Insights Into Algebra 1: Teaching for Learning
Quadratic Functions Teaching Strategies: Developing a Community of Learners
In a collaborative classroom, students are willing to share their opinions. They are enthusiastic about discussing mathematics in the interest of learning. They feel at ease agreeing and disagreeing with one another. All teachers aspire to have a classroom in which students collectively take responsibility for their own learning, but it’s far from common. With careful planning and conscientiously selected activities, however, a teacher can create an atmosphere that cultivates a community of learners.
|Listen to audio clip of teacher
Transcript from Tremain Nelson
I feel that it’s very important that in the classroom we create a sense of community. I believe that there are two things going on: the teacher has a responsibility to teach, of course, but the student has a responsibility to learn. I think it’s very important that we understand our roles and that we understand our roles well. The teacher is a learner, and sometimes the learner needs to be a teacher, and so in the classroom we need to create a community where the teacher is not afraid to not know some things, and the learner is not afraid to take a moment and teach somebody else some things. And so what we do is we create a common goal and that common goal in the classroom is to learn as much about mathematics and to have as much fun learning about it that we possibly can. In the classroom, we create an environment so that we are constantly talking about the math, we’re discussing the math, but we’re also discussing the relevancy of the math, how important it is, and how it relates to their everyday life.
Tony Piccolino, the on-camera commentator for the Workshop 4 video, offers thoughts about how Tremain’s classroom embraces four key features of a community of learners.
Transcript from Tony Piccolino
A major goal of Tremain’s lesson is to develop what we call today “a community of learners.” In a community of learners, you have a standards-based classroom that embraces basically four features. One, the emphasis is on having students value and respect other students’ ideas, keeping in mind that other students’ ideas may potentially contribute to their solving a problem. Secondly, students have autonomy for choosing and sharing the method that they use to solve a problem. Third, students learn to appreciate that mistakes have a value, that students learn from mistakes, and that they can be sources of opportunities to build and construct learning. And fourth, students recognize that the authority to determine whether something is correct or sensible comes not from the status of a teacher or the popularity of a student but from the structure and the logic of the subject matter that they are studying. So that’s essentially what a community of learners is, and Tremain very effectively tries to promote this in his classes by engaging students in the learning and the presentations and assessing one another.
Educator Spencer Kagan offers a simple yet profound argument as to why he believes that cooperative learning is effective. To paraphrase his philosophy, more thinking and learning occurs when nine people are discussing mathematics together (as when students are working in cooperative groups) than when a single person (the teacher) attempts to distribute knowledge to 35 others (the students).
A number of teachers employ cooperative learning strategies in their classrooms because active students appear to learn more than students passively absorbing a lecture. There is a growing body of research to support this idea. In an article entitled “Collaborative Learning Enhances Critical Thinking,” education scholar Anuradha A. Gokhale writes:
According to Johnson and Johnson (1986), there is persuasive evidence that cooperative teams achieve at higher levels of thought and retain information longer than students who work quietly as individuals. The shared learning gives students an opportunity to engage in discussion, take responsibility for their own learning, and thus become critical thinkers.
Read transcript from teacher Tremain Nelson
The focus with cooperative learning is taken off of the teacher and placed on the individual that is truly responsible for the learning, which is the student.
In the essay “What Is the Collaborative Classroom?” presented by the North Central Regional Educational Laboratory, the authors identify four general characteristics of a collaborative classroom:
- Shared knowledge among teachers and students
In a truly collaborative classroom, kids contribute as much as teachers. This, however, represents a significant paradigm shift from the classroom in which the teacher is viewed as an information giver. To foster a cooperative environment, mathematics educators must recognize that students have as much to offer, and sometimes more, than the teacher does.
- Shared authority among teachers and students
The responsibility for what happens in the classroom belongs to students as much as the teacher. Kids help to manage the classroom, decide what types of projects are done, determine how things are graded, and, in some cases, assess their own and one another’s work.
- Teacher as mediator
As necessary, the teacher provides guidance to students and offers as much help as the student needs, but not more than they need. The responsibility for learning falls to students. As a mediator, the teacher directs the activities of the classroom so that students discover mathematics; this is a shift from the lecture type classroom in which the teacher imparts information and informs students of what and how they are expected to learn.
- Heterogeneous groupings of students
If educators believe that students have something to contribute, they group students deliberately so that they are able to learn from one another. Each student has a unique perspective and offers special talents to a group; teachers should carefully group students who can help each other. In addition, students working in heterogeneous groups improve their social skills by interacting with peers other than their friends.
Tremain’s classroom is truly collaborative. Students discuss and disagree with one another about mathematics, and as a result, everyone learns more. Through conscientious planning, Tremain encourages this type of collaboration.
At the beginning of the year, Tremain requires students to discuss problems in two-person teams. Frank Lyman of the University of Maryland developed the two-person construct that is often used in classrooms, known as “Think-Pair-Share.” This low risk structure works well for teachers who are just beginning to incorporate collaborative learning into their classrooms.
In Think-Pair-Share, the teacher poses a problem to the class and allows students to think about the problem for a minute (“think”). Allowing students some time to think individually gives them a chance to struggle with the problem and begin to form their own answers. Then, students discuss the problem with a partner (“pair”). During discussions with their partner, students construct their own knowledge, question their understandings, and find out what they know and don’t know. These discussions make Think-Pair-Share an effective technique – students engage in an active learning process that does not occur during expository teaching. When students have had enough time to discuss the problem with their partner, the teacher should convene the class for a discussion (“share”). Because students share responsibility for incorrect responses with a partner, they are more willing to offer comments during the classroom discussion. In addition, student responses are often lucid and concise because students have had an opportunity to “flesh out” their thinking. Students gain skill in communicating orally, learning how to socialize with a partner and becoming comfortable sharing their ideas with the entire class.
One way to vary the Think-Pair-Share structure is to replace the discussion at the end with other activities. The students could write their thoughts in journals, which the teacher could review to assess the level of understanding and possible misconceptions. Another possibility is to skip the discussion altogether and instead pose a slightly more difficult problem that builds on the first problem; then, perhaps, pose a third problem that builds on the second problem. Allowing students to build their knowledge in this way creates a think-pair-think-pair-think-pair-share structure.
“Trading Papers” is another two-person strategy developed by Williams College math professor Edward Burger, the 2001 Tepper Award winner for distinguished teaching. Although his students are college undergraduates, his idea works well for middle and high school classes. Dr. Burger’s strategy requires the teacher to pose a multi-step problem to the class. Problems that can be solved with more than one solution strategy work best. (A simple problem requiring only one or two steps won’t work.) Students begin solving the problem and writing their solution down on paper. When most students are about halfway through the solution, the teacher stops the class and has them trade papers with a partner. Each student then completes his or her partner’s solution without erasing any of the other student’s work. This strategy has two key benefits. First, a student who may have had difficulty solving the problem receives the beginning of a solution, which may lead him in the right direction. Second, a student who was on the right track may analyze the work of a student who was headed in the wrong direction and be able to rectify misconceptions during a follow up discussion.
By working in pairs, students become comfortable discussing mathematics. In Tremain’s class, two-person activities prepare students for further collaborative work in teams of three to solve problems and prepare presentations.
Transcript from Tremain Nelson
We use cooperative learning every single day. If you look at the organization of the classroom, the physical set up of the classroom, you will see that the students come in and they sit with their teams every day. And so I place “Teams” in front of every desk, so that they are actually numbered “Team 1,” “Team 2,” and “Team 3.” And I’ve been doing that for the last couple of months now – and what I’ve seen is that by using the word “team,” instead of using the word “group,” on a daily basis, they actually function more like a team and less like a group. Because when you think about a group, a group is a gathering of individuals who come together. But a team is a gathering of individuals who come together for a common cause. And having the common cause helps to create the community environment and give the classroom direction.
How do students work in your class – individually, in pairs, or in teams? How does each arrangement affect the amount and type of learning that occurs?
Tremain uses an adaptation of a “fishbowl” to conduct what he calls “table talk.” Just as owners of fish look into a fishbowl to observe their pets, some students learn by watching other students solve a problem or have a discussion at the front of the classroom. Tremain’s classroom is large enough to allow for a separate table at the front of the room, but in smaller classrooms students could push several desks together at the front of the room to create the fishbowl area.
During table talk discussions in Tremain’s class, one student from each cooperative team joins the teacher at the table in the front of the room. While the teacher and these students discuss mathematics, go over instructions for cooperative work, or conclude an activity, the other students in the class gather around and pay attention to the discussion.
Transcript from Tremain Nelson
The idea behind the table talk is we want to get the students to begin to talk as much as they can about the mathematics. A lot of times that can be extremely intimidating for them. The table talk brings the teacher down to their level, so that we’re using everyday language talking about mathematics in a way that the students feel comfortable talking about the subject matter. I find that they don’t mind making mistakes as much, because I’m not criticizing them when they make mistakes. The second goal is to make the classroom more like a community. So we spend time talking about the type of interaction that you would have in a community.
The table talk discussions generally serve one of four purposes:
- To introduce new material
A table talk discussion can give students necessary information before they begin a collaborative investigation.
- To prepare leaders for each of the cooperative teams
The students sitting at the table are implicitly more involved than the observers. They can then lead the discussion when teams return to their seats or go to stations for group work.
- To collect student thinking
If students are conducting an investigation and processing the results, the teacher may call a table talk session to discuss the findings. In this way, the entire class benefits from the collective observations of each group.
- To conclude an activity
After students have spent time in their groups conducting an exploration of some mathematical topic, they can use a table talk discussion to clarify thinking, to rectify misconceptions, or to bring closure to the activity.
In some instances, a table talk discussion will serve several purposes simultaneously. For instance, a brief discussion about quadratics prior to an investigation of vertex form introduces new material and prepares leaders for group work. Likewise, a table talk session held after an investigation of quadratics allows the collective thoughts of all groups to form a more complete analysis; in addition, this classroom conversation will help clarify the thinking of many students as well as provide a summary of the activity.
Transcript from Tremain Nelson
The purpose of the table talk discussions is to eventually pull me out of the picture, so that I’m not so much responsible for the learning that’s going on in the room. Students are now engaged, they are now talking more, they are presenting the material in front of the class. While they are in their cooperative learning groups, they are working on material that they have never seen before. And so they are, in essence, teaching themselves the material. My job then becomes more of a facilitator or a coach – to walk around and check for problems, and to encourage them to go on when they feel like they are getting stuck or frustrated. Hopefully, what you would see is my pulling back further and further and further, until all the responsibility is really given to them for the learning.
A table talk discussion may seem like a big step. The teacher shifts from being an information giver to a mediator of discussion. This new role requires a delicate touch. In an 1998 Phi Delta Kappan magazine article entitled “Inside the Black Box: Raising Standards Through Classroom Assessment,” Paul Black and Dylan William suggest a strategy to make these interactions successful: “?the dialogue between pupils and a teacher should be thoughtful, reflective, focused to evoke and explore understanding, and conducted so that all pupils have an opportunity to think and express their ideas.”
Through these discussions, students develop social skills by interacting with peers, and they are able to clarify their ideas regarding mathematics. Many times, it will be a classmate – not the teacher – who brings clarity to an idea for another student. Eventually, students mature from passive recipients of information to active, independent learners.
|Listen to audio clip of teacher
Transcript from Tremain Nelson
During the table talk, I began by working a complete problem for them. Not really getting a lot of input from them, but just trying to show them the steps that I use to solve a problem. And as we began working more problems in that manner, they began to be a little more relaxed. One of the biggest things that I saw, and why I kept that as part of my teaching style, is that the kids feel really comfortable sitting around, because they are used to sitting around and working on stuff. They felt that they weren’t obligated to a desk – some that felt like sitting in chairs could sit in chairs; some that felt like standing could stand; some that felt like sitting on the table could sit on the table – and they could actually talk a little more comfortably about the problem that we were trying to solve.
Tremain didn’t force students to participate in table talk discussions at the beginning of the year, as he knew it might overwhelm them. Instead, he began with groups of two, then groups of three; at that point, he used table talk more as an opportunity to present information in a different setting. Finally, once students felt comfortable, he used table talk sessions to have mathematical discussions with them. What steps do you take to make sure that students develop into active participants? How can you transform your classroom from a collection of individuals into a community of learners?
Respect for Fellow Students
When it comes to developing a community of learners, the most important ingredient in any classroom is respect. Students have to be respectful of one another, and there has to be mutual respect between teacher and students.
Establishing explicit behavioral guidelines goes a long way toward creating a respectful environment.
In a segment of the video for Workshop 4, Part II, students in Tremain’s class observe other students’ presentations about quadratic functions. The observers must take notes so that they can give feedback to the presenters. Students are supposed to identify parts of the presentation that were done effectively, as well as note areas needing improvement. Requiring all students to take notes ensures that they pay respectful attention to the presenters. Here are some other rules for presentations that Tremain has established to foster respect in the classroom:
- Ask students to share positive comments
In their feedback, students must report what each team did well. This allows students to think about positive elements that they should include in their own presentations, and provides encouragement to the group that presented.
- Solicit constructive criticism
After group presentations, Tremain asked the class to share ideas for “things that you thought that maybe they should do to improve the presentation.” Although subtle, this phrasing provides a different classroom tone than if he had asked, “What things did they not do well?”
- Allow for polite disagreement
Students will not always agree with one another. In fact, it would be unnatural if they did! However, students should express their disagreements in a respectful manner and back them up with reasoning.
By encouraging class members to respect each other, Tremain also fosters mutual respect between himself and his students. In a classroom environment where students and teacher respect each other, positive interactions (including valuable discussions and increased student learning) become the norm, and negative interactions (like disruptive behavior) slowly disintegrate.
Transcript from Tremain Nelson
When I speak to them, I try to speak to them on their level. I try to give commands that sound more like requests. And so they are more willing to participate and more willing to become a part of the classroom. I am not afraid to give ownership to them. I believe that this is their classroom, and so there’s no reason for me to direct. And they get out of it what they put in it.
What rules, activities, or expectations foster mutual respect between students in your classroom? What other strategies could you use to encourage respectful behavior in your classroom?
Workshop 1 Variables and Patterns of Change
In Part I, Janel Green introduces a swimming pool problem as a context to help her students understand and make connections between words and symbols as used in algebraic situations. In Part II, Jenny Novak's students work with manipulatives and algebra to develop an understanding of the equivalence transformations used to solve linear equations.
Workshop 2 Linear Functions and Inequalities
In Part I, Tom Reardon uses a phone bill to help his students deepen their understanding of linear functions and how to apply them. In Part II, Janel Green's hot dog vending scheme is a vehicle to help her students learn how to solve linear equations and inequalities using three methods: tables, graphs, and algebra.
Workshop 3 Systems of Equations and Inequalities
In Part I, Jenny Novak's students compare the speed at which they write with their right hands with the speed at which they write with their left hands. This activity enables them to explore the different types of solutions possible in systems of linear equations, and the meaning of the solutions. In Part II, Patricia Valdez's students model a real-world business situation using systems of linear inequalities.
Workshop 5 Properties
In Part I, Tom Reardon's students come to understand the process of factoring quadratic expressions by using algebra tiles, graphing, and symbolic manipulation. In Part II, Sarah Wallick's students conduct coin-tossing and die-rolling experiments and use the data to write basic recursive equations and compare them to explicit equations.
Workshop 6 Exponential Functions
In Part I, Orlando Pajon uses a population growth simulation to introduce students to exponential growth and develop the conceptual understanding underlying the principles of exponential functions. In Part II, a scenario from Alice in Wonderland helps Mike Melville's students develop a definition of a negative exponent and understand the reasoning behind the division property of exponents with like bases.
Workshop 7 Direct and Inverse Variation
In Part I, Peggy Lynn's students simulate oil spills on land and investigate the relationship between the volume and the area of the spill to develop an understanding of direct variation. In Part II, they develop the concept of inverse variation by examining the relationship of the depth and surface area of a constant volume of water that is transferred to cylinders of different sizes.
Workshop 8 Mathematical Modeling
This workshop presents two capstone lessons that demonstrate mathematical modeling activities in Algebra 1. In both lessons, the students first build a physical model and use it to collect data and then generate a mathematical model of the situation they've explored. In Part I, Sarah Wallick's students use a pulley system to explore the effects of one rotating object on another and develop the concept of transmission factor. In Part II, Orlando Pajon's students conduct a series of experiments, determine the pattern by which each set of data changes over time, and model each set of data with a linear function or an exponential function.