Assessment in a System of Practice
Attending to the details of students’ thinking allows you to assess where students are at both conceptually and procedurally in the midst of instruction. This requires listening to and observing (attending) students’ actions on paper, with their use of models/manipulatives, or their gestures and verbal explanations. The assessment flows from interpreting students’ thinking. Is the student’s explanation based on a nugget of a good idea? Is the strategy a good one but may have a computational error? Is the thinking mathematically sound but partially understood or exhibits a full understanding? What properties of operations might implicitly be understood by the students that you might draw out? The interpretation, the assessment, is based on teacher knowledge of the mathematical content and the progressions of student thinking that students pass through. Deciding how to respond both in-the-moment as well as in one’s planning for the next lesson(s) reflects your adjustments necessary to guide students forward towards comprehending and executing the targeted mathematical benchmarks. Attending, interpreting, and deciding how to respond are the teacher noticing skills (Jacobs, et al., 2010) that allow educators to be responsive to the students’ mathematical understanding as they progress in their learning (Richards & Robertson, 2016).
A System of Practice
The Minnesota STEM Teacher Center website’s new professional development materials outlines A System of Practice that merges ideas from Simon (1995), Hiebert et al. (1997), Jacobs, et al., (2010), and Richards and Robertson (2016) along with the eight instructional practices outlined in NCTM’s Principles to Actions (2024, 2014) and elaborated further in the Mathematics Teaching Framework (Huinker & Bil, Taking Action Series, 2017). Figure 1 captures how assessment, differentiation, language development, and the responsive teacher noticing skills are integrated into the teaching framework. The merged model places a focus on anticipating students’ thinking that teachers need to bring to the planning process (Jacobs & Empson, 2021, Hiebert, et al., 1997, Simon, 1995).
Having a clear mathematical goal for the lesson is the first step of the instructional process. It serves as the goal post as the lesson unfolds. The merged model elevates the importance of planning assessment strategies prior to the selection of the instructional tasks. Anticipating how students might progress, the range of possible learning trajectories, and what student understanding looks and sounds like are thought through prior to task selection. Instructional tasks are shaped around how the evidence might best reveal student understanding. Will ‘Task A’ actually produce the level of student thinking one is anticipating? Moving assessment forward in the planning process shifts the emphasis from a series of pre-determined instructional tasks from which the students might benefit to tasks shaped and modified around what one already knows of one’s students prior knowledge so that that tasks are more responsive to what students already know, can access (low floor, high ceiling), and be adaptable mid-lesson if necessary. 
Figure One
This sequence forms the “system of instruction” outlined by Hiebert, et al. (1997) where teachers select a sequence of problems based on the possible developmental learning trajectories students might move through. The trajectory is the teacher’s vision of the mathematical path that the class might take. It’s based on the teacher’s projection about how learning might proceed along the path, what constitutes student understanding. The trajectory guides the teacher’s task selection, but feedback from students and the teacher’s assessments of the levels of understanding lead to revisions in the trajectory. Tasks are selected purposefully, but the sequence can be revised. (pp. 33-34)
Defining Assessment
Assessment is often confused with the instruments used to collect data; a unit test, an exit slip. Those are instruments to collect assessment data. Assessment is an act of interpretation and the subsequent decision making and feedback resulting from that interpretation. Assessment is built on the information gathered before, during, and after the instructional task(s) are undertaken by the learners. It is the knowledge that the teacher has of the students’ level of developmental understanding of the mathematical concepts based on progressions of strategies, the level of efficiency and fluency by which the strategies were enacted, and the level of precision and accuracy of the work. This involves interpreting students’ thinking and using that information to decide how to plan, adjust, adapt, elicit, differentiate, scaffold, etc. The data is collected prior to students engaging with the task (prior knowledge inventory), while the students are working (formative assessment), and after a unit is completed (summative assessment). The assessment data is the feedback needed to inform instruction in-the-moment, to be used to launch the next day’s lesson, to differentiate a lesson to allow access points for all students, and to provide data for additional support to further learning. It is also the feedback provided to students so that they can monitor their own progress and decide next steps. Instruction and formative assessment in particular are so intertwined that they are at times indistinguishable (Levi & Ambrose, 2018). Having an organized framework by which to interpret students’ thinking and resulting artifacts is key to grounding instructional decision making. Only then does the data become useful. (See Figure Two)
Figure Two
Planning & Assessment Cycle
Assessment flows from monitoring students’ emerging procedural fluency. Determining if the student’s understanding is at the initial and often messy stage, or at intermediate, or refining stages influences the type of assessment data one is looking for/implementing and therefore which tasks are best suited for drawing out student thinking. In the midst of the lesson, monitoring influences the types of questions, prompts or in-the-moment adjustments to a task so that the student(s) can progress in their thinking. From the student artifacts – physical, drawn, written, gestured, and verbalized – the interpretation of students’ progress shapes the planning for the next day’s lesson. The assessment process is iterative: planning, in-the-moment adjustments, interpretation of artifacts, planning… This is a central core element of responsive teaching.
Assessment Feedback
Providing “high information feedback” to students has a statistical effect size of .99 according to a recent meta-analysis (Wisniewski, B., Zierer, K., & Hattie, J., 2020). What constitutes such feedback consists of three types: “feed-up”, “feed-back”, and “feed-forward” (p. 2). “Feed-up” provides students with a clear understanding for the learning goals to be accomplished. “Feed-back” provides students with details about what they have accomplished in relation to the learning goals. “Feed-forward” provides students with a vertical pathway for where the strategies and processes are headed towards. High information feedback helps students “not only to understand what mistakes they made, but also how they made those mistakes and what they can do to avoid them the next time” (p. 12).
This level of feedback aligns with the research on pressing students for the details of their thinking and comparing and contrasting their thinking to those of their peers (Kazemi & Stipek, 2001; Webb, et al., 2014, Bishop, 2021). Look back at the Mathematics Teaching Framework, the blue portion of the System of Practice in Figure One. Posing purposeful questions to elicit and use student thinking, and using and connecting representations is how a rich source of formative assessment data emerges for the teacher to interpret and respond to in-the-moment. This rich source of teacher knowledge of their students, along with analyzing student artifacts after class, provides guidance in planning how the next lesson needs to be shaped, differentiated, and/or enriched, as well as how the next lesson is launched connecting the previous day’s learning with the focus on next steps. Up, back, and forward!
References
Bishop, J.P., (2021) Responsiveness and intellectual work: Features of mathematics classroom discourse related to student achievement, Journal of the Learning Sciences, 30:3, 466-508, https://doi.org/10.1080/10508406.2021.1922413
Hiebert, J., Carpenter, T.P., Fennema, E., Fuson, K. C., Wearne, D., Murray, H. Olivier, A., and Human, P. (1997). Making sense: Teaching and learning mathematics with understanding. pp. 29-41. Heinemann Press: Portsmouth, NH.
Huinker, D. and Bil, V. (2017). Taking Action: Implementing effective mathematics teaching practices in kindergarten – grades 5. Taking Action Series. National Council of Teachers of Mathematics.
Jacobs, V.R. and Empson, S.B. (2021). Profiles of teacher;s expertise in professional noticing of children’s mathematical thinking. Proceedings of the 43rd Annual Meeting of PME-NA, pp. 652-661. Retrieved from: https://www.pmena.org/pmenaproceedings/PMENA%2043%202021%20Proceedings.pdf#page=667
Kazemi, E. & Stipek, D. (2001). Promoting conceptual thinking in four upper-elementary mathematics classrooms, The Elementary School Journal. 102 (1) pp. 59-80.
Levi, L. and Ambrose, R. (2018). Cognitively guided instruction and formative assessment. In E.L. Silver and V.L. Mills, Eds., A Fresh Look at Formative Assessment in Mathematics Teaching. pp 41-60. NCTM: Reston, VA.
National Council of Teachers of Mathematics. (2024, 2014). Principles to actions. National Council of Teachers of Mathematics: Reston, VA.
Richards, J., & Robertson, A. D. (2016). A review of the research on responsive teaching in science and mathematics. In A. D. Robertson, R. E. Scherr, & D. Hammer (Eds.), Responsive teaching in science and mathematics (pp. 36–55). New York, NY: Routledge
Simon, M.A., (1995).Reconstructing mathematics pedagogy from a constructivist perspective. Journal for Research in Mathematics Education, 26, pp. 114-145.
Webb, N.M., Franke, M.L., Ing, M., Wong, J., Fernandez, C.H., Shin, N., and Turrou, A.C. (2014). Engaging with others’ mathematical ideas: Interrelationships among student participation, teachers’ instructional practices, and learning, International Journal of Educational Research, 63, pp 79-93.
Wisniewski, B., Zierer, K., & Hattie, J. (2020). The power of feedback revisited: A meta-analysis of educational feedback research. Frontiers in Psychology, 10, 3087. https://doi.org/10.3389/fpsyg.2019.03087