* In what follows, I share what I typically do in order to co-construct learning progressions using student work. It is important to note that my current thinking is the result of (many) discussions, planning and leading PD with my colleagues- Mélanie, Stéphanie, Karine and Fabrice- at the school board. It also stems from professional learning led either by former school board math leads- Céline and Michelle- or by provincial mathematics facilitors- Maude, Jhonel, Nathalie, Jules and Denise.
A little while ago, I had this little exchange with Robert with regard to math progression documents:
He asked if I had blogged about what I was talking about. The short answer? No. So here it is…
For me, co-creating a progression of mathmatical reasoning using student work is one of the most powerful ways to build a teacher’s content and pedagogical knowledge (also known as MKT knowledge) in a simultaneous -and perhaps iterative- manner. As an example, we often hear of the importance of questionning– an aspect of pedagogical knowledge. I currently believe that open ended questions that produce cognitive dissonance and reorganization of prior ideas in a student’s mind are what we as teachers should ideally be aiming for. However, without having an understanding of what the mathematical content is and how it develops, how can we know what questions to ask? We can ask questions, but to what end? What are we aiming for? We need both content and pedagogical knowledge in order to effectively develop young mathematicians.
Analyzing student and trying to understand student thinking is what teachers must do everyday and in real time. In a PD format, it is therefore important that we frequently practice looking at student work and thinking deeply about it in a collaborative setting.
Here is how I have gone about co-creating mathematical progressions with teachers:
Do the math. No really- Do. The. Math. This is for everyone, myself included. We solve the problem that we will be asking the students to solve. This is a way to engage in the mathematics of a problem and also as a way to anticipate student thinking. I, as a facilitator of professional mathematics learning, aim to guide the subsequent discussion and question teachers about the concepts, procedures, representations, etc. of the problem. This is an inital phase, so I personally tend not to « push » teachers too much at this stage- more a question of building a safe learning environment. As for the problem itself, I want it to be a rich task– mutliple entry points, multiple possible solutions. Here is an example: It doesn’t seem like much, but I invite you to try and find different ways to solve this problem and also to ask yourself « Does this problem offer mutiple entry points? Are there different ways to solve this? How far can the mathematics extend? What can we add, remove, modify to open and extend this problem? ».
Get into a classroom. With copies of the problem in hand, we head to a classroom. It doesn’t matter if the students have learned the math behind the problem or not. The goal here is simply to get student work samples in order to have a range of mathematical thinking. No need for a big explanation of the problem or any hints given before students start. We usually read the problem with the students, clarify any unknown or difficult words, ask them to show their thinking, hand them a blank sheet of paper and let them at it. If a student asks « What am I suppose to do? » or « Do you want me to…? » or « Can I …? » we typically answer with either a shoulder shrug paired with silence or with « Do what you want to do ». To me, it also doesn’t matter if all students finish the problem or not. As mentioned before, the objective is to get a sample and a range of student thinking- fodder for our later analysis.
Here are a few student work samples from grade 4-6 students:
A first look. Once we get back from the classroom, I ask teachers a fairly general question: « What did you notice? ». Again, this is a « gentle », open and easy way to start for teachers. There is no right or wrong answer and everyone, if they choose, can share something they noticed. Also, I have found that if we skip this step and try to jump too deeply into the content, teachers naturally want to talk about what they saw in the classroom. The parallel I draw is that of using manipulatives with students for the first time- if they don’t have time to play with the manipulatives at the start, you’ll be fighting to keep their attention. Just as students with manipulatives, I think teachers need to start by exploring their own noticings and questions without my interference.
After this initial phase, I ask « Take a look at the work samples. Analyze them with a friend and discuss your thinking ». My instructions with regard to « analyze » are purposefully open and ambiguous- and I like it that way (for now). Usually, the much of the analysis is what I would call « superficial »- wrong vs right answers, neatness vs messiness of the writing, strong vs weak students, etc. To me, it’s all good. This part of the process serves me two purposes: 1) it gets those noticings out of the way for a deeper look at student reasoning later; 2) it is a sort of diagnostic tool which gives me insight into teacher thinking, conceptions/misconception about not only the content but also the pedagogy and beliefs about learning mathematics. I tend to listen a lot and ask questions to in order to make sure I understand their thinking. I also tend to mentally « mark » things that teachers said or did during this time. These mental marks help me plan what questions I may want to ask in the next phase or how I want to set things up to surface specific content elements or certain conceptions/misconceptions that teacher may have.
A second look. Now for the deeper dive. « With your friend, place work samples with the same or very similar mathematical reasoning together in the same pile. Then, place the piles in an order that you believe shows a progression with regard to mathematical reasoning. » If you haven’t noticed, I want teachers to look at the mathematical reasoning of the work and not the kind of superficial stuff I mentioned earlier. Depending on a number of factors, this may not be so easy for teachers. This is where I, as a facilitator of mathematics professional learning, come in. This is where I challenge and push with questions- « What kind of reasoning does this work sample show? How do you know?« ; « If the work shows a wrong answer, can we still determine the type of reasoning the student used? »; « If two work samples used different reprentations, can they have the same reasoning? »; « Why did you place this work sample ahead of this one in the progression? Is it possible that they are at the same « place »? ». In my opinion, this second look at the work samples is the meat of this whole process. If all goes according to plan, teacher mathematics knowledge gets challenged, reorganized and added to. This is also the most challenging part for me as I have many decisions to make- what to say, what to question, letting ideas simmer (or not), challenging ideas (or not), « reading » the confort level of teachers to know when to push and when to back off, etc. This is much like a teacher leading a classroom discussion, except my class is made up of adults. In a study (see p.164, section 6.2), Hilda Borko, Karen Koellner and Jennifer K. Jacobs identified this as leading high-press exchanges – a term borrowed from Kazemi and Stipek 2001 (referenced in the aformentioned section). Borko et.al mention that enacting these types of conversations with teachers is a challenge for novice leaders; however, I believe it is a challenge for most if not all facilitators including myself. I have gotten better at this skill, but I am still looking to learn and improve.
Comparing with progression documents. Once we have ordered the work samples into a progression we are happy with, I break out one or a few progression documents that already exist. Here are some possibilities:
I don’t use these as the « right answer ». I use them to compare our progression of student work samples to that of a progression document as a way to compare our thinking, maybe challenge it and to add to it. I have found that by placing this step after we have created our own progression rather than before places teachers in an inquiry stance, one where they are seeking to understand and build upon there knowledge rather than trying to follow the thinking of the author of the progression document.
Documenting the progression. How to capture the co-created progression? To be honest, I’m not sure on this one. Maybe a slide deck could work or maybe a gallery of photos. Scanning them and placing in a folder or even a Prezi could be other options. I’ve created a yet to be used (and openly shared up to now) website that organizes student work samples that my colleagues and I have started to collect:
Not sure where it will end up, but we plan on adding to it and offering it up as a resource for teachers, math leads, collaborative teams and for our use when working with teachers.
Thoughts from teachers Here are some comments from teachers who have gone through this process:
- I really like this. I’ve never looked at student work in this way.
- This helps me understand what comes before and what comes after my grade level.
- We stopped talking about grade level or marks and just looked at the math. We worried less about the curriculum (standards) and thought more about the progression.
- Student thinking can be very similar across many grade levels.
A last note about organizing the physical environment. Having led this process a number of times with my colleagues, we have found through trial and error and collective reflection that having two work areas is essential. The first is where we sit with our computers, personal items and a projector. This area is for some discussion, working individually and going through some slides. This is not a good area for analyzing student work. For this, we have found that a wall or a window works best. Having student work up on a vertical surface allows for easier access to all- the work is public and visible. It also allows for different people at different times to step forward and take a leadership rôle or to point something out. Also, the work can be moved around and placed either in a sequence or for a side by side comparison.
I have also facilitated a variant of this process in order to practice parts of the 5 Practices with teachers. I’m planning on writing a short post about it in the next little while.