TRU-Math Project Summer Academy 2017

This summer’s TRU-Math Project Academy was held during the week of June 26-29, 2017 at Polytech’s Adult Education Conference Center. The academy theme Inspiring Curiosity and Mathematical Intrigue: Promoting Conceptually-Based Reasoning & Mathematical Argumentation fostered the content and pedagogical knowledge and leadership skills of 75 teachers, coaches, specialists, project-based connectors, and higher education partners. The project is funded by a Math Science Partnership grant administered by the Delaware Department of Education.

 Members of the TRU-Math Elementary Math Teacher Leader (EMTL) community invested the four days applying their understanding of whole numbers to operations with fractions. The work was inspired by content from Sherry Parrish’s new book Number Talks: Fractions, Decimals, and Percents as well as the community’s ongoing focus on the use of routines that promote access, reasoning, and student agency. The daily structure provided

opportunities for participants to meet both as a Grades 2-5 community and in smaller grade level specific groups that delved deeper into the content they had learned during their whole group sessions. Goals for the academy sessions included:

• Exploring discrete, area, and number line models to represent fraction operations

• Using story problems to illustrate fraction relationships and operations with fractions

• Extending properties of whole numbers to fractions (e.g., commutative, associative, and distributive properties)

Each day of the academy was devoted to a particular fraction operation. On day 4, TRU-Math EMTL participants engaged in a culminating CCSS Grades 3-5 Number and Operations Fraction Progression activity that was purposefully designed to help participants unpack and apply the fraction work they had been engaging in during the week. Participants also invested time focusing on ratios and proportional reasoning.

TRU-Math Project University of Delaware thought partners Dawn Berk and James Hiebert shared their Learning to See Students’ Opportunities to Learn presentation with the community. The content supported the teachers and coaches in learning more about the Third International Mathematics Study, particularly as it related to the implementation of “making connections” (conceptually-grounded) tasks. The team also shared both local and outside expert Nicole Kersting’s research related to the benefit and use of video analysis in terms of measuring teachers’ content knowledge and correlating the results with teachers’ classroom practices and their students’ mathematics achievement. During the second year of the TRU-Math Project, the entire EMTL community will be engaging in video analysis as part of their monthly conceptually-grounded content-focused work. Feedback from the participants: “5 stars out of 5. It has helped me solidify my understanding of fractions decimals, and proportions. The conversations were so helpful to help me build deeper understanding. I LOVE THAT MATH & MATH WORK WAS THE FOCUS OF EVERY DAY!” “I thought the 4 days were remarkably well planned. Each activity/lesson consisted of tasks that were clearly related and reinforced one another. I thought the work with the number talks, claims, and visual models deepened my conceptual understanding of each of the operations

“Creating and sharing the posters (Fraction Learning Progression) helped clarify the CCSS at each grade level as it pertained to fractions. The share out time was a good way to share BIG IDEAS. It helped seeing the progression of conceptual

development that ends up being the building block for middle and high school.”

(with fractions). Well done! The pacing was quick enough to dive into a lot of content but we had enough time to grapple.” “Every EMTL workshop inspires in me new ideas of how to improve not only my teaching, but also my mathematical leading. I find myself taking a lot of notes on “things to do.” They are things to make, lessons to write, and things to research further. Thank you for making Delaware’s math teachers better!!” “I loved the book choice (Number Talks: Fractions, Decimals, and Percents) and the focus on fractions. Thank you for allowing us time to think deeply and to strengthen our understanding of fractions (especially w/ multiplying and dividing fractions).”

“I love the work we do at EMTL. It pushes my thinking and gives me great ideas to take back to my building. Thank you to the guest presenters! I didn’t realize the power behind such analytical measures to video analysis. It’s inspired me to do some work around that with my observation caseload.”

“I learned a lot over the course of these four days. The activities in the large groups helped me strengthen my own understanding of fractions. The activities provided me with opportunities to experiment with a variety of solution methods. I also really appreciated the opportunity to work in grade level groups for a portion of each day. We could focus on key concepts and common misconceptions at our specific level. I also liked that we had a chance to see how certain routines, lesson activities, story contexts, and visual models can be used to address the

standards for my grade level. I gained a good understanding of how I can improve my own teaching and can see how this will impact my students’ learning, even in future grades.” “I really enjoyed all of the content that was covered in these sessions of EMTL! I felt pushed outside my comfort zone and feel as though I have gained new knowledge that I can use myself and share with the teachers in my building.” “Building the models for visual representation is much more challenging than doing the quick algorithm. I find that I rely on quick algorithm answer to help direct my visual but the visual model is so much harder.”

“The Number Talks for division of fractions was a really useful topic to dig into as a community. Many teachers still have trouble understanding and why division of fractions works. The work we did with visually-based arguments strengthened my own understanding and will provide me with a way to

approach these conversations with my colleagues. Thank you!” “The morning session (on multiplying fractions) was great. I loved how much could be discovered from the “Which one doesn’t belong” activity. This really helped me think a little deeper. I grappled today. The What do you notice? activity with 8 x ¼ and ¼ x 8 was also a lot of fun with having to complete all three models. It was eye opening to see when putting a story context to the problem helped in some cases.” “I have a better understanding of how to visually represent multiplication of fractions. It was interesting to experiment with different models and see which models lent themselves best to various story contexts. I also gained a good refresher of how the product’s value varies based on the value of the factors.”

“The activity with the fraction bars was great and I can’t wait to use it in my classroom-especially the games like “cover up”. It’s great that it hits so many of the 3rd grade standards-all in one activity.” (G3 teacher) “Making claims forces summarization of concepts presented. Visuals can bridge gaps between basic understandings of concepts and ideas that students have.” (Grade 5 teacher)

“Definitely a morning of grappling. So mind-blowing how changing wording is such a tricky thing to do to show groupings with division of fractions. I really enjoyed working with my group to figure out how to solve these difficult story contexts. It was also helpful to make a conjecture with our group. Also liked reading another group’s conjecture and editing it.”

“Jen did a great job of going through the progression of the two number talks. The activity (exit ticket-four square) was super helpful-determining what students know, their misconceptions, what questions you would ask and next steps. I loved the idea of using this activity with my class; analyzing other students’ work… It really helped to show their students. Overall-A+ afternoon! (Grade 4 teacher) “Visual models help students see the relationship between the numbers in problems involving ratios. Some models can be more concrete than others. Ratio tables are another useful problem solving tool. Students can construct various patterns (doubling, 10s, halves, etc.) as they make ratio tables. This allows them to choose the problem solving method that works best for them. The ratio table allows students to take different pathways to reach their solutions.”

Members of the TRU-Math Project Leadership Coaching Lab Academy community also focused on the theme, Inspiring Curiosity and Mathematical Intrigue: Promoting Conceptually-Based Reasoning & Mathematical Argumentation. Participants invested the four days applying the two research principles related to teaching for conceptual understanding and building upon their previous work related to mathematical argumentation and developing a keen eye (video-based analysis). Goals for the academy included:

! Fostering algebraic reasoning and generalization. ! Developing/Applying structures that promote more compelling

representations-based arguments. ! Assessing the extent to which a given set of visual and/or contextual

representations is useful/powerful in promoting conceptually-grounded mathematical generalizations.

! Analyzing teaching episodes in order to identify the learning goals, key ideas and connections and making suggestions related to students’ opportunities to learn.

! Deepening our understanding of how teaching for conceptual understanding and students’ sense of self-efficacy are linked.

The content for the four-day academy was inspired by the following resources: Liping Ma’s book Knowing and Teaching Elementary Mathematics: Teachers Understanding of Fundamental Mathematics in China and the United States, Kazemi and Hintz’ Intentional Talk book, the learning sequence structure from Russell, Bastable and Shifter’s book But

Why Does it Work?, Nicole Kersting’s  Capturing Teacher Knowledge video repository, video from secondary teacher leaders’ classrooms, and routines  from Chris Danielson’s book Which One Doesn’t Belong? On the last day of the academy, University of Delaware thought partners Dawn Berk and James Hiebert engaged the members of the TRU-Math Leadership Coaching Lab community in a presentation designed to help participants take a deeper look at Kersting’s video analysis findings. The experience promoted conversations about potential reasons why teachers who focused on the math (as part of their analysis) and in particular, who provided suggestions for improvement, related to the desired learning goal had higher mathematics achievement in their classrooms.

Few research studies have found a strong correlation between teachers’ content knowledge, classroom practices, and student achievement. Kersting’s study did just that! TRU-Math Leadership Coaching Lab participants seemed genuinely curious about whether their ability to get better

at noticing and citing suggestions for improvement will in turn result in better practices and student achievement performance in their classrooms. This will be the focus of our ongoing research as a community.

During year 2 of the TRU-Math Project, Leadership Coaching Lab participants will continue to analyze “stock” video and focus increased attention on capturing, sharing, and analyzing video from their own classrooms. As a leadership community, TRU-Math Leadership Coaching Lab members will continue to improve their capacity to: identify the learning goal, key ideas, and connections in the videotaped lesson, focus on the pivotal moments in the clip(s), and practice providing even more targeted mathematically-focused feedback related to suggestions for promoting greater progress towards the desired learning goal for the lesson. Feedback from the participants: “The Number Talks for division of fractions was a really useful topic to dig into as a community. Many teachers still have trouble understanding and why division of fractions works. The work we did with visually-based arguments strengthened my own understanding and will provide me with a way to approach these conversations with my colleagues. Thank you!”

“The presentation about leveraging opportunities for learning was very powerful! It was insightful to get the background to our video analysis work. Categorizing the

recorded comments into 1) Pedagogy 2) Math Descriptive 3) Math Critique 4) Math Suggestions definitely helps me understand the “look fors” in the video analysis work.”

“I love how visual models are expected, along with research about tools that promote reasoning and argumentation. Thank you for a great four days of learning!” “Loved having a chance to grapple with the math. During the exploration, we made connections between the generalized formula for the area of a pentagon and the formula for the area of a circle. I so appreciated how our group worked together to compare strategies and consider how our ideas were related and then build towards a better understanding of the generalizations.” “The digging deeper with a chosen concept was exciting! I was able to “master” my concept and focus on the math and visuals to represent the math. I thought the visual of the Twin Primes task was interesting and makes me want to look into it more.” “Grappling with visual representations of twin primes was VERY difficult. I learned how important visual representations are for promoting conceptual connections.” “I really appreciate having the claim conversations and challenge of making claims visually, algebraically, etc. Strong, rich conversations pushed us to share and understand each other’s thinking.” “Great 4 days---I only wish every teacher in our school (district) could attend. So beneficial!” “The kite problem as a lot of fun! I enjoyed coming up with different ways to think about the area of the kite and how each algebraic representation communicated a different way of making sense of the area. The generalizations related to “thinking outside the box” really pushed my thinking. In the end, we were all able to make sense of how the different

ways of reasoning were connected. I felt that if given an opportunity, students could understand that formulas are not so mysterious.” “I really appreciated the opportunity to focus on rough thinking and rough draft talk. I agree that this is an important norm to build in my classroom, particularly as it related to to promoting productive grappling. I found the time invested building a tool of descriptors for actively promoting more compelling arguments very useful to our classroom work!” “Enjoyed the morning session with number talks. Really interesting conversations emerged regarding the struggle with the visual representation as compared to the numerical representations. Felt like while we are getting better with different strategies and pushing for stronger conceptual understanding, yet the visual end was still the more difficult challenge.” “Video analysis was awesome to be led by a peer. The continuous lists for opportunities created and missed opportunities as we watched each video clip helped us unpack the lesson in chunks and make sense of the learning goal.”   “I loved Kathleen’s video analysis piece. I am going to take back the two questions that Kathleen posed to our group to analyze the videos. LOVED IT! Very thankful!” “I have really loved the focus and analysis of video during these sessions. I feel like I have developed a critical eye toward teaching that will be essential to improving my practice.” “Kudos to Kathleen for sharing with us all her video and willing to put herself out there; loved seeing her students sharing out and collaborating-and think lots of valuable discussion using what teachers moved created opportunities and suggestions for improvement-opens up for teacher to feel more comfortable turning class over to their students.”

“The video observation showed a variety of teacher moves and it was really powerful how we thought about the moves from the perspective sense making or “getting in the way of sense making.” This video helped me think about:

• Previewing a rigorous task (Interior angle of a 20-gon) • Making sure kids understand the task (sum of interior angles) • Progression of student-led examples.”

“Doing the corral problem w/ Tom made me wonder about opportunities for students to develop their own formulas. Many students come to us with w/ formulas memorized but I think conceptualization and comparison are so powerful. We need to continue to work with our peers in other grade levels to focus on teaching for conceptual understanding and having students own their formulas and be able to defend their thinking.” “Dissecting Tom’s lesson was powerful for me. Unpacking what I did in this lesson last year and reflecting on that as part of Tom’s video has helped me plan a better lesson the near time I teach it.” “Enjoyed the corral problem and it pushed my thinking. It helped me make connections to trig, Pythagorean Theorem, and algebraic reasoning.” “I love comparing formulas/rules. It really stretches my thinking. Tom did a great job focusing us in on a search for the apothem. I hadn’t about other formulas as deeply as he asked us to.” “Although the n-gon formula problem gave me a splitting headache, I enjoyed working thought the math with my group. They really helped me process my thinking.” “Leadership and building strong teams is such a HUGE think-I think all work toward this can only better education as a whole.”

“Jamila-Thank YOU and all of your team for yet another wonderful learning experience. It never ceases to amaze me that, despite 18 years of experience, I am constantly coming to new realizations about the field of teaching. That's what keeps it exciting!”