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

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The Power of Information

A review of social media reveals many NCCS detractors either do not understand the program, or feel it is a federal takeover of local school systems.  This indicates a need for better communication regarding the goals and components of the program. The majority of Americans are not really sure what the term “Common Core Standards” even means. Unfortunately, this lack of knowledge leads to resentment and distrust.  What should be viewed as a straightforward plan for increasing students’ critical thinking skills and improving education across the United States, is too often seen as a socialist plot to indoctrinate our children.

Organizations such as the conservative Heritage Foundation and Republican National Committee feed into the campaign of misinformation by spreading stories of classrooms where answering a math problem incorrectly is acceptable and the social sciences are redesigned to support a leftist agenda.  Facts be damned; opponents “feel” the Common Core is a partisan plot which must be stopped.  Of course, as is always the case, there are those on both sides of the partisan divide who would scrap the NCCS for various reasons, and others who support the new standards fully.

This is not a new phenomenon.  Every attempt at improving education in this country, whether top down or bottom up, has met resistance. Often, detractors have been correct in their opposition.  Just as often, improvements in method have been abandoned due to lack of understanding and support from parents, organizers and politicians.  Educators themselves have sunk numerous efforts at improvement in the classroom.

While the National Common Core Standards will probably not be a panacea, almost any effort to improve K-12 education in America is worth a try.  Success will depend on buy-in from educators, politicians and citizens of all ages and backgrounds. Cooperative effort will undoubtedly lead to positive results.  Tweaking will be required, but the goal is a good one, and can be achieved.  Partisan bickering and misinformation campaigns can only lead to a continuation of the downward spiral currently leaving so many American students behind.

America’s ability to educate cannot afford to lag as the speed of information technology continues to advance.  High tech societies will not thrive in an environment of low tech workers.  It is inherent on our government and educators to inform the masses, ensuring they understand the NCCS effort, and its desired affect.  Anything less can only lead to another failure.

James Wheeler – commoncoremath.net

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Critics of the National Common Core Standards

 

One of the best things about being an American is being entitled to share opinions regardless of their accuracy.  Many have used this right to criticize the relatively new National Common Core Standards for learning.  While some of this grousing is deserved, there is value in the idea of national learning standards.

Complaints have come predominately from conservatives and tea party activists.  Recently the Republican National Committee has joined the chorus of nays.  The conservative Heritage Foundation has stated;

“One of the primary objections by conservatives to the Common Core standards is the view that the Obama administration is intent on controlling what is taught at each grade level in schools across the United States. According to the Heritage Foundation, the Obama Department of Education “has used its flagship ‘Race to the Top’ competitive grant program to entice states to adopt the K-12 standards developed by a joint project of the National Governors Association (NGA) and the Council of Chief State School Officers (CCSSO).”

This indicates much of the criticism has more to do with the standards being developed and implemented during the Obama administration than with any real complaints regarding the curricula.  Of course, there are also folks on the left who question the validity of the new standards, although the reasoning is generally different.

Common complaints regarding the common core include but are not limited to;

  • A national takeover of education
  • Leftist indoctrination of our children
  • Teaching that answers can be wrong as long as they can be explained
  • A movement away from basic facts and the three Rs

These complaints are driven by a great deal of misinformation and partisan politics. While critics complain of the poor job done by educators out of one side of their mouths, they attack almost any attempt to improve education out of the other.

National standards will give every school district in every participating state a baseline to measure against.  In addition, common standards will tend to lower the cost of education through increased efficiencies provided by centralizing a portion of the process.

Nothing in the common core even hints at “leftist indoctrination.”  The core was developed to enhance critical thinking skills through better reading comprehension and a more thorough understanding of basic mathematic processes.  Reading will emphasize non-fiction, not of the “Communist Manifesto” variety, but of documents such as the Declaration of Independence and United States Constitution; hardly leftist rags.

Wrong answers will continue to be wrong, and students will be tested on their knowledge of basic facts in the fourth grade.  Passing or failing will depend on a student’s understanding of the basics as well as an understanding of why math works the way it does.  Written answers will be graded according to a rubric, which will provide points for understanding a mathematical process as well as the answer.  This is hardly revolutionary.  Rubrics such as these have been in use for many years.

Unfortunately, we as Americans have become a divided society.  Partisanship has taken the place of common sense in a myriad of situations.  Opinions are formed based on the rantings of talk radio and cable tv hosts.  Politicians leap at every chance to smear the opposition.  Worst of all, the vast majority never bother to do any research, such as reading the standards themselves, but instead take their views from the rabble rousers.

While I am not 100% sold on the National Common Core, I respect the effort to improve our system of educating k-12 students.  The country would be much better off if everyone pulled toward this common goal.  My suggestion to concerned citizens would be to put in the work required to truly research this effort, and try to put politics aside.

James Wheeler – commoncoremath.net

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commoncoremath.net

This website is perfect for teachers and homeschoolers.   Thousands of pages of practice and assessment developed to ensure student achievement.

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September 13, 2013 · 4:19 pm

A True Challenge

jim

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

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