What Kind of Problem is This Anyway?

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Anthony Johnson, DeForest, WI  anthonyjohnson@cpm.org

We have all been there: as students read through the next math problem someone will raise their hand and say, “This problem is crazy, why would anyone do this? Can’t they just Google it?” As frustrating as this question may be, we have to pause and appreciate the fact that students are questioning the things they are being asked to do. As teachers, we are not in the profession of creating a society of mindless followers but rather, would love to have each of our students leave our room feeling like they can critically analyze problems and know how best to solve them. Couple this ambition with the work that I have been doing with an amazing group of teacher researchers with TRC, and I have created a perfect storm for student inquiry.

The TRC group that I am working with is attempting to answer the question, “How can we create an environment where students increase their engagement with productive learning practices through increasing their psychological safety?” These productive learning practices include asking questions, sharing ideas, and taking risks. One may ask, “What does that look like in a classroom?” I think the best way to answer that question is to share a story of an interaction I had with a student earlier this year.

The students walked into my Algebra 1 class, sat with their teams, and took out their materials for the day. I introduced the lesson and the students began to work. As I started to make my way around the room, I heard what had come to be a familiar conversation. Cam was talking to his group about how the problem just did not make sense. I did not pay it much thought and continued on my circulation path. When I finally got to his group, Cam looked at me and said, “Mr. Johnson, this problem isn’t realistic. What time period is this from?” The problem in question involved a female worker at a soda bottling plant. She was placing caps on bottles and would make money for every successfully capped bottle but would lose money for every bottle she broke. Cam continued, “These costs just don’t add up. First, let’s say we want her to make anything even close to minimum wage, how many bottles would that have to be an hour?” He calculated it. “Now let’s assume she does break even one bottle, the company can’t deduct that from her paycheck, can they? Isn’t there some kind of OSHA law that is being broken here? And really, does this job even exist? I’m pretty sure machines do this now so we don’t have to worry about any of that.” He raised a lot of really good points, none of which I was able to answer. All I could think to do was acknowledge that he was making some really great points, thank him for being so passionate about the problem, but sadly tell him that I was not sure what we could do about the way the problem was written. Eventually we both agreed that, although the problem is not timely in 2018, he could move forward and use the information to learn the content.

The next day Cam came into class and was still talking about it. He became hypercritical of the problems in the book and continued to bring up issues with their context. The following day Cam walked into the room with a big smile on his face. He told me, “Mr. Johnson, I emailed Leslie Dietiker about that bottling problem.” In my head there was no way that someone in high school would have taken the time to do this, even though I knew that if anyone was going to be crazy enough to do this, it would be a high school freshman. My response was simple: a head shake and a smile. I walked around the room as the class began working for the day and he told me again that he had an email conversation with Leslie. My response this time, “I’ll believe it when I see it.” He pulled out his phone, opened his emails, handed me his phone, and just said, “Here.” I could not believe it. Three emails from him and two responses from Leslie later, and Cam showed me the holy grail of teaching. His ability and willingness to take words on a page, think critically about them, and to feel safe enough to vocalize his thoughts was remarkable. He was not simply trying to get out of doing the work, he was questioning the methods that I was using to help him learn. When I was not able to provide him a satisfactory response, he looked at the front cover, chose a name, determined who that was, found an email address, and did something about it. This student was not complaining for the sake of complaining, he wanted to change the way students interact with these textbooks. In his first email to Leslie, Cam wrote, “I’ve been kinda crazy over this with minimum wage problems and safety for the bottle cappers. And the time period would really help me out a lot.” Leslie’s response to Cam was that most students were not as far along as his, responding to him as though he were a teacher. Cam replied, “Unfortunately I’m just a student who took the time to think about the problem with real life problems. I just wanted to speak with a teacher who helped contribute to the book. I’m currently going overboard with a presentation for my teacher to prove a fun little point. So thanks for replying.

In the moments after reading this conversation, I was in shock. Cam was still sitting, just smiling. I took his phone to a colleague next door and showed her the conversation, explaining that I could not put into words how awesome this was. She came next door and told him how awesome it was that he had reached out to someone to get answers to his questions. Looking back, I have no idea how we should have celebrated how unique this moment was, and I know that we did not make a big enough deal of it, but I have never been so proud of a young student doing such amazing things.

In a room where students feel safe, take risks, and ask questions, they will also flourish and do the most amazing things to learn the math. We, as teachers, should want students to be as invested in their education as we are. CPM strives to have stakeholder involvement as the curriculum evolves. Through TRC, teachers work closely with one another to try out new innovations and tweek older practices to ensure that there is more math for more people. Through this brief exchange, a window opened for students to become equal collaborators about the things that they will be working with that drive their education. I have learned to address the question, “Why would anyone do this?” a little differently now. I ask the students, “Why?” and then I listen to the valid and thoughtful reasons that they come up with and we have a conversation about it. I am not sure if any other students will ever be as motivated to make change as Cam was, but allowing them to have a voice in the classroom and feel like someone will actually listen, even if nothing can be done about it, is one of the best ways to build a safe environment that I have come across so far.

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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

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.