In order to teach for math proficiency, which is a driving reason for
the differentiation discussed in this book, the authors state that the following
National Research Council (2001, 2010) strands must be interwoven into a
teacher’s practice:

1)

**Conceptual understanding**of the core knowledge of math, students, and pedagogy
2)

**Procedural fluency**with instructional routines
3)

**Strategic competence**in planning and responding to problems
4)

**Adaptive reasoning**in justifying, explaining, and reflecting on practice
5) A

**productive disposition**toward math, teaching, learning, and improving one’s practice
In chapter 9, the authors focused on the first and the last of these
strands. In support of strand number 1, Carol Tomlinson (2001) is again referenced –
she believes that gaps in a teacher’s understanding of content is the most significant barrier to
effective differentiated instruction. (Gaps in pedagogical skill and classroom
management are also threats to effective differentiated instruction.)

According to NCTM (2000), teachers at all levels need knowledge about:

1) The whole math domain

2) Curriculum goals and important ideas central to their grade level

3) Challenges that students encounter in learning the ideas

4) How to represent the ideas effectively

5) How to assess students’ understanding

The authors recommend the following sources for clarifying content and
pedagogical knowledge for teaching math:

Tagxedo word cloud of the Charles' Big Ideas |

1) The standards documents of NCTM:

*Principles and Standards for School Mathematics*(2000),*Professional Standards for Teaching Mathematics*(1991) and*Assessment Standards for School Mathematics*(1995)
2) Randal Charles’ “Big Ideas and Understandings as the Foundation of Elementary
and Middle School Mathematics” (2005)

Randall Charles, in his “Big Ideas…” proposed that teachers ground
their math content knowledge and practice around a set of Big Mathematical
Ideas, which by his definition are statements “…of an idea that is central to
the learning of mathematics, one that links numerous mathematical
understandings into a coherent whole.” The authors include Charles’ list of 21
big ideas, which is also available here.

3) NCTM’s

*Curriculum Focal Points*(2006), which designates three key topics and their conceptual connections for each grade level for grades preK -8.
4) NRC’s How Students Learn:

*Mathematics in the Classroom*(2005)
This document outline these three principles of learning:

1) Use of prior knowledge – it can support new learning or be a barrier to
new ideas. It is necessary that teachers are aware of students’ preconceptions
before introducing new ideas.

2) Learning concepts with understanding, which includes creating a
conceptual framework for learning of factual knowledge and using multiple
representations to apply the new knowledge to unfamiliar situations.

3) Importance of self-monitoring, which is the metacognitive approach of
putting children in control of their own learning and taking responsibility for
their own learning.

The authors includes the following teaching strategies that have proven to be effective in the
teaching of math:

1) use of visual, linguistic and contextual supports related to formal
math notation and language

2) encouragement for talking about math thinking and reasoning

3) bridging informal math experiences to formal math procedures

4) following research-based learning paths that engage students’ hearts,
hands, and minds

5) harnessing the power of connections within strands and across
strands using relationships and multiple representations

6) providing multiple opportunities for skill development over time

The authors turn next to strand five, which is the need for
professional development. They support the idea that teachers should deepen
their knowledge of elementary math through their own teaching, always asking
themselves “why?” Teachers should take opportunities to collaborate with
colleagues because learning occurs through reflection and discussion.

How much opportunity do you have to discuss, reflect, and collaborate with your peers?

How much opportunity do you have to discuss, reflect, and collaborate with your peers?

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