Collaborative Investigation and the Flipped Classroom

Introduction
My first year of teaching was in 1995 at an innovative, elite private high school in San Francisco named Lick-Wilmerding. The math department at Lick-Wilmerding employed a method of teaching which was unfamiliar to me and to most teachers at the time. Students worked in small groups of 3 or 4, exploring geometric relationships on a shared laptop and reporting their findings in documents written as if they were lab reports. Studying algebra, students used graphing calculators to explore functions dynamically and again, report their findings in written form accompanied by detailed examples and analysis. Students would receive direct instruction when necessary, but for the most part questions were explored and answered by members of the groups themselves. I adapted hesitantly to this new structure, as I myself had been taught by teachers and professors using lecture combined with repetitive homework. Over my subsequent years of teaching in the states I explored a range of methods tailored to the various cultures of the schools at which I taught. This included straight lecture and homework, a mixture of lecture and independent class work, and collaborative group work supported by a traditional textbook. Finally, now at the American School of Valencia, I’ve developed a suite of collaborative investigations which allow my ninth and tenth grade math students to explore mathematical concepts much as they explore scientific concepts in laboratories. This is supported by online resources which include videos and extra practice problems.


The Philosophy
The American School of Valencia is a college preparatory school whose recently updated mission is to “prepare globally-conscious life-long learners in an innovative, collaborative and caring environment.”  This mission is deeply connected to the International Baccalaureate program, the preeminent curriculum of international schools around the world. The IB curriculum revolves around the learner profile, which features 10 characteristics which we aspire to develop in our students. For example, we hope to develop students who are inquiring, drawing upon their own innate curiosity to explore the world around them and develop and delight in an understanding which persists beyond high school. We aspire to help them become communicators, who work readily and effectively with others to enrich their own understanding and are capable of verbalizing and writing about their ideas. We hope they become deep thinkers who can apply technical skills and conceptual understanding to complex new problems. We strive to help them take intellectual risks, to share novel ideas and pursue creative lines of thinking, and to respond to failure with resilience.

The Math Intuition
Research shows that not only do we have an innate sense of mathematical relationships, but that sense is rather more complex than one might imagine. Indeed, as an example, research suggests that a child’s numeracy is likely logarithmic rather than linear, which means they compare the relative numbers of things by thinking exponentially. More broadly, a young child recognizes that a toy car is the “same thing” as a real car, and his ability to model things demonstrates more than just a sense of proportionality, but also a capacity to represent the real world symbolically. A graph is precisely such a symbolic representation if it is a graph of profits increasing over time, or the curvature of a rocket’s path, or the pace of a heart beat as a runner increases her speed. It is incumbent upon us to harness the natural capacity of students to understand the world through intuitively sensible numerical relationships and modelling.

Memorizing Procedures vs. Understanding
As things stand now, a great many students learn mathematical procedures through nearly mindless repetition without any connection to real world. A precious few of these nevertheless find themselves mastering the discipline and exploiting its full potential in a career requiring it. But a sad majority of students learn mathematical procedures with little sense of the greater representational capacity of these tools. They rightly imagine that mastering mechanics with no appeal to their innate numeracy or to any connection to the real world is pointless unless they are destined for a career that demands these esoteric tools. They may indeed learn these procedures well in the moment and perform adequately on tests that do not demand more than a procedural proficiency, but they are unlikely to carry this learning forward and apply it meaningfully to the world. An oft remarked exemplar of this phenomenon is that a great many adults can remember the quadratic formula but cannot imagine what it is for, much less what a quadratic function might model. And yet daily we all observe and likely comment on changes that are quadratic in nature. (An accelerating car, for example, is modelled well by a quadratic.) We are teaching students who later become adults who lack the confidence and capacity to use the tools they spent 15 years learning. This should come as no surprise--imagine if we taught English as if grammar were the end in itself. Clearly, learning mathematical procedures divorced from conceptual connection to things that change in the real world is not developing inquiring, deep-thinking risk takers who can solve novel problems. Instead, we’ve taught students how to play the notes of Mozart without ever recognizing any melody, and certainly without endowing them with the confidence or capacity to improvise. The failure of this procedural obsession has prompted a wholesale revision of math curriculum in the United States, with a deeper conceptual focus wherein students use math to model real world scenarios.

The Silent Classroom
The American School of Valencia, the IB Program, and the recently adopted Common Core Standards in the United States are also unified around the notion that students need to be able to articulate their understanding to others and take intellectual risks when presented with new problems. But a traditional lecture classroom provides limited opportunities for students to articulate their ideas by virtue of the simple fact that the teacher does most of the talking. Further, there is no intellectual risk involved in listening, and frankly no risk at all in not listening if the ultimate assessment only requires regurgitation of procedures which can be memorized from the textbook. In this environment, students are not engaged unless they are entertained, and even then the opportunities for each student to inquire deeply and creatively are limited to questions here and there. While an earlier generation celebrated children who were seen but not heard, this supposed virtue has little value in our modern world. Rather, a survey of business CEO’s by the Association of American Colleges and Universities shows that  the “ability to work well with others” is the most important skill a college graduate needs to succeed in her career. Also on the list is the ability to read and write well, and the ability to be innovative and solve problems creatively. Other surveys show similar results. The message is clear: we should discourage silence, and instead coach students to produce, share and refine creative ideas.

Collaborative Investigations
The alternative to lecturing about procedures is an extension of the tools I first encountered at Lick-Wilmerding High School in San Francisco. In my class, students work in small groups of three or four to explore mathematical concepts as applied to real world problems. They work on the investigations I’ve designed by talking about the scenarios described therein, and by drawing diagrams and writing equations to attempt to model the given scenarios. Investigations are designed to guide students with leading questions and a logical flow which directs their work to a learning objective. At the same time the investigations are replete with open-ended questions which require students to discuss and decide how best to represent the given situation. They then use their models to solve problems. They must justify their models and their solutions in writing. They teach each other, at a pace determined by their group. Sometimes they will solicit help from another group, or from me. I walk around the class observing and providing hints and guiding questions as needed. My role is that of a consultant, a counselor, a tutor, a manager and a resource, but rarely a performer. Students know that I am looking for determined collaboration and creative problem solving, and that I have created scenarios and problems which admit a variety of solution methods. Indeed, in an appeal to the diversity of skills the students bring to the group, the questions are designed to be answered multiple ways, so that everyone both plays to their strengths and must address their weaknesses. My students also know that I expect, indeed, require that they make mistakes and then correct them, and that their ability to persist is more important to me on a daily basis than their arrival at a correct solution. I assure them that correct solutions can be arrived at by many methods, and that they will arrive eventually through effort, and this has proven to be the case.

The Structure of the Class
Students work in small groups for about two weeks before the group members are shuffled. Each week, they complete several investigations, and at the end of the week they correct any errors in their investigations with the answers I post at the school website. There is no external text, only the hundreds of investigations I’ve authored. Students take individual and group quizzes with considerable frequency. For group assessments, all students must record the same answers, but ultimately I simply grade one student’s assessment at random and everyone in the group receives that grade. Frequent group assessments both reduce the pressure and stress of infrequent high stakes assessments, and allow me to give very demanding assessments--students are able to accomplish more sophisticated modelling and problem solving when working in groups. They also receive frequent individual assessments, which are worth more than the group assessments. Research invariably shows the value of frequent assessment in learning, which is consistent with the idea that students need opportunities to demonstrate their understanding verbally and in writing. A test or a quiz can be a great formative learning tool, particularly if students must correct their quizzes or tests afterwards with answers posted on the school’s website, as indeed they must do.

My classroom environment is one of vigorous collaboration. My teaching is adaptable, improvisational, and ranges from observation of interpersonal interaction to occasional mini-lectures responding to a widespread question, to one-on-one tutoring to help a demoralized student. Students sometimes argue extensively about how best to approach a problem before finally resolving their differences. Sometimes they lose focus and I must guide them back toward working. Sometimes a group has a substantial insight which leads them to solve a problem using a more complex and time consuming method. Sometimes a group smoothly proceeds ahead and needs an extra challenge investigation that the other class members aren’t yet ready for. The fiction of linear, regularly paced ingestion of data isn’t maintained in my class, rather, students learn at a pace guided by the direction of their inquiry. Of course, the particular conclusion--that the slope of a particular line is 3, for example--is not the point. The point is for the student to continue to try to collaboratively and creatively work toward a solution to a novel problem. From my perspective as a teacher, the errors, deviations and arguments, and the group’s reactions to them, are an inherent and integral part of this process--the students’ strengths and weaknesses are quickly revealed to each other and to me, and we work together to move forward.

Flipping the Classroom
Collaborative investigation is not easy. Students are frequently working at the edge of unknowns, and this can be, indeed should be, slightly unsettling for them. To anchor their learning, sometimes students want to review what they’ve learned at home. To serve this purpose I’ve provided instructional videos and practice problems posted at the school’s website. These videos, usually those produced by education innovator Salman Khan of Khan Academy, provide short lectures that students can pause and replay as desired, whenever they are prepared to focus on the lecture. Those who don’t need the entire lectures are free to use them to whatever degree needed. (Feedback on their needs emerges quickly during daily work or frequent quizzes.) The videos allow students to use the classroom entirely as a laboratory of learning fueled by teachers and students working on mathematics. This “flipped classroom” model is precisely the reverse of the traditional model where lectures take place at school and a modicum of repetitive homework done at home. Instead, students direct their learning in school and decide individually what they may need to do at home to be ready for the next quiz.

Recent Results
In my observations, over the course of one year at the American School of Valencia, my students performed well on two standardized tests which ASV uses for tracking student performance--the International School Assessment and Pre Scholastic Aptitude Test. In the case of my ninth grade class, they also proceeded further through the curriculum than was expected. Furthermore, students displayed increasing confidence and facility working in groups and attempting challenging problems, both in a group and an individual setting. Finally, it was clear that they had improved their understanding of deeply complex mathematical concepts and how they could be applied to the real world. This is consistent with considerable research showing that collaborative investigation improves learning.

It is also clear to me that students simply enjoy working in their groups. In surveys I conducted last year, students put investigations and group assessments at the top of their list of favorite elements of the class. I am pleased to see students connecting socially and intellectually with each other in my class. Research also shows that social skills developed during school are among the most important factors in future success, and I am glad to provide them an opportunity to be social and productive members of the school community. For my students, collaborative investigation has been an intellectually enriching, challenging and fun experience, and I think that applies to me as well. Regardless of my small contribution at the American School of Valencia, the collaborative, investigative model supported by flipped classroom tools, promoted through a variety of educational clearinghouses like IB and Common Core and increasingly verified by research, will soon become more prominent in Western education.