When Five Plus Five Equals Eight

For the last several months, I have been privileged to work with hundreds of teachers as they unpack the Common Core State Standards. In each session, which have been held in grade level groups, we have worked through the meanings of the 8 Standards for Mathematical Practice and each time one point becomes extremely obvious.  These eight standards, or measures of student behavior, are inextricably linked.  It is no surprise really, as they were developed using information from the National Council of Teachers of Mathematics (NCTM) Process Standards and the National Research Council’s (NRC) Strands of Mathematical Proficiency.   The point is driven home, however, when teachers work through the process of identifying for themselves what each standard means and does not mean for their classroom.

NCTM’s five process standards include:

Problem Solving

  • Build new mathematical knowledge through problem solving
  • Solve problems that arise in mathematics and in other contexts
  • Apply and adapt a variety of appropriate strategies to solve problems
  • Monitor and reflect on the process of mathematical problem solving

Reasoning and Proof

  • Recognize reasoning and proof as fundamental aspects of mathematics
  • Make and investigate mathematical conjectures
  • Develop and evaluate mathematical arguments and proofs
  • Select and use various types of reasoning and methods of proof

Communication

  • Organize and consolidate their mathematical thinking through communication
  • Communicate their mathematical thinking coherently and clearly to peers, teachers, and others
  • Analyze and evaluate the mathematical thinking and strategies of others;
  • Use the language of mathematics to express mathematical ideas precisely.

Connections

  • Recognize and use connections among mathematical ideas
  • Understand how mathematical ideas interconnect and build on one another to produce a coherent whole
  • Recognize and apply mathematics in contexts outside of mathematics

Representation

  • Create and use representations to organize, record, and communicate mathematical ideas
  • Select, apply, and translate among mathematical representations to solve problems
  • Use representations to model and interpret physical, social, and mathematical phenomena

The five stands of mathematical proficiency are:

(1) Conceptual understanding refers to the “integrated and functional grasp of mathematical ideas”, which “enables them [students] to learn new ideas by connecting those ideas to what they already know.” A few of the benefits of building conceptual understanding are that it supports retention, and prevents common errors.

(2) Procedural fluency is defined as the skill in carrying out procedures flexibly, accurately, efficiently, and appropriately.

(3) Strategic competence is the ability to formulate, represent, and solve mathematical problems.

(4) Adaptive reasoning is the capacity for logical thought, reflection, explanation, and justification.

(5) Productive disposition is the inclination to see mathematics as sensible, useful, and worthwhile, coupled with a belief in diligence and one’s own efficacy. (NRC, 2001, p. 116)

When you combine the two you end up with these 8 Standards for Mathematical Practice:

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

2. Reason abstractly and quantitatively.

3. Construct viable arguments and critique the reasoning of others.

4. Model with mathematics.

5. Use appropriate tools strategically.

6. Attend to precision.

7. Look for and make use of structure.

8. Look for and express regularity in repeated reasoning.

As teachers begin to discuss each of these eight standards they see how the process standards and mathematical proficiencies are interwoven throughout.  Examples of what that looks like for their classrooms and their practice can be found on videos at:  http://www.insidemathematics.org/index.php/common-core-standards

“Sum” thing to think about!

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Persistent Problem Solving

A question I have been asked frequently is, “What should the common core look like in practice?”  In my opinion there is no one right answer to that question.  However, the common core’s eight standards for mathematical practice suggests that it should not be business as usual.

The first of those eight standards requires students to make sense of problems and persevere in solving them.  For me, this suggests the need for problems that are something other than the word problems in textbooks.  They need to be engaging, perplexing, and present a challenge to students.  For older students a good source of this type of problem can be found in Dan Meyer’s blog, dy/dan. His 3 Acts videos entitled Hot Coffee,  Incredibly Shrinking Dollar, and Domino Skyscraper are just a few examples of this type of problem.  Meyer poses non-routine problems in an engaging manner without immediately disclosing every detail necessary for students to solve the problems.  Determining the important information needed to solve each problem should be a part of the students’ discovery process and is a critical component of sense making.

We, at ESSDACK, are looking forward to meeting and learning more about “Real World Math” from Meyer on February 6, 2012.

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All For One and One For All

Seven entities with vested interest in the area of mathematics and common core state standards have agreed to lend their collective expertise to the formation of a new coalition called MC3.  The organizations include the National Council of Teachers of Mathematics (NCTM), the National Council of Supervisors of Mathematics (NCSM), the Association of Mathematics Teacher Educators (AMTE), the Association of State Supervisors of Mathematics (ASSM), the Council of Chief State School Officers (CCSSO), the Smarter Balanced Assessment Consortium (SBAC), and the Partnership for the Assessment of Readiness for College and Careers (PARCC).

Their website, lists the mission of the Math Common Core Coalition as striving to ensure the successful communication, interpretation, implementation, and assessment of the Common Core State Standards.  They will work together to:

  • Provide a means to review, research, nurture and communicate common messages throughout the implementation and assessment of the CCSSM.
  • Provide expertise and advice from the communities of mathematics education content and assessment experts for the development of the content framework of the assessment consortia for the CCSSM.
  • Collect, analyze, and disseminate information about the implementation and assessment processes of the CCSSM that will inform future revisions of the CCSSM.
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Good Advice From the Top

I just read a blog post by the National Council of Teachers of Mathematics President, J. Michael Shaughness, about the Common Core State Standards for Mathematics. In his post, CCSSM and Curriculum and Assessment: NOT Business as Usual, Shaughness shares his thoughts about how the common core movement affords our nation an opportunity to make significant changes in mathematics instruction.  He includes links to assessment sites and his thoughts on curriculum and instruction.  Shaughness encourages schools to begin examining current curriculum to see how it aligns to the common core but warns against states or districts writing their own curriculum. With the majority of the states adopting the common core standards, this is our first opportunity to have a nationwide collaborative community working to improve mathematics curriculum, and instructional and assessment practices.

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Tidbits from NCSM

Last week I attended a meeting of the Kansas State Mathematics Leaders.  A good portion of our day was spent discussing the common core state standards and what is known about the future mathematics assessment.  Several of the members had attended the National Council of Supervisors of Mathematics national convention and reported back to the group.  Here are some of the things they shared.

·      The new state assessment will include the opportunity to learn (OTL) component that is currently available in Kansas at the high school level.  This means a 12-week testing window at the end of each school year will allow for two opportunities to take the state assessment.  Schools will be allowed to re-teach and re-test students who do not reach the level of mastery expected for that grade level on the first administration.

·      The reporting of assessment results will be in the form of a growth model, which may mean each student would have to be pre-tested or screened at the beginning of each school year.

·      Concepts will not be retaught from one grade to the next.  Mastery of grade level content is expected at the end of each year.  Therefore, it is imperative we develop support structures for struggling students early.  They will need access to the regular curriculum and additional support for content not mastered in a previous grade.  RTI and MTSS will play a critical role in the common core standards process.

·      The concerns about the common core for math that were enumerated at the convention included:  a disregard for technology, due to political debates; the overload of material to be covered in sixth grade;  concepts only being taught once; the need for vertical discussions about what and when to teach content; the need for professional development for teachers in both math content knowledge and pedagogical skills;  and the fact that everyone must teach the common core state standards for their grade level or the system will break down.

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What’s Behind the Common Core Math Standards

The release of the Common Core State Standards (CCSS) is a milestone in the standards movement that began more than 20 years ago when the National Council of Teachers of Mathematics (NCTM) published Curriculum and Evaluation Standards for School. NCTM, along with the National Council of Supervisors of Mathematics and other mathematics organizations, supports the goal of the CCSS to describe a coherent, focused curriculum that has realistically high expectations and supports an equitable mathematics education for all students.

The Standards for Mathematical Practice, which describe expertise that math educators at all levels seek to develop in students, is also a key component of the new Common Core Standards movement. These practices rest on key “processes and proficiencies” with longstanding importance in mathematics education, including the NCTM process standards and the strands of mathematical proficiency from the National Research Council’s report Adding It Up. The five process standards which run through all grade levels are problem solving, reasoning and proof, communication, representations, and connections.  The strands of proficiency specified by “Adding It Up” include:  adaptive reasoning, strategic competence, conceptual understanding, proceduaral fluency, and a productive disposition.

Other key elements in a student’s success in mathematics are: • Making sense of problems and persevering in solving them. • Reasoning abstractly and quantitatively. • Modeling with mathematics. • UsIng appropriate tools strategically.  • Attending to precision. • Looking for and making use of structure.

So, what is behind the common core standards for mathematics….RESEARCH!

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Where Do We Begin?

A journey of a thousand miles begins with a single step. ~Lao-tzu Chinese philosopher (604 BC – 531 BC)

With every new task we undertake, there is the question of where to begin. Sometimes the first step is obvious and other times there are various options. Moving to the Common Core State Standards could be equated to the journey of a thousand miles, in that it requires new learning for the majority of teachers in the nation.

Obviously, one of the first steps is to acquaint oneself with the grade level and subjects one is required to teach. However, on a larger scale, it is important to be familiar with all the standards. It is important to understand what content students have already been exposed to, been expected to master, and what they will be learning in the future.

Another critical component to be aware of is the transition timeline we will encounter as we move from our old system of standards to the new Common Core Standards. Without careful investigation and gradual implementation of the Common Core, students will have gaps in their understandings when the new assessments are fully implemented.

The Kansas Department of Education has recommended that teachers of Kindergarten and First Grade begin implementing the Common Core Standards in 2011-2012. The rationale is that those students will never have to take the current state assessment and this would alleviate part of the gaps we might experience when we are fully implementing and assessing the Common Core Standards.

At Essdack, we are working to make teachers’ first steps into the Common Core meaningful by providing days for investigating the content at their grade level. We also recommend investigating the crosswalks between the current state standards and the Common Core. As teachers learn where the Common Core and current standards overlap and differ, they will be able to design instruction that should lessen the gaps that might occur when the new assessments are implemented.

“Sum”thing new to start digging into!

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Focus on Fractions

The Common Core Standards for Mathematics, have focused on fractions, and for good reason.  In its comprehensive report on the state of math education in America, the National Math Panel said understanding fractions is “the most important foundational skill NOT developed among American students,” and is key to learning Algebra.  Since fractions are the gateway to Algebra and Algebra is a required class for high school graduation, elementary and middle school math teachers need to have a clear understanding of fractions and fraction instruction.

Today, in many middle school classrooms, students struggle with fractional concepts.  In the common core standards, students will first be introduced to fractions in the third grade.  They will learn that fractions are numbers, not just parts of cookies or pizzas.  This transition from thinking of fractions as “parts of a shape” to numbers will make it easier for students to comprehend their use in operations.  Students will be expected to work proficiently with fraction operations by the end of sixth grade and understand the relationship between fractions, decimals, and percents by the end of seventh grade.

With the new, and long overdue, emphasis on fractional concepts in the common core standards, it will behoove teachers to examine their own understanding of fractions.  It is often true, we teach the same way we were taught and most of us weren’t taught about fractions from a conceptual standpoint but rather in a very procedural manner.  In response to this new focus on fractions, teachers need to look for professional development opportunities that will increase their own conceptual understanding and provide them with instructional strategies to use to help their students.  Here at ESSDACK, we have established a facebook page called, Geared Up: Common Core Standards where teachers can interact with each other as we move toward full implementation of the Common Core Stndards.  We are also working on a website called , All Things Common Core, where teachers can share resources and ideas.

Just “sum” things I hope you will check out.

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Math: The Universal Language

Let’s face it; math is the same everywhere.  What other subject can say that?

In a very simplistic example the reason for math’s universal nature is easily explained:  one thing is true of all people in all nations of the world.  Barring a freak nuclear reactor incident or in the case of my good friend, Bentley, who lost a finger in a battle with a pig, we all have ten fingers and ten toes.  Our counting system is based upon this number.  Kids the world over are taught to count by raising one finger at a time….watch for it….it doesn’t vary from country to country.  The properties of the geometric shapes are the same the world over.  The fundamental principles for the manipulation of numbers, the order of operations, is an accepted practice throughout the universe.  The words are different, but the math is the same!

So, if all that is true why do we HATE math in America?  It is NOT socially acceptable to be bad at reading ….no one ever says, “I can’t read and I’m doing just fine in life”,  but it is screamed from the mountaintops that people don’t get math, never got math, can’t help their kids in math, hate math, see no reason to spend time doing math, and furthermore,  don’t ever use math!  No one is embarrassed to say these things aloud!  There is no social stigma attached to these statements like there would be if  we were referring to reading.  How has this happened?  We literally have a math pandemic in America and it is hurting us as a nation.

I feel really strongly about the need for teachers to educate themselves in the area of mathematics and the best practices for teaching it to our children.  We, as teachers,  are products of the system of instruction that we lived through. We had math “malpracticed” on us and we (including myself in the early parts of my career) have  carried on the malpractice.   We learned math as isolated skills without connection to other concepts or real-world application and wondered why we were doing it or when we were ever going to use it.  We learned math by memorizing procedures, often without understanding, and then when there got to be too much to memorize, we quit taking the subject.

So what do we do to turn this around?  It’s all up to us: the life-long learners, the people who love working with kids, the idealists who want to make the world a better place to live.  We have  to first recognize in ourselves the need for new information.  We, as professionals, have to be willing to learn new ways to help kids understand math conceptually.  We have to immerse ourselves in math content beyond the grades we teach, so we know where the math we teach leads.  We have to work together with other teachers to develop a cohesive plan of instruction for all grades, based upon research into best practices, a common set of standards and a common vocabulary.

We need kids to believe math is fun!  We need kids to believe math is important!  We need kids to believe math is relevant!  We need kids to believe math is not something to be feared!  In order to make all those things true,  teachers have to believe it first!!!!!!!!

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A Whole New Look

For teachers who are proficient at navigating their current state standards, the Common Core Standards may have a completely different look.  In the area of mathematics, the content of the document is divided into domains, clusters, and standards. Domains are the overarching term and refer to a large group of related standards.  Clusters are groups of related standards.  Because mathematics is a connected subject, standards from different domains and clusters may be closely related.  Finally, the standards define what students should understand and be able to do.  An example from fourth grade would be:  Domain – Operations and Algebraic Thinking; Clusters – Use the four operations with whole numbers to solve problems (three separate standards), Gain familiarity with factors and multiples (one standard), and Generate and analyze patterns (one standard).

Within the document, each grade level does not necessarily have the same domains or number of domains as the preceding grade.  However, the standards are aligned vertically from one grade to the next.  Unlike some current state standards, the Common Core State Standards do not spell out every step in the instructional process and some do not have example problems.  Therefore, educators are going to have to investigate the new standards thoroughly and translate them into new instructional practices.

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