Session Focus

Welcome! We are excited to spend the next 7 weeks thinking about making sense of fractions with you.

Throughout this course, you will have an opportunity to explore how in the Investigations curriculum students make sense of, compare, and compute with fractions using representations and reasoning, and to consider how best to support their learning. This orientation session is designed to help you get started. You will explore the course website, learn how to create your profile and post in the discussion forums, complete an orientation survey and begin to think about addressing issues of equity as you work with students on fractions. You will also have an opportunity to introduce yourself to your fellow course participants and share some of your experiences learning and teaching fractions.

Goals and Guiding Principles

One of the main goals of Investigations is to support students to make sense of mathematics and learn that they can be mathematical thinkers. They learn that they are capable of having mathematical ideas, applying what they know to new situations, and thinking and reasoning about unfamiliar problems.

Read Goals and Guiding Principles of Investigations.

Making Sense of Fractions

For many of us, learning about fractions and particularly learning to compute with fractions meant being taught and then using procedures. There is usually little connection between this and making sense of fractions.

“Premature attention to rules for fraction computation has a number of serious drawbacks. None of the rules helps students think about the operations and what they mean. Armed only with rules, students have no means of assessing their results to see if they make sense. Surface mastery of rules in the short term is quickly lost. When mixed together, the myriad rules of fractions computation soon become a meaningless jumble.” (Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2010). Elementary and middle school mathematics: teaching developmentally. 7th ed. Boston: Allyn & Bacon, p. 161)

Students can and should have the opportunity to make meaning for fractions and to solve problems that involve fractions with understanding.

Watch Investigations author Keith Cochran give an overview of some of the essential ideas students need to develop an understanding of fractions.

Which of these ideas particularly struck you? Which, if any, of these ideas feel particularly connected to your own learning or teaching of fractions?

There are no materials that need to be collected or printed for this session.

In this activity, you will have an opportunity to get to know the layout of this course. You will be guided to set up your user profile and learn about how to access and post on the course discussion forums.

Take some time to get familiar with the layout of the course. Like this session, each session includes tabs at the top of the page: Overview, Activity 1, etc. Click on each tab above to become more familiar with the structure of a session. Go to the course homepage (pictured below) and become familiar with the structure of the page. Each week, a new session - with a description and links to the discussion forums - will appear below where you now see the Orientation Session.

Watch the video tutorial below to learn how to modify your course profile.

Update your profile and add a profile picture.

Discussion Forums

Watch the video tutorial about using the course discussion forums.

Post a message introducing yourself in the Orientation Discussion forum. In your post include:

• Your name and the grade level you teach (or your role in your school/district).
• Explain what motivated you to enroll in this course, what you hope to learn, and how you hope to apply what you learn to your professional life.
• Attach a photo of yourself or of something that represents your interests.

Make sure to read and respond to some of your colleagues' introductions. You can return to the Orientation Discussion Forum using the link above or on the homepage.

Technology Preparation

Web Browsers

On occasion, you may find that certain applications or websites in this course may work differently in different browsers. We recommend that you become familiar with how to use an alternate web browser in addition to the one you primarily use.

PDF Format

PDF (which stands for "Portable Document Format") is a popular format for distributing documents on the Internet. A number of readings in this course are posted in PDF format. To view and print PDF documents, you need Acrobat Reader software, available free from Adobe's web site. If you do not have Acrobat Reader installed on your device or computer, download it from the Adobe site and follow the directions for installation.

This is going to be a busy course with a variety of learning opportunities – some aspects of the course may feel familiar while others may feel new and challenging. Re-read the Welcome Letter as a reminder of how we think about the course.

Guidelines for Participation

In each weekly session you will:

1. Complete 3 activities. These activities often include solving mathematics problems, looking at student work, analyzing classroom video, and reading articles.
2. Participate in the activity forum for the session. Make 3 posts in the activity forum by sharing your own work or thoughts in response to the prompt, and then responding to at least 2 other participants’ posts.
3. Engage in conversations on the session’s discussion forum. Make at least 3 substantive posts in the session’s discussion forum by either responding in a thread started by someone else or starting your own thread. These forums are designed to facilitate meaningful conversation between you and other course participants about the content of each session and to support you in reflecting on your experience as a mathematics educator.

Agreements for Our Learning Community

Being part of a professional learning community can present both exciting and challenging opportunities for learning. Read about math coach Heidi Fessenden’s experience as a participant in a math team meeting. As you read her story, consider what she needed in order to engage fully in the task she was working on with her colleagues, and what her experience taught her about working with students.

Reflect on yourself as a learner.

What do you feel you need in order to engage fully in this learning community? What are some norms for our work together that feel important in order to create a learning environment that is productive and safe for you and for others in this course? You may wish to consider the following in your response:

• Risk taking
• Sharing ideas in progress
• Building on the ideas of others
• Sharing disagreement
• Handling mistakes and revising thinking
• How students are described and discussed

Share your ideas on the Agreements for Our Work Together forum.

Graduate Credit Information

Participants in this course are eligible to receive 3 graduate credits from Framingham State University. This option involves an additional cost, as well as additional weekly readings and writing assignments, and a final paper. Please let your facilitator(s) know if you plan to take this course for graduate credit, or if you have questions about the graduate credit option.

Orientation Survey

Please take a few minutes to complete this survey which collects some background information about your job role, teaching experience, and familiarity with online learning. In addition, it invites you to comment on your expectations and ask any questions you may have about participating in this course.

Consent Form

Please take a few minutes to read the consent form and respond.

Because of the history and continued presence of institutionalized racism and inequality in the United States generally, and within math education specifically, and because “deficit-based thinking is historical, cultural, institutional, ideological, and persistent” (Aguirre, 2019), we think it is important to consider what equity in the teaching and learning of mathematics means and looks like. Our staff has been examining issues of equity, identity, and agency and -- as one aspect of this online course -- we are asking you to examine them with us.

As you read earlier, one of Investigations’ guiding principles is:

Students have mathematical ideas. Students come to school with ideas about numbers, shapes, measurements, patterns, and data. If given the opportunity to learn in an environment that stresses making sense of mathematics, students build on the ideas they already have and learn about new mathematics they have never encountered. They learn mathematical content and develop fluency and skill that is well grounded in meaning. Students learn that they are capable of having mathematical ideas, applying what they know to new situations, and thinking and reasoning about unfamiliar problems.

This and the other two guiding principles for Investigations have implications for equity in the teaching and learning of mathematics. But, as our staff has become more cognizant of current work about equity and identity, we recognize the importance of talking about these ideas more explicitly. What does equity mean in the realm of mathematics learning? And what does it look like in the classroom?

Equity

In order to talk about equitable practices in the math classroom, it is important to consider what we mean by equity. Here is one definition of equity in math teaching and learning that we have found useful: “All students, in light of their humanity—their personal experiences, backgrounds, histories, languages, and physical and emotional well-being—must have the opportunity and support to learn rich mathematics that fosters meaning making, empowers decision making, and critiques, challenges, and transforms inequities and injustices. Equity does not mean that every student should receive identical instruction. Instead, equity demands that responsive accommodations be made as needed to promote equitable access, attainment, and advancement in mathematics education for each student. This perspective on equity challenges common notions that students need to learn math “in spite of” or “regardless of” who they are. We argue that students need to learn mathematics in light of who they are and the diverse gifts that they bring to their experiences every day.” (Aguirre, Mayfield-Ingram & Martin, 2013, p. 15)

What strikes you in this description of equity? Are there other aspects of equity in a math classroom that you think are important to include in a definition?

Mathematics Identity and Agency

In the learning of mathematics, students’ mathematics identities and sense of agency play significant roles in their success with mathematics. Mathematics identity can be defined as: “The dispositions and deeply held beliefs that students develop about their ability to participate and perform effectively in mathematical contexts and to use mathematics in powerful ways across the contexts of their lives.” (Aguirre, Mayfield-Ingram & Martin, 2013, p. 14) Students who have a strong sense of mathematical agency are: “active participants in, rather than passive recipients of, their mathematics education experiences… They can exercise these forms of agency in productive ways – resisting negative identities that are imposed on them, developing mathematical strategies within the context of small-group work, or using mathematics as a tool to understand their life circumstances or events in the world.” (Aguirre, Mayfield-Ingram & Martin, 2013, p.15)

Educators have a significant impact on students’ mathematical identities and their sense of agency. What can we as educators do to support and nurture students’ mathematical identities and their sense of agency and not undermine them?

Equity-Oriented Teaching Practices

Giving all students access to rich mathematics, giving them opportunities to solve problems in ways that make sense to them, and giving them opportunities to share their mathematical ideas are important elements of equity-oriented teaching practices. However, there is more to think about beyond these elements because “many of these students attend schools and sit in classrooms each day where their cultures are not reflected in the curriculum, where their ideas are not taken up in the public space of class discussions, and where they are seen not as individuals with unique identities and cultural perspectives but instead as data points on a measure of underachievement.” (Goffney, 2018, p. 159).

How do we go beyond simply saying that all students’ ideas are valued? How do we make sure we are mindful of our personal biases? How do we work consciously against inequities?

Lens on Equity in this Course

Throughout this course we will take time to consider the teaching and learning of fractions with Investigations through a lens focused on issues of equity. We hope to offer a space for thinking about these important issues along with our focus on the learning and teaching of fractions through the Investigations curriculum.

As a starting place for thinking about race and culture in relationship to the teaching of mathematics, read the article Why Should Mathematics Educators Care About Race and Culture?

In all subsequent sessions you will find a “Lens on Equity” section, highlighted in blue, where you will be asked to consider some questions about connections between issues of equity and the content of the session.

The Forum for Equity in Elementary Mathematics

The Forum for Equity in Elementary Mathematics is a website for reflecting on and discussing equity, access, identity, and agency in the K-5 mathematics classroom. The work is not curriculum specific. Its goal is to provide resources, publications, and professional learning opportunities to broaden and deepen perspectives on equity and to open up discussions among educators as we seriously and passionately pursue equity in mathematics learning for elementary students. Download a framework that supports reflection on equity; read blogs about various aspects of equitable learning communities, and explore tools that support teacher and student reflection.

Additional Resources and References

Below are references for the quotes above, as well as additional readings for those interested in learning more about issues of equity in mathematics teaching and learning.

Aguirre, J., Mayfield-Ingram, K. & Martin, D. B. (2013). The Impact of Identity in K-8 Mathematics: Rethinking Equity-Based Practices. Reston, VA: The National Council of Teachers of Mathematics, Inc.

Ball, D. L. (April, 2018). Just Dreams and Imperatives: The Power of Teaching in the Struggle for Public Education. AERA 2018 Presidential Address, New York, NY. (Ball’s talk starts at about 53:30.)

Funderburk, J & Brown, C. (Hosts. (2023, February 13) Tapping Into Identity and Equity in Mathematics [Audio podcast episode]. In Room to Grow - a Math Podcast. https://www.roomtogrowmath.com//1581535/12219420.

Gutiérrez, R. & Goffney, I. (Eds). (2018). Annual Perspectives in Mathematics Education, 2018: Rehumanizing Mathematics for Students Who Are Black, Indigenous and Latinx. Reston, VA: National Council of Teachers of Mathematics.

Moschkovich, J. (2013). Principles and Guidelines for Equitable Mathematics Teaching Practices and Materials for English Language Learners. Journal of Urban Mathematics Education, 6, (1), 45–57.

Nasir, N. S. (2016). Why Should Mathematics Educators Care About Race and Culture? Journal of Urban Mathematics Education, 9 (1), 7–18.

NCSM & TODOS (2016). Mathematics Education Through the Lens of Social Justice: Acknowledgment, Actions, and Accountability [position paper].

Post your introduction in the Orientation Session Discussion Forum. Include:

• Your name and the grade level you teach (or your role in your school/district).


• Explain what motivated you to enroll in this course, what you hope to learn, and how you hope to apply what you learn to your professional life.


• Attach a photo of yourself or of something that represents your interests. You can learn how to upload a photo here.

After you have posted your comments, take time to read others’ posts and respond to at least 2 of your colleagues.

Return to the Agreements for Our Work Together Forum. Respond to posts of at least 2 other people.