Research in Our Classroom

Structure, Photo by p medved

Structure, Photo by p medved

I have been questioning lately what methods I can use to understand my students better – not just their work, but their experience of mathematics in my classroom and of the subject in general. I’m taking a uniquely structured (I mean this as a good thing!) research methodology class with Dr. Susan Gerofsky and Dr. Cynthia Nicol here at the Department of Curriculum and Pedagogy at UBC (In fact, it can only be characterized as “standard” insofar as it is a course requirement for my program). Our exploration of research methods has been helpful both in learning methods that one can use for academic research and in reflecting on ways I will be able to investigate my practice when I return to the classroom.

I’ve been used to some pretty standard ways of “getting to know” students. We give them assessments to perform – a variety of types of tasks from tests to open-ended, long-term projects – to give us a sense of their understanding of the concepts of the course they’re taking. Throughout the year we might give them written surveys telling us a bit about how they’re feeling about our teaching or about the course’s progress or our teaching subject in general. We likely give formative checks for understanding through observation, a quick chat, an “exit card“, or visible thinking routines. Regardless of how much information that can be gained from some of these standard and non-standard ways of collecting data, might there be something missing? Might there be something to be gained from collecting information through a different medium – and from involving them in the information gathering process?

Freedom, Photo by Josef Grunig

Freedom, Photo by Josef Grunig

In Donal O. Donoghue‘s (2007) article on boys’ masculinity in places outside the classroom, Donoghue uses photography and a/r/tography methodology to create meaning with boys aged 10 and 11 rather than to use them to discover and make truth claims (as most research does in treating research “subjects” as if they are being used to gain knowledge about something). According to Donoghue (2007), “doing research in and through art offers opportunities to capture and represent that which is not always linguistic – that which can be more profitably represented and understood through nonverbal forms of communication” (p. 63). My conflict with this type of research is that I see both sides. I see that it offers a different way to view a sensitive topic – using non-verbal “data” (i.e.: photographs) through the view of 10 and 11 year olds – a  view that has the potential to reveal something never before explored. However, I can see also the risk of  photographs to be open to a much wider scope of interpretation than written data might be. So, based on what this method offers us – and does not offer us – Donoghue’s (2007) words make me both optimistic and nervous: “how we do and represent research is inseparable from what gets communicated” (p. 64).

Comparing our work with interviewing methods for research purposes, and reflecting on similarities and differences with the use of photography as detailed by Donoghue (2007), I notice more similarities between them. As with interviewing, photography has the potential for inviting the participant into the research process, and offers non-verbal representation (interviewing does this through gesture and tone of voice). Both need to be examined within the social and cultural contexts in which the product (speech, photograph, art, etc.) is produced. However, which is liable to produce more accurate interpretation? For example, are we more likely to get an accurate view of what a child thinks of our subject if we ask them to tell us, or if we ask them to take a picture that represents how they feel and discuss the photo choice with us? The old cliché about pictures and words comes to mind, but beyond that, one could argue that reading and re-reading a script made from an interview can continue to create just as many new meanings as can having a look and a second/third/fourth look at a photo. The difference between interviewing and photography that I can see is that interviewing offers the potential for a much more fixed, rigourous process, whereas the use of photography, to a large extent, is itself a commitment to embrace a research method that involves the participant much more in the process.

Regardless of these structural aspects, there might be something to the use of photographs to find out more about students’ thinking. Consider school culture. When you ask a student to write down feedback, like in a survey, this structured written form is similar to what students experience in other parts of school and there may be strong psychological aspects at play governing their answers. However, exit cards are casual and quick, often on 3×5 cards which is not specifically how class tasks are done – which may cause them to open up a bit further. So, if you ask students to send you a digital photo with a description as a way of answering a question (of course, if this is logistically possible in your context), this could provide you with different information that you would have otherwise received using a different format.

References

Donoghue, D. O. (2007). “James always hangs out here”: making space for place in studying masculinities at school. Visual Studies, 22(1), 62–73. doi:10.1080/14725860601167218

Math and Place-Based Education

A scene from Central Vietnam, Photo by Rob DeAbreu

A scene from Central Vietnam, Photo by Rob DeAbreu

Place-based education (PBE) is based on the fundamental idea that places are pedagogical – they teach us about the world and how our lives fit into the spaces we occupy. It began with community education and community-as-classroom – the idea that students could learn by paying closer attention to their community and doing work within it. The idea has since expanded to investigate the learning that happens in field-trips or long-term projects outside of the classroom, to examine the pedagogy of places of all sizes and locations, and to explore the meanings that different people attach to place. One can argue, that – to an extent – there is an activism component against the current state of the education system, which – in most cases – assumes that the school (and the classroom) is the place where learning occurs.

For Dr. David Gruenewald (2003) – who now goes by the name David Greenwood – place-based education (PBE) is in large part a response to standards, testing, and accountability, the threefold education reform movement of the last two to three decades (though grounded in some much older ideologies). As mathematics is the gatekeeper discipline to many careers and university programs – whether with a mathematics component or not – it is a discipline that, it could be argued, is the target of PBE’s response. With this in mind, it is no surprise that Gruenewald/Greenwood (in Green, 2005) expressed his skepticism about the possibilities of developing place-conscious mathematics. However, is mathematics – the very tool incorrectly used to assess students, and thus misunderstood by so many – the ideal vehicle to drive PBE’s response to misguided education reform?

Classroom, by evmaiden

Classroom, by evmaiden

Much has been written about cultural border-crossing in science education – challenges that students come to when negotiating between their life-world and the culture of the discipline of science (Aikenhead & Jegede, 1999; Jegede & Aikenhead, 1999; Jegede, 1995). Similar arguments have been made by Boaler (1993, 1998) and Schoenfeld (1989) that a similar struggle, manifesting in difficulties in knowledge transference, goes on in mathematics education. PBE acknowledges the divide between students’ life world, and the culture of school and mathematics, and Gruenewald/Greenwood (in Green, 2005) cites it as a result of the disconnected place – the school and the classroom – that students are meant to learn in each day. So, PBE can contribute to mathematics education, and mathematics can contribute to the activist elements of PBE. I disagree with Gruenewald’s challenge that place-conscious math can’t exist.  Gruenewald/Greenwood (2003) himself says, “people make places and places make people” (p. 621). PBE embraces our agency to leverage the power of place in our lives and learning just as it acknowledges the influence that place has over our identity. While learning must take place in a physical classroom in most schools, with all the aspects of schools that this entails (timed periods, separate subjects, etc.), it does not mean that we should give up trying to transcend the barriers and isolation that schools can create. In the interview, Gruenewald/Greenwood (in Green, 2005) points out that in the process of “aligning” curriculum and standards, curriculum is treated as a means to an end (to meet the standards) and is forever altered. How do we mediate the two? If we can’t, what changes can we make to enable schools to connect students better with the outside world?

Technology is a given, by Scott McLeod

Technology is a given, by Scott McLeod

One could argue that the infusion of technology in our classrooms further removes us from our world – because technology forces us to perceive our world through a screen and interact with it through a machine. There are others who would argue that technology connects us – like I am connecting with you right now having made my ideas available for comment, or like many professionals and friends connect using Twitter and other social media.  In a different way, a framework like ethnomathematics is one way to enact PBE in mathematics – by inviting students to be aware of other places and cultures that surround us. Perhaps by being inspired by the mathematics embedded in others’ and our own cultural practices, students can transcend the classroom space and acquire the learning that we seek for them. Regardless of what solution is suggested, however, can we transcend place? Or does the fact that students are located in a classroom during the day completely undermine the ability to enact PBE? And, if we can transcend place – that is, if the place they are in (school and classroom) recedes from consciousness as teachers attempt to enact PBE – does this mean that we have enacted PBE successfully or failed to enact it?

I have more questions than answers about this at the moment. One of the purposes of PBE is to catalyze a dialogue about place and education, so perhaps finding “ways” to make it “work” isn’t really the point!

References

Gruenewald, D. (2003). Foundations of place: A multidisciplinary framework for place-­‐conscious education. American Educational Research Journal, 40, 3, 619-­‐654.

Green, C. (2005). Selecting the Clay: Theorizing place-­‐based mathematics education in the rural context (Interview with David Gruenewald). Rural Mathematics Educator. ACCLAIM.

Smith, G. (2002). Going local. Educational Leadership, September 30-­‐33.

Boaler, J. (1993) The role of contexts in the mathematics classroom: Do they make mathematics more ‘real’? For the learning of Mathematics, 13(2), 12-­‐17.

Boaler, J. (1998). Open and closed mathematics: Student experiences and understandings. Journal for Research in Mathematics Education, 29, 41-­‐62.

Schoenfeld, A. H. (1989). Explorations of Students’ Mathematical Beliefs and Behavior. Journal for Research in Mathematics Education, 20(4), 338–355.

Aikenhead, G. S., & Jegede, O. J. (1999). Cross‐cultural science education: A cognitive explanation of a cultural phenomenon. Journal of Research in Science Teaching, 36(3), 269–287.

Jegede, O. J. (1995). Collateral Learning and the Eco-Cultural Paradigm in Science and Mathematics Education in Africa. Studies in Science Education, 25, 97–137.

Jegede, O. J., & Aikenhead, G. S. (1999). Transcending cultural borders: Implications for science teaching. Research in Science & Technological Education, 17(1), 45–66. doi:10.1080/0263514990170104

Math as a Human Activity

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Wool bag woven by hand by Vivian Campbell. Photo by Robert DeAbreu

Considering some of the readings about the role of context in the mathematics classroom, I’ve been feeling skeptical lately that ethnomathematics would work with all students. Jo Boaler (1993) states that choosing contexts for mathematics that replicate the complexity of the real world as much as possible benefits students’ learning. I wondered if Langdon’s suggestion was correct: “students acquire a better understanding of mathematics by discovering that it is already a part of their environment than by studying local cultural examples” (in Boaler, 1993, p. 16). Therefore, while an ethnomathematical approach to the classroom can cause a valuable shift in students’ worldview (Eglash, 2009), are we hindering their mathematical understanding by introducing concepts in a cultural context so radically different from their own?  As Nel Noddings says, “slaving away at someone else’s real-life problem can be as deadly as doing a set of routine exercises and a lot more difficult” (Noddings, 1994, p. 97).

Our Mathematics, Community and Culture class visited the Musqueam Community Centre to learn the mathematics embedded in mat weaving and other cultural practices. My experience there essentially summed up the benefits that various readings expounded. According to Mukhopadhyay et. al. (2009), “ethnomathematics draws attention to mathematics as a human activity” (p. 68). Rather than distancing me from mathematics, Vivian Campbell’s weaving presentation drew me in to learn more about Musqueam cultural practices and the mathematics behind them. Even more striking was that, upon my return home, the experience catalyzed an investigation of photos and video I had taken of cultural practices in different countries during my travels. I sought to see “mathematics as a human activity” in weaving, fishing and canoe making in Myanmar; weaving, rice paper making and rice harvesting in Vietnam; and weaving, wood carving, and drum making in Ghana. By “incorporating the mathematics of the cultural moment, contextualized, into mathematics education,” (D’Ambrosio, 2001), I was inspired to learn more about the culture (Musqueam) being presented, and prompted to further investigate cultures that I had experienced, bringing me a much richer understanding and appreciation of culture and mathematics.

In other words, it didn’t matter that Musqueam culture is so drastically different from my own; learning about it was interesting and caused me to look for mathematics in other things that I had seen. Generally, I’m a curious guy, but this was a cool experience. I would love to explore some of this stuff with students (when i finally have a class of my own again!) as I am intrigued by the benefits that can be reaped by widening the cultural paradigm in the mathematics classroom.

References

Eglash, R. (2009). Native-American analogues to the Cartesian coordinate system. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.). Culturally responsive mathematics education (pp. 281-294). New York: Routledge.

D’Ambrosio, U. (2001). Ethnomathematics: Link between traditions and modernity. The Netherlands: Sense. (Chapter 2).

Boaler, J. (1993) The role of contexts in the mathematics classroom: Do they make mathematics more ‘real’? For the learning of Mathematics, 13(2), 12-­‐17.

Noddings, N. (1994). Does Everybody Count? Reflections on Reforms in School Mathematics. Journal Of Mathematical Behavior, 13(1), 89-­‐104.

Mukhopadhyay, S. Powell, A. & Frankenstein, M. (2009). An ethnomathematical perspective on culturally responsive mathematics education. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.). Culturally responsive mathematics education (pp. 65-84). New York: Routledge.

Culturally Responsive Education in Mathematics

indigenous aboriginal education mousewoman

This artwork is of Mouse Woman, the Narnauk supernatural shape-shifter. This art was used as official artwork of the Aboriginal K-12 Math Symposium, held at the UBC Longhouse. Artist William (Billy) NC Yovanovich Jr.––whose Haida name is Kuuhlanuu––is a member of the Ts’aahl Eagle Clan of Skidegate, Haida Gwaii.

Last week, our Mathematics, Community, and Culture class discussed culturally responsive education – yet another controversial and evolving topic in mathematics education. It seems to me that culturally responsive education is a philosophy whereby a teacher (or a writer of curriculum) attempts integrate culture and curriculum in a way that emphasizes the ways of knowing of different cultures so as to provide a richer educational experience. In the context of indigenous issues consistently under consideration here in British Columbia, many of the members of the class wondered if there was bias in which culture education was meant to be “responsive” to. You can see how there can be a debate here about whether or not culturally responsive education aims to “respond” to a particular culture, or remove some of the emphasis from another.

Due to a prevailing dichotomous view of “dominant culture” and “less-dominant culture”, culturally responsive education can strongly imply a requirement of dominant cultures to recognize cultures that have long been less dominant. I agree that the role of culturally responsive education, according to Mukhopadhyay (2009), should be to treat all cultures fairly rather than equally. Education does need to recognize other less dominant cultures more than dominant ones simply because dominant cultures will prevail in other aspects of a student’s environment regardless of what we do as teachers. This does not mean total exclusion of dominant cultures, but it should mean an effort on the part of the teacher in all subjects (not just mathematics) to respond to many cultures positively and to expose students to many cultural paradigms.

How this is done is incredibly complex and likely looks different in each classroom, but, as was said by Christine Younghusband, speaker at the most recent Aboriginal Math K-12 Symposium (2012), if we want our students to be culturally responsive (or take on any other value), we as teachers need to be culturally responsive (or truly believe in that value). Storyknifing, for example, to introduce the idea of geometric visualization instead of, say, giving a worksheet and some plastic block manipulatives to students is a simple way that a teacher can introduce a mathematics topic (and tick those curriculum objectives!) while giving a strong message to students of the strong mathematical heritage inherent in many cultures. But this message will only be strong if a teacher uses a strategy like Storyknifing within a consistent effort in the classroom to invite students to think critically about culture and the perceptions that prevail. The message will only work if the cultural context of an activity, like Storyknifing, is preserved. One wouldn’t want students to see storyknifing while learning Cartesian Coordinates and get the impression that storyknifing was used in navigation.

If someone were to simply use the Math Catcher videos, for example, as one-off or as one of a few of intermittent cultural resources in their classroom, it would appear to the students as nothing more than a video version of an ordinary word problem stated in a Squamish context. Culturally responsive education is not about using storyknifing or Math Catcher videos in one’s classroom, but about a philosophy of classroom practice to expose students to different perspectives and cultures and to encourage them to investigate and question dominant paradigms.

References

Mukhopadhyay, S. Powell, A. & Frankenstein, M. (2009). An ethnomathematical perspective on culturally responsive mathematics education. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.). Culturally responsive mathematics education (pp. 65-84). New York: Routledge.

Lipka, J. Wildfeuer, S. Wahlberg, N. George, M. & Ezran, D. (2001). Elastic geometry and storyknifing: A Yup’ik Eskimo example. Teaching Children Mathematics. February, 337-343.

 

Cooperative Learning vs. Small Group Method

Cooperative Learning Tags - assigning students with roles for an activity.

Photo: Cooperative Learning by ielesvinyes

I am taking a course, Constructivism Strategies for E-Learning, through the Department of Educational Technology here at UBC, and we’ve been exploring different instructional strategies. It’s funny, but I thought I knew what “cooperative learning” was, but there are so many different definitions of it! I found the exercise in comparing strategies valuable, and decided to share my thoughts.

Cooperative learning and SGAM (small group activity method) are similar in that students discover content and teach it to one another and to the class, with the teacher is “guide on the side.”  The focus of the description of each focuses on the specific set up of the classroom activity, such as number of students in each group, how long each portion of an activity is, etc.

Cooperative learning differs from, and seems more effective than SGAM in that groups in cooperative learning are sustained over longer periods – perhaps working on a problem/project for a whole unit or a whole year, while in SGAM the description seemed only concerned with a 30 minute to 1 hour period. In addition, in SGAM, students are working with others, but the sources we were led to did not seem to be mention a focus on teaching students the skills necessary to work effectively in teams, such as listening effectively, interjecting politely and ensuring everyone has a voice.

Both cooperative learning and SGAM are problematic in that they seem to undermine some key aspects of constructivism. For instance, in cooperative learning, diversity of students is addressed but there is no provision for students to have individual thinking time – or at least this is not documented. I don’t think that a student should simply work alone and they need to learn how to work with others, but what if a student learns best by processing content alone first, then sharing their ideas? How do these students access learning?

Cooperative learning and SGAM each have students working in groups which addresses the social nature of learning and all the aspects of the works of Piaget and Vygotsky that speak to this. However, cooperative learning and SGAM are too prescriptive, which seems to contradict the “guide on the side” persona that the teacher is invited to take. Depending on the amount of control exercised by the teacher, the ownership over learning and the complex process of knowledge construction could be compromised. Hopefully, teachers wouldn’t be too invasive with their interventions and hopefully they wouldn’t just set up the class in groups and give an activity and assume the learning happens as long as the students talk it out. The question that teachers need to ask themselves constantly is where the balance lies between being too controlling and too “hands-off”?

Categories of Research in Education

Photo: Balance, by BCth

Photo: Balance, by BCth

Humans characterize things as a way of gaining a better understanding. Research methodologies are no different. However, as Lawrence Sipe and Susan Constable (1996) point out in their article summarizing research paradigms, we need to beware the oversimplification that characterizations can bring – not so easy when what we naturally like to do as humans is categorize things, especially as we are coming to grips with them. We can’t stop characterizing altogether, but Sipe and Constable rightly point out that categorizations may imply a dichotomy (either THIS way or THAT way) or assume a univocality (neat word, eh?) – that there is only one way to look at a concept. The consequence in research, and anything else in life for that matter, is that unique characteristics get lost.

Sipe and Constable are specifically referring to the categorization of research as either “qualitative” or “quantitative” – but of course, one might use both these strategies to triangulate data, or a more reflective research method that doesn’t really fit into either of these categories. Then we run into the problem that these categorizations mean something slightly (or perhaps drastically!) different depending on the field you’re in. So here we see an example of where categorizations, especially long standing ones, fail us.

One aspect of the article that made me stop was Sipe and Constable’s (1996) point that characterization of a research methodology does NOT imply the use of a particular method. Surely some methods would lend themselves to be used by those utilizing certain research methodologies. For example, a clinical interview method might be used by someone prescribing to a qualitative methodology. However, one would not say uniformly that using one research methodology (i.e.: qualitative) means that the researcher is necessarily using a particular method (i.e.: interview). This makes perfect sense, but is not something I had thought of before reading this article. Quite a few of our guest presenters in my Research Methodology in Education course use multiple methods, each of which supports the use of the other!

Another place that made me stop was the characterization of the relationship between researcher and researched given by Sipe and Constable (1996). I have read enough educational research to know about clinical interviews and the various ways this is done to understand students and teachers and others in the educational research process. However, the characteristics given in the article highlight the importance for me in deciding whether or not the research subject should be informed about the research process or not, and to what extent the researcher her/himself lets themselves be analyzed as part of the research process. While some could argue that research data is confounded if the researcher becomes part of the research, there are other schools of thought – A/R/Tography and Currère as examples – where the subject matter is enhanced by the exploration and questioning of the researcher’s worldview (deconstructivist – see article). If we truly respect reflective practice in our teachers, then surely why not reflective practice in our educational research? I can see, of course, the difficulty this may cause for those in the context of copyright laws and BREB ethics approval applications that demonstrate the serious, disciplined side to research. However, I can see now the arguments that could be given for different forms of educational research and how heated the debate could become!

Ethnomathematics

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Image by Ron’s Iteractions (note: not Ron Eglash). Original photo by NASA/GSFC/METI/ERSDAC/JAROS, and U.S./Japan ASTER Science Team

I’ve just recently been doing some readings on ethnomathematics. From what I’ve been able to figure out, ethnomathematics is the study of mathematics of different cultural groups. Its goal is to teach value for one’s own culture, respect for another’s culture, and curiosity to learn more about different cultures while teaching mathematics in a cultural context. That being said, it is a fairly new and ever-growing and changing field of mathematics education, so the definition can vary significantly at this point, informed by the life experiences and culture of the person giving the definition!

Ethnomathematics is likely controversial because it does not conform to the needs of advocates who support traditional, computational, arithmetic and algebra driven curricula (see math wars). It requires a more exploratory and interdisciplinary approach to the subject. However, if one is to embrace ethnomathematics, one then opens themselves up to examining the way mathematics is done in all cultures, rather than only the canonically respected mathematics that was done in Europe that is still taught today. Often teachers balk at “multicultural mathematics” because it means an awkward application of mathematics to, say, number systems at the beginning of the year that quickly gets seen by the teacher as a waste of time because it doesn’t tick boxes on the list of curriculum objectives. This is an unfortunate misunderstanding.

Ethnomathematics also opens the door to issues of culture and representation in mathematics and in education in general, which many teachers may not be emotionally prepared and/or educationally trained for (or simply not be interested in dealing with). However, Ron Eglash (2009) exposes a way that we can use culture as a bridge to math – and nicely tick some of those curriculum objectives as well – while integrating art and mathematics in exploration of weaving or architecture or religion. While I can’t provide the article due to copyright, check out his TED Talk.

In addition, D’Ambrosio’s (2001) more philosophical piece seems to imply that ethnomathematics is a way to explore the diversity of cultures while simultaneously being something that students can gather themselves around. While cultures, such as Inuit and Navajo and Maya, may have different perspectives on the distribution of time, the heavens, and agriculture due to their proximity to the equator – in essence, they have different ethnomathematics – these cultures are united by the fact that they have come to ways of knowing through interaction with their environment – in essence, that they have ethnomathematics. Both D’Ambrosio and Eglash, it seems, agree on the rich, paradoxical “unity through diversity” that ethnomathematics can bring to the classroom.

This is an interesting area for teachers to explore if they’re looking for interdisciplinary learning to come alive in their classroom!

References

Eglash, R. (2009). Native-American analogues to the Cartesian coordinate system. In B. Greer, S. Mukhopadhyay, A. Powell, & S. Nelson-Barber (Eds.). Culturally responsive mathematics education (pp. 281-294). New York: Routledge.

D’Ambrosio, U. (2001). Ethnomathematics: Link between traditions and modernity. The Netherlands: Sense. (Chapter 2).