Multiple Methods, Representations,

 Teachers lead students to understand why the math works and what it means. Teachers develop this conceptual understanding in many ways, sometimes through direct instruction and other times through the facilitation of student discovery. At times, this is done through the concrete use of manipulatives (base ten blocks, Cuisenaire Rods, pattern blocks, color counters, reallife objects, etc.), the pictorial use of illustrations of the math (picture, bar model, fraction strip, chart, number line, graph) and the abstract use of numbers (algorithms and word problems). There is not a prescriptive process on when to use each method; methods will vary by concept.
make sense of math.Students need more than the procedure to truly make sense of the math. In the Common Core Standards for Mathematical Practice, SMP #4 requires students to model with math.
In the clip on the right, Abbott and Costello debate about the answer to the problem 13 x 7. Both struggle as they attempt to solve with the recalled procedure. Check it out for a laugh, and remember that our kids need more than just tricks and memorized steps to be successful in math. 

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Before you teach a lesson, check it out on LearnZillion.org  you'll find Common Core aligned concepts, shown in 35min videos, with an emphasis on visual modeling. Maybe you'll see a way to represent something visually that you hadn't thought of before!
Check out some of the video examples of multiple representations on the EMath video page.
When you are planning for a lesson, search for the multiple ways you can teach students the concept. For example, if you are teaching how to change an improper fraction into a mixed number (4.NF.3), you can teach students to do this in several ways, allowing them to make connections to bigger mathematical concepts and strengthen overall understanding.
Here's an example of what that could look like. First, teach students to convert from improper to mixed using visuals. Below, you'll see I used rectangles divided into fourths. I shaded six parts and found that there was a total of 1 1/2 shaded parts. Here, a student could clearly see how 6/4 is equivalent to 1 1/2. You may have students explore with different pictures and visual models. You also could help students make the connection to an earlier concept, remembering learning builds on what students know. Next to 2), I decomposed 6/4 to 6 unit fractions of 1/4. I circled a whole (I know 4 fourths is equivalent to one whole) and noticed two 1/4 unit fractions left over. 1/4 + 1/4 =2/4. So 6/4 is equivalent to 6 one fourth parts, or 1 2/4, which is equivalent to 1 1/2. Finally, you could show students the procedure (division in number three), but could tie it back to the models they've seen earlier.
Here, as a student I know what it looks like, why it works based on the math I've learned previously, and a quick, procedural way to solve when I am faced with larger numbers. As a student, I should feel empowered to choose any route (or a different one!) as long as I can explain my rationale and make sense of the answer.
Check out some of the video examples of multiple representations on the EMath video page.
When you are planning for a lesson, search for the multiple ways you can teach students the concept. For example, if you are teaching how to change an improper fraction into a mixed number (4.NF.3), you can teach students to do this in several ways, allowing them to make connections to bigger mathematical concepts and strengthen overall understanding.
Here's an example of what that could look like. First, teach students to convert from improper to mixed using visuals. Below, you'll see I used rectangles divided into fourths. I shaded six parts and found that there was a total of 1 1/2 shaded parts. Here, a student could clearly see how 6/4 is equivalent to 1 1/2. You may have students explore with different pictures and visual models. You also could help students make the connection to an earlier concept, remembering learning builds on what students know. Next to 2), I decomposed 6/4 to 6 unit fractions of 1/4. I circled a whole (I know 4 fourths is equivalent to one whole) and noticed two 1/4 unit fractions left over. 1/4 + 1/4 =2/4. So 6/4 is equivalent to 6 one fourth parts, or 1 2/4, which is equivalent to 1 1/2. Finally, you could show students the procedure (division in number three), but could tie it back to the models they've seen earlier.
Here, as a student I know what it looks like, why it works based on the math I've learned previously, and a quick, procedural way to solve when I am faced with larger numbers. As a student, I should feel empowered to choose any route (or a different one!) as long as I can explain my rationale and make sense of the answer.