Each of the practices of modeling discussed above connected directly to the Standards for Mathematical Practice. Standard 4 of CCSS-M describes mathematically proficient students who can apply what they know to simplify a complicated situation and can “apply the mathematics ... to solve problems in everyday life.” In addition to modeling with mathematics in the four ways presented, the process of selecting and using tools to visualize and explore the task connects with CCSS-M standard 5: use appropriate tools strategically.

Furthermore, as students struggled to make sense of their task by translating within and among multiple representations, they developed an important aspect of strategic competence in mathematics. To help us discuss the important ways that modeling math ideas support teachers’ and students’ strategic competence, we created this visual to show how all these different modeling activities supports the “ability to formulate, represent, and solve mathematical problems.” (NRC, 2001, p. 116).

Through our case study of lesson study and professional development with teachers, we have seen models and modeling of mathematics expressed and strengthened in the different and interrelated ways. In chapter one, we began describing our framework for Developing Strategic Competence through Modeling Mathematical Ideas with an analogy of gears that need to work together in tandem. As we conclude our book, we end with this last analogy. Each of the approaches of modeling mathematical ideas was shown to contribute toward strengthening strategic competence in both students and teachers. Through our professional development model, we were able to expand teachers’ understanding of modeling mathematical ideas and provide them with practical means to do so.

In addition, we offered several effective teaching practices that have supported teachers’ professional practice. First, we planned professional development in vertical team with resource specialists. We have found that having vertical teams allow for a natural dialogue of vertical articulation among professionals around the important learning progression of concepts and students ’ developmental readiness. Collaboration of resource specialists (i.e., math coaches, special educators, English Language Specialists, technology resource teacher, and more) also allow for the diverse expertise to work toward a common goal. It may be hard for one single teacher to attend to all the complexity of the learning environment that makes learning optimal, but with the help of the diverse experts, teacher teams can learn more about how to bring rigor to their lessons, consider cultural relevant teaching approaches while attending to the support needed for diverse learners.

Second, representational fluency can be one assessment of students’ mathematical proficiency. Having flexibility among the different representations can be a good measure of mathematical understanding. Cramer (2003) discusses the importance of “representational fluency” as she refers to Lesh’s translation model and states, “The model suggests that the development of deep understanding of mathematical ideas requires experience in different modes, and experience making connections between and within these modes of representation.

A translation requires a reinterpretation of an idea from one mode of representation to another” (p. 1). One way to think about this is that the more dense the connections between and among the representations, the better the mathematical understanding. This can be thought of being analogous to how the brains synapses become dense as more connections are made and formalize learning makes these connections stronger. This research-based notion not only was supported by two decades of research on the Rational Number Project but also more recently with the visual thinking strategies Of course, there are some representations that are more efficient than others; however, we have noticed that teachers and students who can better interpret diverse representations and strategies have a better understanding of the mathematical concepts.

Representational fluency is not easy to gauge; however, for our algebraic lessons, we were able to use the Modeling Math Mat to capture students comfort with representing their understanding through five representations: words, numbers, pictures, tables, and graphs. In addition, technology can be leveraged in the way that it affords multiple representations in its media format and the affordances and constraints that can help focus and amplify the essential mathematics.

Third, we feel that it is important to approach the mathematics classroom as a place to develop important twenty-first-century skills. Being mathematically proficient will be meaningless unless one can communicate their understanding in a collaborative setting, or apply their knowledge to creative- and critical-thinking situations. In this regards, mathematical modeling tasks that are open ended, real world, and messy can be a great place for students to explore in. The critical component will be for teachers to find mathematical modeling tasks that students care to engage in, find relevant and one that their mathematics background knowledge can provide entry.

Mathematical modeling in the purest sense, as used by industry people and engineers to create predictive models, may be out of reach for elementary students. However, applying the mathematics they know to situations they need to mathematize may be a great precursor for young students. Developing quantitative literacy as students become habitual problem posers and problem solves is an important prerequisite to becoming an efficient mathematical modeler.

Fourth, we spent a lot of time talking about the importance of navigating through students’ diverse strategies and being able to anticipate, monitor, select, sequence, and connect strategies. We feel that naming strategies in this process of the Five Practices was essential to building collective knowledge in the classroom. Providing students the time and space to make sense of the problem in their own respective ways was important to building and adding to an existing schema.

We have learned that many research lessons revealed common misconceptions that were not just blind error but partial understanding or an overgeneralization of a rule or strategy one learned. Understanding how students reason through a problem is pivotal in engaging students in productive discourse about mathematics. Novice teachers do not have the experience base to rely on when planning for potential and common misconceptions; therefore, working with a team of new and seasoned teachers is critical to teachers’ professional learning.

Fifth, analyzing students’ work and their justification as the centerpiece of analysis during a research lesson is important to teachers’ professional learning. Not only does it offer educators the chance to assume the role of learning scientists who can test out their hypothesis for learning but also a way to examine whether or not a certain intervention turns out to be the instrument or strategy that helps students learn a challenging concept. Effective job-embedded professional development models include structures, where professional learning is directly related to the work of teaching include coteaching, mentoring, reflecting on actual lessons (Schifter & Fosnot, 1993); group discussions surrounding selected authentic artifacts from practice such as student work or instructional tasks; curriculum materials (Ball & Cohen, 1996; Loucks-Horsley, Hewson, Love, & Stiles, 1998); and Lesson study where teachers collaborative plan, observe, and debrief (Lewis, 2002; Wang-Iverson & Yoshida, 2005).

We enjoyed the collaborative nature of designing professional development with content-focused institutes in the summer and the school-based follow-up lesson study cycles in the school year that encouraged vertical articulation and practical application of the teaching practices and the fine grain analysis of how students learn. During this joint ventures with teachers, we witnessed teachers grow as teachers as scholars and teachers as learning scientists. In fact, the time we spent with teachers grappling with the gaps in instructions or student achievement allowed us to spend more time with teachers for professional learning.

A research brief from Stanford’s Center for Opportunity Policy in Education released a research brief entitled, “How high-achieving countries develop great teachers,” by Darling-Hammond, Wei, and Andree (2010) that states that “most [professional] planning is done in collegial settings, in the context of subject matter departments, grade level teams, or the large teacher rooms where teachers’ desks are located to facilitate collective work. In South Korea—much like Japan and Singapore—only about 35% of teachers’ working time is spent teaching pupils. Teachers work in a shared office space during out-of-class time since the students stay in a fixed classroom, while the teachers rotate to teach them different subjects.

The shared office space facilitates sharing of instructional resources and ideas among teachers, which is especially helpful for new teachers (p. 3).” This evolution of teachers who professionally see themselves as learning scientists are given the time and space and the respect for professional expertise to make changes in the system which is what we also learned from the Finnish Lesson (Sahlberg, 2011) where we “increase teacher professionalism and to improve their abilities to solve problems within their school contexts by applying evidence-based solutions, and evaluating the impact of their procedures. Time for joint planning and curriculum development is built into teachers’ work week, with one afternoon each week designated for this work (p. 9).”

Strengthening strategic competence in our teachers requires collective inquiry into our instructional practices within a job-embedded professional learning environment. More schools are establishing Professional Learning Communities (PLC) so that teachers can engage in collective inquiry of their practice. With this infrastructure in place at many of the schools sites, we have been able to engage teachers in Lesson Study and Instructional Rounds to investigate and co-design lessons.

Conducting lesson study within our PLC affords many opportunities for our professionals. First, it provides a supportive collegial network, which shares a common vision and goal. Teacher teams allow opportunities for co-planning and peer observation of practice. Second, it provides opportunities for analyzing student work and collect student data. This provides time for teachers to collectively reflect about common misunderstandings and gaps that students have and think about what instructional strategies may work for their students’ populations.

Finally, the PLC can sustain the lesson study as a way to engaged in structured dialogue about mathematics instruction and student learning. This formalized group affords opportunities for teachers to become leaders within their schools and provide opportunities for school-based coaches to lead their colleagues. Russo (2004) describes school-based coaching in this way, “School-based coaching generally involves experts in a particular subject area or set of teaching strategies working closely with small groups of teachers to improve classroom practice and, ultimately, student achievement. In some cases, coaches work full-time at an individual school or district; in others, they work with a variety of schools throughout the year. Most are former classroom teachers, and some keep part-time classroom duties while they coach (p. 1).”

A promising initiative that has supporting many school-based professional development models have been led by mathematics coaches and mathematics specialists. In our work with professional development, we intentionally recruit teams of teachers from the same school site with their mathematics coach or specialist so that we can be more impactful with the professional learning. Having school members participate together in professional development with the support of a mathematics coach has been instrumental in helping the professional learning sustain even after the summer institute and fall lesson study ends.

Our prior work (Suh & Seshaiyer, 2012, 2014) indicated that the learning curve for teachers is the greatest at the implementation stage following their participation in a PD as they engage in new pedagogical practices. Through multiple iterations, emphasizing the importance of social support in PD, we have refined our model to include coach-facilitated PD as a critical component not previously included. The coach-facilitated PD enhances our existing model to capitalize on the social support that goes beyond collective participation.

Coach-facilitated PD leverages the role of a coach knowledgeable in both content and instructional practice. The PD content promoted algebraic connections aligned to the elementary and middle-school curricula and ensured coherence to the standards of learning. We encouraged collective participation by recruiting school teams and sustaining the PD into the fall through Lesson Study. The goal of lesson study was to provide teachers with continued support in learning and implementing algebraic content, supply materials and strategies to the participants, as well as provide

Coach-facilitated PD and collective participation providing social support for PD

Figure 10.1 Coach-facilitated PD and collective participation providing social support for PD.

Source: Authors.

opportunities for vertical articulation between and among grades levels to share ideas/ resources, analyze student learning and work samples (Suh & Seshaiyer, 2014). The Coach-facilitated PD Model (see Figure 10.1) was a critical piece that ensured the sustainability of the professional learning experiences because the coaches were more aware of specific school/district initiatives and could sustain the professional learning throughout the school year to support teachers’ growth.

Through our research, we have identified several affordances of using the coach- facilitated PD model in the school-based PD design. In one of the survey questions, we asked, How did working in a small school-based team help implement content and new strategies in teaching this lesson? Do you have a teacher or a coach from your own school in this course, if so, how did having a colleague enhance your learning? Through participants’ responses, we found that teachers and specialists had developed different collaborative networks through the coach-facilitated school-based lesson study.

We coded their responses, and there seemed to emerge several collaborative teacher networks that had some distinct connections (see Figure 10.2). The first type of collaborative teacher networks that were evident and obvious was (1) the coach and teacher networks, we had coach facilitation built into the fabric of the PD design. However, we were very pleased to see that there emerged other social networks. The other three were (2) multi-district teacher networks with cross grade “vertical” teacher networks and the same grade “horizontal” teacher networks; (3) school-based PLC teacher networks; and (4) resource specialists networks.

Working with coach from their school or their district, there was immediate trust established in our PD efforts. The coaches working with the university faculty and facilitator showed the collaborative nature of our efforts and teachers saw their coaches endorsing our PD. In addition, we formed lesson study teams as communities of practice where teams of general educators worked with ELL and special needs teachers. In this way, we also saw what we called “collaborative coaching” where different participants shared their different professional expertise.

Development of collaborative teacher networks. Source

Figure 10.2 Development of collaborative teacher networks. Source: Authors.

The special educator and ELL specialists shared their expertise on how the task would need modification to provide equal access for all students and helped generalists anticipate how learners might need extra support to navigate through the problem task. The collaborative ownership in the community of practice allowed teachers to go beyond “your kids” vs. “my kids” and change their belief about who can and cannot do math. With our unique school university partnerships, we have been fortunate to work with multiple districts. Having this multi-district teacher network allowed for knowledge exchange among vertical and grade-level teams of teachers across districts, sharing resources, and strategies.

The work with learning trajectories supported vertical teaming by teachers, for it allowed a “chance for teachers to discuss and plan their instruction based on how student learning progresses. An added strength of a learning trajectories approach is that it emphasizes why each teacher, at each grade level along the way, has a critical role to play in each student’s mathematical development” (Confrey, 2012, p. 3). An exciting “Carry Over Effect” was noticed among school-based PLC networks who were able to sustained their professional learning beyond the scope of our PD initiative/ The effect of the PD extended beyond even what we could hope. In fact, for novice teachers, this collective experience provided the teachers a jump start and a chance to get to know their support network. This teacher commented on how she was able to build a resource base.

Being new to the school and content area, a team helped build a resource base. I started the year knowing people I could go to get help. The coach offers onsite (support) and presents questions to explore other options. Having a team of different teachers provided a variety of strategies. By discussing how they use them, it helped me implement them.

As we conclude this book, we strive to work with school-based teacher teams with school-based mathematics coaches or mathematics specialists through lesson study.

We hope that the teachers and coaches who use this book will be inspired to be change agents and “pay it forward” by inviting three more teachers and coaches to embrace the Modeling Math Ideas approach. Like in the movie “Pay It Forward,” a student, comes up with an idea that he thought could change the world. He decides to do a good deed for three people and then each of the three people would do a good deed for three more people and so on.

Before long, there would be good things happening to billions of people using this model. This movie is a great inspiration for teacher leaders to use this as a “math happening” to impact change in our profession. At stage 1 of the process, we can impact three teachers and coaches. How does the number of teachers we impact grow from stage to stage? How many teachers would be impacted at stage 5? The last challenge that we leave you with in our book is to describe a function that would model the way we can inspire our teachers with the Pay It Forward process at any stage. In this way, we can sustain teachers’ professional learning as they strengthen their strategic competence and have this exponential growth in our professional practice so that we can develop more mathematically proficient students in our classrooms.

Think about it!

What are some creative ways we can “buy” our teachers more time and space for collective inquiry on their teaching practices and enhancing student learning?

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