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The best question a student can ask is "Why?," and there is never a bad time to ask it.

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Each of these six procedures represents practice with higher level thinking. This are not exclusive to math, but they must be used in tandem with strategies, skills, and algorithms.

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Thinking logically to find a solution. Problem Solving skills include understanding the problem, formulating plans, completing those plans, and reviewing/extending.
This could include extracting key points, trying new strategies, and others.

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Why does my solution make sense? Is is a structurally sound solution, or did I just get lucky?
Good proofreading and writing skills also help students identify and fix mistakes. Very useful for student motivation, as some of these skills involve perseverance on difficult problems.
Also deepens understanding.

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Learning how to appropriately talk and write about math is a difficult concept at first, but it very influential. It helps keep higher-order concepts manageable. Good communication makes it easier to explain our point of view. Hand in hand with Reasoning and Proofwriting.

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THE FIRST FOUR STANDARDS All students will communicate mathematically through written, oral, symbolic, and visual forms of expression. Communication of mathematical ideas will help students clarify and solidify their understanding of mathematics. By sharing their mathematical understandings in written and oral form with their classmates, teachers, and parents, students develop confidence in themselves as mathematics learners and enable teachers to better monitor their progress.

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Number talks are great ways to get students practicing math communication. They are also useful as starters for higher-level thinking practices.

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Finding connections between formulas and applications, conceptual math and the real world, the inter-relatedness of facts, and old to new problems, amongst many others.

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By NCTM President Linda M. Gojak
NCTM Summing Up, October 3, 2013

One of the most memorable moments I had in teaching mathematics occurred in a fifth-grade class. We began the year using rectangular arrays as a model to develop the concept of prime and composite numbers. We hung student-made posters of the numbers from 1 to 100 with representations of arrays and lists of factors for each number around the room. By the end of that unit all my students had mastered multiplication facts and could factor with facility as we began our work with fractions. The connections among concepts and the use of concrete representations certainly led to deeper understanding. Later that year, students worked with a variety of models to find area and perimeter of rectangles and extended that experience to find the areas of triangles, parallelograms, and trapezoids. Most students were able to generalize a formula, albeit not always the most efficient, for each polygon. One day, a student commented that this was just like what they had studied at the beginning of the year. When I gave a puzzled look, the class pointed to the posters still on the wall from our first unit of study and said, "You know, that factor and multiple stuff." I had a new appreciation for the power of providing experiences that enable students to make connections among mathematical ideas. My students remembered and understood the mathematics that we had studied months earlier!

Since that experience I have given much thought to the Process Standards in Principles and Standards for School Mathematics, and their impact on teaching. With the current focus on progressions and trajectories of content standards, the potential of the Connection Standard (NCTM, 2000) continues to pique my interest. It's a powerful standard:

Instructional programs from prekindergarten through grade 12 should enable all students to-

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.

Too often, rather than making sense of mathematical ideas, students focus on remembering procedures or tricks. For example, how many students learn "flip and multiply" to divide fractions but have no idea why it works? Often those who understand why the procedure works struggle to apply it in problem situations. The procedure alone often leads to misconceptions. Students who work from rote memory often invert the wrong fraction, forget to change operations, or even apply the rule when multiplying two fractions. The meaning of operations doesn't change from whole numbers to fractions. For example, in the early grades, the understanding that students develop of division of whole numbers often rests on the idea that "9 ÷ 3," for example, asks how many groups of 3 are in 9. As students move to fractions, it is important to provide them with experiences that connect this whole-number understanding to similar examples with fractions: "9/16 ÷ 3/16," for example, asks how many groups of 3/16 are in 9/16. In this way, students gain a deeper understanding rather than depending on a memorized procedure and can apply division of fractions to a variety of problem-solving situations and real-world applications.

Many teachers use manipulative materials to introduce a new concept. Manipulatives themselves, however, do not ensure understanding. We must provide experiences that support students' efforts to make connections between what they are doing with the materials and the mathematical ideas that they represent. This takes time and teacher expertise. Algebra tiles are not an end to teaching basic algebra concepts-when used appropriately, they provide students with opportunities to connect their work to the concepts. And it is these connections that enable students to make sense of the abstract representations.

Although it is important to think about the connections among concepts within the grade level or courses that we teach, it is also important to reflect on the connections across grade levels. This work involves thoughtful discussions with other colleagues about the way that concepts are taught and the potential linkages among those ideas. Many of us learned mathematics as isolated pieces of information. Taking a mathematical concept and considering how it originates, extends, and connects with other concepts across the grades will help teachers to develop a deeper understanding. It is then that we can plan instruction that ensures that our students regularly make connections to help them make sense of the mathematics they are learning.

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Making Mathematical Connections - National Council of Teachers of Mathematics

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Linda M. Gojak One of the most memorable moments I had in teaching mathematics occurred in a fifth-grade class. We began the year using rectangular arrays as a model to develop the concept of prime and composite numbers.

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Being able to create representations of concepts not only shows a strong understanding of the concept but also makes it easier to manipulate (for many students). Students who are able to visualize the concepts they are practicing tend to have greater success, in my experience.

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How do we prepare thoughtful questions and make connections between representations?

When students share their brilliance with diagrams, manipulatives, words, or numbers, they share an abundance of information that can be woven into a storyline of reasoning and logic. It takes practice to listen carefully for that brilliance and make connections among those pieces of our puzzle that lead to the deep understanding of mathematics we want for every student, but with time, our pedagogical moves at leading these discussions get better and better. The most important aspect of this skill, however, is that we are using student representations to do so. Otherwise, we are the ones doing the heavy lifting. By using students' representations, their mathematical voice is front and center and they see themselves as doers of mathematics!

When solving a problem, it is common to inherently relate it to others we have previously solved. We begin with proven strategies of success and do our best to find connections between our past understandings and the mathematical challenges ahead. This connection, or bridge, is how we grow our brains and our base knowledge about a topic.

Creating connections among multiple representations, explanations, and expressions or equations is not simple and requires thoughtful questions to consider. The work of Peg Smith and Mary K. Stein in "5 Practices for Orchestrating Productive Mathematics Discussions" illuminates the importance of anticipating how students may reason about a given problem to better prepare thoughtful questioning that challenge our students' thinking. The more prepared we are as educators, the more opportunities we can create for our students to consider the multiple representations produced in each classroom and recognize their similar structures.

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Mathematical Representations: A Window into Student Thinking

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How do we prepare thoughtful questions and make connections between representations? When students share their brilliance with diagrams, manipulatives, words, or numbers, they share an abundance of information that can be woven into a storyline of reasoning and logic.

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There are always going to be unknown situations in Math. Being able to investigate new topics is very useful in grasping that first understanding, instead of waiting for the solution to be presented to you.

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An example of this would be adding positive and negative integers using algebra tiles!

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Structure of Knowledge is the idea that students are able to provide a description of a problem in their own words as a way to start thinking about the problem. This would include understanding what the variables mean, the type of equation one is working with, and more.