Teachers make numerous instructional decisions in each lesson – a common example would be how teachers interpret various non-verbal cues from students to determine if the pace of the lesson is too fast or too slow. However, to promote deep, meaningful learning, teachers use thought provoking activities and questions to elicit student responses. Teachers then listen to and observe student responses to decide on appropriate follow-up questions, and how to adapt the next phase of the lesson. While summative assessment has been described as assessment of learning, and is typically characterized by end of chapter tests and grades, formative assessment has been described as assessment for learning, where instruction is informed by students’ responses to tasks, questions and discussions about important mathematics.
In their seminal study, Inside the Black Box, Black and Wiliam (1998) reviewed over 500 studies related to classroom practices that contributed to formative assessment. From this review, they found that the formative assessment practices had an effect size between 0.4 and 0.7. As they described, an effect size of 0.4 is equivalent to an average student participating in an innovation recording the same achievement as student scoring in the top 35% of students not participating in the innovation (p. 141). One of the most important characteristics of productive formative assessment, as identified by Black and Wiliam (1998), was feedback “about the particular qualities of his or her work, with advice on what he or she can do to improve” (p. 143).
In more recent meta-analyses completed by John Hattie and the Educational Endowment Foundation, they found similar, high-impact results for classroom practices that emphasized the importance of feedback. Essentially, feedback is information that is used by the learner to close the gap between current performance and the intended goal. What is often misunderstood about feedback is the perception that it needs to be frequently provided by the teacher – that more is better. There are two issues with this perception. First, the quality of the feedback is far more important than the quantity of feedback. This is the difference between knowing that something is wrong (and being told such, over and over), and knowing how something can be improved. Second, learners should not need to depend on one source for all of their feedback. Feedback should be multi-dimensional and come from multiple sources, including classmates, available resources, and even oneself. By having students communicate their insights to other students makes learning more visible – it motivates idea formulation for the speaker and serves as potential feedback for the listener. The more opportunities students have to communicate their reasoning and hear the reasoning of others, the more likely they will come to recognize what it means to know and make sense of mathematics. Self-assessment and self-regulation, which is students’ ability to recognize when they understand a concept or skill and when they need to seek additional support, is the goal of productive formative assessment and should inform how classroom instruction and resources are organized.
To increase the quality of formative assessment in your class, it is worth considering the following three questions:
On the surface these three questions seem a bit generic and more of the type of questions you would ask yourself if you were taking a trip – or even worse, lost! With respect to teaching and learning, if the content you teach is organized in an instructional sequence, and the activities and questions are organized according to an underlying conception of how student learning of that content develops over time, then formative assessment is very similar to taking a trip with many decision points along the way.
This is an assessment of students’ prior knowledge related to this goal. In some cases, it may be appropriate to observe how students respond to questions that exemplify the content goal. In other cases, the end goal is a bit further off on the horizon, and it is more appropriate to pose problems that allow students to make connections between their prior knowledge and related skills and concepts in the instructional sequence related to the goal. Essentially, this question motivates teacher inquiry about student learning.
There are numerous opportunities in a sequence of activities to assess progress toward a goal. However, rather than using hundreds of questions and consequently generating too much information to process, it is far more useful to identify a handful of generative, thought-provoking questions for specific concepts in which likely student responses and productive instructional moves can be internalized, and used on the fly. Ideally, when selecting tasks to inform instruction the likely students responses would already be known, as well as the useful instructional moves to support student progress toward the goal.
Effective formative assessment requires the use of authentic information from students about what they know to make instructional adjustments that are responsive to students’ ideas. Of the three questions related to formative assessment this is the most challenging to accomplish, but it is also the most rewarding to experience as a teacher. Developing greater confidence with instructional adjustments requires asking good questions, and posing probing questions and follow up activities that are related to students’ responses to those questions.
For example, if a typical student response to a question involving addition of fractions (e.g, 3/4 + 1/8) is to add the numerators and denominators (e.g., 4/12), a related instructional move would be to ask students to model this problem using a number line or fraction bars, or discuss in groups if the result should be closer to 0 or to 1, and explain why. When these types of problem questions are aligned with students in-the-moment responses, they promote students’ deeper engagement in the content, helps students make connections between other students’ responses, and helps students model strategies of how they could justify their answer.
Organizing instruction to include these in-the-moment formative assessment opportunities has a greater likelihood of supporting the formative assessment practices and quality feedback found by education researchers to be one of the greatest contributors to student achievement.
First, it is important to recognize that some degree of formative assessment is being used in most mathematics classrooms. Adjusting the duration of an activity, responding to students’ questions about homework, or having students share their solution strategy all include aspects of formative assessment that use information received from students to adapt instruction. A distinguishing feature of the last two examples is that when teachers respond to students’ questions or have students share their ideas, these are potential opportunities for students to receive information that can be used as feedback to improve their learning.
One of the most productive ways to improve the quality of formative assessment is to find ways to create opportunities to see and hear student thinking. Capturing students’ ideas visually and showcasing to the rest of the class on a whiteboard, tablet or projector screen, allows students and the teacher to see how mathematical ideas are being communicated using diagrams, graphs, and/or symbols. The format and organization of the work can also communicate mathematical structure – e.g., think of a cluster of numbers compared to those same numbers organized in a sequence, or in a table. Written work can be viewed, analyzed, edited or re-viewed at a later time. It can also provide students who are less adept at listening more opportunity to make sense of what is being communicated.
Productive classroom discourse is also an important feature of feedback rich classrooms. Even though verbal discussions produce information that is fleeting, articulating an argument and listening to mathematics helps to develop an awareness of how mathematical ideas are communicated and justified verbally. Even though this doesn’t have to be the case, written mathematics often represents a close-to-final solution to a problem. In contrast, having students verbalize what they are thinking about can offer the class a window into ideas in process. Also, one students’ ideas can lead to a conversation among several students, promoting greater engagement in mathematical reasoning. It is worth noting that unless students can hear what another student is saying, then asking a student to verbalize is only for the benefit of that student. To enhance opportunities for feedback, it is essential to establish norms where students who are speaking can be heard by others, including the teacher!
Assessment information can be gathered in many ways. However, a range of questions that reveals students’ recall of knowledge, their fluency with strategies and representations, and ways they apply their knowledge in new ways is needed. So, what are your instructional goals? More importantly, what are the questions and tasks that illustrate those goals? What are the skills, strategies and explanations that would exemplify those goals? From this, what are the tasks that would elicit those types of student responses? Once you are able to identify the student responses that exemplify the achievement of those instructional goals, the selection, adaptation or design of good questions can be accomplished with a clear target in mind. Even though this seems counterintuitive – identifying exemplar student responses first before you select tasks – since instructional decisions are based on student responses, this type of planning is more closely aligned with formative assessment practices and student-centered instruction.
David Webb is Associate Professor of Mathematics Education at the University of Colorado Boulder and the Executive Director of Freudenthal Institute USA, an international research collaborative for mathematics education. Dr. Webb’s research interests are in the areas of the preparation of mathematics teachers, classroom assessment, and the design of professional development activities. Recent research projects have focused on studies of teacher change in classroom assessment, the impact of reform curricula on student learning and achievement, and teacher design and use of formative assessment tools. Dr. Webb taught mathematics and computer applications courses at both the middle school and high school levels in Southern California. He currently teaches methods courses for prospective middle and high school mathematics teachers, and graduate level seminars that focus on the nature of mathematics and mathematics education.
Dr. Webb is a member of the American Educational Research Association, the National Council of Teachers of Mathematics, the National Council for Supervisors of Mathematics, and the National Society for the Study of Education.