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A Vertical Approach to Math Instruction

Establishing vertical mathematics teams creates professional development opportunities for teachers while also improving student achievement.
By Linda Gojak
Principal, November/December 2012
Web Resources

In the current era of mathematics stan­dards, whether they are Common Core State Standards or other state standards, effective vertical mathematics teams offer an opportunity for teachers to grow professionally through shared experiences, for leadership to grow among the faculty, and for the school to change its perspective on the teaching and learning of mathematics. At their core, vertical mathematics teams regular­ly bring groups of teachers across grade levels together to discuss content, peda­gogy, and practice in a structured and supportive environment.

Traditionally, undergraduate preservice programs pre­pare future elementary teachers to teach multiple subject areas, often focusing on reading. Vertical mathematics teams can deepen teachers’ content knowledge in the subject, enabling them to hone in on effective teaching practices that affect student learning and align content across grade levels so students receive rich and coherent mathematics instruction. In addition, effective vertical teams provide a structured professional development expe­rience that encompasses the intersection of analyzing and understanding curriculum and content with the practice of teaching focused on student thinking.

Key Strategies

The following are strategies that contribute to successful vertical mathematics teams. The primary goal is for teachers’ mathematical understandings to lead to student growth.

Vertical teams should span three to four grade levels. This provides teachers the opportunity to look at the students’ mathematics experiences across grade levels by examining how topics develop from one grade to the next without making the span so large that the focus of the work gets lost. If there are mathemat­ics intervention teachers or other professionals engaged in teaching the subject at your school, include them in the teams as well. Within a vertical team you might also want to organize grade-level groups that can use the knowledge from team meetings to concentrate on the mathematics con­tent at a particular grade.

The development of mathemat­ics across grades needs to involve more than just building skill. As teachers work together in teams, they should expand their knowledge from being solely skill oriented to building a foundation of conceptual understanding of key mathematics topics. This not only leads to a deeper understanding of the mathematics concepts, but also supports the devel­opment of mathematical skills. Teams begin by looking at how conceptual understanding of a mathematical topic builds across grades, and then teams determining the specific skills that develop from that understand­ing. Following this analysis, the verti­cal team considers both rich mathe­matical tasks that help students build deeper understanding, and support­ing strategies for engaging students in focused skill development.

Build support and leadership with­in your teams. Although it is impor­tant that principals be supportive and provide leadership for teams, it is also beneficial for principals to capitalize on and further develop mathemat­ics leadership among teachers. This may involve a mathematics coach or specialist, but it also should include key classroom teachers and interven­tion specialists. Consider having a teacher from each grade level form the mathematics leadership team that takes responsibility for facilitating the vertical team.

Principals should meet with the leadership team early in the year, but the responsibility for convening vertical team meetings belongs to the mathematics leadership team. Princi­pals should make it a point to attend at least part of each meeting.

Provide time to be a team. For teams to function effectively, they need time to meet and to work dur­ing the school day. Given that time is one of the scarcest commodities in a school, this takes a serious com­mitment. In one scenario a vertical mathematics team would meet once a month for 90 minutes and grade-level teams would meet on a weekly basis during common planning time. Vertical team meeting time could be arranged through a combination of early-release days and in-school activi­ties monitored by faculty not involved in mathematics instruction and assist­ed by school staff.

Develop a mathematics focus. Together, the team determines the topics to explore for the school year. The leadership team then plans the agenda for each vertical team meet­ing and finds materials to support the work of the teachers, including journal articles with research on the topic, professional books, manipulatives, rich tasks, games, and other resources.

Work from expectations to oppor­tunities to implementation. Start by examining state standards to deter­mine expectations for a topic across multiple grades levels. This includes looking at grade levels before and after the span of the vertical team to determine where students are coming from and where they are going with that topic. Analyze both the concep­tual understandings and associated skills that are part of those expecta­tions. This is an opportunity for the collective team to better understand the entire progression of the topic, including concepts and skills, connec­tions between grade levels, and each teacher’s role in supporting student growth. Once essential understand­ings have been identified, the group develops strategies to implement and artifacts to collect relative to their work on that topic.

Simply put, successful vertical math­ematics teams:

  • Meet regularly (preferably within the school day);
  • Consider themselves a learning community;
  • Establish mutual trust and respect among members of the team;
  • Set common goals;
  • Focus on student learning;
  • Have a set agenda;
  • Focus on a topic over time; and
  • Have access to professional resourc­es that inform their work.

The following describes how this might play out for a team in an ele­mentary school.

Sample Team Dynamic

The vertical team is comprised of classroom teachers in grades 3-5 and the mathematics intervention teacher. The first meeting will explore how to help students with basic facts, concentrating on strategies that are currently used to teach basic facts and the expectations around achievement at each grade level. A teacher on the leadership team agrees to facilitate the meeting. This meeting will focus on addition and subtraction facts, because the teachers are concerned that many of their students are still counting on their fingers. Since the operations and facts of addition and subtraction have been introduced at earlier grade levels, this work will not require teaching new concepts.

Two days before the meeting, all teachers on the vertical team receive the agenda, which includes a list of materials to bring to the meeting. The meeting begins by setting goals for that day, after which teachers begin by looking at their state stan­dards to identify those standards related to addition and subtraction basic facts at their grade levels, in preceding grades, and subsequent grades. They learn that fluency with addition and subtraction facts is an expectation by the end of grade 3, which helps them to see the urgency to develop this topic in greater depth. The conversation moves to building agreement on what is considered “flu­ent” at each grade level. After some discussion and consulting several resources, they agree that to be con­sidered fluent, a student should be able to recall an answer within three seconds without counting.

In addition to talking about how they can consistently build skill with these facts, they turn the conversa­tion to how to assess student prog­ress. All agree that tests in which students are asked to recall 50 to 100 facts in a given amount of time is not a good assessment of this skill, and decide they will need to spend time developing strategies for assessing student facility. The leadership team agrees to find articles that will help to inform this work and share them with the rest of the team before the next meeting. In the meantime, an online discussion group is estab­lished so that teachers can reflect on their practice, continue conversations between meetings, and share their observations about student thinking around basic facts.

At subsequent meetings, team mem­bers will read articles about helping students develop strategies leading to fluency with addition and subtraction facts, share games and materials, dis­cuss enlisting parent help, and build models for formative assessments. Teachers continue to discuss this at their grade-level meetings using the articles that have been shared. The timing allows teachers to take what they have learned in the vertical team meetings and adapt it to the specific needs of their students.

By the fourth vertical team meet­ing, the teachers are ready to move on to multiplication and division facts. This does not mean that they have waited until November to work on these facts with their students. Rather, it is the time when they are ready to discuss instruction around this topic as a group, sharing success­es and barriers they have faced with their students.

The method continues until the end of the school year, at which time the team meets to talk about success­es with their work and areas that need further attention. They are satisfied with the progress students have made with mathematics facts. They agree that using similar strategies, expecta­tions, and assessments connects and reinforces learning from year to year. Additionally, this work has resulted in a plan for effective instruction—including student-centered lessons built around rich tasks that support conceptual understanding of the operations and build fluency with facts—and has a specific flow or progression from one grade level to the next. In September, they will be ready to implement this beginning and the team will be ready to take on a new topic.

At the end of the year, the third-grade teachers meet with the second-grade teachers from the “feeder school” to share strategies and resources from their work. Similarly, the fifth-grade teachers will meet with the sixth-grade teachers to ensure con­nections between the elementary and middle school instructional practices.

A Cohesive Math Program

While the work of vertical teams can follow different structures, what is important is that it provide teachers with the professional learning oppor­tunity to build a cohesive mathemat­ics program across grade levels that focuses on students’ development of mathematical understanding. They also increase teachers’ pedagogical content knowledge.

This job-embedded work opens the door to more professional opportu­nities and the shared experience of working together to achieve a com­mon goal: cohesive and successful mathematics learning experiences for students.

Linda Gojak is president of the National Council of Teachers of Mathematics.


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Gojack_ND12.pdf1.6 MB