1. Helping individual students clarify and share their own thoughts
2. Helping students orient to the thinking of others
3. Helping students deepen their own reasoning
4. Helping students engage with the reasoning of others
Students often need to gather their own thoughts before even beginning to clarify them. In an earlier post, we explored the importance of providing think time for students before they engage in discourse (either with each other or when sharing out). Using that think time is an excellent way to help students gather, clarify and share their own thoughts.
In the first couple minutes of the Teacher Channel video below, you can see a nice example of a sequence that encourages productive math talk. The teacher asks students to observe and think, then turn-and-talk, then share:
The student that is first asked to share what patterns he sees seems relaxed, confident and not intimidated. He was given an opportunity to look at the math, to think about it, and then to share his ideas with a peer and listen to his peer share his ideas (turn and talk), all before being asked to share anything with the whole group. Also, students are all seated on the floor, where they have become accustomed to communicating about math as a group.
I have seen some terrific student discourse happening in classrooms I visit. It is late in the school year and students in many classrooms are developing stellar talents for communicating their mathematical thinking. Especially in the K-2 grades, where the new Everyday Math curriculum requires more collaborative thinking and sharing of mathematical thinking than ever before, students are getting used to sharing not just their solutions to problems, but also how they solved those problems. I have even seen students volunteering to restate others' strategies, sharing new strategies, and critiquing each other's strategies. I notice more and more students leaning over their neighbor's desk, walking him or her through a problem to help find an error. They are not just learning how to communicate their mathematical thinking, but they are becoming very good at it too.
Once upon a time, in middle school and high school, collaborative learning opportunities used to be a rare thing in the math classroom. Desks were mostly in rows, and looking at your neighbor's strategies, or sharing your strategies with your neighbor, could have landed you in the principal's office for cheating. Teaching usually involved a lot of explaining and not much discovery. It was not all bad, at least for those students who already had discovered an appreciation for math and had confidence.
A more traditional style of teaching math, at least ideally, went something like this:
First: The teacher talks the class through a new math procedure in front of the class.
Then: The teacher gives the class an example to work through together, with guidance from the teacher.
Last: The students work on a series of problems, similar to the above work, independently.
This approach is commonly referred to as the "I do, we do, you do" approach, and it is a logical way to teach someone how to complete a procedural task. In high school, I was taught how to graph a linear equation that way.
Even as we experiment with different classroom techniques, employing a workshop model in the classroom, facilitating group and partner work, using centers, and incorporating movement into our lessons, the tendency to want to simply demonstrate how something is done remains dominant in most of us. "My students are clearly confused about area and perimeter. I know the distinctions between area and perimeter," we think to ourselves, "so I will just show them so they know too," and we launch into a mini-lecture on the topic, complete with nice diagrams and a laser pointer.
There are two main dangers that linger when this happens.
The first danger is that in order for even a very brief mini-lecture to be effective, you need complete student engagement. As soon as you have to re-direct just one student to face the front or keep his or her hands to his or her self, there is a disruption to your presentation of the topic. If it happens twice, it becomes a burden for those that are listening to remain focused. And this is assuming you began with 100% attentiveness. In the event your students are not all that interested in what you have to say from the start, your mini-lecture is doomed... The more behavioral disruptions the more you lose student engagement, the longer the mini-lecture becomes and the period of time you expect your students to remain focused on you.
The second danger is just as significant as the first, but more simple: Your students may not connect with what you are saying. Any number of them might not understand your way of explaining the concept. Students learn math in a myriad of ways, and your explanation, algorithm or procedure may be lost on one or more of them.
Since I know (from my own experience) how dominant this tendency to want to explain things to the class is, I am not condemning the practice, but I do offer a suggestion; monitor yourself very carefully when slipping into "explain mode." Know that if your explanation lasts more more than a minute, you have likely lost at least a couple of your students. Also, if it happens more than a couple times during a math class, your students may be missing out on more valuable learning and discovery time.
Here is a neat way to think about the learning experience as it pertains to mathematics;
Think of learning math and problem solving as a social experience.
It is like learning a language. If you have ever had the experience of trying to learn a foreign language, perhaps you can relate. You took Spanish I, and got an A+. Como Estas? Muy Bien, gracias! Spanish is easy! You took Spanish II, and it was harder, but you learned a lot! Spanish III? Oh my, accents, novellas, dictations, I sure am getting good at Spanish! Spanish IV made you a master! I must be fluent now! I feel so confident! You took a spring break vacation to Cancun and it was: What are they saying? How do I respond?? Ayudame! Any foreign language teacher of professor will tell you, you don't become fluent until you immerse yourself in the language.
Luckily, we do not need to send our students to Mexico or Spain to immerse them in mathematics. We need to give them opportunities to problem-solve with each other and construct their own solutions together. I used to tell my middle school students scientists and mathematicians rarely make great discoveries in isolation. NASA does not usually send a solo astronaut into space to work on its most precious and complex projects. Problems are sometimes solved independently, but mathematicians and other specialists often work in teams to solve the world's most important problems.
Solving problems in pairs and teams can allow students the opportunity to reach greater heights with their thinking, and celebrate greater victories. Even a classroom open response experience is better off spent tackled with a partner or two. A confident solution with a consensus should warrant a high-five! Eureka! Math is fun.
As a social activity, math is about sharing, about experimenting, about collaborating, about persevering and about finding solutions. Every part of that problem solving process can be a stimulating, engaging, enjoyable experience.
Here are some ways some teachers in RSU 5 are increasing that excellent mathematical discourse that is at the heart of a superior learning experience in the classroom:
- Incorporate turn-and-talk opportunities into every delivery of whole-class instruction, so your students become accustomed to sharing their thinking with each other regularly.
- Get into the habit of incorporating whole-class sharing and whole-class instruction in a part of the room where students can be seated on the floor, close together and close to you. Turn-and-talks are effective in this setting too.
- Practice specific protocols for communicating in groups, and have students do it every day. For example, speaking at an indoor volume, always remain quiet when a group member is speaking, identify and use specific polite words when questioning or critiquing someone else's thinking (like Can you clarify that for me? and I think you made a mistake. Can I show you?)
- Always be looking for opportunities for students to explore new ideas and new concepts, especially when you are tempted to explain stuff to them. Think: How could I get them to work this out without me showing them how to do it?
- Assign roles to group members, such as presenter, note-taker, ambassador, fact-checker, editor, time-keeper, equalizer (making sure everybody is contributing), etc.
- Promote a strictly safe environment for sharing in class to promote confidence and reduce anxiety when sharing. Encourage everyone to take each other seriously so nobody is ever laughed at for giving a wrong answer or making a mistake. This includes the promoting of making "brave mistakes" as the essential element of achieving success!
- Arrange desks in ways that allow students to easily communicate with each other. Tables are best for this, as they can sit across or next to or diagonal from each other and communicate, while having some distance from other groups.
Do you have other ideas or tips for creating an environment for collaborative learning? Please let me know and I will add them to the above bullets!
*The book mentioned at the top, Classroom Discussions in Math, was published in 2013 by Math Solutions of Sausalito, CA, and was written by Nancy Anderson, Catherine O'Connor, and Suzanne Chapin.
Recently published (Fall 2015) articles on Mathematical Discourse:
Orchestrating Mathematical Discourse to Enhance Student Learning
Establishing a Culture of Collaborative Learning
Tips for Launching an Inquiry Based Classroom
Learning Is Loud