You don't get a second chance to make a first impression.
This is naturally true, and although I don't believe that you can build or destroy a successful semester in one class, it is definitely important to put your money where your mouth is on the first day. I have spent the past few days digging around my materials for the best possible starter activities.
I had some very helpful responses from the twitter-verse, and it prompted me to somehow sort out the information being provided to me. In the past, I have had very productive lessons on the first day. Not productive in the "coverage"sense, but rather in the "intriguing" sense. My goal was to define a set of characteristics that, in my opinion, create a suitable opening day problem.
It should be noted that I have somewhat of a loose cannon reputation in my school. I like to keep students off guard (mathematically) in my class. Most returning students come waiting for the next surprise, and I do not like to disappoint.
The following is a list of four (4) attributes that I believe constitute an effective first-day task:
1) The problem should be rooted in real mathematics
2) The problem should be solved in a cooperative spirit
3) The problem should require authentic problem solving and not algorithm execution
4) The problem should be presented in an intriguing setting
It wasn't until I had drafted these 4, that I realized that these criteria could possibly apply to each lesson. I do believe, however, that the importance of numbers 2 and 4 is heightened on the first day. These build the strongest communal bonds between students and teachers. After I had my target attributes, I set out to review my options. Of course, time and materials became limiting factors for me as well. (when are they not?) I was given 45 minutes total for each class on day 1.
I began my search by reviewing the activity that I did for the past 2 semesters. I introduced myself to the class and immediately asked students to put everything away. I handed out a test. The tension in the room rose very obviously once I began to circulate with the papers. On the test, there were questions similar to the one below:
A man bought a pair of shoes that cost $75 and gave the merchant a $100 bill. After the man left with his shoes and his change, the merchant discovered that the $100 bill was counterfeit. What was the total loss to the merchant?
Students completed the exam individually, but I did not discourage "cheating". Without announcing an official end to the test, we began to discuss different answers. As is plain to see, students come up with a variety of answers and rationale. What is your answer? Can you back it up? As the teacher, I kept my answer well hidden and agreed with every argument that made sense. Not only did this cause mass mayhem, students began to vehemently defend their position. I was able to step back and allow the students "do" the math. When I look at the criteria, I see this task meeting numbers 2, 3, and 4. The problem became a group task, the contention was a result of the intrigue, and there were definitely no algorithms to solve the problems. (Although some students created them!).
The issue for me is the fact that the problems weren't really based in math skills but rather in the deft wordplay of the questions. Usually the arguments were not math related but rather "real life" related. Students were amazing at finding the loopholes. Like the merchant didn't lose $100 and the shoes, he lost $100 and the wholesale value of the shoes. Such is a splitting of hairs that I didn't want; I wanted the contention to come over mathematical ideas. I decided to scrap this idea, and try to find a task or tasks that fit all 4 criteria.
The activities come from various places including @jamestanton, A Number Story, by Peter Higgins, and past experiences. I haven't decided which to use yet, but now have a clearer vision.
1) Magic Square Problem
I first saw this problem from a professor. Draw a 5x5 matrix of numbers beginning with 1 at the top left, and continuing across horizontally until 25 is in the bottom right hand corner. The teacher writes a prediction on a piece of paper, and gives it to a trustworthy student (based on first day impressions). Then he asks the first student to pick a number in the first row. Circle the selection and then cross out all other numbers in it's row and column. Continue until 5 numbers are selected, sum them up, and reveal your guess. If they ask you to repeat, do so with a 6x6 matrix. Without fail, some student will say it always works. This is perfect, because now the question is: Why? How does the result relate to the length of the side? Why is the result always the same? Like most of my problems, the secret is in the base-10 number system and the number theory behind it. Encourage students to rationalize and explain to classmates.
2) Box of Toothpicks
I have 39 toothpicks in a box and ask a student to pick out a number of toothpicks and leave any number they wish in the box. To eliminate cheating, I then ask him to take that number he placed in the box, add the digits, and take that sum out of the box. I then guess the number of toothpicks in the box by simply shaking and listening. This can be iterated, but once a repeat comes up, the students become suspicious. Once your cover is blown, ask them to explain why the answer must always be the same? Or does it have to be? A similar example can be found at http://www.albinoblacksheep.com/flash/mind where a psychic reads the students' minds. After enough iterations, predictions will be made. Always ask for follow up explanations.
3) Cats and Dogs
This problem has been solved using formal methods on my blog previously. http://musingmathematically.blogspot.com/2011/06/when-school-math-falls-short.html. The students are given a question that can be solved using sophisticated methods, but they are not apparent. Get students to work in pairs or groups until they guess their way to the answer. You, as the teacher, should elicit strategy as they go. A very interesting mice strategy emerges. Would you ever add 3 mice to the equation? Why or why not? What special quality do the cats have? Is your solution the only one? How can you tell? Can we add an animal at a different price to switch the solution? The problem has many layers, each of which is rooted in real mathematics.
4) Iterated Sharing
This problem comes from James Tanton's book, Solve This. It is the second problem in the book and involves a group of people passing candies around a table. I have not personally tried the experiment, but am intrigued by the problem. I will not disclose the entire problem here to respect James' ingenuity. His book is an excellent resource for those teachers (and learners) interested in moving students (or themselves) outside the box.
You may notice that these problems don't have explicit curricular ties, and that was not one of my goals. Each is rich in mathematics and accomplishes the first day goal of establishing a mathematical ecology in my room. It is a far bigger challenge to design tasks that meet both explicit curricular goals and the four criteria. That is a much larger burden which math teachers collectively bear. Creating an effective environment is very important in mathematics education. A carefully designed first-day task can work wonders.