__Chapter 2 Entry__

Mathematics is more than completing sets of exercises or mimicking processes the teacher explains. Doing mathematics means gathering strategies for solving problems, apply approaches and see if they lead to solution and whether answer makes sense. Doing mathematics in classrooms should be closely model the act of doing mathematics in the real world.

The classroom environment for doing mathematics must have the language of doing mathematics. Children engaged in these verbs and actions in mathematics classes will be actively thinking about the mathematical ideas that are involved. Students begin to take the math ideas to the next level by connecting to previous material, responding with information beyond the required response and conjecturing or predicting. When this happens, students are getting an empowering message that they are capable!

Two things that result in conceptual understanding is making mathematics relationships explicit and engaging students in productive struggle. The teacher's role in making mathematical relationships explicit is to be sure that students are making the connections that are implied in a task. he focus is on students' applying their prior knowledge, testing ideas, making connections and comparisons, and making conjectures. Students must have the tools and prior knowledge to solve a problem and not b given a problem that is out of reach or they will struggle without being productive. When students know that struggle is expected as part of the process, they embrace the struggle and feel success when they reach an answer or solution.

There are two theories which have been developed and are most commonly used by researches in mathematics education to understand how students learn mathematics- Constructivism and sociocultural theory.

Students learn through inquiry and activate their own knowledge and try to assimilate or accommodate new knowledge. The implications for teaching mathematics are:

1. Build new knowledge from prior knowledge

2. Provide opportunities to talk about mathematics

3. Build in opportunities for reflective thought

4. Encourage multiple approaches

5. Engage students in productive struggle

6. Treat errors as opportunities for learning

7. Scaffold new content

8. Honor diversity

What does it mean to understand mathematics? It means relational understanding- know what to do and why and mathematics proficiency.

The benefits of developing mathematical proficiency are:

1. Effective learning of new concepts and procedures

2. Less to remember

3. Increases retention and recall

4. Enhanced problem- solving abilities

5. Improved attitudes and beliefs

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