Though evaluation and assessment are many times used interchangeably, there is a subtle difference. Evaluations are many times viewed as summative, a final product that demonstrates student conception or understanding of a concept. Assessments can be used as a means to evaluate students, but assessments should do more. Assessments should move students forward in their understanding of a concept. Assessments geared toward increasing understanding are commonly referred to as formative assessments. Petit, Zawojewski, and Lobato (2010) describe formative assessment as “a way of thinking about gathering, interpreting, and taking action on evidence of learning by both the teachers and students as learning occurs. (p.68)”

According to Petit, Zawojewski, and Lobato (2010), formative assessments should improve instruction by clarifying and sharing learning intentions and criteria for success, providing feedback that moves learners forward, and activating students as instructional resources for one another. Making expectations and objectives clear can help students attempt to achieve satisfactory results as well as demonstrating how students can achieve these expectations (Petit, Zawojewski, & Lobato, 2010). Teachers who provided feedback to student assessments encourage self-reflection; however, identifying special actions a student can do that can support improvement are even stronger examples of teacher feedback (Petit, Zawojewski, & Lobato, 2010). Mathematical practice standards within the Common Core State Standards for Mathematics (NGA & CCSO, 2010) and the Alabama College and Career Readiness Standards (ASDE, 2010) require teachers to have students critique the reasoning of others, one of Petit, and Zawojewski’s (2010) elements to effective formative assessments. Though these three elements provide an excellent starting point for a teacher to use formative assessments to guide instruction, students can be more active in this process.

Petit, Zawojewski, and Lobato (2010) clarify that formative assessments should activate students as the owners of their learning and engineer effective classroom discussions, questions, and learning tasks that elicit evidence of learning. Activating students as owners of their own learning provide openness (NCTM, 1995) within assessment that is not offered in all mathematical classrooms. Students who complete self-assessment tasks may be asked to create an individual work plan, create problem-solving activities to explore, provide opportunities to evaluate their own performance against their work plan or against jointly created criteria for performance, identify specific problems they are having (Petit, Zawojewski, & Lobato, 2010). The use of assessment portfolios is a good example of this type of formative assessment (Garfield & Chance, 2000). Analyzing discourse within the classroom can provide strong evidence of what is valued within the classroom. Orchestrating productive mathematical discussions is an art that not only provides a means for students to build mathematical power, but also continually assess student understanding in the present(Stein & Smith, 2011). Through monitoring, teachers can provide sequenced explanations that build on each student strengths and help build connections between mathematical concepts (Stein & Smith, 2011).

After incorporating formative assessment in the classroom, it is imperative to know what to look for using the assessment triangle (Petit, Zawojewski, & Lobato, 2010). The assessment triangle’s vertices consist of observation, interpretation, and cognition (Petit, Zawojewski, & Lobato, 2010). The cognition vertex refers to the ways students understand, misrepresent, misunderstand, or use prior knowledge that influences this understanding of a particular concept (Petit, Zawojewski, & Lobato, 2010). The observation vertex refers to the descriptions from research that produces specific responses (Petit, Zawojewski, & Lobato, 2010). Interpretation in formative assessment or in the assessment triangle is the tools and methods used to reason from the evidence (Petit, Zawojewski, & Lobato, 2010). “To develop effective assessments that lead to sound inferences, each corner of the assessment triangle must connect to the other in a significant way regardless of the level of assessment. (Petit, Zawojewski, & Lobato, 2010, p. 70-71)”

Formative assessment is one of the most robust tools teachers can use to improve student learning within the classroom (Petit, Zawojewski, & Lobato, 2010). Though it may seem like more work for a teacher who has not incorporated these strategies in the past, students engaging in self-reflection and critique actually take the burden away from teacher grading and move toward teacher facilitation. Good instruction uses formative assessment in ways that when summative assessment takes place, there is no surprise. It is imperative that teachers use these tools appropriately to improve student understanding within the classroom.

Alabama State Department of Education. (2010). Alabama course of study mathematics: Building mathematical foundations of college and career readiness. Montgomery, AL: Author.

Garfield, J. & Chance, B. (2000). Assessment in statistics education: Issues and challenges. Mathematics Teaching and Learning, 2, pp. 99-125.

National Council of Teachers of Mathematics. (1995). Assessment standards for school mathematics. Reston, VA: Author

National Governors Association Center for Best Practices & Council of Chief State School Officers. (2010). Common Core State Standards for Mathematics. Washington, DC: Authors.

Petit, M.M., Zawojewski, J.S., & Lobato, J. (2010). Formative assessment in secondary school mathematics classrooms. In J. Lobato (Ed.), Teaching and learning mathematics: Translating research for secondary school teachers (pp. 67-75), Reston, VA: NCTM.

Stein, M. & Smith, M. K. (2011). 5 practices for orchestrating productive mathematical discussions. Reston, VA: NCTM.