7. Look for and make use of Structure.
The mathematically proficient student is capable of making close observations to discern patterns or “structures”. These structures form the basics of mathematics. The ability to discern these structures is vital to problem solving. The structures may be as simplistic as a number line; 2 follows 1 and 3 follows 2, or they may be incredibly complex. Mathematical growth requires building an ever greater awareness of structures and interconnectedness.
Specific instances could include recognizing multiplication is a specific number of sets, or that fractions represent the present parts of a whole. Students start by learning to count in order; 1, 2, 3,…. Later students learn more complex structures such as properties of operation, which can be used to add, subtract, multiply and divide. Recognition of even more complex structures allow for solving difficult algebraic equations, creating geometric proofs and statistical analysis.
Finally, students must be able to describe structures in writing, orally and through visual representations. Critical thinking skills and a true understanding of how the structures are formed is required to process such translations.
Gottlieb Wilhelm von Liebnitz
6. Attend to Precision.
The mathematically proficient student will need the ability to communicate precisely. This is true for both verbal and written communication. The core standards for mathematics requires students to do much more writing about problem solving than has been necessary in the past.
While this concept confuses many parents, politicians and even teachers, it is a helpful tool for sorting through problems and remembering the steps taken to solve them. Most classroom subjects require note-taking. The notes can be useful for review and test preparation, but their greatest benefit may be in the memory enhancement provided simply by the act of writing as well as hearing a lesson. Good note-taking requires listening or reading closely, and then separating the truly important information from the rest. The good note taker, along with the proficient mathematician, will then be able to take those nuggets of information and communicate them in complete sentences; a requirement of proficiency standard 6.
Understanding which information is truly “the most important,” and being able to communicate it requires the ability to use clear and accurate definitions. Students cannot clearly communicate multiplication if they do not understand the meaning of the words multiplier, multiplicand and product. They cannot describe measurements without being able to clearly define the difference between area and volume. True proficiency requires the ability to define concepts and processes. Additionally, students must recognize and be able to define or describe the meaning of mathematical symbols. Math is a language filled with symbols. The inability to understand mathematical symbols makes problem solving impossible; just as the inability to understand letters and punctuation makes reading impossible.
Labels are another key to mathematical precision. Maps, coordinate planes, measurements and graphs are all incomprehensible without labels. The labels must be precise, and they must be used in a manner that makes it clear what item is being labeled. Students who fail to use labels appropriately will find it impossible to communicate their work.
Finally, the mathematically proficient student will need to synthesize all of these abilities, along with the ability to calculate accurately and efficiently. Once their problem is solved they will need to be able to communicate their process and answer via carefully formulated explanations. This action alone may point to flaws in their logic, causing a review and reworking of the problem at hand. This is accomplished through the careful examination of solutions through the magnifying lens of the definitions and constraints inherent in the problem itself. When looked at in this light it becomes apparent precise communication is key to not only the fulfulling of an important standard, but to the basics of problem solving using the language of math.