Using Inquiry Principles of Art to Explore Mathematical Practice Standards

Scholarly reports document the ongoing scrutiny of science, technology, engineering, and mathematics (STEM) education in the United States (Dautrich, 2009; Hill & Ball, 2005; International Association for the Evaluation of Educational Achievement, 2011). Reports of diminished returns in the field are reflected in global rankings where U.S. standings fall below that of other developed nations (U.S. Department of Education, 2008). These reports have prompted STEM educators to rethink how we prepare preservice teachers to be effective teachers of STEM. This reality coincides with the advent of the Common Core State Standards in Mathematics (CCSS-M)—a rigorous set of guidelines that hold the potential to create a paradigm shift in the field. Aligned to the CCSS-M, the National Governors Association Center for Best Practices (2010) has published the Standards for Mathematical Practice to establish a set of recommendations for exemplary mathematical performance. This body advocates for skills and proficiencies that include comprehension of mathematical concepts, operations and relations, as well as skills in carrying out procedures flexibly, accurately, efficiently and appropriately. Moreover, these practices aim to promote a view of mathematics that is “sensible, useful and worthwhile, coupled with a belief in diligence and one’s own efficacy” (National Governors Association Center for Best Practices, 2010, p. 6). Whereas the CCSS-M outlines “what” content to teach, the Standards for Mathematical Practice provide insight into “how” to teach that content. These new standards and practices create a set of challenges of their own—namely, educators at all levels are presented with the challenge of how to best implement them.

This challenge is exacerbated by current school funding structures at the middle-grades level which have led to cuts or elimination altogether ofthe arts. Faced with difficult budgetary decisions, many districts have opted to focus on core subject matter to the exclusion of the arts (Davis, 2006; Israel 2009). In an age of increased STEM accountability, the argument to limit arts education is predicated on the belief that mathematics and science instruction should take precedence over music, fine art, and creative composition. This reality is troublesome on multiple levels. First, the arts teach more than self-expression and creativity. Baker (2011), for instance, found middle school students who were enrolled in music courses performed significantly better on high-stakes state tests than students not enrolled in music education while other scholars have linked arts education to improved graduation rates (Israel, 2009). Second, the National Task Force for the Arts in Education (2009) has documented skills and dispositions the arts teach that transcend subject matter. This scholarship highlights the role the arts play in fostering increased communication skills and critical thinking abilities, and in developing a global perspective. These skills are strikingly similar to those called for in the Standards for Mathematical Practice. In contradiction to the notion that the arts are a mere addon to the middle school curriculum, we argue that STEM education generally and the Mathematical Practice Standards specifically might be more fully realized through the integration of rather than the exclusion of the arts.

The research at hand is an extension of our earlier research where we looked at content integration as a means to disrupt the deep-seated beliefs preservice teachers often hold about STEM education. Specifically, we reported on our work with middle childhood mathematics education majors as they developed their abilities to integrate mathematics and art through participation in a yearlong partnership we created with our local museum of art, culminating in a 2-week study abroad trip. (See Douglass, Conley & Trinkley, 2013 for a complete description of this work). The findings from our previous study led us to look more closely at how STEM plus the arts—STEAM—might assist practitioners in realizing the call set forth in the Standards for Mathematical Practice. In light ofthis new-found understanding, we posed the question: In what ways can STEAM inform the way we prepare preservice middle childhood educators to integrate the processes and proficiencies outlined in the Standards for Mathematical Practice? Given that teachers and teacher educators alike are seeking ways to implement the standards effectively, this question is both timely and important.

Our work is grounded in our beliefs as con-structivist educators. Inherent in our work with students is our desire to engage them in meaningful mathematical challenges that push the limits of their thinking and to provide them with the tools to be successful. Knowledge of facts and operations, however important, is secondary to conceptual understanding (Carr & Hettinger, 2003). Whenever possible, we strive to create contexts which share the teaching and learning space with students. Sharing a pedagogical space with students entails more than involving them in “hands-on participation.” This, of course, is not a new ideal in education. For example, in The Effects of a Mathematical Problem-Solving Intervention on the Errors Made by Middle School Students With and Without Learning Disabilities (Sharpe, Fults, & Krawec, this volume), problem solving strategies were employed which challenged the students to understand the mathematics contextually. More accurately, it asks students to be responsible for their own learning, to pose their own questions, and to articulate their thinking so that it can contribute to and be scrutinized against the understandings of others. We believe, as Brooks and Brooks (1999) have described, that learning is about the search for meaning, and meaning entails accommodating new experiences into existing schema. Accordingly, we strive to present mathematical problems that are open-ended, interdisciplinary, and relevant to students’ personal experiences.

The Standards for Mathematical Practice describe the range of performance abilities educators should expect to observe or develop in their K-12 students. The practice standards reflect constructivism in nature, advocating for a variety of classroom processes such as communication, representation, and making personal connections. The practice standards are not intended to be a prescribed curriculum or a one-size-fits-all model. Indeed, educators are encouraged to take students’ developmental needs into consideration:

While the CCSS-M define individual grade-level benchmarks of what students should know at a particular grade level, the Practice Standards offer educators useful support as to how to nurture these competencies in an age-appropriate way. Although interconnected, the standards are developed along eight distinct strands and are consistent throughout all grades K-12:

• Standard 1: Make sense of problems and persevere in solving them

• Standard 2: Reason abstractly and quantitatively

• Standard 3: Construct viable arguments and critique the reasoning of others

• Standard 4: Model with mathematics

• Standard 5: Use appropriate tools strategically

• Standard 6: Attend to precision

• Standard 7: Look for and make use of structure

• Standard 8: Look for and express regularity in repeated reasoning

Like the authors of the standards, we believe learning is a process of moving students from the “known” to the “new.” We also believe that learning is inherently interdisciplinary. In our work with middle childhood mathematics majors, we have found that fostering cross-content connections serves our students well as they grapple with the complexity of the practice standards. Recently, we have found the integration of mathematics and the arts to be particularly useful to us as we construct curriculum. An integrated curriculum model allows us to frame mathematics content in a contextualized manner and it reflects more accurately our world as most problems do not fit neatly into single-subject silos. As we became more interested in integrating diverse fields of study in our middle childhood mathematics methods course, we sought out resources at and began a yearlong collaboration with our local art museum. One of the results of this collaboration was our introduction to the museum’s learning-thinking model: ODIP (observe, describe, interpret, and prove). Upon initial introduction, we reflected on the potential of ODIP to:

• broaden the context in which our students traditionally learn mathematics (classroom spaces).

• assist our students in making real world connections to and application of mathematics.

• explore content integration of mathematics and art.

Only later would we come to examine ODIP’s usefulness in practicing the content standards with our students.

The Columbus Museum of Art takes a con-structivist approach to museum education, believing each patron is as unique as each piece of art in its collection. The museum utilizes a model of art exploration that involves patrons in (1) observing, (2) describing, (3) interpreting, and (4) proving their beliefs and understandings about art. ODIP is a thinking-learning routine that leads small groups of patrons through a four-step process. In each step, patrons are encouraged to utilize prior knowledge and life experiences to draw conclusions about a piece of art. The process is intended to be active and participatory in nature. While Ritchhart (2007) notes the average museum visitor spends between 6-14 seconds viewing a single work of art, the ODIP model encourages longer periods of exploration. It is common for patrons to spend 30 minutes or longer interpreting a single art piece while implementing ODIP. ODIP draws upon the thinking of classic theorists (e.g., Dewey, 1934; Vygotsky, 1978) as well as contemporary thinkers (e.g., Barrett, 2002; Hein, 1998) as it “fosters critical and creative thinking, nurtures rich group dialogue, and is designed to allow for multiple interpretations” (Douglass et al., 2013, p. 110).

The four-step strategy is as follows:

1. Observe: The first step in drawing conclusions about a piece of art is deceivingly simple. Participants are asked to use their senses to formulate their initial perceptions. They often draw upon concrete visuals (size, scale, color, perspective) as well as ephemeral reactions (emotional responses). The key characteristic at this step is to engage in systematic, deliberate study.

2. Describe: Based on their observations, participants are asked to voice the most salient qualities of the piece. Participants are encouraged to focus on data, facts, and truths (generalities and specifics). This step is typically challenging for participants, as it demands the thinker to parcel out what is known versus what is perceived (Ritchhart, Church, & Morrison, 2011). This step is vital, however, in expanding the conversation and broadening the path toward a group conclusion.

3. Interpret: After all descriptions are gathered, the leader asks participants to translate the descriptions into more far-reaching statements about the piece. This asks participants to reframe their thinking based on the input of their peers. This give-and-take process allows multiple explanations to enter into the discussion. Ultimately, participants are asked to address the significance of the artwork.

4. Prove: Participants are asked to justify their interpretations based on previous evidence and statements made by the group. Collectively or as individuals, they draw logical conclusions developed from generalizations collected in Step 3. The goal of Step 4 is to develop a summative statement or argument about the artwork. Participants are reminded that there is no singular right or wrong conclusion to be reached. Rather, their conclusion should be reasonable, thoughtful, and defendable based on the evidence.

Throughout the process, the facilitator may add details about the artwork or artist to foster deeper interpretation. Similarly, the facilitator may choose to ask authentic questions to extend the conversation or provoke new thinking.

ODIP was developed by faculty at the Columbus Museum of Art to enhance interpretation skills and overall patron experience. ODIP is similar in approach to the scientific method in that it serves as a tool to examine new phenomena and correct or integrate knowledge based on experience and previous understanding. The model has interdisciplinary application and has been used by young children as well as medical students. For instance, in their work with fifth grade students, Luke and Yocco (2010) have described the ways in which ODIP is beneficial in the development of critical thinking skills. Similarly, Jacques et al. (2012) has illustrated the similarities between ODIP and the process by which a doctor diagnoses patients.

The 12 student-participants in this study were seeking initial licensure in middle childhood mathematics at a small liberal arts university located in the Midwest. Over the span of three semesters, participants were engaged in mathematics methods courses and mathematics content courses at the university. The seven female and five male participants were White, traditional-age college students who were identified as middle class. In many ways they were representative of the type of student who typically populates teacher education programs. What was unique was their desire to extend their college coursework through a university-sponsored, 2-week study abroad program in Spain. Participation in the study abroad experience was voluntary. Students received one college credit for the academic work they completed in advance of and during the study abroad trip.

The broad goals of the study abroad experience were for students to gain greater cross-cultural aptitudes and to investigate content integration—specifically, the integration of mathematics and art. In order to be better prepared for their time abroad, participants attended two half-day field trips to the Columbus Museum of Art. Their first experience provided them with background knowledge about the museum’s philosophy and approach to education. During this first visit, student-participants were introduced to the ODIP model by museum docents. During their second trip, student-participants practiced the ODIP model by leading one another in discussion and interpretation of selected art pieces. Practicing afforded students the opportunity to ask questions about the ways they were implementing ODIP and to receive feedback from the docents on their abilities to pose thought-provoking questions, foster rich communication, and draw group conclusions about the pieces of art they were investigating. Student-participants had the opportunity to solidify their understanding of the ODIP model in Spain. While in Spain, students took a course on student-centered pedagogies and participated in afternoon excursions to art galleries, landmarks, and cultural points of interest. The afternoon excursions were structured opportunities for students to explore the integration of mathematics and the arts. For instance, we intentionally took students to Gaudi’s La Pedrera to analyze the use of parabolas in the building structure and assisted them in developing a mathematical rationale for their use.

Throughout this inquiry the primary researchers and the student participants composed pieces of writing that took the form of journal entries. While at the university, students completed assigned entries that responded to course readings on content integration and entries that captured their thoughts on using the ODIP model after their trips to the Columbus Museum of Art. While in Spain, students wrote about the ways they utilized the ODIP model to answer mathematical questions/prompts that were assigned to them by their instructors. Students’ journal writing was a primary source of data for this research.

The lead researchers participated in the practice ofjournaling as well. Within the pages of their respective journals, the researchers captured their reflections on student-participants’ writings, questions, and growth. It was in these pages, too, where specific aspects of the study abroad experience emerged. For example, researchers used their journal as a space to generate context-specific mathematics problems that could be posed to students while abroad, and to prioritize locations of interest in Spain where the ODIP model could be implemented. In addition, the researchers kept analytic memos similar to what Richardson (1994) described as the researcher’s personal notes. These memos captured the surprises, frustrations, and unexpected turns the research was taking as well as aspects of the pedagogical practice that was keeping the inquiry moving. For instance, one analytic memo explored how students seemed to be utilizing the ODIP model to move beyond the cross-disciplinary model that had been anticipated and toward what was being conceptualized as a transdisciplinary model. Other analytic memos explored how preservice teachers might advocate for an arts-infused curriculum and how the ODIP model was proving to be aligned to the Standards for Mathematical Practice. All of these writings, in their various forms, assisted in constructing meaning. Accordingly, narrative as data also served as narrative as method.

The narratives that encompassed the data set for this study were twofold. First there were the primary narratives that the student-participants and the researchers wrote. These narratives were viewed as “original” data sources. The subsequent writings that the researchers completed contained a more sophisticated level of interpretation and analysis. Take, as an example, a student-produced journal entry about effective ways to integrate middle grades curriculum. This can be viewed as original data or as a primary narrative. As the lead researchers went about the work of responding to students’ writing, a second level of writing was produced. These secondary narratives were analytic and reflective in form. In her practitioner research, Hankins (2003) illustrated the fluidity between different levels of narrative writing:

Data and method, as Hankins points out, are entangled and inseparable. Richardson (1994) elaborated on writing as method: “we usually think about writing as a mode of ‘telling’ … writing is not just a mopping-up activity at the end of a research project. Writing is also a way of ‘knowing’—a method of discovery and analysis” (p. 516). The line between narrative data and narrative method was not always clearly delineated in this study. Instead, we wrote narratives, shared those narratives, and revised narratives through further writing. This recursive practice propelled our research and moved us to “writing as a method of discovery” (Richardson, 1994, p. 516).

Each of the narratives used in this study has gone through an analytic process. Journal entries were read and responded to as themes emerged from the data. This process was particularly useful in that it allowed us to monitor students’ understanding and shift the direction of the inquiry to meet the needs of the group. As such, our narrative analysis was representative of the multiple perspectives and interpretations of the group of individuals participating in the research (Clark & Moss, 1996; Cochran-Smith & Zeichner, 2005; Seidl & Conley, 2009). Ultimately, our analysis led us to examine the ways in which the ODIP model came to support constructivist mathematics teaching and how it could be beneficial in bringing the Standards for Mathematical Practice to life for our students.

Throughout our time at the university and abroad we provided our students with problems to solve. These problems were context specific and open-ended in nature. In addition, they served to create a pedagogical space to practice the ODIP model and constructivist mathematics teaching with our students. After each problem-solving episode, students reflected on the process and their learning in their journals. Additionally, students were asked to construct viable arguments as to how they interpreted the art or architecture to which they were exposed. However unintentional at the time, we came to see how our work with students was aligned to the Mathematical Practice Standards.

One recurring theme apparent in students’ journal writing on the implementation of the ODIP model to solve mathematics problems was their heightened awareness of constructing meaning through shared dialogue. Take, as an example, Adam’s reflection on using ODIP to address the prompt: Provide a rationale for the design of architect, Antoni Gaudi’s, chimneys on the rooftop garden at La Pedrera.

Adam’s journal writing recounted how he and his peers formulated an argument to address the problem prompt. He highlighted the importance of shared dialogue in this process and suggested that the thinking of others enhanced his conjectures and conclusions. He seemed to value the ODIP model as a tool to foster discussion and to bring multiple perspectives to bear on the problem situation. Finally, he seemed to value the diversity of thought necessary to solve the problem, suggesting that different approaches from “different directions” were necessary to formulate a plausible argument. This awareness is important for future teachers.

In order to teach preservice teachers how to prepare their students for such thinking, we want them to not only know but also be able to use these mathematical practices. If we want mathematics teachers to be able to support the various abilities and levels of thinking inherent in any middle school classroom, then they must first value these differences. Future teachers need to understand that a student’s way of conceptualizing a problem might be more valid than that of the teacher or the textbook. Adam demonstrated that he was beginning to understand Mathematical Practice Standard 3 - Constructing a Viable Argument. He listened to and incorporated the ideas and interpretations of his peers to establish the parameters for a logical conclusion.

During another implementation of the ODIP model, we challenged our students to develop a rich mathematical problem while visiting the Segrada Familia Basilica in Barcelona (see Figure 1). A note taker was assigned in each group to record the process. The prompt they were given was: Use ODIP to develop a middle school appropriate mathematical problem. Focus your observations on one of the stained glass murals at the Segrada Familia Basilica. Record your process and notes in your journal. Mary’s journal entry, below, was representative of how the ODIP model supported students in developing a solution to this challenge.

Mary’s journal captured the initial observations her group made as they went about the work of developing a challenging middle grades mathematics problem. Her writing highlighted basic facts about the stained glass as well as the relationship between the number of shapes in the design. More interestingly, Mary and her peers have provided evidence of their thinking process: how they entered into the problem and how they negotiated, and ultimately selected, one problem over earlier attempts.

By engaging in this process, our students were establishing the characteristics of a good mathematics problem. That is, they considered the degree of difficulty and developmental appropriateness of the problem as well as the materials and extension opportunities necessary to develop conceptual level understanding. While we did not ask them to solve one another’s problems, the process of developing the question involved them in CCSS-M Practice Standard 1: Make sense of problems and persevere in solving them. In Mary’s example, they selected a problem about symmetry over pattern or percent. Mary’s writing highlights the multiple approaches and iterative attempts they made to achieve their goal.

On a road trip from Barcelona to westernmost Spain, we stopped in Cape Finesterre. Prior to Columbus’ voyage to the Americas, Finesterre, as the name implies, was conceived of as the “end of the earth.” We hiked up the cliff to get an unobstructed view of the ocean and to catch a glimpse of the western horizon. There, we contemplated how those before us believed the world ended where the sea met the horizon (see Figure 2). During this contemplation, one student asked how high we were above sea level. Not knowing the answer, another student suggested we use the ODIP model to answer this question. She suggested the prompt: Use your estimation skills to determine the distance from the edge of the cliff to the waves below. Sarah wrote about this experience in her journal. She, like most of her peers, referenced a known quantity to provide her with a reference point to answer the question.

In this implementation of the ODIP model to explain a peer-generated measurement prompt, Sarah used the resources at hand to frame the problem. The tossing of a stone and her perceived abilities to throw a stone began her efforts in deconstructing the problem. “Estimating from an estimate,” however, caused her greater uncertainty than operating from a known standardized measure—namely, that of a soccer field’s length. By manipulating her unit of measure, Sarah was not only applying existing knowledge to the problem situation, but also demonstrating her ability to utilize complementary mathematical abilities and operations. It also revealed her capacity to attend to the meaning of quantities in an applied context.

We characterized Sarah’s writing as an example of Mathematical Standard 2: Learning to reason abstractly and quantitatively. Sarah used the ODIP model to construct a coherent problem solution before utilizing mathematical calculations or operations. This prompt is similar to how a middle school student might approach a conceptual representation of a very large number, as the concept of very large or very small numbers is quite difficult for early and middle grade students. The CCSS-M indicate throughout several middle grade levels (i.e. 4th, 7th, 8th) that estimation in different domains is a “critical area of focus” (National Governors Association Center for Best Practices, 2010).

By applying the ODIP model to estimation, Sarah has gained a tool to analyze abstractly and quantitatively that will allow her to assist her future students as they conceptualize very large or very small quantities.

With teachers, administrators and teacher educators all wanting to know how to proceed with the CCSS-M, this study offers insight into how the Standards for Mathematical Practice might be realized. Our research suggests that important STEM concepts can be more fully illuminated through the inclusion of the arts. Indeed, STEAM is a prudent course of action given the reduction in national middle school funding. The ODIP model provided our students with a structure to approach mathematical problems, and it served as a pedagogical tool that demonstrated alignment to the practice standards. In particular, participants’ implementation of ODIP allowed them to develop skills in formulating problems, reasoning abstractly, and constructing arguments. Our study also adds to the body of literature on learning-thinking models, generally, and ODIP, specifically, to develop more sophisticated levels of thought. Further study is needed to determine the lasting impacts of this experience on student-participants. Whether or not the student-participants are able to continue to develop constructivist-based approaches in STEAM education and relate it to middle childhood mathematics teaching that is aligned to best practices will be based largely on their abilities to seek out future opportunities to build on the experiences described in this study.

Given that this research was conducted with a small sample of students, the findings are not easily transferable to other contexts. In addition, the type of collaboration created by the participants and researchers is not easily reproduced. Rather, the research and learning depended upon the unique individuals involved. Other participants with alternative motivations and life experiences would result in different conclusions. There are, however, some key tenets of this inquiry that are worth elevating for others who are interested in pursuing similar STEAM investigations. First, a key to our success was our student-participants’ willingness to work on this project for a sustained period of time. Traditional mathematics teaching cannot be unlearned in a single-semester course nor can the Mathematical Practice Standards be mastered in 16 short weeks. Other content areas other than mathematics have realized the importance of inquiry, as well, and are veering away from more traditional teaching practices. For example in science, consider Construct Validation of Student Attitude toward Science, Technology, Engineering, and Mathematics Project Based Learning: The Case of Korean Middle Grade Students (Han & Carpenter, this volume). Having experienced years of mathematics teaching as students, preservice teachers need ample time to engage in alternative approaches. Second, this research benefitted from the sharing of expertise. We could not have developed this experience without the artists and academics at the museum who offered us their knowledge and insight. Finally, the “out-of-the-class-room” contexts, both at the museum and abroad, were vitally important to many aspects of the inquiry. Out-of-the-classroom experiences placed student-participants in new contexts where they could encounter new teachers. It was in this context that the participants were able to interact with ODIP and each other in authentic ways and take the risks necessary to expand their understanding of how to implement constructivist teaching and the Standards for Mathematical Practice.