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8. Elicit and use evidence of student thinking. 

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8. Elicit and use evidence of student thinking. 

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To support this principle, I have attached a portion of my diagnostic interview. The student first solved this subtraction problem as an addition problem, until I reminded her it was subtraction. She wrote the story problem based off of one I had shown her earlier on. Then she asked for help drawing the picture. When it came to explaining how she solved it, she did not want to write it so I had her explain it verbally and then I helped her write. She explained that she drew the tens and ones columns, subtracted 5-4 then 3-2. Then she said the answer was 11.
The student had some misconceptions while explaining how she solved this problem. She was slightly confused about the problem and thought that the numbers were 32 and 54, not 35 and 24. She began to explain the solution in this manner until I corrected her.  What I would do next with this student is show her the problems written vertically as well as horizontally so she can see that it is the same problem but just written in a different way.

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7. Support productive struggle in learning mathematics.

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7. Support productive struggle in learning mathematics.

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Supporting productive struggle while teaching mathematics is extremely important, but is often overlooked. In Intentional Talk, the authors discuss a method called "Troubleshoot and Revise". This method of discussion allows the teacher and students to have a meaningful discussion involving revising student work. It is not only used to help find the "correct" answer, but help students understand the answer and to collaborate on and make a decision about a practical solution.
The authors state, "We want students to know that thoughtful mathematicians voice their confusions; thinking collaboratively through errors can help everyone better understand the mathematics" (Kazemi & Hintz, 112).

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6. Build procedural fluency from conceptual understanding. 

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6. Build procedural fluency from conceptual understanding. 

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The article titled Procedural Fluency in Mathematics on the NCTM website discusses the importance of helping students build procedural fluency. NCTM states, "Procedural fluency builds on a foundation of conceptual understanding, strategic reasoning, and problem solving" (NCTM - 2014). Procedural fluency also supports the analysis of a student's, as well as his or her peer's, approaches to solving problems. It is extremely important for students to have a significant understanding of various procedures, as well as when particular strategies should be used. The article uses geometry as an example. NCTM states, "In geometry, procedural fluency might be evident in students’ ability to apply and analyze a series of geometric transformations or in their ability to perform the steps in the measurement process accurately and efficiently" (NCTM - 2014).

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For this principle, I found a video of a teacher doing a Number Talk with her first grade class. She used a ten frame and asked the students to think about how many blocks were shaded in. She first asked students how many they think they see and wrote the numbers on the board. Then she asked "How do you know that is your answer?" and wrote how they solved it on the board. If a student answered incorrectly, she asked if they wanted to change their answer or defend it. If they changed it she asks why they did and how they knew to change it. She also asked them why they defend their answer. She used multiple talk moves including adding on and revoicing with the students. She does not stop after one student shares, she allows all students to share their thinking.
She does not just ask for one answer, she goes further than that. She asks for multiple answers and she asks everyone to state how they knew what the answer was. Also, if they were incorrect she asks if they would like to change their answer rather than stating they are incorrect. The students responded really well to this method.

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4. Facilitate meaningful mathematical discourse.

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4. Facilitate meaningful mathematical discourse.

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I have attached my Number Talk lesson plan for this principle. I came up with several double-digit addition problems to present to the class. I began by reviewing the motions that are used in a Number Talk. Then I wrote the problem, gave students time to think, and then picked a student to state the answer; then I asked how they got to that answer. I used talk moves including revoicing wait time, and reasoning. After a student would explain how they got the answer I would ask the rest of the class if they agreed with the answer. If the answer required revisions I would ask the student to rethink. I also had at least three people solve each problem. This was to show the rest of the class that there can be multiple ways to solve problems.
This lesson relates to the principle because I required the students to use verbal discourse when explaining their answer to the class. Also, I required the students to use written discourse when I asked them to come up to the board to show how the problem is solved.

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3. Use and connect mathematical discourse. 

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3. Use and connect mathematical discourse. 

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In this NCTM publication titled Bringing in the Real World, the author discusses methods of how to incorporate real-life scenarios, diversity, and social issues into mathematics. For example, he states that Leonard and Guha had their students take photos of the community and then had them write word problems based on the pictures they took and what they saw. He states, "[the problems] have the potential to be more engaging because they come from students' environments and because the students wrote them" (Felton, 2014).
This connects to the principle because the students are connecting their word problems and mathematical thinking to things they already know; as well as using written, and most likely, verbal discourse to create and solve these word problems.
(article)
http://www.nctm.org/Publications/Mathematics-Teaching-in-Middle-School/Blog/Bringing-in-the-Real-World/

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2. Implement tasks that promote reasoning and problem solving. 

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2. Implement tasks that promote reasoning and problem solving. 

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I have attached my Community-Based Math Lesson plan for this principle. I wrote story problems for the students to solve involving time and places in their community. For each problem, I picked a popsicle stick to select a student to tell me the answer and show it to me on the clock (both analog and digital). Then I asked them to explain how they knew what time it was or how much time had passed.
Throughout this lesson, I required the students to use problem solving and reasoning. I showed them some problem solving questions, I required them to write their own questions, and then they had to solve another student's. I also asked them to explain how they knew what time it was.

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1. Establish mathematics goals to focus learning. 

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1. Establish mathematics goals to focus learning. 

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I have attached the objectives, standards, and assessment I used in my Community-Based Math Lesson. I showed the students some example story problems regarding time, then I modeled how to write a story problem and had the students write their own story problems. After they wrote their problems they shared them with a partner and had them solve the problems.
This artifact fits the principle because I set achievable goals/objectives for my students to fulfill during my lesson. Then, I assessed my students to see that they met those objectives. I did so by filling out the checklist (also attached) and observing the students as they answered the questions and wrote their own.