What About the Gaps?

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Gail Anderson, Lansdale, PA  gailanderson@cpm.org

In the 1980s, Just-in-time Manufacturing became popular in American companies. Just-in-time is a protocol companies use to get their product to customers right when they need it, instead of stockpiling goods in warehouses that may end up obsolete before they are ever sold. Companies developed a mindset of watching the market, surveying customers, and becoming flexible enough to respond to demand. This mindset allowed them to respond quickly to surprises in the market (such as a pandemic!), while saving money and better supporting their customers.

If you teach CPM, you probably have already done some form of just-in-time academic support with your classes. For example, I remember a time when I noticed most of my teams were stuck on a problem because students forgot how to use trigonometry to determine missing side lengths. I paused the class, asked a team who had solved the problem to briefly explain their solution method to the class, and that resulted in a mini-lecture on sine, cosine, and tangent. The rest of the class was able to immediately apply the “renewed” knowledge in the context of the current lesson, and we were able to move on.

The joint statement released by NCTM and NCSM, Moving Forward: Mathematics Learning in the Era of COVID-19, suggests that this approach also makes sense on a larger scale.

There are better options than using testing at the beginning of the school year to assess a laundry list of prerequisite understandings from previous grades that would consume a significant amount of instructional time. Prerequisite skills or understandings that may have been missed as a result of COVID-19 could be strategically taught right before the connected unit of study or incorporated as spiral review or as part of instructional routines and procedures. Teaching these skills as connected to grade-level or course-level content deepens students’ mathematical understanding.

Studies in colleges have similarly shown that student success rates are higher when students are enrolled in grade-level courses with access to support as they need them, rather than when they are enrolled in a non-credit remediation course designed to “fill the gaps” first. Consider this: would you rather be told “take exit 45, turn left at the light, and then take your third right” in the morning before you leave the house or when you are a couple miles from the exit?

You can count on encountering significant gaps in student learning this fall, even more than in the past. You have a lot of ground to cover this year, and you will likely have quite a bit less time in which to cover it. Every moment you have with your students is precious. You will want to use this time to focus on opportunities for learning, and leave the gap-filling for when it is relevant and necessary.

Consider starting the year on a positive, optimistic note. For example, instead of giving your students a pretest, constructing a list of the gaps in their knowledge, and then spending the first half of the semester (or more) teaching what you wish they had learned last year, look for what they do know and remember. The joint statement by NCTM and NCSM recommends:

Educators should view students in terms of their strengths, not weaknesses, and avoid the urge to immediately reteach all the skills we think students should have learned before arriving at school this fall. It is more productive for teachers to think of learning opportunities that are most important for students in relation to the mathematics learning progressions.

As an alternative to a preassessment, perhaps you can use a team assessment. CPM’s team assessments are designed to give students an opportunity to practice collaboration and communication while applying and connecting mathematics skills they have learned. In the context of the beginning of this unusual school year, a team test at the beginning of the year could give students a shared goal and reason to explain to each other what they remember and know how to do, as well as to look up what they don’t remember or know how to do. Team assessments give the teacher time to listen to the students’ mathematical thinking as they discuss and work out complex problems with their teams. As with all CPM tests, the team assessments are cumulative in nature, so you can use a test from a later chapter in the previous course to get a snapshot of what students remember from the whole year. Sample team assessments for every CPM course can be accessed via the CPM Assessment tab in your teacher eBook (choose Download Sample Tests).

We sincerely hope you will have the opportunity to be with your students in the fall to experience the math with them in person. Suggestions for team assessments are included in the Assessment tab of the Teacher Resources in your ebook. If your classes are virtual, here are some ideas that may work to conduct a team assessment in that environment:

  1. Synchronously: If you can host a virtual class meeting time (or perhaps a couple of different options from which students can choose), assign students randomly to breakout rooms (which are available in Zoom and Schoology virtual meeting spaces, as well as many others). You can monitor the discussions and give teams a shared whiteboard or Google doc on which to collaborate. Students can upload photos of their hand-written work at the end of the class time.
  2. Asynchronously: Allow students to choose two or three other students to form teams they can work with remotely, or have them sign up to individual team time slots, which will fit their scheduling needs. We have created a Desmos template you can use, with or without your own edits and customization, to give students a place to list answers and submit their work, and to give you a place to collect all of the responses in one place. Please read more detailed suggestions for using the template in the Activity Details and Teacher Moves of the template. Another format option is to give the whole class these rich problems, instruct them to talk to each other as much as they want, and document their collaboration and sources as well as their solutions. If you are grading the assessment, be sure to include their collaboration documentation as part of the grade, so students attach a high value to that. Students could use a document such as this Shared Rough Draft Space (gdoc) to share and discuss their thinking process during the test. Another option is to require that each student submit a video of themselves explaining the solutions to the problems (after they have discussed them with their team) using a screen recording on a laptop or FlipGrid, their phone camera, or any of the many video options available.

As students work on the tests at home, you may or may not be able to monitor how much help they get from outside sources. Consider encouraging teams to find sources to help them with the content they do not remember or never learned. Tell them you know that each one of them missed out on a lot of instruction during the previous school year, and you will be working with them not only to accommodate those needs throughout the year as they arise, but also to help them learn to independently find and fill those needs.

By giving a team assessment with rich, complex mathematical problems, you will be able to see the strengths your students are coming in with and work from there to develop a pathway to navigate through a year I am sure will be filled with many surprises.

Please be considerate to all of our teachers and do not post solutions to CPM sample tests or test problems. Copying the problems and posting to any public site is a violation of US copyright law.

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Algebra Tiles Blue Icon

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

This series contains three different courses, taken in either order. The courses are designed for schools and teachers with a minimum of one year of experience teaching with CPM curriculum materials. Teachers will develop further understanding of strategies and tools for instructional practices and assessment.

Building on Equity

In this course, participants will learn how to include equitable practices in their  classroom and support traditionally underserved students in becoming leaders of their own learning. Participants will reflect on how their math identity and mindsets impact student learning. They will begin working on a plan for implementing Chapter 1 that creates an equitable classroom culture and curate strategies for supporting all students in becoming leaders of their own learning. Follow-up during the school year will support ongoing implementation of equitable classroom practices.

Building on Assessment

In this course, participants will apply assessment research to develop methods to provide feedback to students and to inform equitable assessment decisions. Participants will develop assessment action plans that will encourage continued collaboration within their learning community.

Building on Discourse

This professional learning builds upon the Foundations for Implementation Series by improving teachers’ 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 rigorous, team-worthy tasks with all elements of the Effective Mathematics Teaching Practices.