# Tag Archives: Math

## 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.

## 4. Model with Mathematics; MODEL

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

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

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

## 2. Reason Abstractly and Quantitatively; REASON

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

## Standards for Mathematical Practice

Below are the standardized processes for achieving mathematical proficiency.  A close reading reveals the standards not as an effort to “indoctrinate our students,” but a method for increasing critical thinking and problem solving skills.

1.  Make sense of problems and persevere in solving them.

Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, “Does this make sense?” They can understand the approaches of others to solving complex problems and identify correspondences between different approaches.

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.

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.

4.  Model with mathematics.

Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose.

5.  Use appropriate tools strategically.

Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts.

6.  Attend to precision.

Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions.

7.  Look for and make use of structure.

Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression x2 + 9x + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(x – y)2 as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers x and y.

8.  Look for and express regularity in repeated reasoning.

Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (y – 2)/(x– 1) = 3. Noticing the regularity in the way terms cancel when expanding (x – 1)(x + 1), (x – 1)(x2 + x+ 1), and (x – 1)(x3 + x2 + x + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results.

James Wheeler – commoncoremath.net