Friday, October 2, 2015

Common Core - #3 Structure of K-8 Content Standards: Footnotes Matter!

My last two posts focused on the structure of the Common Core Standards of Mathematical Practice. The big idea of those posts was the title of the practice is NOT enough - you need to read the narrative to get to what students should be doing and saying.

The theme of this post is much the same - the Common Core Math Content standard alone is NOT enough to truly
understand what it is students should know and be able to do if they have mastered the content. The content standard, which is often what is posted in textbooks or put on the board, is in fact, a small part of the big picture and without seeing the big picture, we end up teaching isolated skills and facts. Understanding the structure of the content standards provides a big picture at each grade level (focusing on K-8 right now) of where students are going, and how the standards, as a whole, are continuing to develop and expand mathematical content knowledge. It is a "learning progression".I am going to focus on K-8 structure now, as it is the same for all these grades. My next post will focus on the High School Content Standards, which have an additional structure.

For all grades K-8, there is a grade introduction, which gives the BIG IDEA of the whole grade and what, by the end of that grade, students should know and understand about mathematics. I am pulling the introduction from Grade 3 as an example:

In Grade 3, instructional time should focus on four critical areas:
  1. developing understanding of multiplication and division and strategies for multiplication and division within 100;
  2. developing understanding of fractions, especially unit fractions (fractions with numerator 1;
  3. developing understanding of the structure of rectangular arrays and of area;
  4. and describing and analyzing two-dimensional shapes.
  1. 1) Students develop an understanding of the meanings of multiplication and division of whole numbers through activities and problems involving equal-sized groups, arrays, and area models; multiplication is finding an unknown product, and division is finding an unknown factor in these situations. For equal-sized group situations, division can require finding the unknown number of groups or the unknown group size. Students use properties of operations to calculate products of whole numbers, using increasingly sophisticated strategies based on these properties to solve multiplication and division problems involving single-digit factors. By comparing a variety of solution strategies, students learn the relationship between multiplication and division.

  2. 2) Students develop an understanding of fractions, beginning with unit fractions. Students view fractions in general as being built out of unit fractions, and they use fractions along with visual fraction models to represent parts of a whole. Students understand that the size of a fractional part is relative to the size of the whole. For example, 1/2 of the paint in a small bucket could be less paint than 1/3 of the paint in a larger bucket, but 1/3 of a ribbon is longer than 1/5 of the same ribbon because when the ribbon is divided into 3 equal parts, the parts are longer than when the ribbon is divided into 5 equal parts. Students are able to use fractions to represent numbers equal to, less than, and greater than one. They solve problems that involve comparing fractions by using visual fraction models and strategies based on noticing equal numerators or denominators.

  3. 3) Students recognize area as an attribute of two-dimensional regions. They measure the area of a shape by finding the total number of same-size units of area required to cover the shape without gaps or overlaps, a square with sides of unit length being the standard unit for measuring area. Students understand that rectangular arrays can be decomposed into identical rows or into identical columns. By decomposing rectangles into rectangular arrays of squares, students connect area to multiplication, and justify using multiplication to determine the area of a rectangle.

  4. 4) Students describe, analyze, and compare properties of two-dimensional shapes. They compare and classify shapes by their sides and angles, and connect these with definitions of shapes. Students also relate their fraction work to geometry by expressing the area of part of a shape as a unit fraction of the whole.
As you can see, the introduction is important because it gives the critical areas (i.e. BIG IDEAS) that are the focus for the entire grade and all the standards in that grade. The critical areas are highlighted at the beginning and then there is a more detailed description of each critical area. The introduction provides the end result, what students in that grade should be able to say, do, know and understand if all the standards for that grade have been effectively taught and learned. How many people read the introductions? If you just look at the standards alone for a specific grade, you miss many key components - such as, in this example, the idea that you should be developing the idea of multiplication and division using equal size groupings in various arrangements? Or that you begin fraction understanding with unit fractions?

There are 11 domains in the K-8 content standards, some of which appear only in certain grades (i.e. Counting & Cardinality is only in Kindergarten). Domains are the groups of related standards, which help give focus. The domains in K-8 are as follows (with the grades they appear in listed):
  • Counting & Cardinality (K)
  • Operations & Algebraic Thinking (K-5)
  • Number & Operations in Base Ten (K-5)
  • Number & Operations-Fractions (3-5)
  • Measurement & Data (K-5)
  • Geometry (K-8)
  • Ratios & Proportional Relationships (6-7)
  • The Number System (6-8)
  • Expressions & Equations (6-8)
  • Functions (8)
  • Statistics & Probability (6-8)
It makes sense that kindergarten is the only place we see counting & cardinality - remember the more time in kindergarten should be spent on number than anything else. Which means when students leave kindergarten they should have a clear understanding of counting and cardinality that will help them as they move forward into the other grades. Notice that fractions don't come along until third grade - because K-2 you should be building the foundation of whole numbers, operations, algebraic thinking first.  The domains give you a broad sense of each grade levels focus and how the progression of standards is occurring as we move vertically in the grades. Students are building their knowledge as they go through the grades, and even though a domain may not appear in a grade, it doesn't mean that students are no longer using those skills or content - they are applying what they learned in new and different ways.

Clusters group related standards within a domain and are a summary of those standards that explains how they relate. Let's look at the clusters within the Grade 3 domain "Operations & Algebraic Thinking":
Domain: Operations & Algebraic Thinking (Grade 3)
  • Represent and solve problems involving multiplication and division (cluster)
  • Understand properties of multiplication and the relationship between multiplication and division (cluster)
  • Multiply and divide within 100 (cluster)
  • Solve problems involving the four operations and identify and explain patterns in arithmetic. 
So, if we look at the first cluster, taken into account with the domain it falls under, all the content standards within this cluster relate to representing and solving problems that have to do with multiplication and division, which help students understand algebraic thinking (domain) and the operations themselves. 

Content Standards 
Have you noticed all the information that has occurred before we even reach the specific content standards? The content standards are the SPECIFIC skills students should understand and be able to do within the broader context of the critical areas, domain and cluster. Let's look at the specific content standards for 3rd grain under the following Domain & Cluster:
Domain: Operations & Algebraic Thinking (Grade 3)
  1. Cluster: Solve problems involving the four operations and identify and explain patterns in arithmetic
  •  3.OA.D.8  Solve two-step word problems using the four operations. Represent these problems using equations with a letter standing for the unknown quantity. Assess the reasonableness of answers using mental computation and estimation strategies including rounding.3
  •  3.OA.D.9  Identify arithmetic patterns (including patterns in the addition table or multiplication table), and explain them using properties of operations. For example, observe that 4 times a number is always even, and explain why 4 times a number can be decomposed into two equal addends.
As you can see, these specific content standards are not just isolated skills but are supporting the cluster, where students are able to solve problems using the four operations and explain patters, and are also part of students overall operations and algebraic thinking. All of these components in the structure of the standards work together. The standards are specifics that are just a piece of the big idea for the whole grade level.

Footnotes and Examples
Additionally, there are footnotes and examples embedded throughout the content standards, appearing in all parts (Introduction, Domain, Cluster & Standards). These are often overlooked but are key to really helping focus the learning at each grade level.  The examples provide insight into what the standard might look like at that grade. The footnotes have several purposes - referencing the Glossary for terms, or further explanations, or...most importantly, provide additional parameters.  In the example above, you will note that standard 3.OA.D.8 (3 is grade level, OA is domain, D is cluster, and 8 is standard) has footnote 3, which states:
This standard is limited to problems posed with whole numbers and having whole-number answers. Students should know how to perform operations in conventional order when there are no parentheses to specify a particular order.

Um...WOW!!  So, students should be solving two-step word problems with the four operations, use equations, etc. but with WHOLE NUMBERS and with WHOLE NUMBER answers. So, even though 3rd grade is introducing fractional numbers, this particular Grade 3 standard says students should work with whole numbers, as they are just developing fractional understanding. Footnotes are important!  How many teachers are reading them?

I am not sure I need to reiterate, but I will - the content standard is NOT ENOUGH!!  The individual content standards themselves are just a small, specific skills that are part of much larger overarching content. The standards all relate back to the critical ideas for each grade level . It is important to familiarize yourself with your grades' overall big idea, which includes not just the standards, but where they fit into the domain and critical areas (big ideas) for your grade. And don't forget your footnotes!!!

(In my work with The Charles A. Dana Center at UT, Austin, they have this nice visual to the right of the K-8 content standards structure).

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