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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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I believe every classroom should promote productive struggle. Each classroom should be an environment in which students aren’t afraid to make mistakes because their classroom fosters mistakes by having open discussions in which a student can listen to one another and change their thinking if needed. By having challenge and struggle a student can only learn from mistakes and misconceptions when in an environment that allows them to continue to build upon their thinking. The artifact included is a lesson plan in which students who were placed in different ability groups were given different goals and objectives in the lesson. An involved and intentional teacher will recognize the abilities and different goals of individual children in their classroom. Each student should have a goal that is a reasonable challenge for them. Each student should be slightly uncomfortable so they are challenged enough to be pushed further in their thinking. This is productive struggle. In this artifact you will see that I have included different goals for different guiding math groups based on their ability to skip count using coins. They were given greater totals if they were strong in this skill because the greater totals were a much greater challenge than if they were given ones they had already mastered.

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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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When students explain their mathematical thinking, reasoning, and understanding in mathematical problem-solving and discourse, they eliciting their understanding of a concept. Within this, students are reflecting and self-monitoring their mistakes so they may go back and revise their thinking. This was evident in our Intentional Talk book where we learned about trouble-shoot and revise. This strategy was incredibly welcoming of discussing mistakes or changes that could be made to a strategy or way of thinking. Students also had the opportunity to change their thinking to create the most effective and efficient method of problem-solving specific types of problems. I included the diagnostic interview I conducted in my practicum classroom because the student made in-the-moment decisions about what strategies to use and I used purposeful questions to elicit their thinking and understand what the student was thinking. I chose to include the diagnostic interview as my artifact because I was asking the student to explain their thinking and trying to understand how they came to a solution.

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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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Effective teaching of mathematics builds fluency with procedures. Once a student has a conceptual understanding of a mathematical concept, they have the opportunity to use their own reasoning and strategies to solve problems. In a diagnostic interview students do just that. They choose a strategy to problem solve and them must explain the mathematical basis and reasoning behind their thinking. Teachers must be flexible with the use of strategies in their classroom because students are likely to reflect on whatever strategy works best for them for that specific type of problem. Students may use strategies that are most efficient or that cater to their learning style the best. Teachers must provide students with opportunities to build procedural fluency using the conceptual strategy that is most efficient and effective for them.  I used the diagnostic interview I conducted with a student in my practicum classroom along with a video we watched in class because it effectively shows how students use their own strategies and procedures based on their understanding of concepts they have already learned.

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What Number is Below the Arrow?

This video clip provides an example of a teacher and student engaging in the diagnostic assessment. As you view the clip, consider the following questions:

What is the teacher learning about the student's mathematical understanding?

If you were this child's teacher, how would you use this information to change your instruction?

What are the benefits of conducting such an interview?

How will you organize and schedule opportunities to conduct diagnostic assessments? Which of your students are first priorities for the interviews?

What Number is Below the Arrow? (5:17 min)

This clip shows a student responding to three assessment items.

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What Number is Below the Arrow? This video clip provides an example of a teacher and student engaging in the diagnostic assessment. As you view the clip, consider the following questions: What is the teacher learning about the student's mathematical understanding? If you were this child's teacher, how would you use this information to change your instruction?

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Within a number talk, posing purposeful questions is essential for students to be able to understand and explain their own thinking. As a teacher, I will work on guiding questions and revoicing to pose purposeful questions that keep discussions moving forward. Posing purposeful questions allows students to realize where they may be making an error and want to consider changing their thinking. Posing purposeful questions also advances students understanding by creating questions and concepts students can build upon. This causes students to clarify and elaborate more on their thinking. Teacher should make sure their questions go beyond just asking for information and require students to respond with justification. Intentional questions also require wait time so that each student has a sufficient amount of time to think. Purposeful questions will draw information out of students that leads them into a rich discussion with one another. I again used my number talk as my artifact because it is crucial to ask purposeful questions to elicit student thinking, clarify, and drive conversation during a number talk

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4.	Facilitate meaningful mathematical discourse

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4.	Facilitate meaningful mathematical discourse

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In number talks, students are able to reflect and project their own understanding of mathematical concepts in addition to comparing and critiquing their ideas and the ideas of their classmates. A number talk must be a collaborative effort where students lead the discussion and support one another to reach a common understanding of the mathematical concepts. Within a number talk hand signals and talk moves such as reasoning, revoicing, wait time, changing one’s thinking, and making connections are often used. Students express whether they had the same thoughts as others, if they would like to add on to or clarify someone else’s thinking, or if they would like to change their own thinking. A teacher is the facilitator of this discourse because they are able to ask intentional guiding questions that keep the conversation moving forward, allowing students to revise their own thinking, and questioning the students as a whole to either agree or disagree with one another. This discourse has been proven to be extremely effective in allowing students to accept and fix their mistakes and confusion and learn through error and discussion. The artifact included is an example of a number talk I did in my kindergarten classroom during my practicum experience at West Middlesex.

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3.	Use and connect mathematical representations

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3.	Use and connect mathematical representations

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Using and connecting different mathematical representations in compare and connect. The artifacts I use would be one example of doing this. A student may represent their thinking using different materials or ways of thinking. The greatest example of this that I can think of would be expanded notation versus base-ten blocks. The example I use is with the number 132. I think this is crucial when teaching mathematics because it allows students to connect strategies that work best for them. I find it extremely beneficial for students to be able to choose methods that cater to their learning style and by showing more than one way of thinking a student is able to display their true understanding. The artifact I use is an example of two way in which students may present the same number/problem. One student may represent a number using expanded notation whereas another student may use ten blocks. Each of these ways of thinking may be presented by students. One student may like a strategy where they can break down numbers and one may be more comfortable using manipulatives. In a class compare and connect discussion the teacher can facilitate a conversation how students strategies are different but still correct and effective.

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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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A task that promotes reasoning and problem solving would be having a notice and wonder discussion. Notice and wonder is a concept discussed in our Intentional Talk book. Notice and wonder helps students to understand concepts and relationships within a problem. Students recognize multiple strategies for tackling a problem. Students self-reflect and decide which strategy they find most effective to solve the problem. Students in a notice and wonder discussion are ideally in an open classroom environment where they feel free to share their thoughts and explore new ideas. Students notice things about a problem and explain their reasoning behind solving a problem. Students also may wonder about a step or confusion with the concept. The artifact I included is a video that we watched earlier this semester where we see notice and wonder in action within a classroom.

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Notice & Wonder Math Routine: Third Grade

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Teacher and students engage in a Notice and Wonder Math Routine. Using text without numbers, the class tries to make meaning of the content. Aligns with Common Core Standard Math.3.OA.A.3, Math.Practice.MP1.

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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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Every lesson plan has goals, objectives, and measurable outcomes for students. This artifact is part of my lesson plan on money for kindergarten. During this lesson plan, there is a big idea you want your students to take away. Next, your goals should align with your standards. I clearly state the goals I have for this lesson under “Instructional Objectives.” I also listed how I differentiated these goals are for the different ability groups in my “procedures” as well as in the activity where you will clearly see there was a much higher total value that could be reached so that each student was challenged. In my evaluation and assessments I highlight how I determine if these goals were met. This artifact reveals my ability to differentiate within my classroom for each child to feel challenged. I also feel that it is good to observe and refine abilities within groups but also to individually assess child in more focused one-on-one assessment.