Bridging Practices in Argumentation

Jocelyn Dunnack, Mansfield Middle School, Storrs, CT

CPM problems are ripe for teachers to promote the Standards for Mathematical Practice (SMPs), but as I have been repeatedly reminded this year, SMPs (like a lot of other things) take time, effort, and practice. I have been part of a research collaboration called Bridging Practices among Connecticut Mathematics Educators (BPCME), which brought together UConn researchers, graduate students, and teachers from urban, suburban and rural districts to learn about the third SMP: Students will construct viable arguments and critique the reasoning of others. We have developed resources to support teachers and students in the process of mathematical argumentation. Here
are a few things I have learned:

  1. Arguments are not describing your steps or showing your work. Arguments consist of claims, warrants, and evidence. You have to state a claim, and provide enough explicit evidence to prove you are right. It is more like those fun problems that have multiple strategies or solutions, the ones where students have to prove their solution was actually mathematically sound. Argument is about why you are right, mathematically speaking.
  2. At first, students needed to learn how to provide enough evidence. Then, they had to learn how to explain the reasoning as to why their evidence even mattered. That was much harder to do, and much more important.
  3. The CPM teachers in BPCME had no trouble implementing argumentation (and seeing growth in our students’ work). Great tasks were already part of our lessons! However, we did have to devote extra time for these tasks. By digging into a good argument question and skipping a few of the other questions in the lesson, we found deeper understanding than we previously got. It was definitely worth the investment and the sacrifice.

Our work will be compiled and submitted for publication approval by NCTM this summer. Even though I had already been doing argument tasks that existed in CPM, it was very powerful for me to see what happened when I made a conscious choice to explicitly teach a math practice, rather than just knowing it was embedded. Guess what? My kids could not really do it at first, but now they can! I am sure you are not surprised.

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Algebra Tiles Session

  • Used throughout CPM middle and high school courses
  • Concrete, geometric representation of algebraic concepts.
  • Two-hour virtual session,
  •  Learn how students build their conceptual understanding of simplifying algebraic expressions
  • Solving equations using these tools.  
  • Determining perimeter,
  • Combining like terms,
  • Comparing expressions,
  • Solving equations
  • Use an area model to multiply polynomials,
  • Factor quadratics and other polynomials, and
  • Complete the square.
  • Support the transition from a concrete (manipulative) representation to an abstract model of mathematics..

Foundations for Implementation

This professional learning is designed for teachers as they begin their implementation of CPM. This series contains multiple components and is grounded in multiple active experiences delivered over the first year. This learning experience will encourage teachers to adjust their instructional practices, expand their content knowledge, and challenge their beliefs about teaching and learning. Teachers and leaders will gain first-hand experience with CPM with emphasis on what they will be teaching. Throughout this series educators will experience the mathematics, consider instructional practices, and learn about the classroom environment necessary for a successful implementation of CPM curriculum resources.

Page 2 of the Professional Learning Progression (PDF) describes all of the components of this learning event and the additional support available. Teachers new to a course, but have previously attended Foundations for Implementation, can choose to engage in the course Content Modules in the Professional Learning Portal rather than attending the entire series of learning events again.

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Building on Instructional Practice Series

The Building on Instructional Practice Series consists of three different events – Building on Discourse, Building on Assessment, Building on Equity – that are designed for teachers with a minimum of one year of experience teaching with CPM instructional materials and who have completed the Foundations for Implementation Series.

Building on Equity

In Building on Equity, participants will learn how to include equitable practices in their classroom and support traditionally underserved students in becoming leaders of their own learning. Essential questions include: How do I shift dependent learners into independent learners? How does my own math identity and cultural background impact my classroom? The focus of day one is equitable classroom culture. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for Chapter 1 that creates an equitable classroom culture. The focus of day two and three is implementing equitable tasks. Participants will develop their use of the 5 Practices for Orchestrating Meaningful Mathematical Discussions and curate strategies for supporting all students in becoming leaders of their own learning. Participants will use an equity lens to reflect on and revise their Chapter 1 lesson plans.

Building on Assessment

In Building on Assessment, participants will apply assessment research and develop methods to provide feedback to students and inform equitable assessment decisions. On day one, participants will align assessment practices with learning progressions and the principle of mastery over time as well as write assessment items. During day two, participants will develop rubrics, explore alternate types of assessment, and plan for implementation that supports student ownership. On the third day, participants will develop strategies to monitor progress and provide evidence of proficiency with identified mathematics content and practices. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

In Building on Discourse, participants will improve their ability to facilitate meaningful mathematical discourse. This learning experience will encourage participants to adjust their instructional practices in the areas of sharing math authority, developing independent learners, and the creation of equitable classroom environments. Participants will plan for student learning by using teaching practices such as posing purposeful questioning, supporting productive struggle, and facilitating meaningful mathematical discourse. In doing so, participants learn to support students collaboratively engaged with rich tasks with all elements of the Effective Mathematics Teaching Practices incorporated through intentional and reflective planning.