Key ideas for David and Shanna

Cognitive Complexity Level 3 (Strategic Thinking) requires reasoning, planning, using evidence, and a higher level of thinking than the previous two levels. In most instances, requiring students to explain their thinking is a Level 3. Activities that require students to make conjectures are also at this level. The cognitive demands at Level 3 are complex and abstract. The complexity does not result from the fact that there are multiple answers, a possibility for both Levels 1 and 2, but because the task requires more demanding reasoning.

CCSS: Standards for Mathematical Practice

2. Reason abstractly and quantitatively.

Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize—to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents—and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects.

7 Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure.

8 Look for and express regularity in repeated reasoning. Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts.

TPEP: Critierion 4 Providing clear and intentional focus on subject matter content and curriculum.

CEL: CP3: Instruction is frequently consistent with pedagogical content knowledge and supports students in discipline-specific habits of thinking. (Proficient)

CP4: Teacher demonstrates a solid understanding of how discipline-based concepts relate to or build upon one another. Teacher identifies and addresses student misconceptions in the lesson or unit. (Proficient)

Marzano: The teacher demonstrates a comprehensive knowledge of the subject and the standards for the subject.(Proficient)

Pedagogical Content Knowledge

This knowledge is different from the knowledge of a mathematician and also different from the general pedagogical knowledge shared by teachers across disciplines.

It includes knowledge of the representations, tools, and teaching strategies that support development of specific content knowledge, as well as understanding students’ strategies, prior conceptions and potential misconceptions related to this content knowledge.

CCSS content standards
F-BF.1a: Write a function that describes a relationship between two quantities. a. Determine an explicit expression, a recursive process, or steps for calculation from a context.

2. Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms.★

F-LE.1: Distinguish between situations that can be modeled with linear functions and with exponential functions… Recognize situations in which a quantity grows or decays by a constant percent rate per unit interval relative to another.

A-SSE.1: Interpret expressions that represent a quantity in terms of its context.★ (a) Interpret parts of an expression, such as terms, factors, and coefficients.

Noticings / Wonderings
TPEP Criterion
Cognitive Complexity
SMP 2, 7, 8
Lesson Purpose

RAMP-A January 11, 2013