Well-prepared beginning teachers of mathematics analyze different approaches to mathematical work and respond appropriately. |

Well-prepared beginning teachers of mathematics analyze both written and oral mathematical productions related to key mathematical ideas and look for and identify sensible mathematical reasoning, even when that reasoning may be atypical or different from their own. Well-prepared beginners value varied approaches to solving a problem, recognizing that engaging in mathematics is more than finding an answer. They make mathematical connections among these approaches to clarify underlying mathematical concepts. Well-prepared beginners recognize the importance of context and applications in uses of mathematics and statistics. They make connections across disciplines in ways that illuminate mathematical ideas.

The tasks in Figures 2.2 and 2.3, which might be used in a mathematics methods or mathematics content course for teachers, exemplify the level of mathematical analysis expected of well-prepared beginners.

Students are given the following prompt:
Four student responses follow: 1) We should vote on whether we have enough. 2) We should survey teachers about how many they have used in the past. 3) We should determine how many words each pencil can write and estimate how many words our class will write in a year. 4) We should collect data for a week before trying to answer the question. What would you ask to further each student’s approach to developing a model to answer this question? |

*Figure 2.2.* Sample task for Pre-K–5 teacher candidates.

Students are given the following prompt:
Four student responses follow: 1) 30 ÷ 3 = 10, so 10 and 30. 2) 30 ÷ 4 = 7.5, so 7.5 and 22.5. 3) Half of 30 is 15. Half of 15 is 7.5, so go up and down 7.5 from 15. 4) What question would you ask to clarify each student’s thinking? |

*Figure 2.3.* Sample task for middle level teacher candidates.