“The essence of metaphor is understanding and experiencing one kind of thing in terms of another.”
-George Lakoff and Mark Johnson, Metaphors We Live By
Student Perspectives on Harkness Mathematics
What is Harkness Mathematics?
In a recent survey about classroom activities, I asked my students to explain the structure of our class to someone who does not know anything about round table, discussion-oriented (Harkness) mathematics classes. Here are several of the responses:
"A Harkness class is discussion-based class. We put problems on a board and our thoughts on solutions. We go over the solutions as a class and come up with a correct answer based on theories we had learned before, usually with minimum help from the teacher."
"The class environment is very comfortable. It's like a big open discussion about math. We share our problems and discuss how they went. When you have a problem you bring it up and either a peer or the teacher will take time to explain it to you. Speaking freely makes it a more accepting environment, when one person struggles everyone wants to help and when we all understand a problem we get really excited, which I love. It's definitely one of my best experiences in a math classroom that I've ever had."
"During class, we usually put the problem set up from last night on the board and everyone has an equal opportunity to put a problem up each day. Then we discuss/present our problems to the whole class and then the whole class discusses the problem, the ideas presented in the problem, and brings it even further. The class usually involves a lot of participation, discussion, and critical thinking."
"I love our class environment, I believe it is excellent. I always feel like I can contribute to the conversation without being judged. I think that when we are discussing as a class and it is going well, we are able to feed off of each other’s energy and make the conversation great."
"I love our class. I think the environment we create is encouraging and enthusiastic. I feel totally comfortable asking questions or asking for help. I really like that it isn't all work and that we can joke around sometimes but also get our work done. I also really like the Harkness table because it makes the class more open to discussion."
"I love the environment in my class. I am in Geometry 2 [course section] and it is a comfortable environment for me to say what I am thinking even if I am wrong or have a question. I think that working in pairs has been a great aspect of class, but I like working out tough problems as a group. I also really enjoy (even though it is great to have you as our teacher) when the teacher is not able to come to class and we work out the problems on our own. I think it is a great way for us to talk about the problems in a more relaxed setting."
"A Harkness class is a discussion-based class around a topic of interest. Students present their ideas and their peers either agree or disagree, then discuss. The teacher monitors the discussion and makes sure the students are on the right path, and shows us paths to go when the students are lost in their findings."
"All of the students sit around a circular table and figure out the solutions to the proofs together. We build off of one another's ideas. We are respectful of other's thoughts and do not make fun of each other if we do not understand. The teacher leads us in the right direction and gives us a hint if we are stuck. We engage in solving proofs."
"Throughout my educational life up until this point, I thought of math class as a place where students have no say in the way their experience unfolds. I grew accustomed to the very conventional lecture-styled classroom sessions, and accepted that I'd never be a straight-A student because math was something you either got - or you didn't. Harkness-styled class discussions completely changed this. Math became a conversation - and making mistakes was acknowledged as the best route to genuinely understanding the material and growing as a mathematical thinker. I now think of math as a team-effort that may never be considered perfect, but will always be considered something I learn a lot from."
"It's very encouraging in regards to trying new methods, even if you aren't sure if they'll work, as it's a lot more open to discussion than a normal classroom. We can talk about the problems as a class and fully discuss them and ask each other questions when we don't understand, and the Harkness table makes this easier and encourages everyone to contribute new ideas to the discussion. When we're talking about a previous problem set in class, the discussions are very student based but if we can't think of anything new, the teacher will step in and guide us in a direction we hadn't thought of before, and oftentimes then we'll be able to figure it out."
"Before class, we do a problem set that contains instructions. Much of the problem set is new topics. Some can be figured out, but others are very confusing. When we come into class, we put the problems on the board. Then, the person who did each problem puts it on the board. If there is a question or dispute on a problem, then the teacher will step in and ask us questions in order to get to the answer."
"Our class is a casual environment where you can feel comfortable sharing your work to your peers without being afraid of criticism. The way the class is run leads to serious discussions on the material we're learning that help us to better understand the problems. The teacher helps fuel these discussions by providing questions to discuss and answer and also steps in if we're all super confused on something."
"We put our solutions to a problem set on the board, and together as a class discuss what we did to come to that solution. Instead of sitting down each class and being taught all of the material, we go home and try to work through it, and once we get to class we try to come up with a general way of solving the problem. It’s all very free--we don't raise hands to speak, we just kind of talk as if it was a normal conversation between a group of people, just specific to math. The teacher lets us talk and work through a problem without coming out and giving you the answer. In fact, this whole year, I don't think she has given us an answer if we haven't gotten an answer, we will work through the problem for as long as it takes to get an answer, which is really nice because we will never forget how to answer a question after working on it for so long. Of course our teacher will step in to teach us specific things, but it’s definitely more a student lead class, where our teacher kind of leads us through and makes sure we are on the right path, which again, is so much more effective than just giving us answers and telling us how to do things."