Cultivating Teamwork in Math Class

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

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.

What If...?

A teacher could use an agility (speed) ladder for this activity. 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.

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.
• 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.

The NeSA-M Formula Sheet: Helping Students Notice Connections

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.

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.

Problem 2: Find the area of a given trapezoid.

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).

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

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).

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.

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.

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?

Instructional Rounds in Our HS Math Dept

I would love to get some feedback on our second version of the form we use while on instructional rounds at our school. To see the PDF of our form, please click the following link.

So far we have only had one set of formal instructional rounds. Our math department has common planning time as an administrative support from our administrators and counseling department, something for which we are eternally grateful. Our department went to visit two language arts classrooms and one science classroom during our first instructional rounds session. Below is a scanned image of the revisions made to the original form as a result of our conversation after rounds took place.

12-4-14 Instructional Rounds Form Revision

I collaborated with a social studies teacher at our alternative school while constructing the form. We drew from the works of both Charlotte Danielson and Robert Marzano. Our math department has spent some time together in our PLC (Professional Learning Community) meetings revising the form prior to its initial use.

What we have found with both novice and experienced teachers observing expert teachers is that it is really, really easy to be dazzled and get lost in the instruction and action in the classroom. Teachers often forget the reason why they came in the room in the first place. Filling out the form has three purposes:

• Keep the participating teacher focused on observing the actions of the students and the observed teacher. This minimizes the chance the participant is passively captivated.
• Provide a basis for the conversation afterwards. Almost like an autopsy, what did the teacher do that facilitated learning? What did the students do that facilitated the learning?
• Scaffold the experience of being evaluated by an administrator for the participating teacher. The participating teacher learns what it feels like for an administrator to go into someone else's classroom looking for particular things. This in turn reduces the anxiety the teacher has towards having a formal evaluation.

Here's some background information about us. Our math department has nine teachers. One of these teachers spends her day at our alternative high school. The other eight of us are in the same building. One of our eight in the building is on a different floor, but roughly speaking, we are geographically located in the same area in our school building.

We have worked really hard to establish a culture that engages in cross observation on a consistent basis. Our math teachers know that, to improve their own instruction, they must learn from the instruction of others. For a year and a half, our department members have spent portions of class periods observing their peers in the act of teaching about once every two to three weeks. The frequency of the observations usually depends on how busy the teachers are, time of year, etc. But our conversations are always positive and lead back to supporting one another.

I have spent a lot of time visiting with Angela Mosier at Omaha Westside and followed the example set by Kristi Bundy at Ashland-Greenwood within our own state (Nebraska). They have established great cultures within their schools using this as an in-house professional development strategy. After observing the teaching of others, we send out a "positive blast" email to the observed teachers. This email highlights the positive actions and learning we observed in the classroom. Everyone involved learns something about teaching as well as learning in the classroom.

Our goal at SHS is to participate in instructional rounds on a once per month basis this spring.

Here's a video that explains different types of collaborative structures in a middle school setting. Administrative support is essential to creating a culture of collaboration and trust. Cross observation is featured as a professional development tool at the [2:05] time signature.

The Most Visually Appealing “50”

A few years ago, one of our local elementary schools had a 50th anniversary celebration. The principal of the school contacted me regarding an anniversary photo the school wished to take with students and staff. The principal asked our Math Theory & Problem Solving class to come up with the "most aesthetically" pleasing dimensions for the photo. Our class was tasked with using mathematical methods to systematically design the dimensions of both digits.

Here's a quote from the newspaper article about the photo before it was taken:

Browning, who has taught at the school since 1976, starting as a music instructor and then principal in 1998, said Friday would begin with the weekly assembly in the gym.

“Each class will perform a song from each decade, starting with the 1960s,” she said. Browning added the day-long celebration for students would also include special drawings for prizes, carnival games and the group photograph – if it’s not raining. The students and staff will form a “5-0” and have their picture taken by Downey’s Photography from the Scottsbluff Fire Department’s aerial ladder truck.

The principal emailed some information about how many students and staff would be involved in the photo, the approximate height of the aerial ladder, and that the photo would be taken in a field adjacent to the elementary school.

Our class used mathematics to figure out the size of the viewing field of the camera (based on conservative estimates regarding the camera lens viewing angle). The students did some research and found information on the Golden Ratio, a number that appears over and over again in artwork.

The students wrote instructions for the staff members to utilize when organizing the photograph. We discussed the challenge of keeping that many students organized and engaged for a sufficient period of time in which the photographer could take the photo.

Below is a screenshot of the photo taken to celebrate the 50th anniversary of Westmoor Elementary.

(photo by Downey Photography)

After the photo was taken, I asked my students, "If we compute the ratio of the width of each number to the height of each number, how close is the actual value to our recommendation of the golden ratio (approximately 1.618)?"

Below are screenshots from the Geogebra worksheet examining how close the dimensions are to the target value of 1.618.

Screenshot with the initial question. Students can use measurement tools to judge whether or not the 5 and 0 in the photo meet the desired dimensions.

Clicking on the checkbox in the Geogebra sheet (Show / Hide Measurements and Ratios) reveals the details in evaluating how close the ratio of height to width for each digit is to the Golden Ratio.

Here are the details to the Geogebra sheet. If you would like to download the Geogebra sheet and mess with the values to see what happens, you can find the Geogebra sheet here.

Stumbling Into an Interesting Example (Inverse Functions)

I was working on preparing problems for precalculus class on inverse functions. I started to type the following run-of-the-mill problem type:

If f(x) = blah and g(x) = bleh, show f(x) and g(x) are inverse functions.

And the work usually goes something like this:

f(g(x)) = f(bleh) = algebra kung fu happens here = x
g(f(x)) = g(blah) = some more algebra kung fu = x
Then conclude f(x) and g(x) are inverses.

Without thinking about it, I typed f(x) = -x - 6. Then I wrote the statement on my pad of paper, started working... exchanged the y and x, solved for y.... and got the exact same function.

y = -x - 6. Hmmm.

A part of me wondered if I had made a careless error. Double checked. Nope. No error. I wondered if the graph of the function was its own reflection across the y = x line.

Sure enough... Since y = -x - 6 is perpendicular to the line y=x, it will be its own reflection across the y=x line and consequently its own inverse. In fact, this made me think of an interesting question to pose to my students...

"Can you define a class of linear functions that are all their own inverses?"

In hindsight, perhaps I should be more mindful when constructing tasks for my students. But then again, this would be a great discussion to have with my students. Why does the function end up being its own inverse? Can we think of non-linear functions that are their own inverse? Can we define these classes of functions carefully?

Midwest Noyce Conference: “Using Statistics in Math Classes”

This post contains resources for the talk "Using Statistics in Mathematics Classes" given by Jason Vitosh (Falls City High School, Falls City, NE) and myself at the Midwest Regional Noyce Conference on Thursday, October 2 from 2:15 pm - 3:00 pm.

Click on the link below to access the presentation file containing resources, images, and links.

10-2-14 Stats is not Math

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Differentiation in Math Class: Make Students Write

My elective math class has all grade levels represented within it. While not ideal, this is a feature of our scheduling system. So I have freshmen in Algebra I all the way up to seniors that have already completed Calculus AB. This poses a huge classroom differentiation challenge each Monday, Wednesday and Friday we hold class. Here is an instructional strategy I use that gets students writing about the mathematics they do in class.

Students worked on one of four things in the computer lab last Friday.

1. Construct 2013 Probe I Problem 7 diagram (2D)

2. Construct 2013 Probe I Problem 23 diagram (2D)

3. Construct 2013 Probe I Problem 11 diagram (3D)

4. Work on Alcumus problems independently

After we spent approximately 55 minutes in the lab, we returned to my classroom for a writing activity.

Here is the writing prompt I put on the board:
(Writing exercise on a separate sheet of paper to turn in to me)
Think about what you learned about the diagram or diagram(s) you built in Geogebra. Write a letter to the you of October 12. How did building diagrams in Geogebra help you understand the problem better?

I put eight minutes on the clock and informed students they would need to continuously write for the eight minute timeframe. Here are some samples of student writing from the activity. For convenience, I have inserted another copy of the problem. Immediately below each problem appears student writing associated with the problem.

The Students revealed their thinking about these math problems throughout their writing. While some students chose to concentrate on the construction process in Geogebra, others also revealed some of the mathematical structure they encountered while making the diagrams.

Writing samples from students that worked on Alcumus:

These writing samples revealed to me the depth of student thinking going on in the classroom. If I could have a superpower, I would be a mind reader. Then I wouldn't have to guess at what my students are thinking. Having the students write for an extended period of time gives me insight into how they are seeing the mathematics and gives me ideas on how I can help further their understanding and guide them as they struggle.

I collected these writings immediately after students completed them. I ran the pages through my ScanSnap scanner and converted them to a PDF for me to review later. I told students we would get these writings back out in a month's time, emphasizing the need for specificity on what they were working on and what they learned that day.

Going forward, these writings help me be more efficient with respect to differentiating classroom instruction. We don't need to be working on the exact same thing at the exact same time at the exact same pace for the students to engage in meaningful problem solving.

Does the Size of a Fraction Bar Matter?

When we teach children to add, subtract, and multiply, we commonly see a vertical format used in school.

9
+ 6
15

Then, the student learns division. Students usually see the division algorithm represented horizontally.

A student may not, beyond a worksheet with dozens of problems, be comfortable with a vertical format for division.

It almost feels strange to me to the type the problem vertically. Then comes fractions. We did the following exercise in a workshop I led today. I wanted participants to consider the question, “Does the size of a fraction bar matter?”

Below is an image of the problems we worked on in groups. Calculators were allowed. Spirited discussion and abundant disagreement followed.

As the back of the textbook often says, “Answers will vary.” The participants’ responses, split up by problem, appear below.

How can we resolve this conflict? Let’s type it into a reliable computation source, like Wolfram Alpha. It depends on how we type the input.

Above is one way to type the input. Below is another way to type the input.

Why the difference? Because multiplication is commutative, but division is not. 3 x 4 is the same as 4 x 3. But 3 divided by 4 does not yield the same result as 4 divided by 3.

In the original input, where I typed 3/4/1/2, the computer interpreted this as the product of 3 and the multiplicative inverses of 4, 1, and 2, like this:

In the second example, we see the computer associates the fractions ¾ and ½, treating the middle fraction bar as a grouping symbol.

The confusion for me stems from converting between vertical format and horizontal format. We teach students to follow order of operations, to divide in a horizontal statement as they encounter it from left to right. In school mathematics, we commonly write the fractions vertically when we mean to take the ratio between two ratios, specifically ¾ and ½ in this case.

Here is a state assessment practice item, taken from the Nebraska Department of Education website, that demonstrates this understanding of the vertical format when dividing fractions.

Let’s look at this Word document, specifically this problem. The author of the item intends for students to invert 7/3 and multiply. I worry a student may see this problem in a horizontal format as the work shown to the left, which gives a result which is obviously not an available choice, but if we are examining item reliability, I wonder if we should have the conversation about whether this expression should be typed in the following way:

Again, I’m just posing the question about the test item. I want to be careful in how I represent problems with my students. I want my own students to understand the context matters. I want my students to be procedurally fluent with fractions when they solve problems in the world. The conversation in this morning's workshop was great. Participants reflected on how what they say to their students while they teach the students can have unintended consequences on how students view procedures with fractions.

Here’s a link for a great blog entry on this topic:
http://pballew.blogspot.com/2009/01/why-we-flip-and-multiply.html