###### Assessment, Classroom Environment, Mathematics

#### Classroom Assessment in Mathematics: Paying Attention to Students’ Mathematical Thinking

Assessment does not merely occur at the end of a unit or course. Rather, it occurs constantly and is an integral part of daily classroom practice. Assessment is at the heart of teachers’ work as it focuses on paying attention to students’ mathematical thinking and acting accordingly.

Research suggests that assessment should be integrated into all aspects of teaching and learning in a seamless manner to inform instructional decisions in ways that promote student learning (e.g. Carless, 2007). Students’ learning is supported when the moment-by-moment actions and decisions that teachers make during teaching are informed by evidence of students’ understanding (Leahy, Lyon, Thompson, & Wiliam, 2005). However, these actions and decisions require focused attention in order to make students’ mathematical thinking and understanding visible.

There are many ways that teachers provide opportunities to elicit and listen to student thinking, such as observations during problem-solving, informal interviews during class, or using focused questions during mathematical discussions. These methods allow teachers to be responsive to students’ understandings and adjust instruction as well as deal with particular understandings with individual students.

Assessment can be characterized as questioning, listening and responding to student thinking in order to move that thinking forward (c.f. Davis 1996; Suurtamm, Koch, & Arden 2010). Opportunities to elicit student thinking can be incorporated into lessons, even in the planning stages, for instance, by thinking ahead of time of the kinds of questions that could be asked to make student thinking visible. These questions could occur in a whole class discussion, in individual interviewing or in conferencing with small groups as they work on problem-solving. Questions such as “Why does that make sense?” or “Why do you think this relationship is linear?” or “Can you explain your thinking to your partner Emily?” help make student thinking visible and help inform teachers about next steps (Suurtamm, Quigley & Lazarus, 2015). These types of questions also tell students that their thinking is valued and contributes to their sense of themselves as a mathematical thinker.

These methods allow teachers to be responsive to students’ understandings and adjust instruction as well as deal with particular understandings with individual students.

This focus on attending to student thinking appears in many ways in the mathematics education world. One area of focus is called professional noticing which can be defined as “(a) attending to children’s strategies, (b) interpreting children’s understandings, and (c) deciding how to respond on the basis of children’s understandings” (Jacobs, Lamb, & Philipp, 2010, p. 169). Similarly, Silver and Smith (2015) suggest that formative assessment is embedded in Smith and Stein’s (2011) five practices for facilitating mathematical discussions; anticipating, monitoring, selecting, sequencing and connecting. These practices encourage teachers to pay close attention to student thinking and to respond appropriately to that thinking, which are sound formative assessment practices. Furthermore, Liljedahl’s research into what he terms “thinking classrooms” provides another example of classrooms where assessment is on-going and embedded in observations and interactions with students’ collaborative problem solving (Liljedahl, 2016).

In my research observations of classrooms in Ontario, I have seen many instances of teachers engaging their students in activities that help to make student thinking visible. These teachers also have strategies for probing and prompting student understanding and to determine next steps to move that thinking and understanding forward.

Christine Suurtamm is Vice-Dean, Research and Professor of Mathematics Education at the Faculty of Education, University of Ottawa. Her research focuses on the complexity of teachers’ classroom practice and of teachers’ formative assessment practices. She has been the lead researcher on several Ontario Ministry of Education large-scale projects to examine mathematics teaching and learning. She is also Director of the Pi Lab, a research facility supported by the Canada Foundation for Innovation. She was the Canadian representative on the National Council of Teachers of Mathematics Board of Directors, the Co-chair of the Ontario Ministry of Education Early Math Expert Panel, and Co-Chair of several international panels on assessment in mathematics education. She has received several teaching and research awards.

Interested in learning more about Christine Suurtamm? Visit her other work on TLX, such as Leaders In Mathematical Thinking and Thoughts on Teaching and Learning Mathematics.

References

Carless, D. (2007). Learning-oriented assessment: Conceptual bases and practical implications. *Innovations in Education and Teaching International*, *44(1), *57-66.

Davis, B. (1996). *Teaching mathematics: Toward a sound alternative*. New York, NY: Routledge, 1996.

Jacobs, V., Lamb, L., & Phillip, R. (2010). Professional Noticing of Children’s Mathematical Thinking. *Journal for Research in Mathematics Education, **41(2),* *169-202.*

Leahy, S., Lyon, C., Thompson, M., & Wiliam, D. (2005). Classroom assessment: Minute by minute, day by day. *Educational Leadership*, *63*(3), 18-24.

Liljedahl, P. (2016). Building thinking classrooms: Conditions for problem solving. In P. Felmer, W. Van Dooren, & J. Kilpatrick (Eds.). *Posing and solving mathematical problems: Advances and new perspectives*. (pp. 361-386). New York, NY: Springer.

Silver, E. & Smith, M. S. (2015). Integrating powerful practices: Formative assessment and cognitively demanding mathematics tasks. In C. Suurtamm & A. Roth McDuffie (Eds.) *Assessment to enhance teaching and learning*, (pp. 5-14). Reston, VA: NCTM.

Smith, M. S., & Stein, M. K. (2011). *5 Practices for orchestrating productive mathematics discussions*. Reston, VA: NCTM.

Suurtamm, C., Koch, M. J., & Arden, A. (2010). Teachers’ emerging assessment practices in mathematics: Classrooms in the context of reform. *Assessment in Education: Principles, Policy, and Practice, 17*(4), 399-417.

Suurtamm, C., Quigley, B., Lazarus, J. (2015). Making space for students to think mathematically. *What Works: Research into Practice, Research Monograph 59*.

## 2 Comments

## Laura Servage

Interesting stuff. Especially appreciated the reference to “professional noticing,” which has clear applicability to other disciplines. Of course strong formative assessment (and building skills as a teacher toward this) require more curricular and pedagogical “breathing space” than teachers are offered in practice. This isn’t an excuse to give up on improvement efforts, but more a comment on what is possible if that space is valued and protected.

I tuned out of math and sciences in junior high, and avoided them like the plague in high school. I have lasting regrets about this, so I’m always interested in teaching theory in these areas.

## Chris Suurtamm

I agree – teacher’s professional judgment plays a huge role in formative assessment and for that, they need space as well. I often talk about making space for students’ mathematical thinking but I also believe in space for teachers’ mathematical and pedagogical thinking and exploration. I actually think that the curriculum provides some of that space for teachers as teachers can connect curriculum expectations in various ways and design or use tasks that combine mathematical ideas and strands. It takes some experience and perhaps some collaboration and sharing with other teachers but I think that it is possible.