Teacher Resources

I started blogging about my teaching as a way to help me reflect on what I do in my classes. We worked on sequences in Precalc today. Here are three problems we worked on in class... well, at least we did the setup using the graphing calculator. It has been a long time since I have taught sequences, or Precalc for that matter, so I had to give myself a refresher over the weekend on how to use the calculator to facilitate some of the sequence operations we use to solve problems in the world. I had to download the TI-84 user manual and work through some examples to remind myself about these things. The screenshots that appear below I made using Jing and the TI-Smartview emulator. Essentially, this post is a reminder note for me on teaching sequences using the TI-84.

Roberta had $1525 in a savings account 2 years ago. What will be the value of her account 1 year from now,assuming that no deposits or withdrawals are made and that the account earns 6.9% interest compounded annually? Find the solution using both a recursive and an explicit formula.



Define the sequence recursively and graph the sequence
{-4, -8, -16, -32, -64, ...}.



A really big rubber ball will rebound 80% of its height from which it is dropped. If the ball is dropped from 400 centimeters, how high will it bounce after the sixth bounce?


Here is a prime example why Twitter is a great collaboration resource for math teachers. Last night I was killing time while waiting for a haircut. Reading through some tweets, I noticed a chat going on with the hashtag #ggbchat. With some luck, I caught the very end of the session and posted a question about something that's been bugging me about Geogebra.








I use Geogebra to analyze student summative assessment data in my classes. I like to sort the data to guide me when deciding which students I should group together for class activities. Sorting the data inside a Geogebra spreadsheet would eliminate an extra step for me (specifically, entering the data into Excel and sorting prior to copying & pasting data into Geogebra). I would think the software should allow a user to select a list, right-click, and then be given the option to sort the column of data. Here's a solution to the problem I faced, compliments of Geogebra guru John Golden (@mathhombre).



John also followed up with an idea to create a "Sort" button using a script.





Here is a screencapture of John's suggestion for making a Sort button.



I don't know how many elementary school teachers use Geogebra in their classrooms, but I suspect it's probably not many. Yesterday I led a follow-up workshop session for a group of mostly K-5 math teachers. I showed how to use Geogebra to make a number line with tick marks noting common fractions.

A thought experiment: suppose you are a K-5 math teacher and want to show students how to compare the fractions 4/9 an 3/7 on a number line. The decimal expansions show these two values are pretty close to one another. 3/7 is approximately 0.428571 and 4/9 is approximately 0.444444. Drawing these two fractions on a number bounded between 0 and 1 might be tough. They may even appear to be the same if the scale isn't discerning enough. What would a person do to find number lines to make teaching materials? Like many people, he or she might go to Google and do something like this:



We may able to find number lines with sevenths and ninths indicated, but they will probably be separate. It may take some time to dig through these images. Should a teacher be at the mercy of materials made by somebody else? What if the teacher could design his or her own materials instead? Enter Geogebra.

Here's a PDF I made with some basic instructions on how to start building number lines: Creating Number Lines with Fractions in Geogebra

In addition to a PDF showing an introduction on how to make these things, I also posted a Geogebra sheet on the Geogebra community site for teachers to download for free. An image of the sheet I built appears below.




Type I and Type II error are concepts my students have struggled with year after year. This school year, I decided to do something different in AP Stats.

KNEB news story: "Rabies Outbreak Soars in Goshen County"
(this particular county is very close to where we live in western Nebraska)

The news story above was posted the night before I was slated to introduce Type I and Type II error for the first time. The text of the news story caught my eye because something, literally, didn't add up.

...since February 7th of this year, out of 19 dead skunks collected, 16 tested positive for rabies. They also had 1 red fox test positive for the disease... Mills will be retiring next month and has been at the lab since 1984 and has never seen, even in an outbreak situation, this high of a percentage rate of positive samples at 89.5%...

It is interesting in the news story that the reported percentage, 89.5%, does not match the fraction 16/19 (which is approximately 84.2%). The 89.5% figure is a computational error. The person computing it took included the rabid red fox, despite the fact the red fox is a different species, and also did not increase the denominator by one. The 89.5% comes from the fraction 17/19, which is obviously not valid. (We had a nice class discussion over whether this was a computational error or a purposeful error meant to sensationalize the news story.)

When I first saw this story, I decided to scrap what I had planned for the day in AP Stats and made this worksheet in the planning period I had right before class.

APS Rabies Outbreak 3-7-14

I spent about 45 minutes trolling the internet - in particular, the USDA and CDC websites - trying to come up with count data for how many animals are tested for rabies in Wyoming in a given year. My searches took me here and here. I found data from 2010. 32 wild animals tested for rabies in Wyoming in 2010 - 2 were cattle, 12 were bats, and 20 were skunks. I could not locate more recent data or 2010 data on the total number of animals tested, which posed a problem, because we are really interested in the total number of animals tested to inform us on what proportion of rabies cases we expect. I wanted to reverse engineer the actual percentage of rabid skunks we expect in Wyoming... how high does the percentage of positive tests have to be before we release such a warning to the public? 50%? 60%? 75%?

[side note: I learned a great deal about how rabies spreads and which animals are typically affected. See the map below.]









U.S. map showing which animal is the most frequent rabies carrier by region [source: CDC]

I asked the students to use their TI-84 calculator and, through trial and error, to determine the population proportion value p (the actual proportion of rabid skunks) that would lead to NOT rejecting the null hypothesis for each alpha level. Here's a slide with the work a student wrote on the board:



I had students use trial and error, with the TI-84 one-proportion z-test, to determine what the decision would be (whether we would reject or fail to reject the null hypothesis) for different values of the assumed population proportion p.

For example, with an alpha level of .05 and assuming the population proportion of all skunks that have rabies is 50%, we would reject the null - the population proportion being 50% and conclude in favor of the alternative, that the proportion of skunks with rabies is likely higher than 50% based on the sample, since we expect to see a sample this extreme (16 out of 19 skunks rabid) only 14 times out of every 10,000 samples due to chance alone.

Through trial and error, my students found we would not reject the null hypothesis given a sample of 16 out of 19 skunks being rabid if the actual proportion of rabid skunks was 66.37% or higher. 2/3 of all skunks being rabid is pretty scary to think about, but if we are trying to determine why the USDA released a rabies warning, this is useful information for us.

I had students formulate the idea of a Type I error (mistakenly rejecting a true null hypothesis) and a Type II error (mistakenly failing to reject a false null hypothesis) in the context of this problem through discussion.

Type I error would mean the proportion of skunks with rabies is greater than the assumed proportion when in fact the assumed proportion is true. This would mean potentially raising a false alarm, or releasing an outbreak alert when one is not needed.

Type II error would mean concluding the proportion of skunks with rabies is the assumed proportion when in fact the actual proportion of skunks with rabies is higher. This would mean not alerting the public to a potential outbreak when in fact an outbreak is actually going on.

Students collectively agreed Type II error is more serious in this setting. The students discovered through this activity the decision they should make about the alpha level - maximizing the chance of a Type I error in favor of decreasing the chance of a Type II error.

My students breezed through the exam covering Type I and Type II error for the first time in my career. 🙂

I have been in Washington. DC for the past few days at the Teaching & Learning 2014 conference in our nation's Capitol.

When I stood at this podium, our meeting and session with policy makers had just ended. No audience occupied the opposite side of the podium. I began thinking, however, about the opportunity for positive change an educator could bring to a public office. Towards the end of the conference, we had the privilege to hear Angela McLean. Here is an article about Angela.


Below are some of the resources and knowledge from the conference sessions I attended.

Ralph Smith's Grade Level Reading Campaign



Myself, Deborah Ball, and colleague Dan Schaben after Deborah's outstanding presentation.

Deborah Ball on "Safe to Practice"

Linda Darling-Hammond's Getting Teacher Evaluation Right


Sarah Brown Wessling, featured on the site Teaching Channel, helping teachers connect with great instructional examples

I had a conversation yesterday with a strong second year teacher. Afterwards, I thought about what types of things would have been helpful for me to know or think about when I was a second year teacher. What follows below is my view on staff meetings. Suppose we ask some theoretical high school math teacher in the U.S. the question, "Why did you become a math teacher?" Here are some 'said no one ever' types of responses:

  • "I would like to spend the better part of my twenties, thirties, forties, and fifties grading papers."
  • "I would like to teach a class where kids bring negative attitudes toward the subject matter."
  • "I want to teach a class where students will incessantly ask me, 'When will I ever use THIS in my life?' "
  • "I really enjoy staff meetings."

Let's examine the last one for a moment. Staff meetings. Despair.com has some hilarious de-motivational posters. Here's one of my favorites.

None of us is as dumb as all of us.

In my teaching experience in two very different buildings, I have been at staff meetings that were useful and productive. Unfortunately, these tend to be the exception, not the rule. A contributing factor is that the person or persons running the meeting sometimes do not follow good teaching practice. What would students do if the teacher talked non-stop, without asking for feedback, for 45 minutes? For 90 minutes? The human form is not designed to sit for extended periods of time. Staff meetings serve their function of being a way for a person to communicate information to large groups in one shot, but the problem with passive lecturing is just like a broadcast from a radio tower: if no one tunes in, what's the point?

In my building, I teach on the 3rd floor of the classroom wing. It is rare that a person walks up a set of stairs just to say hello. This means there are teachers in the building, on a staff of roughly 60, I often go without seeing for long periods of time. I ran into a teacher in the hallway last night I hadn't seen for a while, and we had a great conversation. I found myself wondering why staff meetings often do not produce high quality conversations that have impact on teaching, learning, and school culture. What prevents us (teachers) from delving into difficult issues or having productive conversations?

  • Ideas have inertia. It takes force to overcome inertia. That force is someone else piping up and saying, 'yes, I agree with that,' before others jump on board. Sometimes it's too early in the day... or too late in the day... or the person that would have agreed wasn't listening.
  • The vocal minority often do not speak for the entire staff but act as if they do. The staff becomes frustrated because the opinion being expressed does not reflect what is typical of the staff.
  • Meetings decompress to the time allotted. Saying 'we have 15 minutes to make a decision' is very different than saying 'we have 45 minutes to make a decision.'
  • Democracy tends to be a very inefficient way to govern. Power dynamics often play  a key role when we open up large group discussion in a staff meeting. The same good idea sounds very different (and garners a very different reaction) when it comes from the mouth of the venerable veteran as opposed to the first-year rookie.
  • And the list... like some meetings... goes on and on and on and on...


So... can we do better? What does better look like? Here are some ideas I have about adapting staff meetings to better serve the intended audience:

1. Seek ideas from sources online. I'm not the only person that has considered this staff meetings issue. Here's a great post on finding purpose in staff meetings. And here's another one. And another one. The World Wide Web is chock full of great ideas. Some are not so great and require a bit of sifting, but mistakes can be made in a good direction.

2. Change our perspective. There's plenty of drudgery in mathematics, for example. Many times, I have worked through exorbitant amounts of algebra to discover I made an early error and wasted an awful lot of time. Sometimes we have to endure in order to be rewarded. Staff meetings can provide valuable insight into teaching and the culture of our school building. The conversation can go places that will benefit students but we must first put ourselves in the right frame of mind.

3. Take information sessions and make them webinars. It is pretty frustrating to teachers when nothing happens to the teacher that failed to show up to the 7:00 am staff meeting. The principal 'confronts' said teacher: "Oh, you missed the meeting? It's ok, I just went over some information, catch up with me later for 10 minutes <when the meeting was 45 minutes long!>" Make a video and send the link through email. Tell teachers to watch it before the meeting. Make the meeting a productive discussion about the information in the video... which leads into my next suggestion...

4. Provide opportunities for the conversation to go where it needs to go. We've all seen the ambitious agenda, the bulleted list the administrator or presenter intends to 'cover,' but our subconscious quietly chuckles in the background knowing full well the presenter cannot possibly span the agenda in the allotted time.

5. 45 minute meeting? Do it in 25. Take the allotted time for a meeting, cut it in half, then add 5 minutes. When I share ideas on Twitter, for example, the 140 character limit pigeonholes me into being succinct. The 140 character limit forces me to focus only on the important stuff. Cutting staff meeting time down is analogous to the 140 character limit on a Tweet. Just the important stuff, please.

6. Invite staff members to speak as often as possible. Think about the typical classroom. Kids often tune out the instructor, but as soon as a kid is called to the front of the room, that kid often commands the attention of the room simply because he or she is a peer, not a viewed authority.

7. Don't waste time. Time is the most precious currency of the school and of the teacher. If a meeting is called, it better be worth it to all parties involved.

8. Have a plan. A bulleted list or an agenda is not sufficient. We should be clear about the purpose and the function of the meeting. What if an administrator walked into a classroom and the teacher was entirely shooting from the hip? Or that the teacher had not anticipated student misunderstandings or misconceptions? We should have the same high expectations for staff meetings, too.

9. Consider the man-hours that go into a meeting. On a staff of 60 teachers, for example, a meeting lasting one hour, at a rate of pay of, say, $30, means the taxpayer just shelled out $900. What was the outcome of said expense? Was it worth the expense? Did it improve learning outcomes for students? Did it improve the culture of the school? Was progress made? Ultimately, we are accountable to our customer base. In public education, it is the taxpayer.

10. Teach participants how to interact. This goes beyond 'establishing norms' to ensure respect and professional decorum. What I'm talking about is that teachers often do not receive training on group dynamics or on the psychology of group work. Understanding group dynamics can produce better outcomes from group work and alleviate some of the social friction we sometimes see after meetings conclude.

Have other suggestions? Please share.






This semester I decided to try something different with our approach to the first unit exam in our Trigonometry class. Typically, I give students in all my classes an "exam objectives sheet" prior to the exam. The sheet has a collection of objectives the students should master. Basically, I am telling them about what appears on the test and in what order. That way, there are no surprises or "Gotcha!" moments on test day. Here's a sample of a subset of exam objectives from an exam objectives sheet from AP Stats to give you a sense of how these sheets look in my classes:

In my teaching experience, students have struggled at times with being able to dissect the verbal instructions and figure out what potential exam problems might look like. I decided for the first Trigonometry test to run this process "in reverse."

Here's the exam objectives sheet students received Monday:













On Monday, I had students work in self-selected small groups of 3 to 4 students for 22 minutes (timer on the board) on writing the objectives from the problem sets. I modeled on the board how to write the first objective for section 1.1 problems 1-8.

Here's a sample problem from section 1.1:

Find the domain and range of each relation.
#4: { (2,5), (3,5), (4,5), (5,5), (6,5) }

I talked students through writing an objective for items of this type: "Given a relation, state the domain and range." Several students questioned why I did not start with "given a set of ordered pairs." I told them this choice was purposeful because in practice exercises, students were also given relations in graphical form. We discussed that starting the objective with the phrase "given a relation" captured both types of problems.

Students worked in small groups and wrote objectives for EVERY item. I circulated the room and confirmed EVERY student had written objectives for EVERY item. Then, we spent the remaining half of class discussing common misconceptions and errors on problems, addressing why each error occurred and what the student making the error would be thinking, along with why the thinking was erroneous. Here's the data for one of my classes (n = 16)

Before I start jumping for joy, it's probably a good idea to consider my other classes. Here is a comparison of the three sections I have.

The other two sections did not fare so well. The much larger dispersion among scores in the classes labeled B and C concerns me (s = 13.0229 and s = 11.3002). On our grade scale, the median in all three sections is an "A."

Teachers have to be data detectives to diagnose what students do or do not know. The three outliers above had some misconceptions, evidence to suggest little to no work outside class is taking place. As a practitioner, I also have to think about how effective I was or was not teaching the material. Obviously something different is going on in the class labeled "Column C."

When I compare the global mean and median across my three sections this year (mean = 92.0351; median = 96) to last year's data (mean = 85.61; median = 87), I am pleased to see a dramatic improvement on the assessment (yes, this is the exact same test I used last year). I will likely take the same approach to preparing students for the second exam and use data analysis to determine if the approach is contributing to the improvement in scores.

My self-imposed holiday blogging break is over. <cracks knuckles>

The holiday break affords teachers the opportunity to reflect. One dimension I think many overlook in teacher reflection is to sit and think deeply about one's content area. I would even be bold enough to say the best math teachers are mathematicians at heart. When I sit down to think about mathematics, as a high school math teacher, I like to approach it like a person that can solve a Rubik's Cube. When you watch this feature below on a speed solver, Chester Liam makes the following claim about speed solving a Rubik's Cube:

As a speed solver? No, there is no math involved, no thinking involved. It's just finger dexterity and pattern recognition. There is nothing, no thinking involved in the entire solving process. [1:02 - 1:14]

We can unpackage Chester's thoughts about the Rubik's Cube and how he solves it so quickly by applying mathematical structure to the cube. However, we should ask: what is the objective? What are we trying to do? There's gobs and gobs of mathematics wrapped up in speed solving a Rubik's Cube, but if Chester were to pay attention to the procedures he is applying, it would inhibit his ability to solve the cube quickly.

But consider this: what if Chester attempts to teach how to solve the cube to another person? What would he have to do? What examples, explanations, and demonstrations would he utilize to teach his pupil speed solving? What implications does this thinking have on teaching mathematics? There would be many features of speed solving the learner may not perceive until Chester brings it to the learner's attention. And if Chester's choices are calculated, deliberate, purposeful... the learner may be helped or hindered dependent upon Chester's ability to communicate his thinking which leads to the automaticity of the procedures he applies to solve the cube. Just like learning how to read or learning how to drive a car, we want to teach learners how to do these tasks so well they become 'automated' at some level.  To understand mathematics deeply, I believe it is often necessary to unpackage some of these automated tasks. I will share an example of such a mental exercise below.

Today I've been thinking about procedures we accept as true while doing math at the high school level, algebra in particular. As an example, suppose we want to determine the location of the x-intercept for the line 5x - 3y = 7. We might approach this 'task' in the following way:

And we might even graph the line to confirm our solution...

Yep. There it is. The x-intercept at (1.4, 0). As a student, we might simply yawn and move on to the next exercise. The student must recognize the y-coordinate of the line will be zero when the line crosses the x-axis. Yes, we have a solution, but I'm not so sure as a math teacher I'm satisfied to stop there. What if we take a different approach? Let's turn the Rubik's Cube and look at the problem another way. A 'typical' algebra student might subscribe to the church of y = mx + b and do the following...

Just another equally valid path to the value of the x-intercept. I'd like to focus on something in one of the above lines.

This statement says five-thirds of some mystery number is seven-thirds. So, what's the mystery number? I think if we asked a room full of high school math teachers to draw a diagram explaining why we multiply each side of this equation by the multiplicative inverse of 5/3, namely 3/5, we would get some really surprising results. It's not an indictment of teacher education. Rather, it's to say some teachers may not have considered the how's and why's of this procedure before for the same reason someone learning to speed solve a Rubik's Cube may miss a key structure. They may not have the experience of needing to know why it works. Rather, it was more important that they can find the x-intercept of the line; the multiplication by the multiplicative inverse was viewed as "below" the task or level at hand. Perhaps a student never asked "why?" at the critical moment to give the teacher pause.

I struggled to produce the corresponding fraction diagram without trying to reverse engineer the solution. When I write this blog, I often worry I might make a mistake that will be indelibly written into the electronic space of the Internet. But this worry violates the spirit of my blog's theme: that we need to make mistakes in a good direction to evolve our mathematical understanding. Here's the images of my failed by-hand attempts to generate this fraction diagram:

ChickenScratchesPage_1Chicken scratches, page 1















Chicken scratches, page 2

I had trouble thinking about making the diagram because 5/3 and 7/3 have the same denominator, so I wrote out some equivalent fractions. Then, I wanted to use "ninths" because that made sense to me in terms of the grid on the graph paper I had cut apart. But, I then realized I would need to cut fifths to find the mystery number, and I was REALLY struggling with trying to free-hand cut fifths with the grid in the background. That led me to coordinatize the points of the polygon. Then I had a problem with relationships between the linear units on the horizontal axis and the area (the fact the polygon is not "one" unit vertically... which is basically the notion of a unit fraction we see emphasized in CCSSM). So I abandoned the paper approach in favor of Geogebra because I could generate better precision and be more efficient with respect to time. <Sorry for the sloppiness of this paragraph, but it does describe my thinking and the mistakes I made.>

Below is an image of the fraction diagram I constructed using Geogebra.

When stating the equation , we should think of it as a verbal statement: "Seven-thirds is five-thirds of what mystery number?" Well, if the green polygon represents a whole, then the orange polygon is one-and-two-thirds of that whole. Then the green area of 1.4, which equals 7/5, corresponds to the solution. Mentally, how in the world did I end up with fifths, then? How did I know to cut the horizontal into fifths using vectors and vertical lines in the coordinate plane?

We can think of a fraction in the most basic way. Consider 5/3. If the denominator indicates the number of pieces we partition from a whole, and the numerator indicates how many pieces we possess, then I knew we needed to cut the orange rectangle in a way that would make five pieces. It gets to the root cause of WHY we invert and multiply. The numerator 5 becomes the desired number of pieces, considered in a denominator.

For the sake of time, I will stop my mental exercise there. Between working the problem, generating the fraction diagram by hand and on Geogebra, and typing this up, I've spent about two hours roughly on this article. This professional development is incredibly powerful for me as a teacher, and it's absolutely free (well, not quite free, I do pay for the website hosting, but you get the idea).

Let's end by stirring the discussion pot. Consider our understanding of how to find the x-intercept of the given line. Does our understanding, or lack of understanding, of the fraction diagram and how to construct the fraction diagram (essentially "invert and multiply" in many high school classrooms) inhibit our ability to solve the original problem? Is it still possible to understand the original solution without knowing all the nuts and bolts of the fraction procedure? Stephen Wolfram argues in favor of using computers to automate trivial computation procedures to help us access problems in the world outside school.  How will our teaching of mathematics change as computers continue to become faster and more powerful?

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BlueCowSuppose for a moment a parent shows a child images of 10,000 blue cows. Yep. No typos. Ten. Thousand. Blue. Cows. We are talking blue. Like the cow pictured at the left. 10,000 is a healthy number of cows. This would amount to showing the child one blue cow every second for roughly 2 hours, 45 minutes. The child might conclude, given this overwhelming evidence, every cow is blue. We could really rock the child's world by introducing an image of a white cow or a brown cow or a black cow. This counterexample would stimulate the child to re-evaluate his or her conception about what makes a cow. The hope would be that the parent would help the child understand better the definition of "cow." What makes a cow? Four legs? Not necessarily, the cow could be an amputee. An udder? Not necessarily - couldn't the cow be a bull? Horns? A tail? A particular set of adenine, guanine, thymine, and cytosine? Do we need to be that specific? We collectively have a definition we use for the animal "cow," and it's often based on experience. As you are reading, you may have even conjured up, in your mind's eye, a picture of a cow or two. Specifically, that definition of "cow" may vary according to the context in which we are operating.

Take the above situation and replace every word "cow" with "mathematical example." If developing students with strong mathematical understanding is the goal, we must be wary of how to model for students how to behave when the white cow or brown cow or black cow comes along.

My experience with teachers and students tells me identifying the domain of a function is often tricky business for students. What types of functions does a typical Algebra 2 book examine when looking at the domain of a function? They look at lines, parabolas, and cubics for sure. A student sees the domain of a run-of-the-mill linear function and a run-of-the-mill cubic function are the same. They see the domain for a parabolic function is also the same. The student's mind attempts to search for some sort of pattern. The student may formulate misconceptions that do not generalize. It is a challenging task for the teacher to develop a robust understanding of domain in students. We hope the student understands the notion of a function, input values, binary operations that are not closed in the real number system, etc.

I was helping a former student yesterday with some material for an upcoming College Algebra exam. We came across the following problem.

There are many things to like about this question. The student must have a pretty solid understanding of what linear functions look like. The student must understand A(t) does not mean the product of quantities A and t. The student must recognize the quantity t/3 can be rewritten as (1/3)*t by leveraging the distributive property to combine the like terms 4t and t/3. The student must also recognize 8 can be rewritten as 8*t^0 to explain why 8 is not a like term with the others. Our work for the problem is below.

No trouble, run of the mill example (blue cow).


Now for the brown cow.

What a GREAT QUESTION. <Trumpets herald from the heavens>

Here's some initial work on the problem, which looks an awful lot like the student that does not see the problem for what it is - something different.

As a teacher, I was thinking of how to leverage other connected ideas to help the student deepen his understanding of linear functions. Heck, we even graphed it in Geogebra and the silicon genie confirmed the student's suspicions about linearity with a picture.

I put all that other stuff in the sheet (the slider and the point A whose x-coordinate is governed by the value of the slider) after the fact. The student was convinced the function was linear and was ready to move on.

My background knowledge from modern algebra and rings and binary operations and all that jazz let me see the problem for what it was. I was trying to think through how to meet the student at his level to help him develop an understanding that would allow him to identify functions that may appear linear but have potential domain issues. One could argue the function is "linear" everywhere except at x = 0. I asked the student about the operations he saw going on within the original function statement.

Multiplication by x, the difference between 2 and the quantity 3/x, and the quotient of 3 and x. The following exchange ensued.

Mr. A: "What do you get when you add two real numbers?"
Student: "A real number."
Mr. A: "What happens when you subtract two real numbers? What do you get?"
Student: "A real number."
Mr. A: "What happens when you multiply two real numbers?"
Student: "You get a real number."
Mr. A: "So what do you think about division?"
Student: "You get a real number."
Mr. A (with poker face): "Is what you said always true? Will there ever be a case where you divide two real numbers and you do not obtain a real number as a result?"

Then we did a little work to back it up.

I even went so far as to write this statement on the board:

The student immediately questioned this claim, since we had worked this problem earlier:

We had mutually agreed the domain of t in the function above would be numbers greater than or equal to zero, since it wouldn't make sense for the variable t to take on negative values, as this would correspond to times before the billing cycle began. From the perspective of polynomials, however, the function C(t) = 0.1t + 11 has a domain of all real numbers. We choose to impose the restriction that t must be strictly greater than or equal to zero in the context of the problem. The student was drawing from a previous experience in trying to understand the domain issue we faced. The fact we disallowed negative values appeared to be a contradiction to the student because the purple sentence I wrote contained the phrase, "does not allow."

My thoughts turned to polynomials and how I could help the student move forward. I asked the student, "What is the definition of a polynomial?" My hope was that he would respond with something about the operations allowed on the variable. No such luck. After a long pause, I said, "Well, let's do what any sensible person would do. Let's look it up."

This definition didn't help unmuddy the waters. I am a firm believer in not clicking on the first link in Google immediately and helping the student reason through which links to use. After some digging, we eventually clicked the first link and came across this definition:

Indirectly, this definition states the operations addition, subtraction, and multiplication on the variable are permitted. I wanted the student to also recognize there are different classifications into which functions could fall, like rational functions or radical functions.

We used the examples towards the bottom to address different possibilities for powers that may lead to division by a variable or taking the root of a variable. This rich discussion took roughly forty-five minutes to run its course, with some pauses for questioning, thinking, and doing some algebra. I finished our discussion of this problem with some other looks this problem might present.

I hope you enjoyed this article. I would argue one of the fundamental purposes of schooling in mathematics should be to help students develop a rich understanding and to know how to behave when a cow of a different color comes along.

This is an exciting time to be a math teacher. New technologies give us the chance to refine the art and science of teaching through video study. I spent a semester last fall looking into ways to create and to use classroom video of my own teaching to improve my instruction. I'll share a few things I learned in my travels below.

I think teachers can learn a lot from NFL players. NFL players have much easier access to massive collections of film. Sportscasters and fans alike circulate tales of how the best NFL players are students of the game and spend massive amounts of time in film study. In an article entitled A Former Player's Perspective on Film Study and Preparing for an NFL Game, there's a line that jumps off the page:

The position an NFL player plays also determines how much film that player needs to watch.

Suppose this idea generalizes to the population of teachers. Does the physical education teacher need to study more film or less film than the social studies teacher? While the metaphor may break down, it stands to reason a mathematics teacher - sometimes faced with daunting differentiation loads and a compressed time schedule which sometimes leans more towards content coverage than understanding - would probably need to watch a lot of film... especially when it comes to asking good questions, maximizing class time, and identifying and redirecting quickly students that are off-task.


The cognitive demand on math teachers to adjust instruction based on events in the classroom is high, particularly for novice teachers with little experience from which to draw. The book Mathematics Teacher Noticing: Seeing Through Teachers' Eyes provides wonderful insight on how teachers can use video to improve their ability to pay attention to classroom events, specifically those events that occur during instruction. This book is a collection of some of the best field research currently available. From my perspective as a classroom teacher, you will be pleasantly surprised at the style of writing in the book... this isn't dry academic canon. The authors of these articles breathe life into the classrooms and the dialogues ongoing between teachers and students. If you haven't heard of it, and you teach mathematics, I'd highly recommend picking up a copy. There's a reason why I only see 2 available on Amazon this morning.

Suppose a teacher is thinking of filming his or her own classroom for the purposes of reflection. There are several things to consider before a teacher can just plop a camera in the corner. First, how do we address student privacy? Often, school policy lags behind technology. A parental permission form will be needed (of some kind) for every student. Here is a simple example from the New York City Department of Education. The teacher will also need a plan if a student opts out of being filmed. It probably isn't acceptable to force a student to sit in the corner if he or she is the only one not granting permission to film.

FlipSecond, what technology do we use for filming the classroom? In every case, it is critical we obtain the highest quality image and audio possible. Many teachers use Flip cameras. These are quick and easy to use. Simply charge the camera and move files by plugging the camera into a USB port. I personally use my 64 GB iPad2. Regardless which type of camera a teacher selects, a wide angle lens is absolutely essential for capturing classroom action, along with a tripod for stabilizing the image. There is a pronounced difference between using a wide angle lens and not using a wide angle lens. If a teacher uses an iPad, one solution I have come across that I have been very satisfied with is the Makayama Movie Mount.



Third, how will the teacher address sound issues? It depends on the purpose of the video. If the teacher is using the video strictly for personal use, adding subtitles for student speech is a possibility. Through trial and error, I have found with the iPad it is best the person filming circulates the room to follow action. The built-in microphone on the iPad 2 captures sound pretty well within an 10-12 foot (3.05m - 3.66m) radius. Entities like the National Board for Professional Teaching Standards and the Presidential Awards for Excellence in Mathematics & Science Teaching do not allow subtitles to be added to video. This is a common practice among entities reviewing classroom videos for teaching awards. If anyone out there finds a strong, functional sound solution for the iPad, please let me know. If the teacher uses a Flip camera or a traditional video camera, the place to start would be to look for an omnidirectional boundary microphone.

Fourth, what are the risks to the teacher? If the video is shown to others, the teacher may want to consider the implications. Teaching is an unbelievably personal act. The teacher would need to be comfortable with the other parties viewing the video. The working environment would need to focus on teacher growth and reflection - and not run the risk of being punitive either explicitly or implicitly to the filmed teacher. I have not done a lot of study in this particular area since my studies in filming the classroom have been limited to myself.

Fifth, what if a teacher isn't ready to film his or her own classroom? Not a problem. Others are doing it now and sharing their videos. Check out the Teaching Channel website, where phenomenal teachers are posting videos of teaching from their classrooms. Teachers can reflect on the practices they see in these classrooms. A teacher could start a video club, perhaps during a common planning period or PLC time, to discuss the effective practices demonstrated in these videos and how to incorporate those practices into instruction.



In the twenty-first century, film study is essential in teacher preparation programs. Here is a statement below from a replication study using classroom film study with preservice teachers, based on Sherin's work, at Harvard:

We found that preservice teachers began the methods course with relatively poor observational skills, and after a course focused on improving their ability to notice a full range of classroom events, preservice teachers were better observers of both mundane and important events.

You can find a summary of the study here. When I think of my own daughter and the future of her math education, my hope is that her future teachers, likely early in their careers and in methods classes now and for the next few years, are studying video and using video to become more effective teachers.