1. 1 ELL and Secondary Mathematics Linking Mathematical Language and Symbolism to Conceptual Understanding

2. 2 Mathematical Proficiency: 5 Interwoven Competencies i) Conceptual understanding—comprehension of concepts, operations, and relations ii) Procedural fluency—carrying out procedures flexibly, accurately, efficiently, and appropriately iii) Strategic competence—ability to formulate, represent, and solve problems iv) Adaptive reasoning—capacity for logical thought, reflection, explanation, and justification. v) Productive disposition—habitual inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own capacity From National Research Council (2001) Adding It Up

3. 3 The Power of Conceptual Understanding It may not be an accident that conceptual understanding is the first component listed. Conceptual understanding is the foundation on which the other 4 components are grounded.

4. 4 The Power of Conceptual Understanding view an idea/concept from multiple perspectives; make critical connections to other fundamental concepts/ideas; and recognize and understand subtleties in language, symbolism, and representation These do not occur in isolation. The three will often overlap/connect. Conceptual understanding enables us to

5. 5 Beyond Vocabulary Conceptual Understanding Conceptual understanding goes far beyond “how to.” With conceptual understanding students know what a concept really is, why it works, how it connects to other concepts, and what it looks like symbolically and graphically

6. 6 Beyond Vocabulary Conceptual Understanding Define slope and give a real life example. Only with a high level of conceptual understanding can a student visualize and comprehend 3 as∆ 3 4∆ 4

7. 7 Subtleties in Language, Symbolism, and Representation Conceptual understanding enables students not only to decipher the meanings of words in mathematics from the meaning in standard English, but also to decipher the subtle meanings of words in different mathematical contexts. Ex: One pig grew from 5 pounds to 10 pounds. Another grew from 100 to 108 pounds. Which pig grew more?

8. 8 Subtleties in Symbolism and Representation Conceptual understanding is especially essential to recognize subtleties in mathematical symbols and representation. 25.6 8.25% 8 1/4 %

9. 9 Subtleties in Symbolism and Representation Trace how the “whole” changes in this scenario: Stevie has 1/2 gallon of gasoline in his container. He uses 2/3 of it to mow the grass today. How much gas did he use?

10. 10 Subtleties in Symbolism and Representation Traditional instruction often minimizes the symbolic connections through shortcuts, does not sufficiently address symbols, and often is not cognizant of the critical role of symbols and representation.