Explain:Provide thorough and justifiable accounts of phenomena, facts, and data. Interpret:Tell meaningful stories, offer apt translations, provide a revealing historical or personal dimension to ideas and events; make subjects personal or accessible through images, anecdotes, analogies, and models.Apply:Effectively use and adapt what they know in diverse contexts. | Have perspective:See and hear points of view through critical eyes and ears; see the big picture. Empathize:Find value in what others might find odd, alien, or implausible; perceive sensitively on the basis of prior indirect experience.Have self-knowledge:Perceive the personal style, prejudices, projections, and habits of mind that both shape and impede our own understanding; they are aware of what they do not understand and why understanding is so hard. |

Their discussion of these facets may help to guide and focus the discussion of what teachers mean when they say that they want their students to "understand" the concepts that they are teaching. While all of the facets are important, for this particular project my primary focus will be on three of the six facets: Explain, Interpret, and Apply.

Explain: Students will be asked to explain and justify their thinking as they solve problems and make connections.

Interpret: Students will be asked to interpret situations and scenarios as they look for ways that the mathematics inform their world.

Apply: This particular facet of understanding is especially pertinent in the mathematics classroom as it is vital that students are able to put theory into practice. As the CCSS states, mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

Explain: Students will be asked to explain and justify their thinking as they solve problems and make connections.

Interpret: Students will be asked to interpret situations and scenarios as they look for ways that the mathematics inform their world.

Apply: This particular facet of understanding is especially pertinent in the mathematics classroom as it is vital that students are able to put theory into practice. As the CCSS states, mathematical understanding and procedural skill are equally important, and both are assessable using mathematical tasks of sufficient richness.

## Connections to (revised) Bloom's Taxonomy

Many of these facets are connecting to the ideas in Bloom's taxonomy. Displayed in the classroom will be Bloom's Taxonomy and the Standards for Mathematical Practice as all of these constructs share similar language ad raise the bar for students with regards to high-order thinking.
Remembering: Can the student recall or remember the information? define, duplicate, list, memorize, recall, repeat, reproduce state Understanding: Can the student explain ideas or concepts? classify, describe, discuss, explain, identify, recognize, report, select, translate, paraphrase Applying: Can the student use the information in a new way? choose, demonstrate, dramatize, employ, illustrate, interpret, operate, schedule, sketch, solve, use, write Analyzing: Can the student distinguish between the different parts? appraise, compare, contrast, criticize, differentiate, discriminate, distinguish, examine, experiment, question, test Evaluating: Can the student justify a stand or decision? appraise, argue, defend, judge, select, support, value, evaluate Creating: Can the student create new product or point of view? assemble, construct, create, design, develop, formulate, write (Overbaugh and Schultz) |

While Wiggins and McTighe see explaining, interpreting, and applying as facets of understanding, Bloom sees them as falling within the spectrum of levels for intellectual behavior. No matter how you slice it, the stage is set for educators to distinguish between the types of tasks they are engaging students in and level of critical thinking that is expected. Coupled with the Standards for Mathematical Practice, educators can teach their students metacognition and challenge them to maintain a presence in the higher ends of the spectrum; a skill that is vital for their success in the 21st century.