Not “Location, Location, Location!”  But “Closure, Closure, Closure!”

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Karen Arth, Fresno, CA

For the old folks like me, you might remember Bob Dylan’s song from 1964, “The Times They Are A Changin.” (Hopefully you have at least have heard of Bob Dylan!) That song’s title is still relevant and its applicableness continues today, and, perhaps, now more than ever in mathematics. The 21st Century Skills desired by employers (effective communication, collaboration, critical thinking, and creativity), along with the power of technology, require us to teach mathematics differently than in the past.

With the CCSS, students now have two sets of standards, the Content Standards and the Standards for Mathematical Practice (SMP), the what and,the how. How mathematics needs to be taught is the big change for many teachers.  SMP #1, Make sense of the math and persevere in solving, and #3, Construct viable arguments and critique the reasoning of others, require students to grapple with the mathematics of the lesson. The Teaching Practices, as outlined in NCTM’s Principles to Actions, state the importance of productive struggle. Both documents ask students to take on some of the math authority. But this is the start, not the end of the story. Lesson closure is equally important for student success. Students might see/understand problem 52, parts a, b, and c, but, are they truly making the connections that are necessary to see the big picture, the overall story?  Maybe some students are, but definitely not all students.

I believe that lesson closure is as important as the SMPs and productive struggle for student success. However, many teachers admit that they seldom get to the closure, either they lose track of time or run out of time. There are different ways of developing the habit of closure. One way is the use of a timer. If lesson closure will take five minutes, consider setting the timer for seven minutes before the end of the lesson. The teacher can instruct teams, “If you haven’t finished problem 72, Facilitators, make sure your team reads through all of it now, because we will be closing the lesson in a few minutes.” If you have a clock on the wall that you tend to check, you might have a sign above it that says “Closure!!!” or, you can put a sticky note in your teacher edition that you move from lesson to lesson as a constant reminder. You can even have your class remind you at a given time each period.

An effective closure can be as difficult as it is important. Closure is not just a restatement of the lesson objective, but rather a bringing together of the specific mathematics from the day. One way to think of what an effective closure might look like is to imagine two students returning from a field trip. They both have missed the math lesson, but are there in time for closure. Student A’s teacher summarizes the lesson by saying, “Today we learned about multiple representations of a proportional relationship. We started with a word problem from which we were able to create a table, a graph and then a rule.  Tomorrow we will continue looking at proportional relationships as well as relationships that are linear, but not proportional.” Student B’s teacher closes the lesson by having a discussion. “Today we learned about multiple representations of a proportional relationship. How can we describe the graph?” (Students answer and teacher sketches on the board.) “What does the table look like?  As the students answer, the teacher adds to the board with a sketch of each representation, making connections among them:

Which closure is more powerful? The second one takes a little more planning, but if we as teachers struggle with, and hence end the class without a closure, how can we expect students to pull the mathematical ideas from the lesson together when we struggle with it ourselves?

So make a commitment to have a lesson closure each day. There is a suggested closure in the teacher notes, discussion questions at the beginning of the lesson that can be referred back to, student work that can be presented, and many other ways to effectively bring together the mathematics of the day. Although teaching mathematics and mathematical thinking are not easy, when done thoughtfully and effectively, they each can be one of the greatest gifts we give our students.

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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.