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.
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
Reflection on the challenge posed by implementation of the National Common Core Standards seems an appropriate first entry. It WILL be a challange.
Nationwide, textbooks based on the common core will be missing from most classrooms. This issue could go on for years in some districts. Teachers and homeschoolers will need to read and comprehend the unpacked standards. This will give them an opportunity to develop concrete objectives.
“Multiple resources” will be required. Students need access to accurate background information, meaningful practice and objective driven assessments. Teachers need the progress measurement data provided by objective based assessments. Additionally, logs of lesson plans, materials used, classroom activities, practice opportunities and assessments will be important for communication with students, parents and administrators; in an era of student performance based evaluation.
Books geared toward “teaching the test” will be thrown together quickly. The publishers may find it difficult to actually teach critical thinking; a vital component of the common core.
I am part of a website team at commoncoremath.net. The site is committed to providing teachers a laddered approach to teaching the common core math standards. At commoncoremath.net standards have already been unpacked and objectives developed. Critical thinking and writing skills are woven throughout the curricula. Classroom differentiation is facilitated by the availability of all materials, across all grade levels for all members, (membership is only $20.95 per year.)
commoncoremath.net provides thousands of pages of lessons, practice materials, assessments, tracking forms and other resources for print, computer or smartboard use. It is the ultimate “teacher’s helper” for these changing times.
Good luck from a sixth grade teacher!
James Wheeler – commoncoremath.net