7. Look for and make use of Structure; Structure

Aryabhata

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.

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6. Attend to Precision; Precision

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.

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5. Use Appropriate Tools Strategically; Tools

Euler

5. Use Appropriate Tools Strategically.

Math proficiency standard 5 requires students to use appropriate math tools and use them strategically. Tools used for mathematics are varied and include pencil and paper, calculators, computer software, Ipads with programmed instruction lessons, rulers, compasses, protractors, manipulatives and countless other items. Students should show proficiency using all applicable tools. Showing proficiency requires realizing what tools can be used, and how they should be used dependent upon the problem being solved. Students should recognize the tool’s value and limitations.

Tools help students better visualize and understand problem solving by providing a deeper level of learning. Students are able to look at problems in different ways and compare solutions by using the data provided by utilizing mathematical tools. This supports learning at higher levels, and facilitates a more comprehensive approach to solving both mathematical and everyday problems. Carpenters, tool-makers, scientists and countless other professions require the proficient use of mathematical tools to complete projects.

While many bemoan the fact today’s students utilize computers and calculators, like the slide-rule and trigonometric tables of old, these tools allow students to solve problems using extremely large numbers or great amounts of data.  The graphing calculator and internet allow students to explore mathematical content that in the past would not have been possible until much later in the educational process.

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4. Model with Mathematics; MODEL

File:Scuola di atene 07.jpg

Euclid

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

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3. Construct Viable Arguments and Critique the Reasoning of Others; ARGUMENTS

File:Einstein 1921 by F Schmutzer.jpg

Einstein

3.  Construct Viable Arguments and Critique the Reasoning of Others.

Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and—if there is a flaw in an argument—explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments.

Being able to utilize a common algorithm to solve a simple problem remains a skill to be taught and practiced in the classroom. Many parents and teachers are confused regarding this issue. Students must know basic facts and algorithms prior to completion of the fourth grade. This has not changed due to implementation of the Common Core for Math. Being able to solve simple, one-step problems by using common algorithms is only a part of understanding mathematics. A true understanding, and the ability to apply math requires much more.

Solving a real world problem generally requires multiple steps, and can often be approached in more than one way.  This standard encourages students to look at problems in different ways, and occasionally find solutions using unconventional methods.  The ability to apply previous knowledge to problem solving is essential, as well as the ability to solve problems in different ways utilizing analytical reasoning.

The main thrust of the third standard is “argument.” Students need to be able to construct viable arguments of why their problem solving processes work. The ability to argue for a method requires the ability to recognize counterexamples that lead to the same solution. Finding flaws in method is equally important. Those proficient in mathematics must be able to justify their methods and critique those of others. This can often be accomplished through the use of “concrete referents”, such as objects, drawings, diagrams and actions. The ability to reason and write mathematical arguments has becom a necessity for today’s student. Critical thinking, a requirement for today’s information driven world, is being practiced through this method.

James Wheeler – commoncoremath.net

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2. Reason Abstractly and Quantitatively; REASON

Portrait of man in black with shoulder-length, wavy brown hair, a large sharp nose, and a distracted gaze

Newton

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

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1. Making Sense of Problems and Persevering; SENSE

File:Domenico-Fetti Archimedes 1620.jpg

Archimedes

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

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