Online Task Force - Vignette: Alg & Geom Thinking

Algebraic and Geometric Thinking in the PreK-6 Classroom

Stephen Pape, Johns Hopkins University

Setting the Context

This lesson was developed as part of a course titled Algebraic and Geometric Thinking in the PreK-6 Classroom, which was part of a 6-course sequence of courses. The program courses sought to support the development of mathematics content knowledge and pedagogy as well as instructional leadership skills. This program may lead a student to an endorsement as an instructional leader for the PreK-6 context.

Course Description

The goal of this course is to support PreK-6 content knowledge for teaching related to the following topics: patterns; numbers and operations; measurement, statistics, and probability. Connections of these topics to an integrated approach to curriculum and instruction are emphasized. The course will model the process standards of problem-solving, reasoning and proof, representations, connections, and communication within the context of algebraic and geometric thinking (NCTM, 2000).

Course Objectives

By the end of this course, you will be able to:

Relate NCTM's Principles for School Mathematics and Standards of Practice to InTASC Standards;
Describe algebraic thinking in terms of patterns, relations, and functions;
Represent and analyze mathematical situations and structures using algebraic symbols;
Use mathematical models to represent and explain quantitative relationships;
Analyze change in various mathematical contexts;
Analyze characteristics and properties of two- and three-dimensional shapes and develop mathematical arguments about geometric relationships;
Specify locations and describe spatial relationships using coordinate geometry and other representational systems;
Apply transformations and use symmetry to analyze mathematical situations; and
Use visualization, spatial reasoning, and geometric modeling to solve problems.

Lesson Development and Implementation

The Backwards Design+ Framework* supported the conceptualization, design and implementation of this segment of instruction. The remainder of this document is framed around the stages of the BD+ framework.

*Dean, C. (2019). Backward Design Plus: Taking the Learning Context into Consideration. In S. Carliner (Ed.), Proceedings of E-Learn: World Conference on E-Learning in Corporate, Government, Healthcare, and Higher Education (pp. 1180-1183). New Orleans, Louisiana, United States: Association for the Advancement of Computing in Education (AACE).https://www.learntechlib.org/primary/p/211225/.

Step 1: Determine the Learning Objectives for the Lesson

Lesson Objectives

Model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
Make connections between the NCTM’s Principles for School Mathematics and the Standards of Mathematical Practice and InTASC standards and how they can be used to support student acquisition of patterns, relations, and functions.

Step 2: What Will the Assessment Look Like?

Envisioning the Assessments

There were two types of assessments within this lesson, discussion within the online forum and an activity that was ongoing across several sessions. Students were asked to make their mathematical thinking explicit within this latter assignment.

The Discussion for the week asked students to consider an activity in which they engaged (see Activities 2 & 3 below). Students were asked to reflect on these activities and consider how the activities allowed for the following:

support, represent, and analyze patterns and functions, using words, tables, and graphs;
represent the idea of a variable as an unknown quantity using a letter or a symbol;
express mathematical relationships using equations or model problem situations with objects; and
use representations such as graphs, tables, and equations to present your conclusions.

The second assessment was a content reflection within their online journal. In these content reflections, students were required to consider the readings and activities within the session and provide evidence of their learning. The instructions for this journal entry included the following:

At the end of each session, create a reflection that demonstrates your content understanding and classroom connections that could be made when teaching the content to your students.
To help frame the reflection, students are encouraged to use the four features of a reflection (see Ryan, 2011). The four stages are (1) recount, (2) description, (3) explanation, (4) discussion.
Your reflection of your new learning may be either written or in video format. If you opt to write your reaction, please do not write more than 2-3 paragraphs. If you choose to video your reflection, please do not record more than 2-3 minutes of reflective dialog.

Step 3: Identify Appropriate Online Tools (including, if applicable for your context, asynchronous or synchronous)

Tools to be Utilized

Tool 1: Online simulation.
Tool 2: Panapto video technology
Tool 3: Online media – Youtube
Tool 4: Professionally developed, open-source activities (Annenberg)

Step 4: Implementation

Each session in this course (roughly) followed a four-step approach: (1) content introduction, (2) anticipatory set, (3) content discussion, and (4) classroom connection (see Pape et al., 2015). The student-facing online content included the following sections:

Content introduction
- Overview and objectives
- Checklist – a guide to the tasks students were to complete during the session. This was intended to orient the students to the work within the session and to make the tasks students were to complete explicit.
Anticipatory Set/Activity
- Investigation – a mathematical activity used to engage the participants initially to stimulate their thinking about the topic and to reveal their initial mathematical thinking at an adult level.
- Presentation – an instructor or professionally developed presentation related to the content.
Content Discussion
- Reading – one or more readings from teacher-focused journals relative to the content.
- For Your Professional Library – additional readings, materials, or online resources that provided the participant additional content to support their learning or to support them in constructing materials for their class.
- Mathematics Activities – one or more mathematics activities intended to support students’ learning of the mathematics content.
- Discussion Forum – a space to discuss one or more aspects of their engagement with the content.
Classroom Integration/Content Application
- Content Application/Reflection – a space for the participants to explicitly apply their learning and reflect upon this application.
- Assignment – an application project with which participants engaged their students across multiple sessions.

Section 1 Introduction

Introductory PowerPoint

[Slides To Be Posted]

Overview and Objectives

The slide presentation (above) will provide you with an overview of the main ideas for the week, it also will provide you with the objectives that will guide your work. To advance the slide show, click anywhere on the slide.

Click the arrow to view instructor commentary.

Instructor Commentary

Each session within this online course began with an overview and session learning objectives as well as a checklist to guide the work that students were required to complete during the session.

The Overview and Objectives were presented in two formats, a slide presentation and a PDF (see Appendix A). The overview provided students with a description of the content of the session. The purpose of this material was to orient the students to the content they will explore during the session.

The objectives outlined the learning that students would engage in or the skills they would be able to do by the end of the session.

In addition, this material for this session included the NCTM standards for Algebra (https://www.nctm.org/Standards-and -Positions/Principles-and-Standards/Algebra/).

This section oriented the participants to the session content to be learned and attended to the

Mathematics Teaching Practices

Establish mathematics goals to focus learning.

Standards for Preparing Teachers of Mathematics:

C.1 Mathematics Concepts, Practices, and Curriculum

This standard holds that well-prepared beginning teachers of mathematics should be able to “read, analyze, and discuss curriculum, assessment, and standards documents as well as students’ mathematical productions”. Thus, the students in this course were consistently exposed to the NCTM Principles of School Mathematics.

Related QM Standards:

QM General Standard 2: Learning Objectives (Competencies)

QM 2.2 The module/unit-level learning objectives or competencies that are measurable and consist with the course-level objectives or competencies.
QM 2.3 Learning objectives or competencies are stated clearly, are written from the learner’s perspective, and are prominently located in the course.
QM 2.4 The relationship between learning objectives or competencies and learning activities is clearly stated.
QM 2.5 The learning objectives or competencies are suited to the level of the course.

QM General Standard 4: Instructional Materials

QM 4.3 The course models the academic integrity expected of learners by providing both the source references and permissions for use of instructional materials.

QM General Standard 8: Accessibility and Usability

QM 8.4 The course provides alternative means of access to multimedia content in formats that meet the needs of diverse learners.
QM 8.5 Course multimedia facilitate ease of use.

Weekly Checklist

Activities and assignments are due by midnight, Eastern Daylight Time, on the day or date posted in the checklist. All readings and viewing of content should be completed by Thursday midnight each week.

Tasks and Due Dates

Investigation: Function Machines (due prior to beginning the rest of the session materials)
Content Activities (due Wednesday)
Readings and Media (due Mid-session Thursday)
Discussion Session 3: Understanding Patterns, Relations, and Functions (Initial Posting due Thursday; Final Responses due Sunday)
Activity: Content Application/Reflection (due Sunday)
Assignment: Topic Summary and Student Activity Plan 1—Algebra (due Sunday, Week 6)

Click the arrow to view instructor commentary.

Instructor Commentary

Mathematics Teaching Practices:

This section oriented the participants to the session content to be learned and attended to the Mathematics Teaching Practices:

Establish mathematics goals to focus learning

Related QM Standards:

QM General Standard 8: Accessibility and Usability

QM 8.1 Course navigation facilitates ease of use.
QM 8.2 The course design facilitates readability.

Section 2: Anticipatory Activity

Investigation: Function Machines

As you continue to explore algebraic thinking, you are encouraged to explore the notion of a connected function machine, which is simply a single function that takes an input, runs it through the network inside, and produces an output.

As you complete this online interactive activity, you will control a network of function machines by controlling the input and the operation performed by each function machine.

As you complete a series of activities exploring connected function machines, draw a function machine network or print out a copy of the function machine network provided on the website to help you answer Problems C1 - C5.

In order to access this simulation, you will need to navigate to E-reserves and select Session 3, Part C: Function Machines from the Session 3 list. Because this is a Flash-based simulation, you will need to open it in Foxfire or Edge, and activate Flash.

Annenberg Learner. (2017). Session 3, Part C: Function Machines. Retrieved from https://www.learner.org/courses/learningmath/algebra/session3/part_c/function_flash.html

Click the arrow to view instructor commentary.

Instructor Commentary

The next segment of the session asked students to engage with a mathematics task. This established the mathematics goals for the session to focus learning by providing students an initial experience with the mathematics content. This engagement with a mathematics investigation as an anticipatory activity began participants’ thinking about the mathematics content they would explore during the session. It was also intended to reveal their initial mathematical thinking at an adult level.

This activity supported the following Standards:

Mathematics Teaching Practices:

Establish mathematics goals to focus learning
Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Elicit and use evidence of student thinking

Standards for Preparing Teacher of Mathematics:

C.1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know relevant mathematics content

P.2 Opportunities to Learn Mathematics

P.2.1 Attend to mathematics content relevant to teaching
P.2.3 Provide sustained, quality experiences

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 4: Instructional Materials

QM 4.1 The instructional materials contribute to the achievement of the stated learning objectives or competencies.
QM 4.3 The course models the academic integrity expected of learners by providing both the source references and permissions for use of instructional materials.
QM 4.4 The instructional materials represent up-to-date theory and practice in the discipline.

QM General Standard 5: Learning Activities and Learner Interaction

QM 5.1 The learning activities promote the achievement of the stated learning objectives or competencies.
QM 5.2 Learning activities provide opportunities for interaction that support active learning.

QM General Standard 6: Course Technology

QM 6.2 Course tools promote learner engagement and active learning.
QM 6.3 A variety of technology is used in the course.

Section 3: Content Discussion

Faculty Presentation

Overview

Although algebra is a word that has not commonly been heard in elementary classrooms, the mathematical investigations and conversations of students in these grades frequently include elements of algebraic reasoning. Algebraic reasoning involves students forming generalizations from their experiences with numbers and computation and formalizing their ideas with the use of a symbol system and exploring the concepts of patterns and functions (Van De Walle, Karp, and Bay-Williams, 2018). To develop algebraic thinking, it is important for every teacher to provide students with multiple opportunities to use algebraic concepts to investigate situations and solve mathematical and real-world problems using numbers, words, and symbols. The more time students spend developing meaning for the arithmetic operations of addition, subtraction, multiplication, and division the better sense of the uses of mathematical concepts they have leading to greater success in algebraic thinking and functions in the upper grades.

Directions

Watch the instructor’s weekly video announcement. For optimal viewing with interactive capabilities, click on the full-screen icon (lower right corner of the embedded video). The presentation will then open in a new window where you can view an interactive transcript, take notes, and discuss the content.
If you are unable to view the embedded presentations, watch them directly on the Panopto site [link to URL]
After completing the presentations, proceed to the remaining sections.

Click the arrow to view instructor commentary.

Instructor Commentary

During each session, the faculty provided an overview of the content with which students would engage. This overview was in the form of a brief written description of the content and a video presentation. The overview for each segment of the session was intended to further orient the students to the content of this segment (e.g., Presentation, Reading, Activities).

This activity supported the following Standards.

Mathematics Teaching Practices:

Establish mathematics goals to focus learning
Build procedural fluency from conceptual understanding

Standards for Preparing Teachers of Mathematics:

C.1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know relevant mathematics content

C.3 Students as Learners of Mathematics

C.3.1 Anticipate and Attend to Students’ Thinking About Mathematics Content

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 4: Instructional Materials

QM 4.2 The relationship between the use of instructional materials in the course and completed activities is clearly explained.

QM General Standard 8: Accessibility and Usability

QM 8.4 The course provides alternative means of access to multimedia content in formats that meet the needs of diverse learners.
QM 8.5 Course multimedia facilitate ease of use.

Reading

Overview

This week’s readings will be focused on children’s understanding of patterns, relations, and functions. You will have an opportunity to concentrate on your grade level. These videos will build upon the instructor’s weekly announcement where examples from your online weekly journals will be shared as a way to continue to support your development of algebraic thinking.

Directions

Complete the following reading.

Van de Walle, J. A., Karp, K. A., & Bay-Williams, J. M. (2018) Elementary and middle school mathematics: Teaching developmentally (10th ed.). New York, NY: Pearson. (Chapter 22)

Click the arrow to view instructor commentary.

This course required students to read a textbook. Other sessions incorporated teacher-focused journal articles relevant to the content.

This activity supported the following Standards.

Standards for Preparing Teachers of Mathematics:

C.1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content

C.3 Students as Learners of Mathematics

C.3.1 Anticipate and Attend to Students’ Thinking About Mathematics Content

P.2 Opportunities to Learn Mathematics

Attend to Mathematics Content Relevant to Teaching

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 4: Instructional Materials

QM 4.1 The instructional materials contribute to the achievement of the stated learning objectives or competencies.
QM 4.2 The relationship between the use of instructional materials in the course and completed activities is clearly explained.
QM 4.3 The course models the academic integrity expected of learners by providing both the source references and permissions for use of instructional materials.
QM 4.4 The instructional materials represent up-to-date theory and practice in the discipline.
QM 4.5 A variety of instructional materials is used in the course.

For Your Professional Library

Media Overview

As you continue to learn about algebraic thinking, consider the following video, which provides an overview of what it means to:

understand patterns, relations, and functions;
represent and analyze mathematical situations and structures using algebraic symbols;
use mathematical models to represent and understand quantitative relationships; and
analyze change in various contexts.

While you watch the videos, think about Bloom’s Taxonomy and how it can be used to develop a deeper understanding of mathematics. This session’s primary video is a recorded lecture from Dr. Bill Selak exploring and unpacking NCTM’s Algebra Strand.

Media

Selak, B. (2011). NCTM Algebra Strand [Video file]. Retrieved from https://youtu.be/hPn06DGsBDw

Directions

Review the NCTM Algebra Strand Lecture in this window by clicking in the following presentation.
To view the video in a new window, NCTM Algebra Strand lecture [Video file] or copy and paste the link https://youtu.be/hPn06DGsBDw into a new window.

Click the arrow to view instructor commentary.

Instructor Commentary

This activity supported the following Standards.

Mathematics Teaching Practices:

Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Build procedural fluency from conceptual understanding
Standards for Preparing Teacher of Mathematics:

C. 1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content

C.3 Students as Learners of Mathematics

C.3.1 Anticipate and Attend to Students’ Thinking About Mathematics Content

P. 2 Opportunities to Learn Mathematics

P.2.1 Attend to Mathematics Content Relevant to Teaching

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge
P.3.2 Provide Foundations of Knowledge About Students as Mathematics Learners

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components.

QM General Standard 4: Instructional Materials

QM 4.5 A variety of instructional materials is used in the course.

QM General Standard 8: Accessibility and Usability

QM 8.4 The course provides alternative means of access to multimedia content in formats that meet the needs of diverse learners.
QM 8.5 Course multimedia facilitate ease of use.

Section 4: Classroom Integration/Content Application

Content Activities

Activity 1: Orange Problem

Directions

Solve the following (remember to capture all your thinking/work – this should be uploaded into your weekly online journal).

A grocer was asked how many oranges he had sold that day. He replied: “My first customer said I'll buy half your oranges and half an orange more.” He then said, “My second customer said the same thing… I'll buy half your oranges and half an orange more.” Then he stated, “My third customers said the same thing... I'll buy half your oranges and half an orange more.” Finally, he stated, “When I had filled all three orders I was sold out and I did not have to cut a single orange all day." How many oranges had the grocer sold in all?

Building from the scenario above, answer the following questions:

What if there were four customers?
Five customers?
Ten customers?
Any number of customers?

Source

Barry, R. (n.d.) Orange Problem. Wisconsin Mathematics Council. Retrieved from http://www.wismath.org/resources/Documents/RBerry-Handouts.pdf

- - -

Activity 2: Graphing Patterns

Directions

Open the Patterns, functions, and algebra for elementary school teachers available in the E-reserves area found in the course navigation panel.
Review the sample activity found on pages Graphing Patterns - 169 and Graphing Patterns - 170.
Complete all examples on page Graphing Patterns - 170.
Submit your work to the online Journal as part of your weekly reflection.

Source

Office of Elementary Instructional Services Virginia Department of Education. (2004). Patterns, functions, and algebra For elementary school teachers. 169-170. Richmond, VA: Virginia Department of Education. Retrieved from https://mason.gmu.edu/~jsuh4/impact/PFAEntire.pdf.

Click the arrow to view instructor commentary.

Instructor Commentary

In this fourth section of the session, students continue to explore the mathematics content by engaging in problem solving, representing mathematical relationships, and struggling with mathematics content through engagement with open source mathematics activities. Some of these activities were submitted for instructor or peer review or discussion. This section was intended to build students’ mathematics content knowledge and content knowledge for teaching.

This activity supported the following Standards.

Mathematics Teaching Practices:

Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Build procedural fluency from conceptual understanding
Support productive struggle in learning mathematics

Standards for Preparing Teacher of Mathematics:

C. 1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content
C.1.2 Demonstrate Mathematical Practices and Processes
C.1.6 Use Mathematical Tools and Technology

C.3 Students as Learners of Mathematics

C.3.1 Anticipate and Attend to Students’ Thinking About Mathematics Content

P. 2 Opportunities to Learn Mathematics

P.2.1 Attend to Mathematics Content Relevant to Teaching
P.2.2 Build Mathematical Practices and Processes

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge
P.3.2 Provide Foundations of Knowledge About Students as Mathematics Learners

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 4: Instructional Materials

QM 4.1 The instructional materials contribute to the achievement of the stated learning objectives or competencies.
QM 4.4 The instructional materials represent up-to-date theory and practice in the discipline.
QM 4.5 A variety of instructional materials is used in the course.

QM General Standard 5: Learning Activities and Learner Interaction

QM 5.1 The learning activities promote the achievement of the stated learning objectives or competencies.
QM 5.2 Learning activities provide opportunities for interaction that support active learning.

QM General Standard 6: Course Technology

QM 6.1 Th tools used in the course support the learning objectives or competencies.
QM 6.2 Course tools promote learner engagement and active learning.
QM 6.3 A variety of technology is used in the course.

Discussion

Directions

Click on Discussions in the course navigation panel or the Click to Launch link to go to the main discussion area.
Go to the Session 3: Understanding Patterns, Relations, and Functions discussion forum.
Select Create Thread and consider how this week's activities allowed you to:
- support, represent, and analyze patterns and functions, using words, tables, and graphs;
- represent the idea of a variable as an unknown quantity using a letter or a symbol;
- express mathematical relationships using equations or model problem situations with objects; and
- use representations such as graphs, tables, and equations to present your conclusions.
Throughout the remainder of the session, continue to engage in meaningful dialogue with your colleagues. Post your final responses by the end of the session.

Note: Remember to be positive and constructive in your responses. For assessment and grading criteria, refer to the Online Interactive Discussion Rubric found on the syllabus page of the course site.

Click the arrow to view instructor commentary.

Instructor Commentary

Following engagement with the mathematics activities, students were required to engage in a discussion about their experience. This discussion was an initial reflection on the activities, their learning within the activities, and focused on the potential for these activities to support their students to learn important mathematics content. Throughout the remainder of the session, students were to come back to this discussion forum to continue the conversation with their colleagues and instructor.

This activity supported the following Standards.

Mathematics Teaching Practices:

Use and connect mathematical representations
Facilitate meaningful mathematical discourse
Support productive struggle in learning mathematics

Standards for Preparing Teacher of Mathematics:

C. 1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content
C.1.2 Demonstrate Mathematical Practices and Processes
C.1.5 Analyze Mathematical Thinking

C.2 Pedagogical Knowledge and Practices for Teaching Mathematics

C.2.4 Analyze Teaching Practice
C.2.5 Enhance Teaching Through Collaboration With Colleagues, Families, and Community Members

P. 2 Opportunities to Learn Mathematics

P.2.1 Attend to Mathematics Content Relevant to Teaching
P.2.2 Build Mathematical Practices and Processes

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 4: Instructional Materials

QM 4.1 The instructional materials contribute to the achievement of the stated learning objectives or competencies.

QM General Standard 5: Learning Activities and Learner Interaction

QM 5.1 The learning activities promote the achievement of the stated learning objectives or competencies.
QM 5.2 Learning activities provide opportunities for interaction that support active learning.
QM 5.4 The requirements for learner interaction are clearly stated.

QM General Standard 6: Course Technology

QM 6.1 The tools used in the course support the learning objectives or competencies.
QM 6.2 Course tools promote learner engagement and active learning.

Activity: Content Application/Reflection

Overview

Throughout Session 3, make a new content reflection in your online journal of the lesson topic based on your in-class experiences and learning. As a reminder, this reflection should be brief (either 2-3 paragraphs or 2-3 minutes), and be completed after participating in the discussion for the week.

Directions:

Download and review the Content Application/Reflection Assignment Directions and Rubric located on the syllabus page of the course site.
Access the Algebraic and Geometric Thinking Class Notebook from the course navigation panel in the "In the Classroom" section.
Navigate in your class notebook to the Journal section, and add a new page entry titled Session 3.
1. Reflect on how you will model problem situations with objects and use representations such as graphs, tables, and equations to draw conclusions.
By Sunday, post either a written or a recorded journal entry reflecting on the session’s impact on your professional practice or a reflection of your new learning.
1. If you opt to write your reaction, please do not write more than 2-3 paragraphs.
2. If you choose to video your reflection, please do not record more than 2-3 minutes of reflective dialog.

For more information on how to add a page to your OneNote class notebook, navigate to Create and Organize Notebooks, Sections, and Pages (web link).

Click the arrow to view instructor commentary.

Instructor Commentary

This activity was an ongoing reflection that students were to complete after each session (see Appendix B). In this reflection students were to demonstrate their content understanding and draw connections to their classroom practice. This activity could be completed in either written or video format. Students were provided a research-based description of an academic reflection with the intended purpose of supporting content acquisition, problem solving, and application. The learning objectives aligned with the assignment are clearly described in the assignment description (see Appendix B), and a rubric is provided to guide student completion of the assignment and instructor assessment of the students’ work.

This activity supported the following Standards.

Mathematics Teaching Practices:

Use and connect mathematical representations
Pose purposeful questions

Standards for Preparing Teacher of Mathematics:

C. 1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content
C.1.2 Demonstrate Mathematical Practices and Processes
C.1.3 Exhibit Productive Mathematical Dispositions

C.2 Pedagogical Knowledge and Practices for Teaching Mathematics

C.2.4 Analyze Teaching Practice

P. 2 Opportunities to Learn Mathematics

P.2.1 Attend to Mathematics Content Relevant to Teaching
P.2.2 Build Mathematical Practices and Processes

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge

Related QM Standards:

QM General Standard 1: Course Overview and Introduction

QM 1.1 Instructions make clear how to get started and where to find various course components

QM General Standard 2: Learning Objectives (Competencies)

QM 2.2 The module/unit-level learning objectives or competencies that are measurable and consist with the course-level objectives or competencies.
QM 2.3 Learning objectives or competencies are stated clearly, are written from the learner’s perspective, and are prominently located in the course.
QM 2.4 The relationship between learning objectives or competencies and learning activities is clearly stated.
QM 2.5 The learning objectives or competencies are suited to the level of the course.

QM General Standard 3: Assessment and Measurement

QM 3.1 The assessments measure the achievement of the stated learning objectives or competencies.
QM 3.3 Specific and descriptive criteria are provided for the evaluation of learners’ work, and their connection to the course grading policy is clearly explained.
QM 3.5 The course provides learners with multiple opportunities to track their learning progress with timely feedback.

Assignment

Topic Summary and Student Activity Plan - Algebra

Overview

Continue making progress on The Topic Summary and Student Activity Plan 1—Algebra, due Sunday of Week 6.

Additional information about this assignment can be located by selecting Syllabus in the course navigation panel of Blackboard and selecting Topic Summaries and Student Activity Plan Assignment Directions and Rubric.

Click the arrow to view instructor commentary.

Instructor Commentary

Students in this course were required to construct an ongoing content analysis (see Appendix C for assignment description). This assignment required students to consider the mathematics content, historical perspectives, the interconnectedness of the concepts revealing how they built upon one another, and a reflection on their learning process. They were also required to provide evidence of the connections to the NCTM Principles for School Mathematics, the Standards of Mathematical Practice, and the InTASC standards. Finally, this assignment required them to plan an activity they would enact with the school students attending to the diversity within their context.

The assignment description page included a clear alignment to the course and program learning objectives as well as a rubric to guide students’ completion of the assignment and instructor assessment of the students’ work.

This activity supported the following Standards.

Mathematics Teaching Practices:

Implement tasks that promote reasoning and problem solving
Use and connect mathematical representations
Pose purposeful questions
Build procedural fluency from conceptual understanding
Support productive struggle in learning mathematics

Standards for Preparing Teacher of Mathematics:

C. 1 Mathematics Concepts, Practices, and Curriculum

C.1.1 Know Relevant Mathematical Content
C.1.2 Demonstrate Mathematical Practices and Processes
C.1.3 Exhibit Productive Mathematical Dispositions
C.1.5 Analyze Mathematical Thinking

C.2 Pedagogical Knowledge and Practices for Teaching Mathematics

C.2.1 Promote Equitable Teaching
C.2.2 Plan for Effective Instruction
C.2.4 Analyze Teaching Practice

P. 2 Opportunities to Learn Mathematics

P.2.1 Attend to Mathematics Content Relevant to Teaching
P.2.2 Build Mathematical Practices and Processes

P.3 Opportunities to Learn to Teach Mathematics

P.3.1 Address Deep and Meaningful Mathematics Content Knowledge
P.3.4 Incorporate Practice-Based Experiences

Related QM Standards:

QM General Standard 2: Learning Objectives (Competencies)

QM 2.2 The module/unit-level learning objectives or competencies that are measurable and consist with the course-level objectives or competencies.
QM 2.3 Learning objectives or competencies are stated clearly, are written from the learner’s perspective, and are prominently located in the course.
QM 2.4 The relationship between learning objectives or competencies and learning activities is clearly stated.
QM 2.5 The learning objectives or competencies are suited to the level of the course.

QM General Standard 3: Assessment and Measurement

QM 3.1 The assessments measure the achievement of the stated learning objectives or competencies.
QM 3.3 Specific and descriptive criteria are provided for the evaluation of learners’ work, and their connection to the course grading policy is clearly explained.
QM 3.5 The course provides learners with multiple opportunities to track their learning progress with timely feedback.

APPENDICES TO BE ADDED

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