Establishing community in the classroom can be a challenge. Here's an activity my students participated in on the first day of school. I learned about this activity while participating in the Advanced Educator International Space Camp in Huntsville, Alabama. The objective is for two crews of astronauts to exchange positions in cramped quarters when a new crew shows up to relieve the old crew at the International Space Station.

How the Game is Played

SSC_StartingPosition

Players must follow these rules:

  1. Only one person can move at a time.
  2. Only movement forward (in the direction a person faces) is allowed. In the above diagram, orange players can only move right, while blue players can only move left.
  3. A person can move to an empty space in front of them.
  4. A person can jump an opposing team member in front of them.

What Does It Mean to Win the Game?

The teams win the challenge when they have exchanged their original positions. See the ending position diagram for an example of what this looks like.

SSC_EndingPosition

 

What If...?

A teacher could use an agility (speed) ladder for this activitTapeOnFloory. Or if a ladder isn't handy, use tape. This is a shot of my classroom the first day. It's pretty unlikely a teacher would have only eight students. I put down three tape ladders in my classroom on the floor. If the number of students is not a multiple of eight (like my class was), the teacher could place the extra students on the side as coaches. To up the responsibility of the coach, add the rule that no one inside the ladder can talk to anyone else.

Teacher Moves

While this activity was going on, I floated between the groups and listened very carefully. I wanted to learn about which of my students would step up and take initiative; which would be a leader; which would be concerned about the frustration of others and take action to minimize other students' discomfort/anxiety. This activity helped me better understand how to assign groups for course work in a meaningful way.

Examples of Student Moves

Below is an example of what some students might do.

ExampleOfMoves

 

If the third blue player from the right jumps the lone orange player, the blue team has a problem. With two blue players in adjacent cells, the game is gridlocked and ends.

 

Computational Thinking

Once the students came up with the solution, I gave them the sequence "1-2-3-4-4-3-2-1" and asked them how it relates to this situation. Think of this sequence as the answer key.

1: Orange moves first
2: Blue moves next - twice
3: Orange moves three times
4: Blue moves four times
4: Orange moves four times
3: Blue moves three times
2: Orange moves two times
1: Blue moves one time

Low Entry, High Ceiling (Extending the Task)

  • Ask the students to come up with some pseudo-code to describe how they would build this game on a computer using programming applications.
  • Ask the students whether the strategy remains the same if there are teams of five? Or if there are two empty middle squares? Three empty middle squares?
  • Ask the students to write a program that allows the user to watch the game. Then ask the students to write a program that allows the user to play the game.ExampleOfMoves
  • Example from my classroom I had two students come up with different lines of thinking for coding this game on a computer. One student thought of a number line to label each cell, using the values -4, -3, -2, -1, 0, 1, 2, 3, 4. Another student thought of simply have numbers represent each student. The starting configuration would be
    1 2 3 4 _ 5 6 7 8. Then, each move would be a shuffling of the sequence. The second row would be 1 2 3 _ 4 5 6 7 8. The third row would be 1 2 3 5 4 _ 6 7 8. The fourth row would be 1 2 3 5 _ 4 6 7 8. We had a really spirited discussion of the issues that could arise from each organizational coding strategy.

 

Like many Nebraska high schools, we are trying to improve our students' scores on the NeSA-M, our state's 11th grade standardized math test. We have a 35 minute period used to stagger our lunch on Tuesdays-Fridays in which teachers teach courses for enrichment or remediation. What follows is the description of one such day in my class for juniors designed to help prepare them for the NeSA-M.

I should also note that from a philosophical perspective, I think it's way more important that I teach my children mathematics well as opposed to teaching them just enough to get through a standardized test. Our department has taken the position that we could really care less about the test. Sure, we care if our students do well on the test, but it's way more important for us to focus on the 1,080 hours of instruction students will get in the classroom rather than one snapshot of what they remember on a single day.

FormulaSheetWithArrows

We spend some of our time in "NeSA-M CATS class" talking about the formula sheet students get to use on the state test. Our state test does not allow for calculator usage. I have some strong opinions on this, but that's a story for a different day. Because students can't use a calculator, I try to be intentional about pointing out opportunities to leverage the distributive property, factoring, association, subtraction, etc. to make the mental arithmetic a little less problematic. The three red arrows on the screenshot of the formula sheet indicate the computations I wanted students to focus on while we discussed the problems below. Students interacted in small groups and discussed methods of solution. I called students to the front of the room to present their thinking at the Promethean board. Then, we discussed as a class whether we agreed or disagreed with the logic presented by the students. Below is a brief summary of the highlights of each problem.

 

Problem 1: Given two ordered pairs, find the midpoint

The endpoints of a line segment are (12, 3) and (10,5). What is the midpoint of the segment? 

With my students I emphasize that when we wish to find the midpoint of two ordered pairs, we should split this task into the average of the x's and the average of the y's. I used the metaphor of streets in a city to communicate this point. For example, if you live on 5th street and I live on 11th street, then we should meet at the 8th street cafe for lunch. But if you live on 5th street, and I live on 14th street, then we will have to meet in the middle of a block rather than on a corner for lunch - since the average of 5 and 14 is 9.5. Below is a screenshot from what I put on the board after a student showed her work at the front of the room, work that was simply substitution into the given formula. I simply wanted the students to see beyond the exercise and to understand the concept of finding an average and its application to computing distances.

2_Problem_1_Work

 

 

 

 

 

Problem 2: Find the area of a given trapezoid.

3_Problem_2

I am bothered when I ask students, "How do you find the area of a trapezoid?" and they reply with something like, "I don't remember the formula." Students spent time working in small groups computing the area; many were successful. I think their success could be attributed to this being a tidy exercise in substitution. We spent time kicking the expression provided by the formula sheet around algebraically. I wanted students to notice the connection to the previous problem, that we can shift our thinking once again to thinking about averages. We can think of the area of a trapezoid as "the average of the bases multiplied by the height." (see screen capture from Promethean board below).

4_Problem_2_Work

 

Problem 3: Find the area of a rectangle with information about the rectangle's perimeter.

5_Problem_3

This is the problem I want to feature in this post. I had students work once again in small groups to solve this problem. Keep in mind many students in the room were in math classes below Algebra 2 (as high school juniors) and many of these students struggle with math and math efficacy. After three minutes working in groups, I asked if any student that hadn't gone up yet wanted to share their solution method. I had a student go to the front and write the following. (It's in my handwriting, because he quickly erased it since he thought he was incorrect).

6_Problem_3_Work_from_Student

 

I experienced tunnel vision myself because I was so used to working problems like this a particular way. At first, the student wrote down the two division problems, as they are shown above, and then simply circled 98 as the answer. With a little coaching, the student wrote the rectangle on the left and labeled its dimensions. I asked if any other student could go to the front of the room to explain the student's thinking. A girl quickly said, "I've got it!" She, too, struggles in math class, but went to the board anyway. She said it was like the averages we did earlier. I still couldn't reconcile the comment and what the students described. After a minute or so of heated discussion, we understood what the boy was trying to say.

7_Problem_3_Student_Work_Formalized

 

The problem was "like the averages" because we can think of three sections of fencing material, each with an average length of 14. In other words, the two lengths of the rectangle form the two segments 14 cm long, but the third segment of 14 is really the sum of the two widths. A formalization of the student's work reveals he chose to use the length as his single variable.

I couldn't immediately understand what he was doing because I was so focused on using the width as the single variable. This approach is really a personal preference, because using the smaller quantity as the single variable helps us potentially avoid fractions. (The constants in this problem are written in a way to facilitate mental arithmetic, but we can't always assume the constants will be so pretty). Another student wrote up on the board the solution method using the width.

8_Problem_3_Using_Width

 

Even though the math here isn't tough, I felt very proud of myself as a math teacher during this lesson. Because I chose my words carefully and never confirmed nor disconfirmed whether the student was right or wrong, the students in the classroom responded by unpackaging their thinking. The students had a conversation filled with respectful disagreement and clarifying questions like, "what makes you say that?" This was early in our rotation, so I knew very few of the names of the students in the room. The smiles on the kids' faces were great to see. Just before the bell, a student said, "I wish I could learn math this way all the time."

So, the challenge for me as a math teacher and a leader of teachers is to figure out how to honor the student's comment on a large scale in our school. As the students filed out of the room, I spun in my desk chair, opened my fridge, grabbed a yogurt, and thought about what would need to happen in all math classrooms to make this type of discussion possible.

I keep going back to the idea that the teacher is so critically important. The teacher poses the task. The teacher asks the questions. As teachers, we have ample experience with students asking underdeveloped or vague questions. Low quality questions get low quality answers. On the other hand, high quality, well-formed questions get high quality, well-formed answers as time allows.

So, here's the question: What professional development sessions, teacher collaboration sessions, or activities already exist that help teachers refine their ability to ask high quality questions and cultivate an environment facilitating mathematical discourse?