Four Innovations that Support Developing A Culture of Investigation

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This is the second part of the TRC report on a culture of investigation

Pam Lindemer, Ionia, MI pamlindemer@cpm.org 
Jennifer McCalla, Parma, OH jenleeford@aol.com 
Christy McConnell, Whitefish, MT christymcconnell@cpm.org


My Favorite No

We incorporated a strategy we saw on the Teaching Channel website (www.teachingchannel.org) called My Favorite No into our classroom as a single event. We asked the students to solve a problem as an exit ticket and then used their work to discuss our favorite misconceptions the next day. We carefully chose a few of our favorite solutions that had common misconceptions embedded into them and asked the students to consider what the misconception was and why that student (or students) had made one of our favorite mistakes. This strategy allowed us to focus on misconceptions in an anonymous way. There was no threat to the students that they would be scolded in front of their peers for making a mistake. In fact, we were celebrating the mistake and thanking them for making it so we can all learn from it! We found this strategy to be very impactful and easy to incorporate, so we continued to use it in a variety of ways. Whether we were checking the student’s homework solutions, reviewing an assessment, or observing conversations and solutions while teams were working on a classroom problem, we would stop and have a discussion about some of the favorite mistakes we were seeing and debrief the mistakes with the entire class. These variations of My Favorite No reinforces the important idea that making mistakes is an integral part of learning. Without the discussions about why we think the particular mistakes were made, the strategy would not have been successful. Also, when students recognize the mistake as their own, they are much more likely to find the work relevant and engaging.  We found that after the first several uses of this strategy, many students started to volunteer. They let the class know that it was their mistake we were looking at  –  they were owning their mistakes! At the end of the year, we even had students saying things like, “I love it when I make mistakes. I learn so much when I make them.

One of us created a variation of the My Favorite No strategy using multiple-choice questions. Students were challenged to look at a problem and try to discover what mistake was made to arrive at each of the distractors. Students were engaged and found that many of the distractors provided them with a deeper understanding of the problem.

Teamwork Rubric (Jen McCalla)

At the beginning of the 2nd semester, I wanted to revisit the team norms and focus more on classroom culture. Some students were sitting back, allowing other students to complete the work for them, but it is important that all students need to be held accountable for using their team roles. After researching rubrics on teamwork, I had students individually rate their teams using the rubric. Then they discussed their ratings with their teammates and came to a consensus. I noted that they needed to be able to support their ratings with specific examples. The teams were asked to identify strengths and weaknesses and to create a team goal. Then each teammate was asked to write personal goals related to their team goal (i.e. How will you help your team reach their goal?) We revisited their team goals two weeks later and went through the same cycle to set new goals.

At first, many teams just rewrote their weakness as a goal without much thought. For example, many teams wrote, “We need to communicate better.” When I asked the teams how they plan on communicating better, they added things like, “We’ll use our team roles more.” Other responses were typical. I could see that the steps that each team needed to take to reach their goal were still not as specific as I would like them to be, but it was a good start.

One team determined that some of their teammates did not speak up as often as they would like, with team discussions typically led by two of the four members. Their team goal was, “We will communicate better.”  I was happy to see that some of their individual goals were written to support their team goal. One student, who participates frequently, stated, “I will ask everyone what they think the answer might be, even if it’s wrong.” This was a strategy that she would use to help her team reach their goal.

Working with the teamwork rubric helped our TRC research group realize that the following changes would be necessary before using it again next year. The first change is that the wording in the Communication strand needs adjustment. Currently, the descriptions seem unrealistic when compared with the actual experiences of a first-year CPM teacher or a student who is new to team work. For example, we assumed that a score of 4 on this strand meant that the students were digging in deeper, justifying their thinking, building on each other’s thinking, and extending their own ideas and work. However, as a fellow TRC adviser pointed out, the way it is currently worded provides a different picture, one where the focus is more on taking turns rather than thinking, connecting, and extending. Should we change the Communication strand to include the verbs construct, justify, build, and maybe extend? Possibly, a score of a 4 on the Communication strand should read as follows: “We can construct arguments to justify our ideas and build on each other’s thinking as we collaborate.” The second change that our TRC team wrestled with was the On-task strand. We realized that, as adults, we typically have several sidebar conversations while we are working on anything in a group setting. We talk to each other about off-task topics during staff meetings, professional development gatherings, etc., but can still focus on the meeting’s agenda and complete the task at hand. After thoroughly discussing this do-as-we-say-not-as-we-do discrepancy between teacher expectations and teacher practices, we came to a consensus and realized that sidebar conversations are important and maybe even necessary to build trust, form relationships, and allow us to function better as a team. However, the rubric does not allow for sidebar conversations, as a score of a 4 on the On-task strand states that the team must keep the conversation on math. This strand seems to be tricky to reword, as we want the team building skills to develop, but we want the math to be the primary focus. Should we take out the phrase, “keeping our conversation on math” and replace it with, “Our team stays focused on math, making sure everyone is on the same problem and finds the same solutions even though our work may differ.”  How can we rephrase a score of 4 on the On-task strand to accomplish what we want?

We definitely thought that the teamwork rubric warranted some additional research. If further investigation is warranted, we would like to consider the following questions:

  1. Is the rubric suitable for introduction at a veteran’s workshop or during Stage 2 of a CPM workshop as it could be overwhelming for a beginning CPM teacher?  
  2. If the rubric is for the students, should it look different for CC1 students as compared to calculus students?
  3. Should the rubric be part of the resources for teams, similar to team roles?
  4. Should a video of what the strands of ‘4’ look like in practice be provided for use in classrooms?
  5. If a school is just starting CPM or is using CC1, is there a specific time in the year to introduce the rubric?
  6. Should it be used at the beginning of a chapter, then midway as a tool to measure growth?

After spring break, another investigator noted that the teamwork in her class was faltering. She had switched teams and no one was happy. They were all complaining that their current teams were not going to work; everyone wanted to blame someone else for the poor teamwork and no one wanted to take responsibility and improve their position. She decided to pair teams up and gave them the teamwork rubric and used the fishbowl strategy, where one team observes the other team and gives feedback using the teamwork rubric as a guide. The next day they switched roles. This exercise reminded students what effective teams should look and sound like. All teams found a way to work together and were motivated to finish strong. Students enjoyed the fishbowl and said we should have done one at the beginning of the year.

Number Talks (Pam Lindemer)

Jonathon is in my Algebra 1 Math Lab. He has struggled with mathematics throughout middle school and came to high school with negative feelings about the subject. I remember the first number talk I did with the class; the subtraction expression was 15 – 11. Jonathon was the first to volunteer to share, and I tried to hide my disappointment as he described an all too familiar algorithm. He had little to offer as an explanation for anything he was doing. “That’s the way you do it.” It was obvious to me that his understanding of the mathematics involved was very procedural, and I wondered how long it had been since math had made sense to him.

At this point I backtracked with my number talks and started over with dot talks. It was fun to see the students begin to open up and share their ideas. Our conversations included more and more students as time went on. I was no longer the only one asking questions; everyone wanted to understand what was being shared. The students were beginning to appreciate that there were different ways to see each dot pattern and they considered it a challenge to come up with unique ways to describe the dot pattern of the day. I often had to cut our discussions short!

After some time, I felt that student confidence was improving and I decided to try another number talk. This time I used the multiplication expression 15 19. After some sharing by other students (including the traditional algorithm), Jayden described a two-step strategy using friendly numbers:

  1. 15 20 = 300  “I added 1 to 19 and multiplied by 20, then…
  2. 300 – 15 = 285  “I removed the one group of 15 I added in step one to get the answer.

Jonathon was hooked! He asked questions of Jayden until he understood the process fully and then wondered out loud, “Will this work all the time?” He wanted to do another number talk to try the strategy…so we did. This time we tried 15 18. Jonathon was eager to share his thinking and see if others agreed. We went through his reasoning and everyone agreed that it seemed reasonable. Another student shared his thinking and when they arrived at the same solution, everyone was convinced. I then decided I had time to try one last number talk that day… 15 21. Jonathon thought about the problem and then finally let me know he had an answer. He did not share his thinking on this problem, but he did voice his agreement with another student who suggested using friendly numbers:

  1. 15 20 = 300  “I subtracted 1 from 21 and multiplied by 20, then…
  2. 300 + 15 = 315  “I added 15 to get the answer. This was the 1 I removed from 21.

From this exchange, I realized that a “one and done” approach to number talks would not be enough for these students. They would need more number talks to practice a new strategy. The more consistently I used number talks, the more successful my students were at retaining and using the strategies. I found that two to three number talks per week were necessary to build fluency, and that I needed to periodically use number talks that might revisit a strategy to help with retention.

As the year went on, I found that Jonathon’s confidence increased and his ability to solve one and two-variable equations improved. He is now successfully solving one and two-variable equations and factoring quadratics!

Resources for number talks are abundant at the K – 8 level, while opportunities for their use at the 9 – 12 level remain virtually unexplored. Future work might include:

  • Placing number talks within the CCA curriculum to support student learning.
  • Creating videos of teachers and students engaging in number talks.
  • Considering curriculum coherence across number talks and between number talks. 
  • Investigating dot talks as a way to introduce and foster a growth mindset.
  • Using number talks as a vehicle for the professional development of teachers.
  • Using Pattern Talks to support algebraic thinking (see diagram below)
  • Investigating the sequencing of number talks.
  • Developing a progression or thread for number talks.

Goal Setting (Christy McConnell)

At the beginning of the year, I wanted my students to experience failure and persistence outside of the arena of math, similar to the concept of the backwards bicycle (Video). So we changed our dominant hand for a few weeks. This experiment leveled the playing field, thereby providing a common experience that was low risk, allowing students the opportunity to transfer both struggles and successes to another area, math. The three positive byproducts of the experiment were that students: 1) became better team members, 2) enhanced their relationships with one another, and 3) had more patience with their teammates. Additionally, it was beneficial for students who do not normally struggle with math to actually have an activity to struggle with and either persevere or give up.

A poster that says "How do mistakes contribute to growth".

As second semester began, I repeated an activity from last year, goal setting. I was curious to see if conducting the My Favorite No activity, and the work we did with highlighting, accepting, and learning from our mistakes had any impact on the students. As we started, I quickly realized my students did not have much experience with making resolutions, so I assumed they did not have much experience with academic goal setting either. I needed to give them an example, so I shared that I was thinking about running a marathon in two months. Since that caught their attention, I continued, letting them know I am not an avid runner, so my goal was a bit unrealistic. However, with the right planning, I could probably be ready to complete the Leprechaun Lope, which is a 5K. As I looked at them, I could see the wheels turning. They were ready to start. Step one was an individual one-minute brainstorming session listing both school and personal goals. The students selected four academic goals and two personal goals. We discussed creating realistic, yet specific and reachable goals. Most of the kids did a great job writing specific goals. The second task was to write action plans that included three or four steps that students could complete to make each goal successful. Now it was back to my goal of a 5K. The students had all kinds of ideas for steps I needed to take in order to build enough endurance to complete a 5K run. After writing my steps, they began to write their own. I was surprised at how many of the kids mentioned mistakes as one of those three or four steps. Some kids were going to pay more attention to their mistakes, some were going to ask a friend/teammate or the teacher about mistakes they did not understand, some were going to keep papers with mistakes until those problems were corrected and understood. Additionally, several students mentioned trying to fix mistakes so they were not repeated. The last task was to make our goals public. Students shared two of their goals and action plans with their teammates. As I circulated, it was fun to hear that some had the same goals and would be supporting each other.

The plan was to have students evaluate their goals every two weeks. During the first session, the majority of my students remembered that we set goals and had been working on them. But for a couple it was a big flashback  –  they had completely forgotten. If this was the case, the student and I looked over each goal and the steps for success. A few students realized some of their goals were unattainable; therefore they had let them slide. I called these my favorite mistakes! They gave us an opportunity to analyze the unattainable goal, talk about what would be realistic, and write a new goal. The end realization was this mistake provided another opportunity for a second chance. After students evaluated their goals on their own, we shared our success/progress in our teams. I smiled as I heard sincere compliments, encouragement, and saw some high fives.

As we did our goal setting, we were able to experience perseverance, celebrate successes along the way, and look at mistakes and/or failures. Mistakes were viewed as goals that were forgotten about or goals that were not written in a way they could be achieved. These mistakes allowed us to analyze what was wrong and what should be modified in order to achieve success, or ask, “Was the goal unattainable?” Sometimes just breaking down the steps for goal achievement differently provided a different plan of attack. However, other times it took honesty to say, “At this late point in the game, what you want is not realistic. What can we do to make the best of this situation, and allow you to turn things around?” This concept of focusing on mistakes and resiliency has made my students more reflective and more insightful.

At the beginning of 4th quarter, the plan was to have the students write one goal (pushing to finish strong) incorporating the components of homework, class work/teamwork, and tests. Under each of these strands they would write their steps to success. I was ecstatic when the students whined about focusing on only one large goal; some of them already had started writing several goals. We compromised—they had to have one math goal and at least two other goals.

Some possible goal setting ideas for further investigation include:

  • Should students write goals that narrow the lens to focus on test mistakes?
  • Is the goal setting process only for younger students, older students, or do all students need to practice goal setting?
  • What about the level of the student  –  honors vs. non – honors students? Is there a difference between their goal setting skills going into this process? Are honors students at an advantage because they may already have some of these soft skills, or are they at a disadvantage because they are not used to failing?
  • Could their reflection on whether or not they met their goals be in the form of a learning log?
  • Should the teacher provide a lens to focus their goal-setting, or should it be open for students to decide what the focus should be? Should the teacher provide some exemplars?
  • Should a template be used to complete this process? Would it help for students to have something in writing about goal setting and have to respond to prompts to determine what their goals should be?
  • Would it be more appropriate to look at the goals before beginning a chapter, mid-chapter, or as part of post-chapter assessments? Does the timing even matter?
  • Should we assign accountability partners, i.e., having another student offer support and encouragement while working towards goals?

Final Thoughts

Overall, our innovations seemed to have the most impact when there was a solid relationship, based on mutual trust, between the teacher and the student. Likewise, trust within a study team was found to be essential ingredient for effectiveness. Students need to feel safe in taking risks. If students feel safe in making mistakes, they will be more likely to participate and be engaged in the lesson. We found that students who were only interested in grades and showed little concern for developing a deeper understanding of the mathematics tended to be the least resilient when it came to failure. These students found it very difficult to recover from setbacks and learn from mistakes. They were not academic risk takers and felt less smart because of a mistake.

Our investigation proved that it takes time, effort, patience, and ongoing support to move students from a fixed mindset to a growth mindset and develop a culture of investigation. We learned that our innovations seem to have minimal or no impact when trust had not been established, especially at the beginning of a chapter when new teams had formed. We believe that parents with a fixed mindset may have a contradictory influence that negates any effort to help students change their beliefs about themselves. Minimal impact was also observed for students who were not present  –  physically and mentally.

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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,
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  • 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.