What is a Harkness Math seminar?

A Harkness Math seminar is a teaching and learning method that is based on the Harkness table method. It was developed by Phillips Exeter Academy, a private school in New Hampshire, USA. In a Harkness Math seminar, students sit around a round table and engage in discussions and problem-solving activities led by a teacher or facilitator.

The Harkness Math seminar is a student-centered approach to learning in which students engage in productive discussions around mathematical concepts and problem-solving. The following are some of the protocols and procedures typically used in a Harkness Math seminar:

• Group size: The ideal group size for a Harkness Math seminar is between 6 and 12 students. This ensures that everyone has a chance to participate and share their ideas.
• Round table: The students sit around a round table to promote equality and facilitate eye contact and conversation.
• Moderator: A moderator, usually the teacher or a student leader, guides the discussion, asks open-ended questions, and encourages all students to participate.
• Preparation: Students are expected to come prepared with their thoughts and ideas about the problem or topic of discussion.
• Participation: All students are expected to participate in the discussion by sharing their ideas, asking questions, and challenging their peers' ideas.
• Respect: Students are expected to respect each other's ideas and opinions and to listen actively to their peers.
• Feedback: The moderator provides feedback to students during the discussion, such as acknowledging their contributions, asking follow-up questions, or providing clarification.
• Reflection: At the end of the discussion, students reflect on what they have learned, what questions they still have, and how they can apply their knowledge in future situations.

These protocols and procedures are designed to create a supportive and engaging learning environment that promotes critical thinking, problem-solving, and collaborative learning. By following these guidelines, students can explore mathematical concepts in depth, build their confidence and skills, and develop a deeper understanding of how mathematics applies to real-world situations.

The Harkness Math seminar is based on the philosophy that students learn best when they are engaged and actively involved in the learning process. It provides a collaborative learning environment that encourages students to take ownership of their learning and to develop critical thinking and problem-solving skills.

In a math seminar, students can ask a variety of questions to deepen their understanding of mathematical concepts and to engage in productive discussions with their peers.

Here are some examples of questions that students might ask in a math seminar:

1. How can we apply this concept to real-world problems?
2. What assumptions are we making in this problem, and are they valid?
3. Can we approach this problem from different angles or using different methods?
4. How does this concept relate to other mathematical concepts we have learned?
5. Can we generalize this problem to other situations?
6. What are the potential errors or pitfalls in this solution, and how can we avoid them?
7. How can we break down this problem into smaller, more manageable parts?
8. How can we visualize this problem using graphs or diagrams?
9. How can we use technology to help us solve this problem more efficiently?
10. Can we create our own problem using this concept and challenge our peers to solve it?

These are just a few examples of questions that students might ask in a math seminar. The key is to encourage students to think critically and creatively, to challenge their assumptions, and to engage in collaborative problem-solving.

The Harkness Math seminar is often used in schools that follow a student-centered approach to learning. It can be applied to a wide range of math topics and is suitable for students at all levels of math proficiency.

The Harkness Math seminar encourages student-centered learning and promotes collaboration and critical thinking. Students are encouraged to ask questions, share their ideas and insights, and work together to solve math problems. The teacher or facilitator serves as a guide, helping students to stay on track, providing feedback, and offering support when needed.

A simulated Harkness Math seminar that demonstrates how this approach can be used to engage students in a math problem-solving activity:

The problem: A triangle has side lengths of 5, 12, and 13. What is the area of the triangle?

The Harkness Math seminar:

The teacher/facilitator introduces the problem to the students and invites them to share their ideas and insights. The students sit around a round table, and the teacher encourages them to take turns leading the discussion.

Student 1: "I remember that the area of a triangle can be calculated using the formula A = 1/2 bh, where b is the base of the triangle and h is the height. But I'm not sure how to find the height of this triangle."

Student 2: "Well, we know that the length of one side of the triangle is 5, and that side is perpendicular to the height. So we can use the Pythagorean theorem to find the height."

Student 3: "Right. So if we call the height h, we can use the Pythagorean theorem to solve for h: 5^2 + h^2 = 13^2. That gives us h = 12."

Student 4: "Now that we know the height, we can use the area formula to find the area of the triangle: A = 1/2 bh = 1/2 (5)(12) = 30 square units."

Teacher/facilitator: "Great job, everyone! Can someone explain why we used the Pythagorean theorem to find the height?"

Student 2: "We used the Pythagorean theorem because we knew that the side length of 5 was perpendicular to the height of the triangle. So we could use the Pythagorean theorem to solve for the length of the height."

Teacher/facilitator: "Excellent explanation! Does anyone have any other insights or questions about this problem?"

Student 5: "I was wondering if there are any other ways to solve this problem?"

Student 6: "Yeah, can we use Heron's formula to find the area of the triangle?"

Teacher/facilitator: "Those are great questions! Let's explore those ideas together."

The students then work together to explore alternative methods for solving the problem, such as using Heron's formula or applying trigonometry concepts. The teacher/facilitator guides the discussion, provides feedback, and encourages students to think critically and creatively.

As you can see, the Harkness Math seminar encourages students to take ownership of their learning, to collaborate with their peers, and to engage in productive discussions that deepen their understanding of mathematical concepts. It's a powerful approach that promotes student-centered learning and can be applied to a wide range of math topics and skill levels.