About

The Compare and Discuss Multiple Strategies (CDMS) routine, as well as the curriculum materials for algebra courses that teachers can use to implement CDMS, were developed over the past decade by researchers at Harvard University and Vanderbilt University. This work was supported by grants from the National Science Foundation (DRL1561286 & DRL0814571) and the Institute of Education Sciences (R305H050179). 

Discussion lies at the core of the CDMS routine. There is widespread belief and strong research evidence that engaging students in mathematical discussions plays an important part in the learning process. Mathematical discussion can provide students with opportunities to develop a deeper understanding of concepts and make sense of new ideas. Mathematical discussion involves listening to peers’ contribution, where students can learn from multiple perspectives and make connections to their own understanding. Furthermore, communicating mathematically can enhance student engagement and interest in mathematics.

But mathematical discussions can be difficult for teachers to implement, for a number of reasons. First, orchestrating a productive discussion requires teachers to listen to and respond to students’ contributions, knitting together what students offer in ways that move the lesson forward towards its objective — and this is very challenging to do in the moment. Additionally, the success of a mathematical discussion often rests on the tasks that students are engaged in, as some tasks seem more amenable to productive discussions than others. Finally, discussions are made easier when teachers are familiar with structured instructional routines that can help them begin and maintain mathematical conversations. 

CDMS is a structured teaching routine that is designed to help teachers engage their students in productive mathematical discussions. In CDMS, students first use comparison to identify similarities and differences between aspects of a mathematical task; they then engage in a mathematical discussion about what they have noticed while comparing, with a particular focus on understanding the ‘why’ behind the procedures or methods used in the task. 

CDMS begins with comparison. Students compare and contrast two methods that have been used for solving a problem. One or both methods may be student-generated, or the methods may be provided to the students by the teacher in the form of a solved example. The teacher engages students in considering any similarities and differences that students notice about the two methods. For example, how are the two methods similar or different? Did both methods arrive at the same solution? How is the reasoning used in each method similar or different? Do students perceive each of the methods to be familiar, useful, effective, and/or efficient? 

After engaging in the comparison of the two methods, CDMS moves into a mathematical discussion. The discussion centers on an instructional objective that is specific to the two methods that have been compared, but generally related to the ‘why’ behind the procedures used in the two methods. In some instances, the discussion may focus on the different mathematical concepts that are evident in the two methods. In other cases (perhaps when the two methods arrived at different final answers) the discussion may engage students in considering which method yielded the correct answer and why. 

The CDMS routine generally can be conducted in 20 minutes, with about 8 minutes spent comparing and 12 minutes spent in discussion. The figure (to the left) shows a typical plan for how this routine can be implemented. CDMS begins with students’ comparing and contrasting the two methods. This 8-min comparison phase begins with students examining each of the two methods one at a time, in order to understand the reasoning behind each method. We refer to this part of the routine as Prepare to Compare. Teachers provide students with questions to engage in, including “What is happening in the first method?” and “What is happening in the second method?” The class then moves to comparing and contrasting the two methods, considering teacher-initiated questions such as, “Why do both methods work?” and “Which method is correct?” as well as other questions that students may generate themselves.

 After engaging in about 8 minutes of comparison, CDMS moves next to discussion. In order to prepare to engage in the discussion (Prepare to Discuss), students first are asked to think independently and then to talk with a partner about questions that emerged from the comparison. Students’ ideas are then shared and discussed by the whole class (Discuss Connections). Here the teacher facilitates a whole-class conversation where students are encouraged to share their ideas and where the teacher may re-voice or summarize student contribution to keep the conversation going and to involve as many students as possible.

Additionally, the teacher asks follow-up questions to help students understand the ‘why’ behind the procedures and also hold students accountable for listening. For example, the teacher might ask students to further provide reasoning to their explanation by asking, “Why do you think that’s true?” or ask peers to make connections from what they have heard to their own thinking by asking, “Do you agree or disagree? Why?” Finally, the teacher guides students to identify the main take-aways (or Big Idea) of the activity in the students’ own words.