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.
4. Model with Mathematics.
Modeling may be the mathematical skill most useful in the everyday world. The proficient student, or skilled problem solver needs the ability to utilize the math they already know to solve real world, everyday problems. This ability is a necessity for business leaders, political leaders, military leaders and homemakers. In the classroom the student must have the ability to turn a real world problem into a written equation or inequality, thus describing the situation. Outside the classroom, the same method is used, although the process may vary. Some may solve the problem via mathematics, but many will come to conclusions without realizing they needed math to reach their conclusion.
In order to change a real world situation into an equation or inequality the student needs to be able to use reason for analysis. This will enable him or her to understand and describe how one quantity depends on another. The typical “teeter totter” metaphor used in many classrooms to describe how an equation works is a great example.
Analysis will often include making assumptions or estimating values to determine the validity of a possible solution. When the estimated solution is obviously far from the mark the student must have the ability to revise using a guess and check, half-split or other problem solving process. These methods require a clear understanding of the important elements of the problem. Students must be able to clear the chaff and get to the root prior to finding an answer.
Finally, learning occurs as the student begins to master the necessary techniques. Application of problem solving processes to similar problems requires an ability to reflect on, and recall the steps taken. Only then will the similarities of various real world situations, and their solutions, begin to become clear. This is the reason for the common core requirement to write steps used to find solutions. The act of writing lends itself to reflection, and this reflection completes the learning cycle.
James Wheeler – commoncoremath.net
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.
This is a skill we all use every day. Some are much better at quantitative reasoning then others. The ability to think in the abstract, and then apply the knowledge gained to manipulate representative symbols is required in almost all aspects of daily life; imagine the tool maker, programmer or cook. Without this skill we could not accomplish tasks as varied as driving a car or operating the ever present electronic devices filling our lives.
Copernicus, Galileo, Descartes, Newton, Darwin and countless others throughout history have moved Western Man from the static “Dark Age” to a time when science and knowledge are moving so fast we can barely keep up. Their accomplishments depended upon quantitative measurement and reasoning, and then contextualizing knowledge gained to shape a new world view based on phenomenology. Suddenly the secrets of the universe started to be revealed at an ever increasing pace.
Our students will need to continue this accelerated pace utilizing the same skills. The future will depend on their ability to solve problems in the abstract and then apply the knowledge gained. Revealing a “Grand Unified Theory” will lead to technological advances that will make today’s world seem like the dark ages.
It is important to recognize the varied abilities of students when teaching. Many will find such abstract methods of critical thinking incredibly difficult. Finding methods to accurately measure the skill will also be a challenge. The “one size fits all” method of standardized testing will undoubtedly fail to live up to the promise. Those creating the standards and assessments must take this into account. The policy makers must recognize the limitations as well.
James Wheeler – commoncoremath.net