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.
1. Making Sense of Problems and Persevering.
Step 1 of the proficiency standards requires making sense of mathematical problems and persevering in solving them. There is a reason for listing this process first. Classroom experience shows many of today’s students are unwilling to think for themselves. They want answers to be straightforward and easy to find. This is no surprise considering the world they inhabit. The internet has brought the world instant access to any information one could possibly seek. Students are adept at “googling” anything they want or need to know.
Solving problems without the aid of the internet, calculators or a teacher’s answers is almost an alien concept at this point. While the information age is making the need for the memorization of certain facts obsolete, there is still a need for critical thinking and problem solving. According to the Psychological type theory of psychiatrist/philosopher C.G. Jung whose work was furthered by the work of Myers/Briggs, some students will be more adept at using thinking skills, while others will be better with feelings. This means many students and adults will find the task of thinking through a problem extremely difficult.
Students will not fit into cookie cutter molds, but the ability to think through problems is a nessecity. Since many who enter the teaching field are “feeling types”, the job of passing along critical thinking skills becomes even more difficult. Most teachers enter the profession because of a desire to help others. Obviously the low pay, long hours and lack of respect today’s teachers face make the job less than desirable for most college graduates.
While develping critical thinking skills is a challenge, it is not impossible. Following best practices in the classroom, constant professional development, maintaining high expectations and modeling appropriate skills can help all students become better thinkers and problem solvers. Unfortunately this cannot happen overnight. Perseverance and patience will be key for all stake holders.
James Wheeler – commoncoremath.net