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 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.

How to Learn Math – Jo Boaler Course Begins

Boaler – Session 01 – 1.1 – Math Perception Concept Map – DEABREU

I’ve just been able to start the Jo Boaler course – getting married this summer is exciting and means I’m not going to be able to give this course the attention I would like. On the other hand, I’m getting married this summer to a truly wonderful woman who I can’t be thankful enough for. 🙂

The course has started out great. Lots of data has been shared already about perceptions of mathematics and the journey people have taken with the subject. And it’s just beginning.

In session 1.1, Boaler asks students to make a concept map of the comments made by people interviewed about their experiences with mathematics. Here is that concept map.

Boaler - Session 01 - Math Perception - DEABREU


Groupwork on the Ground and in the Sky

Today was the first day of a five-day workshop with Karen O’Connell and Jess Griffin called Designing Effective Groupwork in Mathematics. As I’ve spent a year doing my M.Ed in cerebral mode, it was refreshing to talk with teachers about the nuts and bolts of how this might look in a classroom – the practical implementation of it all. I wanted to share here some of my initial take-aways and questions I still have.

I’m also planning to use this space to share my thoughts as my group and I move through the design of a groupworthy task. Watch this space! Your comments and feedback and questions are totally appreciated during this time as they always are!

Setting and Reactions

Karen and Jess have both been complex instruction (CI) practioners for a number of years and have a tremendous amount of experience to offer. In our opening activity, Jess led us through an origami box making task, with us in the role of students in groups of 4. The tables were cleared (KEY move), the task was introduced, roles (two of the many descriptions of roles: way one and way two) were introduced and explained (groupings were randomly assigned by the teachers), and a list of the abilities needed to complete the task successfully was presented. This was a key moment which I had been reading about – the first of two CI treatments called the multiple abilities treatment –  but seeing it in action stated with conviction while playing the role of student really hit home for me:

“Take a look at this list of abilities. There are quite a number of them that are needed to do this task. What are you bringing to your table? No one has all of these abilities, but each of you has at least one of them to offer.”

While doing the task, I immediately found myself with a role (resource monitor – and I love ticking lists and asking questions!), and ways to contribute to the task. I also found myself earnestly supporting others where I could, or being more verbal about my appreciation of other’s work.

As we were all doing the activity, Karen and Jess circulated, encouraging us to continue using “because” statements and asking good questions – like a coach, pointing out when a student is on the right track – while encouraging the group to recognize when a member was struggling with an idea or needed a voice. Yes, they were helping to move the math along, but they were also helping to move the talk along – talk which thereby facilitated the math moving along. This is the second of two CI treatments called assigning competence. Also incredibly key.

The task culminated in our making a “stand-alone” 1 page demonstration of our strategy and our prediction (for what the volume of a box made from a 20-inch sided square piece of paper might be given the four other boxes we had made, measured and analyzed). This was interesting – we couldn’t tell people about it. When showed to the class, no comments were allowed. Our paper explanations had to “stand alone” – and be understood just as they were. What a simple yet powerful idea as a way of presenting finished mathematics products to the class.


I already wrote about a number of “take-aways” throughout the “play-by-play” of the task, above. Here are some more.

There are some incredibly simple tweaks that Karen and Jess made to the task to encourage us to interact. First, there were four different sized pieces of paper (ergo, we needed to make 4 boxes), but only two task sheets and not enough cm cubes and beans to each have a sufficient supply for estimation. Thus we needed to share these latter two resources. The simple act of giving each student a task sheet or enough cubes for each student to work with might have kept us from talking until later in the task. In addition, the folding instructions (to move paper–>box) were naturally tough for some to follow, resulting in many group members needing help with others offering it.

One of our group members, Cameron, pointed out that the resource manager’s job had one crucial addition – to ask the teacher questions. Huge. Other group members were dependent on them as the way to communicate their questions to the teacher, thus the resource manager remained important throughout the duration of the task. Not including this has a student get resources, then (possibly) back right out of the task – “that’s it! Job done!” In addition, it implied that the teacher was just one of the many resources available in the room. Not the resource, but just one of them. A powerful implication. Instead of an “ask someone at your table then ask the teacher” rule, which could imply “don’t bother me” or “the teacher might be more useful (or more important) than the students,” we have a strong subtle implication of the equalized value of all in the room. Epic.

What strikes home most powerfully after today, however, is the self-similarity that is necessary for this kind of teaching to really be successful. We have all heard and likely agree that teachers must model for students what they want them to learn, yet we have all seen how teaching can sometimes be a bit lonely – whether self-imposed or not. The classroom door shuts, literally and figuratively. However, if we want our students to collaborate, must we not also collaborate? Must not our department meetings not be times for us to share things we are doing in our classrooms and get feedback? Must we not design tasks with our teaching partners to gain multiple perspectives on an activity in preparation for presenting it to students? Teaching is an incredibly creative profession, and creativity needs expression to be moulded, to evolve, to improve. (Many of you are likely already thinking the word “time!” over and over again in your heads. I recognize the practical and am indulging in a bit of optimism here 🙂 )

Questions Still Niggling

I have a TON of them, but my top 2 are:

1) How does CI look in a class with english language learners (ELLs)? CI is language (as in language of the classroom) heavy. Students are expected to talk in groups, to record findings understandably for others, to report their own learning in individual reports throughout units. How can we keep ELLs from losing status in the face of the great challenge they may appear to pose to their group members? How can they succeed and contribute?

2) CI works beautifully for exploratory tasks like the origami task. But CI practitioners readily admit we can’t do those all year long. How does CI look for more abstract and calculation based concepts like algebra, logic, functions, and the like? How could the teaching of this look different and be more engaging?

On From Here

Our task this week is to design a groupworthy task in a group of 3-5 people on a math topic – a task that we will then “micro-teach” during a 20 minute session. Fitting, especially considering my comments about self-similarity. As Lotan (2003) says: the creation of a groupworthy task is itself a groupworthy task. My group is pumped and ready for action! We have challenged ourselves to come up with a groupworthy way of involving students in learning algebra concepts connected with completing the square. Wish us luck!

Resources from Today

The book we are reading for this course is Smarter Together: Collaboration and Equity in the Elementary Math Classroom. Written by practitioners and researchers of CI, it’s already reaping great rewards in our explorations.

A colleague, Kate, suggested we look at Lab Gear, a manipulative designed for teaching algebra, during our lesson design. Have a look at Henri Piccioto’s site (he’s the creator) for a summary of what it can do and some free resources for how it can change teaching.

See the Lotan (2003) article that we read today – a short and sweet summary of “look-fors” when designing (or modifying old tasks/questions to make) groupworthy tasks.


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.


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!