- Utilizing ongoing formative feedback is an essential strategy that we as educators use to push the mathematical thinking of our students. Luckily, we have powerful tools and learning continuums to help us identify our students understanding, see where they sit in the larger scope of mathematical development, and support us in planning how we will move them to the next phase of the development. Cathy Fosnot's "Landscapes of Learning" and Dr. Alex Lawson's "Student Continuum of Numeracy Development" are two resources that have been developed to support educators, and both have found wide audiences due to their usability. In her book entitled "What To Look For" Dr. Lawson provides continuums for early numeracy development, both for addition and subtraction and also for multiplication and division. She identifies strategies that students use at different developmental phases of mathematical thinking and provides examples of what students might do or say when using those strategies. Also included are the key ideas that underlie the strategies that students use. These key ideas start at basic one to one correspondence and work their way through to more complex ideas around place value. Dr. Lawson's continuums are clearly laid out, and while they appear to be linear at first glance, we quickly see that several strategies might be used at each stage of development and that students might move both forward and backward as they experiment and test new strategies. Eventually we want students to move from direct modeling and counting strategies through to proficiency, but it's important to understand that each student may favor different strategies and demonstrate understanding at different points on their individual journeys towards proficiency. Cathy Fosnot takes a different visual approach to her landscapes of learning. Her continuum is laid out in a way that makes it clear that students will work their way through stages of development in generally the same way, but the paths they take between the strategies they employ and the tools they utilize may be very different. Fosnot's landscapes also incorporate the big ideas that students will be working on at different points in their journeys. Both resources are great to choices to use for the basis of our ongoing formative assessment, and both seek to help educators answer the three essential questions. What do my students know? What do my students need to know? And how do I get them there? First, educators need to uncover what our students understand and how their thinking, actions, and strategies demonstrate that understanding in class. Learning continuums provide us with key look-fors to determine our student's current level of understanding. For example, a student who is asked to add eight plus seven might decide that they would rather add five plus three, that's the eight, and five plus two, that's the seven. Lawson calls this strategy decomposition using five as an anchor. After noting this and perhaps asking the student to explain why they chose that strategy, we would be able to infer some of their key understandings. Likely, they have a solid grasp of hierarchical inclusion. The student knew that a five lived within that eight and could be separated out to be used. Once key strategies and key understandings have been identified, we can pinpoint where they are in relation to the larger developmental continuum. Knowing where our students are in their learning is essential, but so too is knowing what comes next and where they need to get to. Learning continuums allow us to put the individual strategies that students might use into context. Back in the previous example, the student who decomposed numbers using fives as an anchor appears to be farther along the continuum than the simpler counting strategies. We would want to give several more opportunities for them to demonstrate that they are fully past that phase before moving on. After confirming the student's location on the continuum, it's time to reflect on the questions we can ask and the activities we can present to help move this student to their next phase. Continuums give a great visual for determining that next step, and the resources that contain the continuums offer suggestions on games and activities to move students forward. Grouping students who are at a similar position on the learning continuum can be an efficient way to individualize instruction. Truly, basing formative feedback on learning continuums is a great way to ensure that each student in the class receives the instruction that they need in order to move forward in their learning.