Their intended workshop goals were:
- For us all to consider the unwritten curriculum of our classes and schools, paarticularly around "math smarts".
- For us all to view, experience, and evaluate a math teaching method that is both challenging and empowering to students.
- For us all to leave with concrete next steps for creating an empowering unwritten curriculum in math.
In order to empower students in the math classroom a teacher must incorporate many parts:
- Instruction, Curriculum, and Assessment Practices
- Classroom decorations and physical set-up
- Consensus is the Answer Key
Consensus in the math classroom comes with it's own set of challenges. For example, students who have spent the majority of their time in traditional, teacher-centered classrooms are not used to this new process. Jason and Sarah often found that students clung to the traditional textbook driven classroom, and they admitted because it was easier for them. Also, students coming from traditional classrooms are often coming from places that foster competition. Because of this, it is essential to create a safe classroom culture. Another challenge is that teaching in this way is counter-cultural for a teacher. Constructivism has begun to enter mainstream education and teacher education, but it is still not the way that many of us were taught. Creating a culture where student discovery is commonplace is not just about noninterference by the teacher. Teachers must create rigorous academic environments by asking LOTS of questions and not just stopping when they hear a "correct" answer.
Jason and Sarah found that structure is helpful in building this consensus mathematics classroom. They have a normal classroom routine which consists of students arriving, getting into groups and discussing their solutions to the previous night's homework. Teachers also spend time giving direct instruction on behavior and mindset: reminding students that mistakes help us learn (mindset) and teaching behaviors that create a safe environment, such as the role of the presenter and participants. Students learn the structure of discussing solutions: solutions are presented in logical steps. Questioners do their job respectfully and listen while information is being presented. There is no compromise of the safe environment! Also, the teachers structure in time for presenting unfinished work. They give only enough time so that no one reaches an answer and discuss it so that the students have space to examine process outside of the restraints of solutions. Jason and Sarah also recommend giving specific feedback to students, instead of "good job!". This helps students assess their strengths and weaknesses. Another structure they use is a standard approach to problem solving that they introduce at the beginning of the year. This is a series of questions, which lead students through the process of searching for solutions. This process includes rewriting the question, creating a problem statement, stating any equations given, guessing an original estimate, etc... As the year goes on, the teachers allow the students to adjust this process so that it works best for them.
They provided the following tips for implementing "Consensus is the Answer Key" so that it can be more liberating for students:
- Use direct instruction to teach the skills necessary for Consensus is the Answer Key.
- Because this structure goes against our dominant schooling culture, students will not adopt these habits by accident. It is important to directly teach procedures such as steps for presenting and discussing solutions. It's helpful to post a script or guidelines for what presenters and participants should do. Correct students language if they stray from these guidelines. One common example is that students often begin the school year using language like, "You're wrong." Instead, have them say, "I disagree." This leads much more naturally into an open discussion without defensiveness. You should also directly teach empowering beliefs about learning, such as mistakes being good, helpful things.
- Model the behavior and language that you expect of students.
- Direct instruction most be coupled with modeling. You are working to create an environment where it is good to make mistakes and question each other. Therefore, you must be open to students correcting you or giving you feedback. Always thank them for it, even when they don't phrase things well. Be open to getting authentic help from the students. For example, if you struggle with spelling, ask them to check your spelling. In addition, never put yourself down by saying things like, "I can't draw" or "I'm bad at spelling." We want to break the norm of students saying, "I'm bad at math," but we cannot ask that of them if we're modeling something else. Instead of putting yourself down, express that things are difficult but that you are confident you can grow: "I'm having trouble making a diagram that looks like a football field - can someone tell me what I need to do here?"
- Be uncompromising in creating a safe classroom environment
- In order for this process to work, students need to feel safe to take risks, say incorrect things, and make mistakes. Clearly set up expectations at the beginning of the school year around how to treat each other. Then, do not let any infractions pass, even when students say that they're just joking. We don't allow any put-downs in our classroom, even joking ones. We don't allow the jokes because we think that the butt of the joke is often more hurt by it than s/he claims. In addition, negative joking sets a tone in the rest of the class so that students who may not be involved in a particular joke will feel nervous about becoming the next butt of a joke and will therefore speak less and take fewer risks.
- Insist on rigor, not just hearing initial ideas
- Many classrooms that choose to emphasize student voice encounter a common pitfall. They often do a great job of having students express their ideas and thoughts, but stop at this point. Then, students are left with only their initial, unrefined ideas; they have not learned anything about determining the quality work and thought. When students share ideas, it is important for them to get feedback on those ideas and abandon or refine those which don't hold up to scrutiny. Some teachers feel uncomfortable with the thought of criticizing students' thoughts. Critical feedback is one of the greatest gifts that we give our students because it sends the message that we know they can grow. It's not ok for students to walk out of a math classroom believing that a square has three sides because the teacher just asked them to express what they think of a square. Of course, using Consensus is the Answer Key means that idea refinement occurs primarily within discussions among students. Therefore, this rigor is developed through the norm of questioning and proving answers as a class.
- Be deliberate about the feedback you give individuals
- Feedback is an essential part of learning and growth, but it is not a cultural norm to give feedback that is empowering to students. Too often, students receive most of their feedback from teachers in the form of praise or criticism which makes them dependent upon the teacher for affirmation. Empowering feedback helps students grow, not just feel good or bad about the teacher's opinion of them. Within the Consensus is the Answer Key model, students should ultimately get most of their feedback from sources other than the teacher. Other sources include fellow students, their own internal compass, logic and proof, and measuring their work and actions against clear goals or expectations. This process generally needs to be scaffolded and modeled.
- There are a number of specific ways to give feedback to individuals. Firstly, it is important to avoid feedback about students' abilities or attributes. These types of comments contribute to a fixed sense of intelligence and identity, which leaves students feeling without agency to control their achievement. Instead, give feedback that is about students' observable actions. To help students internalize their own evaluation skills, provide observations of student actions and then ask students to interpret their value or meaning. The last step of this scaffolded process is to ask students to both make the observations and the interpretations themselves.
- Sometimes students solicit your affirmation through specific questions. It's always a balancing act to determine whether it's a case of them needing your expert opinion or whether it's a case of them wanting teacher affirmation (or a shortcut to knowing if they're right.) If a student asks you, "Is this right?" or "Is this good?" never respond with a yes or no. You can ask them if they've checked it with their classmates, ask them to explain what they did and why, or let them know that you don't answer that question. Sometimes students seem to need information from you in order to calibrate themselves to understanding what quality work looks like. In this case, then give very specific feedback or move students to the more internalized levels of feedback described in the previous paragraph.
- When facilitating discussions, do not use students' names attached to their solutions
- Oftentimes, two students will write different solutions to a problem on the board. If you are facilitating the class discussions to determine which (if either) is correct, do not call them "Susie's solution" and "Junior's solution." Calling them by individual's names takes away from the group process and also makes students invested in "their" answer rather than in finding the correct answer. In addition, throughout the discussion, numerous other students should be free to come to the board and edit what is written, so Susie's solution could become Susie's, Juan's and Mirabel's solution. The goal is to come to consensus about what answer makes sense and why, not to figure out who got the right solution. Instead of using students' names for solutions, label them A and B or give them a meaningful name such as "addition method" and "area method.
- Assign super-challenging problems or give students short work time before discussion
- Students have generally been taught that math is primarily about getting the right answer. This is a challenge to Consensus is the Answer Key because students often want to skip the process and get straight to the answer. By assigning problems that are too challenging for anyone to solve on their own, and/or by discussing those problems before anyone has a final solution, you give students practice in discussing the process. In addition, this creates a classroom norm that students can present unfinished work, work that they think may be incorrect, and questions that they have. These discussions tend to have much richer dialogue than discussions when someone just presents a clear correct process and solution.
- Assigning super challenging problems also helps address a common class dynamic in which students label each other as smart or not and begin to place one of the "smart" students in the expert role which the teacher has vacated. If problems are too simple and have only one right answer, there may be a student in your class who quickly and reliably solves problems, and students will begin using that student as their answer key rather than engaging in rich dialogue. Big, challenging problems are beneficial for the student who has been labeled "smart," too. this frees that students from having to defend a fixed identity of being smart and allows her to engage in the kind of challenging work that will actually develop her intelligence.
- Question all answers
- Whenever a student gives an answer, ask how s/he got it. Many teachers are in the habit of asking students how they got incorrect answers but do not ask questions about correct answers. Questioning only wrong answers is a problem for a few reasons. First, students are usually in the habit of looking to the teacher for affirmation. If the teacher has a pattern of questioning wrong, but not right, answers, students will be able to use this information as clearly as if the teacher said something was right or wrong. Secondly, it is very important for students to explain how they got correct answers. Sometimes they get correct answers through lucky guesses and they haven't actually learned the material. Also, the process of explaining how they got an answer helps students to understand the material in more depth, remember it better, and develop clear communication skills.
- Use appropriate wait time
- Do not rush to fill a void of silence. Students need time to think. In addition, students may be nervous about speaking and want the teacher to speak instead. If it's clear that the teacher will not fill the silence and will really wait for an answer, then student will generally speak up.
- Students should be up at the board and the teacher should be at the back of the room as much as possible during whole-class discussion.
- Physical spacing in a classroom is an essential part of the room's culture. At the most advanced level of Consensus is the Answer Key, the teacher is practically non-existent. Students should be in the front of the room, presenting their work. They should feel free to go up to the board to explain their ideas without asking the teacher's permission, especially mid-discussion. From the back of the room, the teacher is free to observe all the students in the class. In addition, students often seek out the teacher as their primary audience when presenting at the board. When the teacher is far away, the students are pushed to direct their presentation to other students. At times, the teacher can emphasize this by even stepping out of the room momentarily or looking down at papers instead of directly at the presenter.
- Be prepared for anger, opposition, and frustration
- When Consensus is the Answer Key is introduced to a new class or school, there is a very predictable period of student being frustrated and angry. Obviously, there are steps you can take to minimize this, such as the direct instruction of the process and an explanation of why you use it. Still, this is a new, uncomfortable process for most students. They are not used to this much responsibility and independence of thought. They will consciously and subconsciously pull out all the stops to try to get you to give them hints, leak an answer, or even abandon the whole enterprise. Often, students are very scared that they will not be able to do well in this unfamiliar format. If you stand firm with not giving answers while providing necessary support, there will be huge breakthroughs that speak for themselves. De-brief that process to point out the initial frustration and the huge sense of accomplishment at the end.
- You may receive opposition from other parties besides students, such as parents and administrators. Be ready with a clear rationale of why you are using these practices and what the benefits are. Accumulate good supportive documents (such as the NCTM math standards which emphasize problem-solving and communication skills or the research of Carol Dweck and Alfie Kohn).
- Give students information that is not discoverable through investigation
- Consensus in the classroom is NOT the answer key for mathematical conventions and definitions. The teacher must furnish this information. For example, students are able to discover that the opposite over hypotenuse ratio is the same for all right triangles, but they cannot discover that this is called "cosine". It is important that students leave our classrooms with the ability to communicate their learning and understanding to the rest fo the world - this means knowing agreed upon language and conventions.