**Instructional Activities Using Routines as Tools for Teacher Education**

• Choral counting: The teacher leads the class in a count, teaching different concepts and skills by deciding what number to start with, what to count by (e.g., by 10s, by 19s, by 3/4s), whether to count forward or backward, and when to stop. The teacher publicly records the count on the board, stopping to elicit children’s ideas for figuring out the next number, and to co-construct an explanation of the mathematics that arises in patterns.

• Strategy sharing: The teacher poses a computational problem and elicits multiple ways of solving the problem. Careful use of representations and targeted questioning of students are designed to help the class learn the general logic underlying the strategies, identify mathematical connections, and evaluate strategies in terms of efficiency and generalizability.

• Strings: The teacher poses several related computational problems, one at a time, in order to scaffold students’ ability to make connections across problems and use what they know to solve a more difficult computational problem. This activity is used to target a particular strategy (as compared to eliciting a range of strategies). For example, posing 4 × 4, then 4 × 40, and then 4 × 39 is designed to help students consider how to use 4 × 40 to solve 4 × 39, developing their knowledge of compensating strategies in multiplication (Fosnot & Dolk, 2001).

• Solving word problems: The teacher first launches a word problem to support students in making sense of the problem situation, then monitors while students are working to determine how students are solving the problem, gauges which student strategies are best suited for meeting the instructional goal of an upcoming mathematical discussion, and makes judgments about how to orchestrate the discussion to meet those goals.

[a fifth IA in recent articles is Quick Images: The goal of this activity is to build students' ability to visualize a quantity.]

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**A Focus on Instructional Dialogue**

A relatively recent focus of Leinhardt’s work on teaching routines has been how they are used in “instructional dialogue” (Leinhardt & Steele, 2005), a practice we would consider to be the centerpiece of ambitious mathematics teaching. In this kind of teaching, an explanation is co-constructed by the teacher and students in the class during an instructional conversation. Maintaining a coherent mathematical learning agenda while encouraging student talk about mathematics is perhaps the most challenging aspect of ambitious teaching. In their study of teaching through instructional dialogues, Leinhardt and Steele (2005) demonstrated the use of eight kinds of “exchange” routines in this kind of teaching to accomplish explanatory work, including maintaining mathematical clarity in the face of student inarticulateness, fixing the agenda of the class on a single student’s idea, making it safe for students to revise incorrect contributions, and honing students’ contributions toward mathematical accuracy and precision. The exchange routines that Leinhardt and Steele (pp. 143–144) identified include the following:

• The call-on routine, which is initiated by a rather open invitation to discussion and has two separate components: the initial identification of a problem and the speaker who responds, followed by a second part in which the class is prompted to analyze, justify, or critique the statement given by the first speaker or another speaker in the discussion.

• The related revise routine in which students were asked to rethink their assertions and publicly explain a new way of thinking about their solutions.

• The clarification routine “which was invoked when a confusion arose regarding an idea or conjecture volunteered into the public space, which in turn involved understanding the source of confusion.”

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