Learning to make border patterns will connect students with an authentic Yup’ik cultural activity and with the mathematics in a concrete way (e.g., what are the properties of a triangle, what makes a geometric pattern symmetric?). Students who understand mathematics as something that is a part of their own culture and heritage are more likely to be successful in learning mathematics (Barnhardt, 2007; Delpit, 1995; Legaspi & Rickard, 2005; Sleeter, 1997).
As a faculty member in the UAF SOE who is a mathematics educator, I focus on helping students in the K-8 elementary education program learn mathematics and learn about teaching mathematics in ways that address the K-8 mathematics standards adopted by the Alaska Board of Education and Early Development (ABEED, 2012). Consistent with the Alaska Native Knowledge Network cultural standards (ANKN, 1998), which have inspired the SOE mission, my colleagues and I also seek to prepare our students to be culturally responsive teachers. At about 16% of the overall population of Alaska, and about 20% of the K-12 student population, Alaska Natives are clearly among the largest stakeholders in Alaska schools (Kisker, Lipka, Adams, Rickard, Andrew-Ihrke, Yanez, & Millard, 2012). It follows that culturally responsive, effective teachers in Alaska’s schools must address both the specific needs and strengths of Alaska Native students as well as teach academic subjects (e.g., mathematics) as part of meeting the learning needs of all students. This paper provides one example of how the mathematics component2 of the UAF SOE undergraduate program in elementary education prepares prospective K-8 teachers to teach mathematics as a culturally-responsive educator.
1 This pattern is often referred to as “pretend mountains” – see Watt, Lipka, Parker-‐Webster, Yanez, Andrew-‐Irke, and Adam (2006) for more about Yup’ik border patterns and teaching about shapes and patterns for grades 3-‐5. 2 The mathematics component of the UAF SOE undergraduate program in elementary education consists of two content courses (Mathematics for Elementary Teachers I and II) and a K-‐8 mathematics methods course focusing on planning, teaching strategies, assessment, and use of manipulatives in the mathematics classroom.
Teaching Prospective Teachers about Mathematics and Culture Page 5
Building Better Mathematics for Alaska’s Schools
Alaska Native Students and K-12 Schools
Alaska Native students have generally been underserved by the traditional U.S. model of schooling, historically scoring below the average of all Alaska students on standardized mathematics assessments (Barnhardt, 2001; Rickard, 2005). Mounting evidence suggests that a lack of connection to the communities and culture of Alaska Native students contributes to the persistent gap in mathematics achievement by making learning in K-12 schools, and learning mathematics in particular, less accessible (Barnhardt, 2007; Kisker et al., 2012). These findings are consistent with research that shows that when the culture of schools and teaching is disconnected from the culture of students, those students face additional obstacles to their learning (Delpit, 1995).
In light of these results, education reformers in Alaska have argued that bringing the culture of schools and schooling in Alaska into closer alignment with Alaska Native culture has the potential to improve teaching and learning for all students in Alaska, particularly Alaska Native students (e.g., ANKN, 1998; Barnhardt, 2007). Such reform efforts have included integrating indigenous knowledge into the K-12 curriculum and implementing inquiry-based teaching strategies that more closely resemble traditional forms of teaching used by Alaska Native elders (Lipka, Jones, Gilsdorf, Remick, & Rickard, 2010; Watt et al., 2006). Moreover, addressing Alaska Native culture in K-12 classrooms does not undermine academic subjects or non-Native students; evidence shows that all students benefit from inquiry-based instruction that provides rich opportunities for learning within diverse cultural contexts (Adams & Lipka, 2003).
Math in a Cultural Context
Math in a Cultural Context: Lessons Learned from Yup’ik Eskimo Elders (MCC)3 is a series of K-7 mathematics modules that are intended to supplement a complete K-8 mathematics curriculum (see Lipka et al., 2010). MCC is the result of collaboration between Yup’ik elders and teachers, mathematicians, mathematics educators and other university-based researchers, and rural and urban teachers in Alaska. This collaboration of diverse stakeholders in mathematics education began in the late 1980s, intensified through the 1990s, and continues today to fulfill a vision of the elders – the inclusion of Yup’ik culture in the school curriculum so that the next generation would learn about traditional practices and knowledge. Development of a supplementary K-7 mathematics curriculum became the vehicle through which to implement this vision. A related goal of MCC is to address the persistent gap in mathematics achievement between Alaska Native students and, more generally, linguistically and ethnically diverse students, and middle and upper-middle class Caucasian students. Thus, MCC seeks to address two key issues
3 Support and funding for MCC has been provided by the University of Alaska Fairbanks School of Education, the National Science Foundation, and the U.S. Department of Education. MCC is an ongoing project directed by Dr. Jerry Lipka, Professor of Education at UAF. Visit www.uaf.edu/mcc to learn more about MCC.
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– aligning the culture of the school more closely to the culture of the community, and improving students’ mathematics performance, particularly Alaska Native students.
MCC modules also seek to implement the vision of the National Council of Teachers of Mathematics (NCTM) for school mathematics as described in Principles and Standards for School Mathematics (c.f., Adams & Lipka, 2003; NCTM, 2000) and the Alaska mathematics standards (ABEED, 2012). As “standards-based” mathematics curricula, MCC modules are problem-centered and engage K-7 students and teachers with mathematically rich experiences to build conceptual and procedural understanding of important mathematical ideas (Rickard, 2005). In addition, MCC embeds mathematics within the context of Alaska and the cultural knowledge and practices of the Yup’ik Eskimo people of southwest Alaska. For example, in one of the sixth-grade MCC modules, students develop their understandings of perimeter and area concepts by posing, exploring, and refining conjectures about area and perimeter of rectangles as they design a fish rack. Yup’ik fish racks, which have a rectangular base, are part of traditional Yup’ik subsistence and are used for drying and preparing salmon (Adams & Lipka, 2003). MCC modules, therefore, are vehicles for implementing the NCTM Standards and the Alaska mathematics standards and for addressing multicultural education in a way that connects learning traditional (i.e. Western) mathematics with learning about Alaska and Yup’ik culture.
Teaching about Culture and Mathematics: Measuring Kayak Outlines
I am familiar with the MCC modules4 and I use them in my mathematics and mathematics methods courses at UAF. I believe the modules serve the dual purposes of helping my students, who are all prospective K-8 teachers, learn about teaching mathematics in the context of Alaska Native culture and develop a deeper understanding of mathematics and its applications. Both of these intents are consistent with the mission of the UAF SOE to prepare culturally responsive, effective practitioners. Moreover, there is substantial research that standards-based mathematics curriculum materials like MCC can be used both as vehicles for reform in the K-8 classroom and for teacher learning and development (e.g., Kisker et al., 2012; Remillard, 2005; Remillard & Bryans; 2004; Rickard, 2010). In terms of standards used to guide teaching in Alaska’s K-12 schools, this approach addresses both the cultural standards (ANKN, 1998) and the mathematics standards (ABEED, 2012).
One of the MCC modules that I use in my courses is Kayak Design: Scientific Method and Statistical Analysis (Lipka et al., 2010). The module was developed for use in grades 5-7 and I use it with my undergraduate students to help them learn about teaching mathematics and about Alaska Native culture. Kayak Design helps grades 5-7 students learn about how to design an experiment, collect data by conducting trials of the experiment, and then use concepts and procedures from statistics to analyze and interpret
4 During 2004-‐2009 I worked as a collaborator with MCC, conducting research and professional development for teachers using the MCC modules in their classrooms. I am also a coauthor of several of the MCC modules.
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the data. Regarding the mathematics of the Kayak Design module, the authors explain that:
students are able to investigate the relationship between the kayak’s shape and its function. By investigating these relationships, students are guided into a series of purposeful mathematical investigations, using the scientific method – controlling variables in a systematic way. They collect, organize, and analyze data by developing tables, converting tables into graphs (line, bar, and scatter plots) interpreting graphs, and using basic statistical techniques (mean, median, mode) to determine if there is a relationship between form and function (p. 1).
Explaining how the Kayak Design module addresses Alaska Native culture, the authors note that students:
will learn about Yup’ik Eskimo culture of southwest Alaska through stories and myths as told by Yup’ik elders… . This module attempts to convey some of the Yup’ik people’s ingenuity, creativity, and wisdom as demonstrated through their design and construction of the kayak… . In this module, we will analyze three Yup’ik watercraft designs. Students will build and test model boats based on the three traditional designs and collect data about their speed, load capacity, and stability5 (p. 1).
In my UAF courses with prospective K-8 teachers, we unpack both the mathematics and cultural connections of the Kayak Design module, including that the module addresses the statistics and probability domain of the Alaska mathematics standards and the Alaska cultural standards as well (cf., ABEED, 2012; ANKN, 1998). We also work through activities in the Kayak Design and other MCC modules6 to model how mathematics may be taught in the context of Alaska Native culture.
Yup’ik Body Measures
One activity from the Kayak Design module that my students and I work through together uses traditional Yup’ik body measures to construct an outline of our own single-person kayak. Like many indigenous people, the Yup’ik people of southwest Alaska have traditionally used body measures for determining lengths – e.g., arm lengths, arm spans, height (see Kisker et al., 2012; Lipka et al., 2010). As the Kayak Design authors
5 The three traditional Yup’ik kayak designs are v-‐shaped bottom, flat bottom, and round bottom. By building and testing model kayaks for each design, students will find, for example, that, compared to the other two types of kayaks, the v-‐shaped bottom kayak is fast, has moderate load capacity, but is the least stable of the three. These characteristics of the v-‐shaped kayak form are consistent with its function, which is for use by skilled kayakers for hunting, who need speed over stability. 6 There are currently 10 modules in the MCC series for grades K-‐7 (each intended to comprise 2-‐3 weeks of classroom instruction) as well as supporting materials. See www.uaf.edu/mcc for more information about MCC modules.
Teaching Prospective Teachers about Mathematics and Culture Page 8
explain, the use of traditional Yup’ik body measures to determine the dimensions of a single-person kayak makes sense so that the kayak is custom fit for the individual person who will use it:
Traditional Yup’ik kayaks are made for a person using her or his own body measures. This ensures that the kayak will “fit” that person and that he or she can effectively pilot and use it. If the kayak is too small for a person, they won’t be able to comfortably fit into it, and if it is too big they won’t be able to pilot it well (Lipka et al., 2010, p. 39).
In making kayak outlines, the Kayak Design module addresses Alaska Native culture by engaging students in an authentic Yup’ik activity and also addresses the mathematics of contrasting standard and nonstandard units, as specified in the Alaska mathematics standards7 (see ABEED, 2012).
Kayak Outlines
The Kayak Design module shows how traditional Yup’ik body measures are used to make the kayak outline (Lipka et al., 2010, p. 40):
7 Standard and nonstandard units are included in the measurement and data content domain of the K-‐8 Alaska mathematics standards (see ABEED, 2012).
Teaching Prospective Teachers about Mathematics and Culture Page 9
With my students, we use the above reference chart to each create our own kayak outlines. With a partner, we measure the kayak dimensions, using our body measures, to construct our kayak outline with masking tape on the classroom floor. Here is an example of a typical kayak outline made by one of my students in my MATH 206: Mathematics for Elementary Teachers II class:
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We then compare the kayak outlines and note how, by using Yup’ik body measures, each of our kayaks is proportional to our size (e.g., shorter students will have kayaks that are not as long as the taller students’ kayaks). We also discuss how to use this activity to help our grades 5-7 students compare their kayak outlines to understand that a strength of
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Yup’ik body measures as nonstandard units is that they allow the dimensions of a “custom-fit” kayak to be easily determined.
After making the kayak outlines, we discuss how we can help our grades 5-7 students connect the (nonstandard) Yup’ik body measure units to standard units by using a measuring tape to find the dimensions of the kayaks (e.g., using inches). The Kayak Design module provides an activity sheet to help grades 5-7 students organize their work, and my students complete this also – students measure their own Yup’ik body measures and kayak dimensions in standard units (inches) (Lipka et al., 2010, p. 54):
Teaching Prospective Teachers about Mathematics and Culture Page 12
Connecting the nonstandard Yup’ik body measure units with standard units helps students understand the different uses for both. For example, everyone’s kayak, at its widest point, has width Ikuyegarnerek Malruk (i.e., a nonstandard unit, see figure above) because all kayaks use this Yup’ik body measure to determine the width of the kayak hatch. However, the kayak hatches are of different sizes because people are different sizes, and this characteristic can be measured and compared using inches (i.e., a standard
Teaching Prospective Teachers about Mathematics and Culture Page 13
unit). In this way, nonstandard units are understood as having an important role in Yup’ik culture and have the advantage of allowing for efficient measurement of single-person custom-fit kayaks. Standard units are also understood as allowing for direct comparison of measurements using a uniform and consistent unit that is absolute and does not change with the person doing the measuring.
Conclusion
Standards for K-12 teaching in Alaska include guidelines for addressing culture as well as academic content in classrooms. Cultural standards for Alaska’s schools note that students’ communities and culture should be factored into instruction, and Alaska standards for teaching mathematics content specify parameters for what mathematics to teach at each grade level K-12 (cf., ABEED, 2012; ANKN, 1998; Barnhardt, 2007). Part of the mission of the UAF SOE is to prepare K-12 teachers for Alaska’s schools who are culturally-responsive, effective practitioners and, as such, are able to address both cultural and academic standards in teaching Alaska’s students. Prior research has established that teaching about mathematics and culture together in the classroom makes sense from both mathematical and cultural perspectives. For example, cultural practices can show how mathematics is used by real people and provide meaning to abstract concepts, and studying the mathematics embedded in cultural contexts can yield deeper insights into cultural practices (cf., Adams & Lipka, 2003; Delpit, 1995; NCTM, 2000; Rickard, 2010; Sleeter, 1997).
To prepare prospective teachers to address mathematics and culture in their K-8 classrooms, I utilize MCC modules as a way to acquaint prospective teachers with potential resources for their future classrooms and, in my classroom, as vehicles for learning about teaching mathematics. Working with innovative mathematics curricula has been shown to help teachers and prospective teachers learn about teaching mathematics and about how to implement the reforms with which the curricula are aligned (Legaspi & Rickard, 2005; Remillard, 2005; Remillard & Bryans, 2004). Specifically, research on teachers’ use of MCC modules has shown their potential for supporting teachers in learning about teaching K-7 mathematics, Alaska Native culture, and increasing the mathematics achievement of all students, particularly Alaska Native students (e.g., Kisker et al., 2012; Rickard, 2005). All this evidence underscores that using MCC modules in a K-8 teacher preparation program as one tool to help prospective teachers learn about how to address cultural and mathematics standards in the classroom is a reasonable approach.
The example provided in this paper illustrates one activity from one MCC module that is used in a mathematics course for elementary teachers at UAF. Using traditional Yup’ik body measures, prospective elementary teachers measure and construct outlines of their own kayaks on the classroom floor using masking tape. Each student then records the dimensions of her or his kayak and the corresponding body measures using inches. In this way, the dimensions of the kayak are measured using both traditional Yup’ik body measures (nonstandard units) and inches (standard units). For the prospective elementary teachers, this activity provides an experience showing how aspects of both the Alaska cultural and mathematics standards may be addressed together in the classroom, enhances their own learning about mathematics and culture, and helps
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them to formulate ideas for their own practice. The mathematics of nonstandard and standard units is given concrete meaning through the kayak outline activity. Moreover, the rich cultural practice of traditional Yup’ik kayak construction in southwest Alaska helps connect K-8 students with the mathematics, and the prospective teachers have participated in an experience that can help them frame teaching their own students in their own classrooms.
The kayak outline example of teaching prospective teachers about mathematics and culture is intended to be illustrative, not prescriptive. The very nature of multicultural education – i.e., drawing on specific and diverse characteristics of students’ culture and their communities in teaching academic content – resists prescriptions for practice. However, helping prospective teachers develop professional dispositions of practice to learn how to address the cultures and communities of their students through teaching academic subjects in a cultural context makes sense. Participating in an activity that seeks to accomplish this, and learning about and from curriculum materials that can provide support for their own teaching practice, enables prospective teachers to see the viewpoints of both students and teacher. The kayak outline activity, and similar experiences with other MCC modules, illustrates one strategy for preparing prospective K-8 teachers to teach good mathematics in meaningful ways for Alaska’s students.
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