For all of you problem solvers out there, here’s a classic math problem:
Suppose there are 4 people in a room. How many handshakes would there be if everyone in the room shook each other’s hand once? How about 100 people in a room?
As you are working out the answer, let us provide some background. At a recent board meeting, two of our esteemed SFDS math teachers, Loren Moyé and Tom Keller, led a discussion on mathematics and concluded the presentation with a homework assignment – the “handshake” problem. The handout with the assignment tasked the board members to determine the answer by drawing a picture to show their thinking and then to make a table to record the number of people and the number of total shakes. Then, they were asked to identify the resulting pattern.
Ok, ready? Here’s the answer: If there were 4 people in a room, each of those 4 would shake hands with 3 others. Thus, we have 4×3. However, this would count the handshakes between each pair twice, so we have to divide by 2, resulting in 6 handshakes. You could also map out the answer by counting the number of hands each person can shake, for example:
Person A shakes 3 hands.
Person B shakes 2 hands since this person has already shaken hands with Person A.
Person C shakes 1 hand.
There are a number of ways we can express this formula:
4×3/2 OR, 3 + 2 + 1 = 6 OR, n (n-1)/2
Why is this relevant, you ask? In today’s world of Google and smart phones providing easy answers at the touch of a button, skills such as reasoning, critical thinking and problem solving are increasingly important and highly prized. Loren and Tom touched on how mathematics can improve this type of skill set for our students at SFDS.
This past summer, the mathematics department worked with Nancy Lobell, a STEP (Stanford Teacher Education Program) Clinical Associate for Mathematics. With Nancy’s support, our teachers identified key goals for this year, some of which include the following:
- Build critical thinking and problem solving skills
- Learn to collaborate, communicate and work together to solve mathematical problems
- Develop more conceptual understanding, recognize “big ideas” and make connections to different concepts and topics
- Build habits of mind that will help build success in math
- Willingness to take risks and make mistakes
- Comfort with mistakes and recognition that they are learning opportunities
- Sense of enjoyment and of embracing challenges
- Confidence in themselves as thinkers
- Perseverance, self-reliance and an awareness of how one learns best
- Ability to ask questions and extend problems form the specific to the general
In order to meet the goals outlined over the summer, the faculty also worked together to introduce and establish specific classroom norms. These norms include the following:
- “Mistakes are expected, respected and inspected.”
- There are multiple ways to solve a problem. Students will show respect for others’ ideas, questions, and solution strategies.
- All opinions, thoughts, questions and answers are valued. No “poof-ing.”
- Math is the authority; the teacher is a facilitator, a “guide on the side” rather than a “sage on the stage.”
- Students will be able and willing to defend their conclusions with evidence, data, examples, counterexamples, and words.
- The process of solving a problem is as important as the correct answer.
- “Convince yourself…convince a friend…convince a skeptic.”
- Understanding takes time; it’s not “all or nothing.”
- Thinking is easier than memorizing.
- Success in math class is related to what you learn and how you progress.
As a way to integrate these lessons into the classroom, the faculty also leveraged the “five strands of mathematics proficiency,” which provides classroom structures to promote conceptual understanding, procedural fluency, strategic competence, adaptive reasoning and productive disposition. These structures can include activities such as peer coaching, poster proofs and “convince me,” where a student outlines each step of their thought process so that they are able to clearly articulate their solution with solid reasoning. Drawing on these kinds of lessons helps our students to obtain a firmer grasp on math and a deeper appreciation for the skills that math can help develop. When faced with a seemingly complex task (such as the Handshake Problem), our students will not be confined by technology for answers, but instead will be able to work together to problem solve and determine a viable solution.
Back to our original question: If 100 people in one room shook hands, how many handshakes would there be? If you answered 4,950, congratulations! You’ve solved the Handshake Problem.
