One thing I have learned about effective formative assessment is that it is not done on the fly. Sure, I may be in the middle of a lesson one day, and realize that a good number of students have a misconception, or there is some key, prior knowledge that may be missing, and decide that I want to stop and give a quick formative assessment before I move on with instruction. This is what observant, knowledgeable teachers do. However, for formative assessment to have the greatest impact, it needs to be planned BEFORE the unit begins. If I am very clear about what my students are likely to already know (on a spectrum), what the next level of understanding looks like, and which activities will help students reach that next level of understanding, half the battle is won. 

Many of us believe that we know these things intuitively, since we have been teaching for so long, and therefore, we don’t need to spend too much time planning. However, let’s take mathematics as an example. If we are being honest with ourselves, how many of us are true mathematicians? I will be the first to admit that I am NOT. I always did pretty well in math throughout my education. I learned the algorithm, than I applied it–no big deal, right? Well, math isn’t taught that way anymore, and besides, being able to DO math, and UNDERSTANDING how mathematical concepts develop are two different things. My school has hired a math tutor for this specific purpose–to help teachers gain a better understanding of how students learn mathematics. Our tutor has been an invaluable resource. Unfortunately, not all schools have access to this type of expertise. So, how can schools empower teachers to become more knowledgeable about mathematics learning?

I have done a good deal of research on the topic of learning trajectories, and I am convinced that they are an integral part to the planning of any unit, whether it be math, reading, writing, social studies or science. A learning trajectory has been defined as  “a sequence of successively more complex ways of thinking about an idea that might reasonably follow one another in a student’s learning”(Smith et al., 2004, cited in Graf & Ariele-Attali, 2015, p. 196). Basically, I would describe it as a plan for learning which takes into account all possible levels of student understanding. 

To develop a learning trajectory you need to start with your standards or scope and sequence documents. First, choose an outcome that is small enough to work with. My example is: “represent and solve problems involving multiplication using efficient mental and written strategies”. Next, you may need to do some research to figure out how student understanding of this outcome grows. Research could include reading, but it could also simply be consulting with a math tutor, or other expert, doing an internet search, or engaging in a brainstorming session with your team members. The goal is to map out, from beginning to end, how students’ understanding develops. Once you have accomplished this, you will want to determine which learning engagements will help students move from one level of understanding to the next. This is going to pay off big time, as your planning will basically be taken care of for the remainder of the unit. 

Once you have the trajectory complete, you will need to preassess students, and place them on the trajectory based on what you see. Then, plan for small group instruction. This is the most important part of the entire process. I cannot stress enough how essential differentiation is. You need to find a fun, easy activity that the whole class can engage in without teacher assistance, so you can work with small groups. I usually choose a game that can easily be made more challenging by choosing bigger numbers, or doing an extension. In the meantime, I work with the small groups, including my advanced groups, on developing the skills and understandings they need to move them to the next level. A side note here. It may seem like a lot of time is being wasted as students play a game, while the teacher is working only with a group of 4-5 students. In reality, a lot more time is wasted when you teach the entire class the same concept, and the concept is really only appropriate for, at the most, 30% of the students. I would rather have my class playing a fun and engaging game, which gives them practice, while I work with a small group on exactly the concepts they need and are ready for. 

Finally, as the unit progresses, you will need to periodically revisit the trajectory, and make adjustments as needed. This is why learning trajectories are often called ‘hypothetical’. They are, and should be, a work in progress, which is constantly being refined. Your first draft is merely a hypothesis, and your work with students will inform the changes you make. 

So, what does this have to do with formative assessment? The real purpose of formative assessment is, or should be, to guide and differentiate instruction. This is precisely what the trajectory accomplishes. It is basically a way of breaking down the curriculum into bite sized chunks so that you have a better understanding of what students need to know, and how you can help them acheive that knowledge. To put it bluntly, formative assessment is ineffective if you, as a teacher, don’t understand what you are teaching. 

Now, I know that this all sounds like a lot of work, and it is, but I promise you, it is work that has big returns, and the more you do now, the less you will do later. This school year, I have very little planning to do when it comes to my multiplication unit. I have my trajectory, which includes student understandings, as well as instructional approaches, and I simply need to administer a preassessment and make my groups. Of course, I will continue to tweak the trajectory, and my small groups, but now I can put more of my energy into my students, and less into planning. Also, I recently presented on this topic at the Collaborative for innovative Education (CIE) conference (my school is one of six schools which belong to this collaborative), and a couple of teachers sitting at my table told me that they have created learning trajectories for other mathematics outcomes and they would be happy to exchange (unfortunately, I never heard from them, and I didn’t get their contact info.–call me!). The point is, if we share the work, we will save so much time!

So, this is where I share the work I have done on the multiplication trajectory with you 🙂 If you have done any work on trajectories, please share it with me as well. Also, if you would like to know more about the research on learning trajectories, please read the essay I wrote as part of my Master’s work for the University of Bath, entitled Using Learning Trajectories to Support Formative Assessment in Mathematics: An Example From a Grade Three Unit on Multiplication.

I am curious about what you have to say about learning trajectories. Have you used them? What do you see as the benefits or drawbacks? Do you have any advice for those trying to develop a trajectory?