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!

 

 

 

A Reflection: Roles and Complex Instruction

Think First. Photo by Jason Devaun

Think First. Photo by Jason Devaun

A few months ago, I wrote a post entitled Roles and Complex Instruction: Getting the School Year Started. It was written during a period of particular optimism and excitement as all we teachers feel at the start of a new school year. After returning to this post and having a re-read, I had some thoughts that I wanted to get down about how it has all turned out so far. As always, I am very much open to comments from anyone who might have some suggestions for how I might improve learning experiences for my students!

I came into the year with a goal to start my grade 9 students off with the language of functions that they would need to be successful during the school year. Our school offers all three IB programs and so their MYP program requires them to investigate patterns, apply mathematics to real world situations, and communicate themselves effectively. It seemed logical to apply complex instruction in this case to get them verbalizing their thought process while gathered in dialogue around mathematical concepts.

Unfortunately, I haven’t gotten roles to work as well as I’ve wanted as students tackle a task together in groups. It often seems that students have so much language to deal with as English Language Learners (ELLs) that focusing on the roles has gotten in their way of getting the task done. The purpose of the roles is to help them work better together on a task as the roles themselves are interdependent rather than acting as a division of labour. Getting them to follow the roles has been more challenging than I expected, and it has a lot to do with how much attention I give this during a lesson. In the end, I just want them to gain understanding and whether they follow the roles or not becomes secondary as valuable seconds of a lesson tick away. Naturally, they have been very quick to figure this out!

Another challenge I face is that a lot of my students’ conceptual work together takes place in their Mother Tongue (often Korean, though some Chinese students attend our school) as it needs to for maximum learning gains (much research has shown this, but here is one example). So the bigger challenge that I face is how to get the students speaking in English and learning academic language that I require of them in English. And when in their conceptual process do I do this?

Looking back, I should have had students learning math language and group work language in smaller chunks. I wonder if I introduced the roles too soon as well. Having a unit where I simply focused on the mathematical language was a good idea, but it also came at a time when a whole host of other words needed to be in their vocabulary too. That was just too much.

My plan now is to take a step back and deal with math language. I will not disregard the roles, and when a task really calls for them, I will use them. So far, however, a strength in my classroom is that the tasks I have been able to present are group-worthy, and so I am getting the students together in productive discussion about mathematics, which is an achievement. I will continue to work to make tasks that are group-worthy, and I will use more visible thinking routines to make the key concepts from lessons more explicit. I have provided sentence frames for students at their tables, but these need to be a more central part of each lesson as they have been often overlooked. I think these things and my expectation that they speak in English at appropriate times in the lesson will go a long way to bringing up their use of academic mathematical language.

By the way, if you’re looking for some examples of math group-worthy tasks, check out NRICH.org. They have a great selection of resources for mathematics teachers, making learning tangible and encouraging students to gather together around a great problem.

I want my classroom to be a place where students are building confidence in their problem solving skills and their intuition for innovation. Math is, after all, where you can learn how to work within parameters to make your own possibilities, not follow a set of rules that someone else gives you. It sometimes feels like the steps I’m taking to get my classroom there are SO small! But I have to remember that it’s the taking of the steps that matters.

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!

Change in Schools and Complexity

In a 2003 paper*, Brent Davis and Elaine Simmt write about the application of principles of complexity to the teaching of mathematics. Complexity science is essentially the study of living systems that are adaptive and emergent – and so it is a wide-ranging branch of scientific study. I enjoyed their article as their consideration of a school, of a classroom, of a department – of learning in general – as a living system really highlighted some important aspects of teaching and learning for me. Again, I know that this article is over 10 years old, and this may be considered “old” information, but I think the article falls into the “oldie but a goodie” category and I wanted to reflect on it here because it is such a good read.

Davis and Simmt outline several conditions that must be present for a complex system to emerge. While reading these, think of the different layers of a school – a concept to be learned, a group of four students working on an activity, a classroom as a whole, a mathematics department, a grade level team, a middle-high school faculty, an entire school staff (K-12), a school community as a whole, the administrative team, etc.:

1)   Internal Diversity: members of the system must be diverse enough to be able to contribute different things to their purpose.

2)   Redundancy: members must have enough in common in order to interact more favourably.

3)   Decentralized Control: members’ results are collective, not the result of one member or a central “leader”.

4)   Organized Randomness: Proscriptive rather than prescriptive. Members come to a result by “living in the boundary defined by the constraints, but also using the space to create something greater than the sum of its parts” (Johnson, 2001, 181 – quoted in Davis and Simmt).

5)   Neighbour Interactions: Members must be allowed to interact with one another for new results to emerge.

Concerning change, there are so many things that come out of this article for me. One of the main thrusts of the article is that complex systems need to be self-similar – such a sweet idea that is incredibly mathematical. In the context of a school and educational change, for me this means that all layers of a school, from administration to faculty and staff to students to parents, need to be operating in self-similar ways. If teachers are meant to be distributing leadership to students to develop their confidence and independence, then so should department heads and grade level leaders distributed leadership for this purpose, and so should administrators do the same for grade level leaders and department heads – and so should school structures be set up to develop the confidence of community members such as parents to contribute their expertise to the learning process. If we wish for students to respect the value of working together with one another in groups, so should teachers and administrators and staff and the school community also demonstrate their respect for this through meaningful action. In short, my feeling is that meaningful change does not happen unless the school as a complex system is self-similar.

My feeling also is that change is most ideally achieved if a school becomes a complex system as Davis and Simmt outline. A school working towards positive change is able to maintain a balance of redundancy and diversity so that people will be motivated to come together yet there will also be enough diversity among people to allow them to develop new ideas. Members of the complex system will be allowed to freely interact so that new ideas can be developed and they should be free from outside control to do this. We can see this latter characteristics of a complex system in the common complaint from teachers about the lack of planning time with other teachers to properly put together a unit of inquiry, or to work in grade level teams on interdisciplinary work.

The most challenging of these aspects – and where the article hits home regarding educational change – is the idea of control. Too often, change makes people nervous and can rattle their confidence. Using a teaching example, often when changes take place or a teacher is not feeling confident or comfortable, they will seek to gain control in other ways, and this is usually when one might see in their classroom more teacher-centered lessons and other such decisions that take independence away from students. Interestingly, this in turn takes confidence away from students. If a teacher doesn’t let students answer a question, or doesn’t let them freely explore a problem, maybe this means the teacher feels they can’t do it. And so we see here that the desire for control in education can powerfully usurp positive change.

Sorry this isn’t a completely thought out post – your thoughts on how to grow this idea are appreciated!

 

*Davis and Simmt. Understanding Learning Systems: Mathematics Education and Complexity Science. Journal of Research in Mathematics Education. 2003. Vol. 34, No. 2, 137-167.

Making Groupwork Happen

I’ve been investigating complex instruction (CI) over the past few months as part of my M.Ed at the University of British Columbia. CI offers an approach for teachers to use in their classrooms to temper the status differences that inevitably arise in group work situations. I first came across the approach when doing some further research on a school called Railside, a name given by Stanford mathematics education professor Jo Boaler to an ethnically diverse, urban school in southern California. Boaler conducted a longitudinal study there and at two other local area schools to study learning gains and found something else.

At Railside, all of the teachers in the mathematics department were using CI, and their students not only demonstrated great learning gains, but showed an appreciation for the power and beauty of mathematics that teachers yearn to pass on to their students and a desire to improve they way they worked in groups so that they could sustain the learning community that had evolved in their classes. Intrigued, I decided to investigate further.

CI explores access issues that take place when group work is implemented. We teachers have all seen students who were too shy to contribute, or who were deemed unable to do the task, or who simply sat back and let others do the work while the rest of their classmates got frustrated. However, thinking of it in terms of an access issue, if we place students together for the purpose of learning and only some students do the work while others are forced out or choose not to participate, not all students have the same access to the learning that is meant to take place in groups.

For many teachers, group work is daunting to implement because of these and the plethora of other problems that can come up. How do I ensure that students truly work together to create a group product that they all contributed to? How do I ensure individual accountability for the contributions students make in their group? How do I ensure students are learning? Naturally, I was skeptical of this new approach. After all, if it is based on over 20 years of classroom research, and two books have been published, why isn’t it already widespread?

Components of Complex Instruction

The answer to that final question still escapes me. CI seems to have all the bases covered. CI starts with a multidimensional classroom – one where academic success is measured on many different abilities, such as coming up with different solutions, explaining solutions, justifying solutions, using different representations, making a model of your solution, asking good questions, and so on. Quite simply, more students have success because there are more ways to have success.

Tasks and group roles are structured to be “group worthy” – so that students have to work interdependently to complete the task successfully. The roles also enable the delegation of authority so that the class can achieve a state of decentralized control. This allows the teacher to move around to assist and prompt students as needed.

Two treatments are recommended as the teacher is circulating. First, the multiple abilities treatment involves the teacher continuing to reinforce – in words as well as through classroom structure – that no one will have all of the abilities to complete a task themselves, but everyone in the group has at least one of the abilities. Second, through the assigning competence treatment, teachers listen intently to group discussions and interject to purposefully raise the status of something a low-status student has shared in a group.

CI is incredibly ambitious in what it sets out to achieve. CI seeks to improve student achievement, collaborative skills, metacognition, equitable participation, student autonomy, and approaches to learning. That’s just about everything that any teacher could possibly hope for their class of students!

My Contribution

Founders of CI state very clearly that all of this hinges on the task that students gather around. So, for my final M.Ed project, I will be investigating what a CI task looks like and design my own task (perhaps a whole unit) and reflect on this design process. I have the fortune of being able to attend Designing Effective Groupwork in Mathematics, a workshop offered by CI practitioners at the University of Washington. As I move forward to teaching at Branksome Hall Asia next year, I am interested to move from theory to practice and examine the feasibility of the use of CI in my classroom, and will post my thoughts as I go here.

Resources

Free mathematics CI tasks are available on the NRICH and the Complex Instruction Consortium websites, and Dan Meyer intends that CI be used for his Three-Act Math Tasks. See a CI lesson presented by Dan Meyer from his talk given at Cambridge University in March of this year. This gives you a great look at some of the ideas I’ve discussed above.

Also, if anyone is interested, Boaler is offering a free online course for teachers and parents through Stanford University called How to Learn Math. It’s available July 15 to September 27, 2013. Pass this on to interested parents and teachers!

 

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”?