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Establish Mathematics Goals to Focus Learning

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Establish Mathematics Goals to Focus Learning

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Definition: "Effective teaching of mathematics establishes clear goals for the mathematics that students are learning, situates goals within learning progressions, and uses the goals to guide instructional discussions," (Huinker, Leinwand, & Brahier, 2014).

Artifact: Lesson plan - standards aligned with objectives

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Establish Mathematics Goals to Focus Learning - Lesson Plan Artifact

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Establish Mathematics Goals to Focus Learning - Lesson Plan Artifact

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Establish Mathematics Goals to Focus Learning - Lesson Plan Rational 

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Establish Mathematics Goals to Focus Learning - Lesson Plan Rational 

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Description: The lesson plan establishes clear goals, also referred to as objectives. These goals are specifically derived from the standards being addressed within a lesson. Each particular lesson plan has different goals, as they align with the Common Core and/or Pennsylvania Department of Education's standards, which focus student learning. The lesson's instruction and development are guided by these goals, which promote the students' learning progression. The goals established are the mathematical ideas being implemented in the lesson, with aim to enhance students' learning and knowledge of mathematical concepts and strategies. Having goals listed on my lesson plans helps me to focus on specific targets to reach in each lesson. These will help me to stay on track during the lesson and accurately assess it afterward.

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Implement Tasks That Promote Reasoning and Problem Solving

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Implement Tasks That Promote Reasoning and Problem Solving

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Definition: "Effective teaching of mathematics engages students in solving and discussing tasks that promote mathematical reasoning and problem solving and allow multiple entry points and varied solution strategies," (Huinker et al., 2014).

Artifact: Math Talk - lesson plan and reflection

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Implement Tasks That Promote Reasoning and Problem Solving - Math Talk Artifact

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Implement Tasks That Promote Reasoning and Problem Solving - Math Talk Artifact

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Leading a Mathematical Discussion Lesson & Analysis - Grade 1 PDF

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Implement Tasks That Promote Reasoning and Problem Solving - Math Talk Rational

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Implement Tasks That Promote Reasoning and Problem Solving - Math Talk Rational

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Description: The Math Talk lesson plan describes a brief discussion that aims to engage students in solving mathematics problems. In the discussion, students are asked to describe how they used various mathematical strategies and ideas of reasoning to complete a task, problem solving. The Math Talk offers multiple entrance points for each problem being discussed. Students are asked to explain the specific strategy they used in solving a problem and why. Further, students are introduced to and being asked to use Talk Moves (rephrasing, revoicing, adding-on, turn and talk, repeating, and waiting). Students are asked to use these Talk Moves to further elaborate or clarify a peer's thinking or explain another possible strategy that could be used to solve the same problem. This lesson shows that I am capable of introducing and implementing a classroom discussion that focuses on specific content and allows the students to build from peer ideas. This will help me to introduce Talk Moves, which can later be used in other content areas. Further, this will allow me to see how my students think and how well they can describe their methods. This will show me the strategy that works best for the majority of my students, as well as the ones that don't. This will help drive my instruction in other mathematics lessons.

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Use and Connect Mathematical Representations

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Use and Connect Mathematical Representations

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Definition: "Effective teaching of mathematics engages students in making connections among mathematical representations to deepen understanding of mathematics concepts and procedures and as tools for problem solving," (Huinker et al., 2014).

Artifact: Math bowling

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Use and Connect Mathematical Representations - Math Bowling Assessment Artifact 1

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Use and Connect Mathematical Representations - Math Bowling Assessment Artifact 1

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Math Bowling Assessment - Addition 4 sets

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Use and Connect Mathematical Representations - Math Bowling Rational 

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Use and Connect Mathematical Representations - Math Bowling Rational 

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Description: Math bowling is a problem solving group activity that can be implemented during centers or as a review game. The teacher draws ten "pins" on the chalk board (for a review game) or individual dry erase board (for centers) in the shape of a triangle, resembling the set-up of pins in a bowling alley. There is an example of this triangle on artifact number two. The students are given two dice to roll. With the numbers rolled, the students need to try to knock down as many pins as they can. The goal of the game is to knock down all of the pins each time the dice are rolled, getting a "strike". In the first assessment piece, the format only allows for students to use addition in the game. This would be a beneficial starting platform to use when introducing the game. This will help students to understand the idea of the game before incorporating multiple operations into their game. The second assessment serves as a way to dig deeper, developing student uses of multiple mathematical procedures. In this piece, the students are provided with an empty box where they can write multiple equations for one roll. This is expanding on the first sheet with only two options for each roll. Further, the second assessment sheet will allow students to incorporate the use of various operations, where the first sheet prompts only for addition equations. The students will be able to make connections from the first sheet to the second as they will still be allowed to use addition as a operation in the second assessment. When using this activity in my class, the second assessment sheet allowed me to introduce multiplication as "groups of". This concept builds on the students' knowledge of addition. For example, if the students rolled a 2 and a 3, we would multiply them and say, "We have one group of three and another group of three. If you add them together, you have two groups of three. Two groups of three adds up to be one group of six. This means two multiplied by three, two groups of three, is the same as six." This activity helped me to see student's individual abilities and assess their base knowledge. If a student did not understand addition, [s]he would not take easily to the idea of "groups of". This shows me where I need to be more direct in my instruction and cautious with my word choice and phrasing. Along with this, I have learned how to manage small groups in a classroom through the efforts of an engaging worthwhile task.

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Definition: "Effective teaching of mathematics facilitates discourse among students to build shared understanding of mathematical ideas by analyzing and comparing student approaches and arguments," (Huinker et al., 2014).

Artifacts: Math Talk - lesson plan and reflection; Discourse worksheet

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Facilitate Meaningful Mathematical Discourse - Math Talk Artifact

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Facilitate Meaningful Mathematical Discourse - Math Talk Artifact

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Leading a Mathematical Discussion Lesson & Analysis - Grade 1 PDF

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Facilitate Meaningful Mathematical Discourse - Discourse Worksheet Artifact

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Facilitate Meaningful Mathematical Discourse - Discourse Worksheet Artifact

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Facilitate Meaningful Mathematical Discourse - Math Talk Rational 

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Facilitate Meaningful Mathematical Discourse - Math Talk Rational 

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Description: The Math Talk lesson plan describes a brief discussion that aims to engage students in solving mathematics problems. In the discussion, students are asked to describe how they used various mathematical strategies and ideas of reasoning to complete a task, problem solving. Students are asked to explain the specific strategy they used in solving a problem and why. Further, students are introduced to and being asked to use Talk Moves (rephrasing, revoicing, adding-on, turn and talk, repeating, and waiting). Students are asked to use these Talk Moves to further elaborate or clarify a peer's thinking or explain another possible strategy that could be used to solve the same problem. This allows for students to analyze, compare, and argue differing approaches to problem solving. In doing so, students will build a shared understanding of mathematical ideas and strategies. This will help me to introduce Talk Moves, which can be used in other content areas. Along with this, I will gain opportunities to explore the thinking of my students. It will also help them to begin understanding how to explain their ideas. This will help me to see which methods work the best and worst among the majority of my students, which is helpful for instruction implementation in my future mathematics lessons.

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Facilitate Meaningful Mathematical Discourse - Discourse Worksheet Rational 

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Facilitate Meaningful Mathematical Discourse - Discourse Worksheet Rational 

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Description: This worksheet can be used as an assessment pieces in various mathematics lessons. On this worksheet, students are asked to illustrate and describe how they used two different mathematical strategies to solve one problem. Students are asked to explain the specific strategy they used in solving a problem using words and numbers After independently completing the worksheets, students could use a Talk Move, turn and talk, to discuss their different strategies. This allows for students to analyze, compare, and argue differing approaches to problem solving. In doing so, students will build a shared understanding of mathematical ideas and strategies. This will help me to gain opportunities to explore the thinking of my students. It will also help them to begin understanding how to explain their ideas. This will help me to see which methods work the best and worst among the majority of my students, which is helpful for instruction implementation in my future mathematics lessons.

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Definition: "Effective teaching of mathematics uses purposeful questions to assess and advance students' reasoning and sense making about important mathematical ideas and relationships," (Huinker et al., 2014).

Artifact: Math Talk - lesson plan and reflection; Discourse worksheet

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Pose Purposeful Questions - Discourse Worksheet Artifact

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Pose Purposeful Questions - Discourse Worksheet Artifact

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Pose Purposeful Questions - Math Talk Artifact 

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Pose Purposeful Questions - Math Talk Artifact 

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Leading a Mathematical Discussion Lesson & Analysis - Grade 1 PDF

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Pose Purposeful Questions - Math Talk Rational 

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Pose Purposeful Questions - Math Talk Rational 

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Description: This lesson plan describes the implementation of a Math Talk. The Math Talk is a class discussion that asks students to explain their mathematical reasoning techniques used for problem solving. During this talk, the teacher asks students to explain their strategies. These explanations will be responses to purposeful questions. After asking the students to explain their reasoning and strategy, the teacher will be able to assess the students' thinking. Upon this, the teacher can reflect in ways that will help advance the students' knowledge. The teacher will be specifically noticing the students' abilities to make sense of the relationships and ideas of mathematics. After reviewing various strategies, the teacher could, additionally, add a "What's Best and Why," prompt to close the discussion. This section would ask the students to evaluate the effectiveness of the strategies used and create logical reasoning to support an argument. This principle and artifact help me to see how to challenge my students at a level that keeps them engaged without discouraging them. Math Talks have taught me to value each student's thinking. As I continue to pose purposeful questions, I will be helping my students to make sense of mathematics, aiding development rather than memorization. Asking these questions will help my students to explore their reasoning methods and how to verbally communicate their ideas. Purposeful questions allow me to elicit student thinking, which I can use for later instructional adjustments as needed.

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Pose Purposeful Questions - Discourse Worksheet Rational 

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Pose Purposeful Questions - Discourse Worksheet Rational 

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Description: This worksheet asks students to explain two strategies they use to solve one problem. Once they have completed this independently, the students will have time to turn and talk. During the peer discussions, the students will explain their reasoning and strategy. Here, the teacher will be able to assess the students' thinking. Upon this, the teacher can reflect in ways that will help advance the students' knowledge. The teacher will be specifically noticing the students' abilities to make sense of the relationships and ideas of mathematics. After reviewing various strategies, the teacher could, additionally, add a "What's Best and Why," prompt to close the discussion. This class discussion would ask the students to evaluate the effectiveness of the strategies used and create logical reasoning to support an argument.

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Build Procedural Fluency from Conceptual Understanding

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Build Procedural Fluency from Conceptual Understanding

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Definition: "Effective teaching of mathematics builds fluency with procedures on a foundation of conceptual understanding so that students, over time, become skillful in using procedures flexibly as they solve contextual and mathematical problems," (Huinker et al., 2014).

Artifact: Base 10 blocks - days of school (to unifix cubes)

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Build Procedural Fluency from Conceptual Understanding - Base 10 Calendar Artifact 1

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Build Procedural Fluency from Conceptual Understanding - Base 10 Calendar Artifact 1

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Build Procedural Fluency from Conceptual Understanding - Base 10 Blocks to Unifix Cubes Rational

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Build Procedural Fluency from Conceptual Understanding - Base 10 Blocks to Unifix Cubes Rational

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Description: Next to the classroom calendar, the teacher keeps a plastic pouch with three individual pockets. The pockets represent three different place values: ones, tens, and hundreds. The teacher records the the number of days that the students have been in school. The students will place a unit, rod, or flat in the appropriate when the days change. For example, on the first day of school, the ones pocket would hold one unit while the tens and hundreds pockets remained empty. The students participate in this counting sequence each morning. Once they get to the 9th day of school, the teacher must explain how to regroup. This will continue throughout the year, ensuring that the students learn about regrouping numbers and exchanging blocks, units for tens and tens for hundreds. This idea can also help develop students' understanding of proportions. They will see that one rod is made of ten units and one flat is made of ten rods. Therefore, they will eventually reach the point of seeing that one flat contains 100 units. The students will then be able to use this idea with unifix cubes. They will use each single cube as a unit. Once they have connected ten units, they will have made a rod. Since the unifix cubes have holes on five of the six sides, the students will be able to connect the rods when they rotate the one side of the cube, which has the attaching piece, toward another rod. They will then be able to make their own flats with unifix cubes. I can use this idea to help my students develop their knowledge of relationships between numbers. (One more than, three less than, six more than, etc.) Along with this, my students will have their own manipulatives to work with. This will help me provide differentiated instruction. This takes my lesson from using two instructional methods; visual and verbal, to using three; visual, verbal, and kinesthetic. Along with this, I will be able to assess my students one-to-one correspondence as they count and build with their unifix cubes. Having the school days count pouch on the wall will allow my students to have a visual reminder of these ideas at all times. After developing their knowledge of place value, the students will have a better understanding of how to regroup numbers when presented with mathematical equations to solve on a class worksheet or test. They will develop fluency from this repeated instruction.

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Support Productive Struggle in Learning Mathematics

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Support Productive Struggle in Learning Mathematics

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Definition: "Effective teaching of mathematics consistently provides students, individually and collectively, with opportunities and supports to engage in productive struggle as they grapple with mathematical ideas and relationships, "(Huinker et al., 2014).

Artifact: Stacking cups activity

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Support Productive Struggle in Learning Mathematics - Stacking Cups Artifact

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Support Productive Struggle in Learning Mathematics - Stacking Cups Artifact

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Support Productive Struggle in Learning Mathematics - Stacking Cups Rational 

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Support Productive Struggle in Learning Mathematics - Stacking Cups Rational 

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Description: During this activity, the students are offered the opportunity to work individually or collaboratively. Each group or individual is given a set of cups and asked to stack them. The goal for the students is to create the tallest tower. The students will create a tower one way, believing that it is the best way until their tower falls or they see another student or group with a taller one. These struggles will encourage the students to think more deeply into their design ideas. Although there is a time limit on this activity, the students are not required to use the entirety of it. Meaning, the students will build until they think they have made the tallest tower, which is when it will be measured with measuring tape. After all of the students have completed this activity, the winning student or group will be asked to share their tower design with the class. Following this, the class will discuss their findings of the methods of greatest and least success. One question will specifically address what did not work well for the students. Following this, they will share their thoughts as to why they believe it was not a productive attempt. Using this activity allows me to support students' individualized ideas. During this activity, I can provide individualized and differentiated instruction to the students. Further, I can provide the students with an engaging activity that allows them to visualize and experience the benefits of productive struggle. For example, if a tower falls, the student(s) and I could discuss why the tower fell and what could be changed in the next building attempt to keep the tower from falling again.

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Elicit and Use Evidence of Student Thinking

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Elicit and Use Evidence of Student Thinking

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Definition: "Effective teaching of mathematics uses evidence of students thinking to assess progress toward mathematical understanding and to adjust instruction continually in ways that support and extend learning," (Huinker et al., 2014).

Artifact:  Math Talk - lesson plan and reflection

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Elicit and Use Evidence of Student Thinking - Math Talk Artifact

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Elicit and Use Evidence of Student Thinking - Math Talk Artifact

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Leading a Mathematical Discussion Lesson & Analysis - Grade 1 PDF

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Elicit and Use Evidence of Student Thinking - Math Talk Rational 

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Elicit and Use Evidence of Student Thinking - Math Talk Rational 

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Description: The Math Talk lesson plan describes a brief discussion that aims to engage students in solving mathematics problems. In the discussion, students are asked to describe how they used various mathematical strategies and ideas of reasoning to complete a task, problem solving. The Math Talk offers multiple entrance points for each problem being discussed. Students are asked to explain the specific strategy they used in solving a problem and why. Further, students are introduced to and being asked to use Talk Moves (rephrasing, revoicing, adding-on, turn and talk, repeating, and waiting). Students are asked to use these Talk Moves to further elaborate or clarify a peer's thinking or explain another possible strategy that could be used to solve the same problem. Following this lesson, the teacher assesses its success. Along with this, the teacher assesses the students' progress toward developing and deepening mathematical understandings.  Following these assessments, I reflect upon them. This reflection allows me to develop new ways to adapt or modify lesson instruction. These adaptations and modifications will help me to further extend my students' mathematical learning.

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Definitions for the Eight Principles of Mathematics

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Definitions for the Eight Principles of Mathematics

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Huinker, D., Leinwand, S., & Brahier, D. (2014). Principles to actions: Effective mathematics teaching as the core for student learning. Teaching Children Mathematics, 20(9), 533-537.

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Pose Purposeful Questions -  Math Talk Video Example

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Pose Purposeful Questions -  Math Talk Video Example

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Link: https://www.teachingchannel.org/videos/student-participation-strategy

Description: This video provides an example of a classroom Math Talk that incorporates the use of Talk Moves to encourage student participation and mathematical connections.

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Use and Connect Mathematical Representations - Math Bowling Assessment Artifact 2

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Use and Connect Mathematical Representations - Math Bowling Assessment Artifact 2

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Math Bowling Assessment - Equation 4 sets

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Build Procedural Fluency from Conceptual Understanding - Base 10 Calendar Artifact 2

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Build Procedural Fluency from Conceptual Understanding - Base 10 Calendar Artifact 2

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Build Procedural Fluency from Conceptual Understanding - Base 10 to Unifix Cubes Calendar Artifact 3

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Build Procedural Fluency from Conceptual Understanding - Base 10 to Unifix Cubes Calendar Artifact 3