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
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