Teaching for understanding is a difficult task. First, it is difficult to construct appropriate and relevant representations of mathematical concepts. Second, many math teachers have been traditionally taught; therefore, they find it difficult to instruct in ways that promote mathematical understanding and reasoning while situating math in a familiar context.

Ball asserts that learning mathematics with understanding involves making connections between informal and formal learning contexts. Effectively teaching children math for understanding requires using familiar objects in students’ domain.

Helping students develop mathematical knowledge depends on knowing students and knowing how they learn.

Different math topics require different teaching strategies.

Teachers are not observers of student development but participants.

Helping teachers develop the skill of teaching for understanding requires a respect for the complexity of such a type of teaching and also this development requires taking teachers seriously as learners. Not only must a teacher understand this type of teaching; they should be willing to modify their instructional strategies as needed.

In order to create and orchestrate fruitful representational contexts (i.e., something that is relevant to students), a teacher must consider:

1. the mathematical content

2. the students – what they are cognitively capable of knowing?

3. the thinking and learning space – offered by the representational contexts

4. teaching as inquiry – constantly re-evaluating content and learning and altering it as necessary

Ball gives numerous classroom examples of attempting to teach for understanding in her third grade classroom. When mapping out how she was going to introduce fractions, she came up with four guidelines that can be generalized to any subject. She considered:

1. content issues – she looked at state and district objectives for teaching fractions

2. students’ prior knowledge – what topics did they know that were related to fractions?

3. link to their prior knowledge – how could she make a connection with fractions?

4. students’ out-of-school experience with the content

Throughout the mathematical vignettes, Ball infuses many pedagogical tips. On p. 173, she states, "Looking at rational numbers form the perspective of a 9-year-old whose familiar mathematical domain is being stretched and transformed, I saw aspect of rational number thinking that I had not noticed before." On p. 174, she talks about students having more visual knowledge of fractions rather than principle knowledge. On p. 175 she states, "Together, students and teacher must develop language and conventions that enable them to connect and use particular representations in situations." Teaching needs to be more collaborative on the part of teacher and student rather than dictatorial. On p. 175-6, she discusses teacher’s dilemmas between allowing students to discover on their own compared to helping students develop mathematical ideas.

She concludes with five "lines of inquiry" regarding the development of new practices of mathematics teaching:

1. Need more theoretical and empirical research on representation in teaching math content.

2. Need to understand the process of pedagogical deliberation in teaching math. What kind of dilemmas and issues arise?

3. Need to better understand the role of mathematical understanding in teachers’ pedagogical reasoning.

4. What knowledge about learners (and learning) contribute to teachers’ ability to teach mathematics for understanding?

5. How do traditionally taught math teachers learn to teach math for understanding in a more non-traditional way?

1. How can Ball’s conclusions be generalized to other content areas?

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