9+6, the Journey #mondaymathbite

“It is good to have an end to journey toward; but it is the journey that matters, in the end.” 
― Ernest Hemingway

Welcome to the new and improved Adventures in Common Core! I'm so excited to have a new design and can't wait to share more math with you!

As math problems and videos continue to go viral, I feel compelled to step up to the plate to address some of the misconceptions. The angry rants about Common Core ruining our children are not letting up, and it seems that the basis of many of these oppositions is that Common Core makes math harder rather than easier, that it has convoluted mathematics somehow.  This couldn't be further from the truth.  Many parents who are anti-Common Core have the best of intentions; they want what is best for their child, and when they see something outside of their own comfort zone, they jump to their child's defense: "This is too hard! My child has always been proficient in math and now they're struggling! They got an answer; isn't that enough!? This is math, not writing! If I can't do it, they can't do it!" 

Here is one of the latest videos to get some heat from opponents, with angry headlines like "Ridiculous!" and "Common Core Teachers Kids New Way to Add 9 + 6 That Takes 54 Seconds!"  

Let me break it down for you. Here are some big ideas, strategies and models in this video that support the journey of learning 9+6:

·        Big Idea: base-ten number system - our number system works in a base ten, with ten ones comprising a ten, ten tens comprising a hundred, ten hundreds comprising a thousand, etc; starting in Kindergarten, students gain foundations for place value by counting, "bundling" into groups of ten, and seeing teen numbers as made up of a ten and "some more" (K.NBT.1); 

      The role of 10 and its relationship to place value continues through 5th grade in the domain in the standards called Numbers and Operations in Base Ten;

·        Big Idea: understand "why" - in the Standards for Mathematical Practice in the CCSS, students to expected to justify their reasoning ("construct viable arguments") with evidence, which could include the use of objects, drawings, diagrams and actions, and be able to communicate this reasoning to others. 

Get your set of awesome free posters for the Standards for Math Practice from https://www.teacherspayteachers.com/Store/Got-To-Teach

     Understanding "why" the math works is far more rigorous than just getting an answer, and being able to engage in explaining their thinking supports students on their way to more complex math topics;

·        Strategy: friendly numbers - "friendly" numbers are so called because they support students in making computations easier. In first grade, students add and subtract within 20 using many strategies, including "decomposing a number leading to a ten" (see below for more about decomposing) and "creating equivalent but easier known sums" (1.OA.6). While, students may have varying opinions of what numbers are "friendly" (your friendly number may not be my friendly number!), in general we work with fives, tens ("their friend 10"), and hundreds to make problems friendlier; 

Friendly 5!

·        Strategy: decomposing numbers - "decomposing" is a fancy word for "breaking apart" numbers in order to use them flexibly. In this video, the teacher decomposes the numbers 6 into a 1 and a 5; 1 and 5 can be said to be parts of 6. The word "decompose" is all over the K-5 math standards, beginning in Kindergarten where students decompose numbers less than or equal to 10 into as many pairs as they can (e.g. 10 can be decomposed into 5+5, 6+4, 3+7) (K.OA.3), and connecting to work with fractions starting in 3rd grade. Students who can compose and decompose numbers are well on their way to fluency.

·       
Model: number bonds - a number bond is a part-part-whole model that shows number relationships; in this video, the teachers decomposes the number 6 into a 1 and a 5 using a number bond. Here is an example of a number bond for 5:  

      Many curriculums use number bonds as early as Kindergarten as a model for decomposing numbers, including Singapore Math, engageNY, and Eureka Math.

·        Big Idea: related facts - related facts are the artists formerly known as "fact families." In first grade, as previously mentioned, decomposing numbers in addition equations allows students to create equivalent equations (such as 9+6 = 10+5 as seen in the video) that are easier to solve (1.OA.6). Initially, students reason about these equivalents with concrete manipulatives and models like tens frames:

Beginning with double tens frames showing 9 and 6, students can create an equivalent sum by composing a ten.

     Additionally, as early as 1st grade, students are beginning to observe properties of numbers (e.g. the sum for 8+3 is the same as 3+8) and make generalizations about the relationship between addition and subtraction.

The major work of grades K-2 is addition and subtraction, and that doesn't mean that students memorize the 200 (200!!!) discrete single digit facts and magically have a deep understanding of these operations. In fact, it is not until 2nd grade that we expect students to take any less than 56 seconds to solve an addition problem within 20, like 9+6=15!

All the work with composing and decomposing numbers to 20 with concrete manipulatives, representations, and mental strategies in Kindergarten, 1st and 2nd grades culminates at the end of 2nd grade, when we expect students to "know from memory all sums of two one-digit numbers." "Knowing from memory" is not a result of rote memorization through diligent flash-card use or timed tests. In fact, contrary to popular belief, "computational fluency extends far beyond having students memorize facts or a series of steps" (Baroody, 2006; Griffin 2005) 

Fluency is a destination reached through a carefully crafted mathematical journey, during which students are learning concepts and not just getting answers. Is the answer important? Of course it is. Accuracy, efficiency and flexibility are the backbones of fluency, and are the goals at the end of the road. The Common Core standards frame this journey toward mathematical proficiency for us, providing a balance of understanding, procedural skills and fluency, and application.