Mathematical concepts build upon each other cumulatively like the pieces in a Jenga tower. If students don’t have a solid understanding of the day’s lesson, this confusion could undermine their learning down the road—much like a missing piece in its foundation can topple a stack of Jenga blocks.
The right formative assessment helps properly trained teachers identify gaps in students’ understanding, enabling them to adjust their instruction in real time and clear up students’ misconceptions before these become long-term obstacles to learning.
When teachers are empowered with early feedback, instead of waiting to collect it from an end-of-unit exam, they can be far more effective in meeting students’ instructional needs.
But formative assessment only works well if teachers know how to take full advantage of it.
Here are three important pieces of advice for transforming formative assessment in mathematics into a powerful tool that drives better student achievement.
At its core, effective formative assessment is not a single event, tool or data point—it is an ongoing instructional mindset. True formative assessment is far more than a static data point; it is the “relentless attention to evidence of student thinking” (Popham, 2012) that transforms a classroom into a responsive environment.
It begins by unearthing pre-existing knowledge to guide unit planning, but its real power lies in how that evidence is “elicited, interpreted, and used by both teachers and learners” (Wiliam, 2011).
When a teacher uses an intentional system of embedded probing, the resulting analysis doesn’t just sit in a gradebook—it triggers immediate instructional modification. This support enables students to identify their own gaps in real-time, replacing blind progress through a unit with feedback that is specific, actionable and delivered when they are ready to make a pivot.
When assessment is used this way, it becomes a live map for student success. With that purpose in mind, the first step toward stronger formative assessment is reconsidering what we pay attention to when students respond.
Shifting focus from answers to thinking
In math, students might get the answer to a problem correct despite faulty reasoning—and an incorrect answer can easily be derived from spot-on reasoning marred by a simple procedural error. This is why teachers should look beyond right or wrong answers and focus instead on the reasoning students use to solve a problem.
Teachers can make students’ thinking and reasoning processes visible by asking them to show or explain their work. That’s not to say the answer itself doesn’t matter—only that it’s one small piece of the puzzle in determining how well students understand key concepts.
Using strategy analysis to inform instruction
Identifying the strategies that students use to solve problems offers invaluable insight into their level of conceptual understanding and where they are on the continuum of math reasoning.
Suppose students are given the problem: “Coral and Sean are counting crayons. Coral counted 15 blue crayons. Sean counted nine red crayons. How many crayons did they count together? Show your work.”
One child might draw 15 blue crayons and nine red crayons and then count them all to get 24, indicating they’re using a “counting all” strategy. Another might draw a number line and count from 15 to 24, suggesting they’re using a “counting on” strategy.
The student who is still counting all probably needs instruction in concepts such as cardinality and hierarchical inclusion before they can progress to counting on. For instance, they would have to understand that the number 15 represents the entire quantity of blue crayons. T
hey would also have to understand that numbers “live” inside of other numbers and that the numbers one through 14 don’t have to be observed and counted because they’re implied.
If teachers understand additive, multiplicative, fractional and proportional reasoning and how these concepts build upon each other in a logical progression, then by identifying the strategies used to solve problems, they can pinpoint where a student might be stuck and deliver precisely the instruction needed to move that child along.
Guiding students toward more effective mathematical reasoning
Often, teachers think of problem-solving strategies as a menu of choices—and students can use their “favorite” strategies or any techniques they’re comfortable with as long as they get the right answer. This student-friendly way of thinking sounds nice in theory, but it’s not practical for helping students advance their math skills.
As the numbers students are working with become larger and the problems become more complex, the strategies that suggest a student’s math reasoning skills are stuck at a basic level won’t work effectively.
Students might like drawing 25 circles and 25 squares to count to 50, and that might lead them to the correct answer. But as teachers, we should be pushing them to use more sophisticated problem-solving strategies, because eventually students will hit a barrier in their math abilities where the strategies, they’re using just aren’t good enough.
Formative assessment plays a key role in math instruction, and it takes a lot of skill to develop. Teachers must have a solid understanding of math reasoning and developmental progression to do it well.
Identifying where the real hurdles exist in a child’s progression requires more than just exit tickets. Can you observe what a student is thinking in solving a math problem, either through live observation or on paper, and from this observation, diagnose precisely what skills or instruction the child needs for success?
If you can, then a student’s math abilities will move forward rapidly.
