When mathematics students and teachers are able to deepen their relationships with the curriculum and with each other, they are more likely to teach and learn in ways that promote sustained, connected, meaningful understanding. Stories of teachers’ engagement with their own curriculum through mathematics discourse, students’ connection with teachers through personalized pedagogy, and students’ commitment to the curriculum through personally meaningful research and opportunities for revision illustrate some ways that Coalition and other like-minded schools are making math personal.

Teachers’ Relationships with the Curriculum and with Each Other

Responding to unsatisfactory 2001 district-wide math scores, Shelley Schneider, Assistant Superintendent of Curriculum and Instruction for the Millville, New Jersey schools, turned to the Center for Effective School Practices (CESP), New Jersey’s CES regional center. Schneider focused her attention on strategically improving teaching, knowing that ultimately improved test scores can be a side effect of instruction that is flexible, mathematically rich, and student-centered. “What we really want is to see our students thinking at a higher level,” said superintendent Schneider. “We want our students to be productive in their world. Math is all around us. The best gift we can give our students is to feel comfortable and competent.”

CESP mathematics coach Shelly Berman responded with an offer to observe teaching and design professional development that would create more meaningful learning. At Millville’s Lakeside Middle School, Berman’s coaching centered on improving classroom discourse about math. “Discourse is what you do as a teacher to help the kids make mathematical meaning out of tasks,” explains Berman. “If kids are rolling a die 1,000 times, it doesn’t mean much, but if you put in the right prompts and ask the right questions, they begin to find sense and significance.” Discourse in math classrooms makes thinking and meaning public, allowing students to connect mathematical language with their own vernacular. Mathematics discourse also allows teachers opportunities for multiple informal assessments to check students’ understanding. Rich mathematics discourse in classrooms depends on teachers’ flexibility with the material; different students will learn at different rates and connect with the big ideas within the curriculum in various ways, and teachers need considerable aptitude and skills to create context in such constructivist environments.

Now in his second year of modeling mathematical discourse and focusing on supporting techniques, Berman says “I’m seeing Lakeside teachers asking more questions, resisting the habit of giving away answers for free. They’re using prompts, practicing wait time, doing groupwork, and helping kids understand that though it feels fun, math time really is not social time. Most importantly, they are modeling thinking; they’re making thinking visible.” Lakeside’s example demonstrates that CES pedagogy of teacher as coach-student as worker characterizes effective classroom discourse.

NCTM Standards for Teaching Mathematics: Discourse

In 1991, the National Council of Teachers of Mathematics released the Professional Standars for Teaching Mathematics, which pushed the profession to develop a more constructivist and student-centered approach. Two of the six Professional Standards center on discourse.

Standard 2: The Teachers’ Role in Discourse
The teacher of mathematics should orchestrate discourse by-

• posing questions and tasks that elicit, engage, and challenge each student’s thinking;
• listening carefully to students’ideas;
• asking students to clarify and justify their ideas orally and in writing;
• deciding what to pursue in depth from among the ideas that students bring up in a discussion;
• deciding when and how to attach mathematical notation and language to students’ ideas;
• deciding when to provide information, when to clarify an issue, when to model, when to lead, and when to let a studen struggle with a difficulty;
• monitoring students’ participation in discussions and deciding when and how to encourage each student to participate.

Standard 3: The Students’ Role in Discourse
the teacher of mathematics should promote classroom discourse in which students-

• listen to, respond to, and question the teacher and one another;
• use a variety of tools to reason, make connections, solve problems, and communicate;
• initiate problems and questions;
• make conjectures and present solutions;
• explore examples and counterexamples to investigate a conjecture;
• try to convince themselves and one another of the validity of particular representations, solutions, cojectures, and answers;
• rely on mathematical evidence and argument to determine validity.

See http://standards.nctm.org for links to these standards along with supporting material, examples, vignettes, and background information.

As teachers begin to engage with mathematics in deeper and more complex ways, they also may confront more directly their own attitudes about mathematics. Genuine ardor is contagious: in many classrooms, students benefit greatly from teachers’ love of math. But sometimes, teachers struggling with their own anxieties may inadvertently transmit ingrained dislike or fear of math. John Belcher, a program associate and math coach at the Center for Collaborative Education’s Systemic Initiative in Mathematics and Science Education, focuses his work with teachers on helping them identify where their own weaknesses limit students. “Your own familiarity with the content places a ceiling on where students can go,” Belcher observes. “Your understanding of the content and what it connects to – horizontally within math and vertically across the disciplines – is an active, ongoing, dynamic process.”

Math coaches like Belcher or CESP’s Shelly Berman can help teachers increase their own mastery and comfort within the range of mathematics that they teach. Additionally, coaches can help teachers refine their teaching techniques as they adopt curriculum frameworks that move from projects and exploration to specific skills. The time-honored CES tradition of teacher collaboration also supports teachers as they transform and refine their practices.

Along with coaches, teachers supporting each other are their own best allies in adapting and refining their teaching to maximize discourse and create curriculum and conditions for project-based, constructivist math learning. In The Teaching Gap: Best Ideas from the World’s Teachers for Improving Education in the Classroom, James W. Stigler and James Hiebert analyze data from the 1999 Trends in International Mathematics and Science Study and other research efforts, concluding that mathematics and science teaching benefits from methods that turn practitioner knowledge (what teachers know works in their classrooms) into professional knowledge (the public sharing and documentation of collected experience and wisdom). Collaborative structures such as critical friends groups, teacher research focused on essential questions, and lesson study – a collaborative curriculum refinement process on which Stigler and Hiebert focus – all serve to create sustained professional knowledge for improved mathematics teaching and learning. (For more on the concept and practice of lesson study, see the Resources section.)

Students’ Relationships with Teachers

Erin Levine still can’t believe that she succeeded in calculus during her senior year at Sammamish, Washington’s Eastlake High School. While her mathematics mastery is due to a number of factors, especially her own hard work, Levine attributes her accomplishment to her relationship with Jane Hunter, her math teacher for her three years at Eastlake. Levine was part of Eastlake’s Extended Core Program (ECP), a school-within-a-school of 120 students in grades 10-12. Each grade level of forty attends classes in small groups, studying an integrated curriculum with a team of teachers. The teaching team often loops with the students for the three years of ECP, creating the conditions for teachers and learners to know each other exceptionally well. After her sophomore year, Hunter encouraged Levine to accelerate her mathematics studies. Levine, quick to say, “I’m not a math person,” was reluctant but accepted the challenge, succeeding with more challenging math curriculum. “Ms. Hunter knew I could do it,” Levine recalls.

Jane Hunter also believes that her students’ persistence and skills result in great measure from personal bonds. “Being able to teach these kids for three years is the most incredible thing I’ve done in my life. Kids stick it out because they love me and love being with each other. And they learned a lot of math along the way. Many of them, like Erin, did math senior year though they didn’t have to.” Certainly, Hunter’s warm personality and the small-scale design of Eastlake’s ECP program created foundations for the kind of personalized approach that fostered students’ desire to stick to it and do well, making it possible for them to leave high school with the math skills that predicate success in future academic and professional endeavors.

Hunter and other Eastlake ECP teachers are finding other benefits in their transformed curriculum, which features intensive cross-subject integration and an experiential, project-based approach. “I have felt for a long time that if there isn’t a personal connection or buy in, teaching discrete skills doesn’t mean much,” says Hunter. “Kids have heard us say far too often, ‘You’ll see the application for this someday.’ With us, all of the skills and concepts come from the labs; the students have more connection and more buy-in.” Reflecting on her years of mathematics teaching, Hunter sees the project-based approach as a profound shift, noting that she – and most math teachers – based their curriculum around teaching explicit skills, hoping to pull in applications to boost relevance when possible. Now, Hunter says, she and other Eastlake ECP teachers teach from hands-on experience, meaningfully contextualizing the skills and concepts outlined by district, state, and National Council of Teachers of Mathematics (NCTM) standards.

Students’ Relationship with the Curriculum: Projects, Interdisciplinary Connections, Politics, and Revision

At Wildwood Secondary School, a 326-student independent middle and high school in Los Angeles, teacher Mike Conway is also feeling the effects of a forceful shift in his mathematics pedagogy as the Wildwood mathematics faculty creates a project-centered curriculum. Conway and his colleagues are in a new school – Wildwood’s secondary campus opened in 2000 and will graduate its first senior class in 2004 – and have the opportunity to craft a new vision of mathematics teaching and learning. “The biggest shift from my prior teaching experience is that I have the chance to step away from the numbers and equations and look instead at understanding,” reflects Conway. “Before, math was all about delivering content, and now it’s about creating habits to enable students to understand concepts.”

Wildwood’s ninth grade math students focus on being organized and clearly showing work. These habits establish the basis for a mastery of mathematical language and make it possible to work effectively in groups. Wildwood’s mathematics pedagogy is organized around groupwork, and this collaborative emphasis dissolves students’ self-perceptions as “good” or “bad” at math. When a number of students work together on a project, they experience a wider range of opportunities for accomplishment. “As a teacher, I help them see their strengths,” says Conway. “If you are a social and vocal person, then you will help other students get the wording just right. Maybe that verbal person won’t be the first one to discover the concept, but the verbal person is also learning the math. That’s small school stuff; in order to do this, you have to know your students really well.”

>Examples of Hands-On Math Labs at Eastlake High School’s Extended Core Program

Jane Hunter and her colleagues use labs, with focusing essential questions, to arrange experiences that create meaningful mathematics learning. Here a series of labs designed to be run multiple times by teams of students in order to produce data sets:

Barbie Bungee Jumping Lab (Linear Model). Essential Question: How many rubber bands will it take for Barbie to have the best (and safest) jump of her life off the balcony? It’s what you imigine: Barbie dolls diving and ascending.

Lid Todding Lab (Quadratic Model). Essential Question: What size lid will not hit any edges of the one-foot-square tiles 40% of the time? Students toss several round lids of varied diameters onto a square-tiled classroom floor.

Volume Lab (Cubic Model) Essential Question: Is there a relationship between the height of a pyramid and the volume of water it can hold? Chemistry-lab beakers represent pyramids in this series of trials.

In addition to creating a project-based curriculum, Wildwood’s mathematics faculty is also working on creating interdisciplinary connections. The World War I Probability Project asks students reading Erich Maria Remarque’s All Quiet on the Western Front to design a simulation that, through 200 trials, explores the probability that an individual at the front would be killed, wounded, taken prisoner/go missing, or escape harm. Students work in pairs, designing the simulation, performing trials and recording results in a frequency table and a histogram, and presenting the results visually and orally. As Wildwood’s project-based approach evolves, Conway hopes to build some additional “real world” community-based data collection and analysis projects at all levels of Wildwood’s secondary curriculum.

For teachers like Conway seeking to move their project-based curricula beyond traditional, “safe” areas of study, Eric Gutstein, part-time middle school math teacher and Associate Professor of Math Education at the University of Illinois-Chicago, suggests that useful projects come from students’ lives and experiences. At their primarily Latino, urban middle school, Gutstein and his students did a project entitled “Mortgage Loans – Is Racism a Factor?” in which they analyzed mortgage rejection rates for African American, Latino/a, and white applicants. Explaining the motivation for the project, Gutstein notes, “These kids had experiences of home desiring; the issue of housing and home ownership is central in people’s lives.” Gutstein affirms and embraces the overtly political nature of the project. “If we’d done it more blandly, if we’d studied other, less controversial statistics, it still would have been rich mathematics. But then students wouldn’t realize that math is a tool that can analyze political realities, and they don’t become prepared to use math to think about deeper questions.”

Eastlake High School’s Jane Hunter suggests that allowing opportunities for revision is another way for students to connect more meaningfully to the curriculum. Eastlake has a policy of allowing students to retake math tests – a one-time option that requires teachers to create two versions of each test. Students must be completely current with class project work and homework, and they must accept the second test’s result. Retesting and rescoring is a lot of work, acknowledges Hunter, especially with the essay-laden assessments that she uses. But the retake policy, she says, “allows students the ability to learn from mistakes. Kids come back from college and say that they really understand because they were able to more deeply engage with the content and skills. I can see that it really makes a difference cognitively.”

Mike Conway agrees with the revision approach, especially in classrooms where students know themselves and their learning styles well. Wildwood’s curriculum leaves room for re-entry and refocusing as projects evolve. “When there’s no opportunity for revision, you either get it or you don’t. You’re not given an opportunity to build understanding.” Conway continues, “Revising makes students stop saying, ‘I’m no good; I can’t do this’ and helps them starting saying, ‘How am I going to work on this? Try harder? Collaborate? Use diagrams because I am a visual person? Build a model because I am more kinesthetic?’ Allowing revision gives opportunities for success. It turns things around in a math class right away.”

Personalized Math Depends on Teacher Autonomy

Coalition schools trust teachers to create learning environments that encourage investigation, discovery, discourse, and revision while balancing the demands of standards-based curricula. Educators who deeply know both their students and the curriculum are able to capitalize on teachable moments, merging discovery with the progression of skills and concepts that they aim to cover.

Teacher autonomy also creates the conditions for the development of structured project-based curricula, which compel students to learn actively, increasing their chance to engage in meaningful and enduring ways. When teachers have the time, structures, and support to work collaboratively as professionals, refining the curriculum and their pedagogy, they are able to transform personal knowledge into their school’s cache of professional expertise about how to teach for understanding.

Related Resources

For an in-depth analysis of discourse at work in mathematics classrooms, along with video illustrations, see “Encouraging Mathematical Thinking: Discourse arouna a Rich Problem” by the Math Forum’s Bridging Research and Practice Group at www.mathforum.org/brap/wrap/index.html

Horace 8.3, “Math and Science in the Essential School” portrays the benefits and the obstacles inherent in integrating math and science, includes examples of several successfully integrated math and science programs. www.essentialschools.org/cs/resources/view/ces_res/165

In “Home Buying While Black or Brown,” (Rethinking Schools, Volume 18 No. 1 – Fall 2003) Eric Gutstein documents his students’ mortgage rate-racism project. www.rethinkingschools.org/archive/18_01/brow181.shtml

Horace 15.2, “Math Programs” reviews a range of math resources that support critical thinking and deep mathematical understanding. www.essentialschools.org/cs/resources/view/ces_res/91#2