Next Top Model: Quadratics in all their Glory

 

Watery Parabola by Martin Kenny

Watery Parabola by Martin Kenny

I’m currently refining a unit of study about quadratic functions that was co-developed last year with Kristina Sharma at Branksome Hall Asia. Our intention for the unit last year was to emphasize students seeing mathematics in the world around them and making connections between mathematics learned in class and real-life contexts in which that mathematics is used. We had some successes for sure, but were well aware that this unit – as any unit – was in need of some changes.

With the help of Will Percy, our Digital Technologies Coordinator, and Yumi Matsui, our EAL Coordinator, I’ve embarked on a journey of sorts to amp up the great work Kristina and I did last year. Students seemed to enjoy the final task last year that had them making video of themselves playing a sport in PE, then turning this into a still image much like videos of Will It Hit The Hoop? of Dan Meyer fame. Students then imported the image into GeoGebra, and used this program to help them model the path of the ball through the air with a quadratic function.

The unit is quite language heavy and, as our school population is mostly composed of English Language Learners (ELLs), I am keen this time around to provide them with more support. Yumi has provided excellent support for us to follow up from our EAL professional training with Dr. Virginia Rojas, and I’ve adapted her language supports and sentence frames for use in various activities. Here is one example of resources Yumi has shared with me that my students have found particularly useful for having constructive classroom conversations (and see Jeff Zwiers‘ work should you be interested in more of this great stuff!).

Sentence Starters for Building Ideas

My challenge right now is to retool the students’ final assessment for this task. I would certainly like to rewrite the questions, but my other goal is to put it in a format that would allow students a wider audience for their work. Last year’s task was rich, but the audience for their work was me, their teacher. Fine, but not terribly exciting. An idea that Will has suggested is to have students do podcasts as formative tasks and for students to complete a talking Pages document with a combination of video/audio and written text. This might be a manageable next step, setting me up for really making this unit solid in its next iteration. Attached is last year’s assessment for anyone’s perusal:

Sports Next Top Model Task NO RUBRIC

I am incredibly keen for anyone’s feedback about this as it is developed. Exciting things will be happening in the coming weeks!

 

 

 

Roles and Complex Instruction: Getting the School Year Started

small group work by susan sermoneta

small group work by susan sermoneta

This past week, I have started preparing for a new school year and reflecting on some of the classroom structures I want to refine. While I’ve studied complex instruction during my M.Ed, I am still very new to implementing it in the classroom so I thought I would throw out some ideas here about group work structures that I’m thinking about implementing. Particularly, I’m wondering about some of the specific aspects of these structures and whether they may have impact on positive outcomes I’m trying to achieve. In this post, I will particularly discuss the administration of roles in group work and complex instruction. Please post suggestions or thoughts below in the comments.

First, I’ve been reading Smarter Together with interest, and have decided to implement their suggested roles for group work: Facilitator, Inclusion Manager, Recorder/Reporter, and the Resource Manager.

  • The facilitator’s role will be to encourage the completion of the task by getting the group off to a quick start, and checking if all of the group members understand what is going on along the way.
  • The inclusion manager will oversee the behavior in groups, keeping people on task, keeping discussion focused on the task at hand, and ensuring students play their roles.
  • The recorder/reporter will make sure data is being recorded and will present group findings at the end of the class.
  • The resource manager will obtain and put back resources that are needed for each task, supervise clean-up of the group’s table, and will bring group questions to the teacher.

I’m keen to use this structure as these roles and the responsibilities are not completely separated. There is enough overlap to avoid confusion among students regarding what each is supposed to do, yet the responsibilities are structured so that students remain interdependent throughout the task ahead of them. I’m planning to rotate these roles to give each student a chance to fulfill different group work responsibilities and build different skills needed in group work – and so students each have access to all the roles. I won’t rotate mechanically but rather plan to use a developmental framework developed during the Assessment and Teaching of 21st Century Skills project coordinated by Patrick Griffin, Esther Care, and others at the University of Melbourne. I hope to use this developmental framework to help me decide when to place students in particular roles, and what kinds of interventions are needed to help them build collaborative problem solving skills. It’s my first time through trying this out after their MOOC this summer, and so wish me luck!

Which way for happiness? by Andrea Marutti

Which way for happiness? by Andrea Marutti

Last year, I struggled with deciding when to rotate groups and their members. Some of the literature I read suggested randomly choosing members for groups and reshuffling (again, randomly) groups and members after every two weeks, enforcing the fact that students couldn’t just change groups and had to learn to work with those they were paired with. However, I found that the group folders – where students put their collaborative work for reference at a later time – broke down as a system since students might have been part of two or three groups during an entire unit of study. My feeling this year is that I will only randomly choose group members after a unit of study has been completed and each of the groups has gone through that complete journey together. I wonder, though, whether this will cause conflict for students if they end up in groups with someone they don’t work well with. It remains to be seen what issues come up and I welcome any ideas about this.

A second wondering I have is how to teach my students about these roles and structures when all of them have the added challenge of being EAL/ELL learners to varying degrees? Specifically, I am thinking about the plethora of information that I will need them to take in and act on: roles and their responsibilities, norms for behavior when working in groups, language they should use to perform each role effectively, language they should use in general when working through a math problem (making observations, analysis, etc.), and the language of the developmental framework so they can understand how to develop their collaborative skills. When I consider as well the other classroom administrative vocabulary that come along with teaching in the MYP and communicating changes for Next Chapter, the challenge becomes so much more daunting. Of course, I don’t expect them to “get it” right away, but I don’t want to throw so much at them that they just turn away from it completely. I plan to put posters up, and refer to them often as I roll through activities at the start of the year particularly directed at helping to make these roles and group work behaviours explicit.

For all learners, skills to promote successful collaboration are essential to learn for many aspects of living in the world. For EAL/ELL learners, collaborative activities can create the need for them to practice vocabulary and promote language acquisition at the same time. However, I am keenly aware that piling too many challenges on students creates the danger of doing many things a satisfactory level instead of doing a few things well. Still, one can’t develop a system that works in their classroom without some thoughtful experimentation!

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