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Students need to be able to make sense of the problem before they can reasonably enter the problem. Once they are engaged in the problem they must have the mindset of perseverance to self-correct initial misconceptions and ultimately check their thinking.
-First Grade

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This practice fosters critical thinking and student independence. Students need to be able to transfer their problem solving skills to real-world situations.
-Tippett

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Students are able to understand the meaning of a problem and find an entry point to begin solving the problem.

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Making sense of problems are so important to build that  understanding.

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They aren't afraid to make mistakes and have a desire to figure out how to correct them. Students demonstrate a growth mindset by tackling new problems.  -Tippett

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Younger students are able to solve the problem in several different ways. They might use manipulatives to help them. They are also able to try a different method and ask "Does this make sense?"

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Concrete models or pictures
Hearing things like, "This makes sense" or "I don't get it."
-First Grade

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This is important because students need to be able to make sense of problems and find that entry point.

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IMO, this is the most important MP for my kindergarten students. The idea behind manipulating numbers and working with quantities is soooo abstract for their little brains, and it is the modeling with these early numeracy skills which helps them understand mathematics.
-Tawnya McLain

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It is important because students have to listen to each other in order to critique each other's answers. They also have to have a strategy ready to go to prove their thinking.

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Teachers can model this by showing the thinking use in understanding what a problem is asking. Also, teachers can show multiple different ways to solve the problem.

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Construct viable arguments and critique the reasoning of others.

This practice challenges each other and their thinking.  The cognitive load is on the students.  Students challenge each other's thinking.

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Younger students might use concrete objects or pictures to show the action of the problem. They will use what they already know to help them solve the current problem.

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Teachers can model this by thinking aloud about an entry point to a problem, model using models or pictures, orgo down a wrong path and coming to a conclusion on why it does not make sense. Teachers can model checking an answer multiple ways to see if you always arrive at the same answer. Celebrate students who have demonstrated making sense and perseverance.
-First Grade

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The way that numbers are modeled can help students understand the quantity, as well as the multiple ways that numbers can be composed and decomposed.
This practice goes beyond modeling numbers, to how to model problems and tasks to help solve or approach problems.

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We do this daily when we build and represent on ten frames, when we use objects to reenact story problems, and when we draw pictures to support our thinking.
-Tawnya

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This practice is important as it allows students to support their understanding by discussing it with their peers. It is not just giving the answer, but being able to explain it in detail.

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The teacher can "think out loud" during the direct and guided instruction. Model the language and steps used for problem solving. Ask questions that deepen thinking. -Tippett

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This practice is crucial for visual learners that need to show a variety of ways to reason/prove his/her thinking. Especially when learning new concepts, this practice allows students to identify quantities in a contextual situation.

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"I respectfully disagree, Can you explain that please, have you thought of, have you tried."

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This practice is important because If one way doesn't make sense they are able to find a different solution that does make sense and keep going.

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This could be students using ten frames or a number rack to represent numbers. This could be various pictures that students draw to represent a number or the place value. This could also be students drawing a model to represent what is happening in a problem. This could be an open number line, a picture to compare quantities, distances, fractions, etc.

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I support this by showing how I can utilize pictures and objects to help me think about problems and solve them.

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Other than during designated math instruction times, critical thinking and problem solving should be embedded in every aspect of the students' day. -Tippett

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This is such an important skill to teach in order to have a voice not only in math but life. We live in a world where it is ok to disagree respectfully and back it up with evidence or thinking.

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Constructing an argument and critiquing other's reasoning is  important for students to accept that there are different ways of thinking and "doing" math.
Kristen

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The students are working together and asking questions. They are not just agreeing with the other, but seeking understanding.

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The teacher can model by using the same language but with the studetns.

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Students are able to work the problem in different ways and can use different strategies in order to check their work.

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Recently, we have been using this practice in class to repeat our reasoning in order to understand the concepts regarding fractions and decimals. Students will be using repeated reasoning communicating with classmates during a fraction/decimal number talk. They will use repeated vocabulary and explanations to state thinking and build on each other.

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The teacher can think out loud while solving a problem and model how to find an entry point.

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In kindergarten, students are given the opportunity to model with mathematics multiple times throughout the day.
-Tawnya

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Number Talks, Lessons, independent work, Number Corner, Workplaces.

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Teachers can use a variety of ways to represent quantities. Numbers can be grouped by 5's, 10's, or fractions based on how you want students to work with these numbers or the patterns that they need to recognize.
Number talks are an amazing time to demonstrate how to write the thinking that students are sharing.

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Number talks, science lessons, read aloud...

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Teachers can model this by sharing different strategies with students and also by calling on students who have the same answer but used different strategies to show their work and then have a discussion around checking our work.

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Mini-lesson and Work Places

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By demonstrating the types of questions and responses used. Referencing sentence stems to help guide the conversation

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Recording different strategies and discussing if the strategy worked and comparing to others. "I like how AH used a picture to solve the problem, I used a number line. I think my way works better for me."
Kristen

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Students are justifying conclusions and communicating them to others.

Students look at data and make arguments.

Students read or listen to the arguments of others and decide whether they make sense. Students ask questions to clarify or improve.

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Problem Strings, Number Corner

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Bridges has done a beautiful job of weaving modeling throughout our second grade units. Numbers are represented in various ways with the structures of 5 and 10 repeated in various ways. The tasks have exposed students to various ways that they can use an open number line to demonstrate the strategies that they are using.

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Ask clarifying questions.

Model justifying conclusions.

Sentence stems

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Number Corner, work places