Kristin Klingensmith

IFL Mathematics fellow

Criteria charts that spur students to engage with the content matter actively are more likely to align with the Principles of Learning, so analyzing the criteria is critical. We have to consider if the combination of expectations results in active engagement with mathematical ideas that leads to deep understanding or simple compliance of learned procedures and rules.

Take a moment to review the two quality work criteria charts and consider the differences in their potential impact on student engagement with the mathematics they are studying.

A quick search on the Internet surfaces hundreds of suggestions for getting students engaged in mathematics classrooms. In this article, we are going to take a closer look at one of the ideas, the use of criteria charts.

Criteria charts are not new to mathematics classrooms. We have seen criteria charts for student work, for rules/procedures for problem solving, and for ways of working or talking during a mathematics lesson. Regardless of the focus, the chart serves as a physical embodiment and public naming of expectations.

Quality Work Chart A

1. Name is at the top.
2. All parts of the question are answered.

3. Number model is included (expression, equation, inequality).
Other representations are optional.

4. Description of how you arrived at the answer.
5. Evidence that the answer has been checked is included.

At first glance, it seems as if criteria charts would align with three of the Principles of Learning: Clear Expectations, Self-Management of Learning, and Recognition of Accomplishments. And though criteria charts have been leveraged by students to manage their learning and by teachers as guides to help decide what to recognize as an academic accomplishment, it is not that simple. It is not that simple because central to the Principles of Learning is the active pursuit and use of knowledge on behalf of the learner. Inherent to this is the idea of active engagement, where the learner actively uses their existing knowledge to construct and refine their understanding of a concept. (Be sure to look for future articles related to Academic Rigor in a Thinking Curriculum, another of one of the Principles of Learning.)

As you likely noticed, the criteria in Chart A is about compliance. Chart A criteria focus on rote application and completion. Students can meet all of the criteria on this chart without having to think critically about the mathematics they are studying.

In contrast, the criteria of Chart B sets the expectation for students to be actively making sense of and exploring mathematical ideas. These criteria, framed through a series of yes/no questions, have students analyze their work. The criteria target specific practices that students who are working as mathematicians should employ, such as creating and connecting mathematical representations. When students connect representations, they have to “translate” the mathematical idea in different ways. There is an explicit expectation that students refer to their representations in their explanations and include mathematical reasoning. The criteria also signal to students that they have to extend their thinking beyond the problem to look for similarities to other problems and mathematical ideas. In short, the criteria of Chart B is more likely to result in students interacting with and making sense of mathematical concepts in ways that require actual engagement with the math rather than just compliance.

Quality Work Chart B

1. Have I responded to all parts of the problem?
2. Does my response include at least two different representations (words, equations, diagrams, graphs, tables) that can help people understand the work?
3. Are the diagrams, tables, and/or numbers in the equations labeled so others know what is being represented?
4. Have I made connections between representations?
5. Does my written explanations refer to the story problem, equations, graph, tables, and/or charts?
6. Does my written explanation contain my mathematical reasoning?
7. Have I referred to other similar problem or mathematical ideas?

Having established that the Quality Work Chart B contains a combination of criteria designed to support students’ active engagement with the mathematics they are studying, let’s think about how such a chart can be used to support the three previously mentioned Principles of Learning: Clear Expectations, Self-Management of Learning, and Recognition of Accomplishment.

Clear Expectations: The expectations messaged by the criteria of Chart B are clear. When such criteria are publicly posted and regularly discussed, students have a means of judging their work and the work of others. The criteria also establish for students the practices they should use when engaging with mathematical explorations.

Self-Management of Learning: By using the criteria of Chart B, students can actively monitor and revise their thinking. We have to keep in mind that hanging a well-designed criteria chart on the wall does not automatically mean that students will use it to manage their learning. Teachers need to refer to the criteria regularly, using it as a tool to support students as they work. In this way, the criteria provide scaffolding for students, which they use less and less as they internalize the practices and expectations.

Recognition of Accomplishment: The criteria provide teachers an outline of what to look for in mathematics classrooms. Because the criteria cover a range of practices, teachers can highlight incremental steps in student performance to increase the number of criteria evidenced in the student work. Teachers may recognize some students for creating multiple representations, others for the connections they make between representations, and still others for the mathematical reasoning they provide.

Additionally, teachers can publicly share examples of student work that meets specific criteria, which not only serves to recognize the student’s accomplishment, but also provides a model from which their peers can learn. Acknowledging students for the real work of making sense of and sharing their understanding of mathematical ideas is essential to promoting active engagement.