Teacher Think-Alouds

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?



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.

MonaLisaGoldenRatioOur 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.

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?

2014-10-02_1234This 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












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.

I was trying to write an item for an assessment where I would give a student a graph of a piecewise function and ask them questions about the domain, range, and to evaluate the output value for a specific input value - for example, find f(-3). The purpose of this post isn't really to pit Geogebra and Desmos against one another; rather, I want to make note of some of the things I was thinking about as I tried to make a piecewise function graph in each program. (Disclaimer: I am not an expert at this. I have much more experience with Geogebra than with Desmos. I want to see what the differences are between the programs to figure out when to use each of these powerful tools to enhance my instruction in mathematics. I am sharing my thinking about this task.)

Here is the graph I made using Desmos: Desmos_Open_Circle_Piecewise

As I typed the function syntax into Desmos, I thought the editor was a little more user friendly than Geogebra. I typed <= and the editor automatically generated the less than or equal to signs for the restrictions on x. When I wasn't sure what to type, I browsed the examples of projects submitted by Desmos users found on the Desmos homepage. Ideally, I want to capture this graph and place it on an assessment. A photocopier may not pick up on the sections of the function on the graph given the lack of density (being able to make the segments and curves thicker). The way to make these curves denser was not immediately obvious to me. OpenCircleProblemThe circle centered at (1, 1/3) with radius 1/10 is my effort to place an open circle on the graph. To clean up the image, one thing I could try is modifying the restriction on x [for example, writing 1.2<=x<=4 instead of 1<=x<=4] so the user does not see the part of the function jutting into the circle shown at the left.

Here is the same graph I made using Geogebra:

GeogebraPiecewise After making the graph in Desmos, I assumed I could use similar syntax to make the graph in Geogebra. Using similar syntax, I had a problem with the restrictions in Geogebra. Each function has a default y-value of 0 for values of x outside the restriction. Pictures are, after all, worth a thousand words... here are the three functions shown individually. Take a look at the x-axis. Red_Parabola Blue_Line_with_Positive_Slope   Orange_Line

Here is the exact syntax I typed into the Geogebra input bar for each of the above pictures.
(1 / 4 (x - 1)² + 1) (-3 ≤ x ≤ -1)
(x - 2) (-1  <  x  <  1)
1 / 3 x (1 ≤ x ≤ 4)
I incorrectly assumed the syntax would be similar to that of Desmos. I knew from experience I could clean this issue up by using Condition to Show Object in the Object Properties menu if I had to, but I couldn't remember exactly how. I went to Youtube and found a video on graphing piecewise functions in Geogebra:

Below is an image of the corrected Geogebra graph using the appropriate If[ ] commands to define the rules f(x), g(x), and h(x). GeogebraPiecewiseCorrected
Here is the corrected syntax I typed into the Input Bar to define f(x), g(x), and h(x):
If[-3 ≤ x ≤ -1,1 / 4 (x - 1)² + 1]
If[-1  <  x  <  1,x - 2]
If[1 ≤ x ≤ 4,1 / 3 x]
This approach eliminated the x-axis issues from the improper syntax I used at first. These graphs show some of the thinking I do day-to-day as a mathematics teacher trying to construct examples to display in class and problems to use in assessments. If somebody reading this has any advice that could help me become more effective with using Desmos or Geogebra for this purpose, please email me at saaberg@sbps.net or find me on Twitter (@ShelbyAaberg). **Update! See below for additional support on Desmos use. Thanks to Eric Berger (@teachwithcode) and Desmos.com (@Desmos). Endpoints_In_Desmos  


Here is the additional resource from @Desmos. Desmos_Tweet

Disclaimer: The following is an idea I have been thinking about this week. I have absolutely no idea whether it would work. I don't know if anyone anywhere is approaching principal work in this way. Some of this has been daydreaming or thinking during a long drive to a conference. These ideas may seem disconnected, but I will try my best to explain the relationships I see between these ideas.

We had some professional development days to start this week. I enjoyed two presentations Monday by David Webb from the University of Colorado Boulder and the Freudenthal Institute. His morning and afternoon sessions focused on formative assessment in mathematics. When he saw many morning participants planned to stay for the afternoon session, he quickly talked about something he and his colleagues use to teach early computer programming concepts to middle school students.

Dr. Webb posed this question to the audience: how do we design intelligent ghosts that will actually chase Pac-Man? The mathematical process, known as collaborative diffusion, describes a possible method for programming ghosts to effectively chase Pac-Man. Here's a link to an academic paper on collaborative diffusion by Alexander Repenning. A screenshot from the paper appears below.


Think of the spaces around Pac-Man as the yellow mountain above. The ghosts want to climb the mountain - and effectively destroy Pac-Man - by climbing to the top of the mountain as quickly as possible. I was thinking about this idea of how the ghosts are chasing down Pac-Man. Then, I thought about how we often in school try to chase down behavior. For example, when a teacher is in the hallway greeting students, sometimes amorous couples try to hide from the teacher's line of sight. If the teacher has to help a student in the class, and cannot man the hallway post, then the threat of punishment is gone. Speeding tickets then came to mind. I thought about how punishment rarely works well as a behavioral deterrent. Drivers may choose not to speed when a police officer is nearby, but once the police officer leaves, look out.

To tie this stream of consciousness back to teaching, think about how often teachers must identify, on the fly and while making mental decisions regarding content delivery, students misbehaving in the classroom. Proximity works well as a deterrent - walking near the student, pointing at the open book on the student's desk as the teacher walks by - but this technique also has its limitations. As soon as the teacher walks away, the student may misbehave again.

Then I thought about how tough it can be to be a principal. Here's a great post on how to navigate the frequent interruptions a principal faces. The principal position can sometimes be very similar to the function of a police officer - a deterrent. But, as a principal leaves, so does the threat of getting into trouble, and the idea is the same as the teacher that walks away from the student's desk. How do we address this behavioral piece while teaching? How do we keep students on task?


The possibility of being called on randomly. 

While thinking about police officers, I thought back to another article I read in a grad class about The Santa Cruz Experiment. The article, which appeared in Popular Science, described predictive police work. <think Minority Report> A mathematician designs an algorithm based on data for allocating patrols. Though random phenomena may be wildly unpredictable in the short term, long terms trends and patterns emerge.












Tying this back to the principal idea... if the ghosts chase Pac-Man... doesn't the principal chase the behavior? Suppose we try to incorporate a random mechanism into the principal's behavior in an effort to make chasing this behavior - just like the patrols in Santa Cruz - more efficient. Let's set up an imaginary simulation. We will declare the following events as things the principal could do.

0 = observe 1st floor hallways
1 = observe 2nd floor hallways
2 =  observe 3rd floor hallways
3 = observe 1st floor classrooms
4 = observe 2nd floor classrooms
5 = observe 3rd floor classrooms
6 = observe school entrance / parking lot exit
7 = monitor stairwells
8 = monitor cafeteria
9 = monitor library

Then, we could use some sort of random process to generate a random behavior for the principal.


Looks like today's focus will be first floor classrooms. Because all outcomes are equally likely, we now have a mechanism like the Popsicle sticks in the classroom. This will be a more efficient approach to deterring negative behaviors among students as well as teachers. This would also give the impression to students that the principal could be anywhere. Thinking back to collaborative diffusion, and Pac-Man emitting a scent that can be chased down by ghosts... the metaphor places data in the role of the scent. We have plenty of sources of data on student misbehavior. Also consider certain events more likely given certain days of the week and months of the year. Isn't a student more likely to get a discipline referral close to a vacation, after a long block of no days off from school, because teachers' behavioral tolerance is lower? Isn't a staff member more likely to violate dress code on a Friday? Aren't students more likely to be off-task close to passing periods? We could use data (and a different random digits assignment scheme) to make an attempt at 'predictive principalship' much like the predictive policing in the Santa Cruz Experiment article.

It would be interesting to see whether this is a viable strategy for administrators to use.

P. S. If you've made it this far in the article, please be sure to read the disclaimer at the top of the article a second time.

P.S.S. I know the title doesn't quite jive with what was discussed here... since the metaphorical principal is the ghost and the metaphorical Pac-Man is the behavior... but the title is way more catchy this way.