## Formative Assessment in Mathematics

“Formative assessment is an active and intentional learning process that partners the teacher and the students to continuously and systematically gather evidence of learning with the express goal of improving student achievement.” (Moss and Brookhart, 2009)

This definition implies three phases of formative assessment.  In order for the assessment to be intentional and systematic, there must be a planning phase.  Once planned, we move into the evidence-gathering phase, making sure that the methods used are active and produce clear indications of the level of student understanding.  Finally, the data must be used to provide feedback and adjust instruction in order to improve overall student achievement.

So how does this process of formative assessment work for a math teacher?  Let’s take a look at how formative assessment can be imbedded throughout your unit planning to maximize student success.

Planning

First, take a look at your topic, and ask yourself three questions:

• What are the essential understandings that students should take away from this lesson or unit?
• What are common misconceptions that will need to be addressed?
• What is the learning target for lesson or unit?

An example might be adding and subtracting fractions with unlike denominators.  Students need to understand the importance of writing fractions with a common denominator before adding or subtracting.  It is common for students to think that they would then just add the “tops” and add the “bottoms” in order to arrive at their answer.  A clear learning target for students (such as “I can add fractions with unlike denominators”) is critical for the formative assessment process.  Once you and your students have a clear understanding of the goal and the common missteps along the way, you can begin to plan the learning experiences and formative assessment checkpoints that will lead to this goal.

Evidence-Gathering

As the learning process proceeds, it quickly becomes time to gather evidence of student learning.  This evidence can be formal or informal, verbal or written, or linguistic or non-linguistic, but it must be planned for, gathered, and used.  In mathematics, there are generally two forms of knowledge that you may be formatively assessing: procedural and conceptual.  As you introduce the concept of a common denominator, it is important to gauge student understanding before you proceed.  Some ideas for formatively assessing conceptual mathematical knowledge may include:

• Hand signals, such as thumbs up/thumbs down, 1-2-3-4-5, etc…
• Timed writing in response to a question or prompt, perhaps including a sentence stem such as “I can’t add ½ and ¼ without finding a common denominator because ________.”
• Colored response items, such as red/yellow/green triangular prisms, blocks, cups, or plates
• Timed pair share, giving students a chance to explain the concept to a partner, and also listen to their partner’s explanation

Once students have some conceptual understanding, you will then spend time helping students understand the process of getting a common denominator and using these new fractions to add and subtract quantities.  This will create new opportunities for formative assessment.  For procedural knowledge, some methods to gather evidence of student learning may include:

• “What’s Wrong?” analysis of incorrect work to help students begin to identify common errors and misconceptions
• Response Cards, where students hold up their answer to either a multiple choice or free response question simultaneously, using colored paper, white boards, or perhaps an electronic response system
• Take and Pass, where one student begins the work and then passes the work to another student who does the next step, who then passes to another student, etc.  Could be done as a relay race as students become more proficient
• Scavenger Hunt (also called Around the Room or I have/Who has), a circular set of problems, where one answer leads to another question and the student ends where they began
• Side-by-Side Problems, in which two students work two columns of different problems that have the same result (For example, the answer to #1 on both columns might be 5, but the two problems are different.)

Notice that both the conceptual and procedural formative assessments provide quick, timely feedback to both you and the student about their level of understanding.