In lessons such as Mr. Angelo’s, mathematics entails following rules and practicing procedures, often with little attention to the underlying concepts.5 Procedural fluency is given central attention. Adaptive reasoning is not Mr. Angelo’s goal: He does not offer a justification for the rule he is teaching, nor does he engage students in reasoning about the structure of the place-value notation system that is its foundation. He focuses instead on ensuring that they can use it correctly. Other aspects of mathematical proficiency are also not on his agenda. Instead, Mr. Angelo has a clear purpose for the lesson, and to accomplish that purpose he controls its pace and content. Students speak only in response to closed questions calling for a short answer, and students do not interact with one another. When a student gets an answer wrong, Mr. Angelo signals that immediately and asks someone else to provide the correct answer. The lesson is paced quickly.

We turn now to our second teacher, Ms. Lawrence, who is working with her fifth graders on adding fractions (Box 9–3). Ms. Lawrence’s goals are different from Mr. Angelo’s. Although she also structures the lesson to accomplish her goals, unlike Mr. Angelo, she emphasizes explanation and reasoning along with procedures. The pace of the lesson is carefully controlled to allow students time to think but with enough momentum to engage and maintain their interest.

Box 9–3 Ms. Lawrence— Teaching Fifth Graders About Adding Fractions

After a few minutes in which the class does mental computation to warm up, Ms. Lawrence reviews equivalent fractions by asking the students to provide other names for She asks the class what fractions are called that “name the same number.” On the chalkboard she writes a problem involving the addition of fractions with like denominators:

She asks the students how to find the sum. One student, Betsy, volunteers that you just add the numerators and write the sum over the denominator. “Why does this work?” Ms. Lawrence asks. She asks Betsy to go to the board and explain. Confidently, Betsy draws two pie diagrams, one for each fraction, and explains that the denominator tells the size of the pieces and the numerators how many pieces all together:



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