# Broken Pipe in a Classroom: Student Work (Sequel)

Here is a link to the original post on the broken pipe in a classroom problem.

This is the email I received from a teacher at a different school:

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

Some of the outcomes of this activity:

• Students gained a stronger understanding of Excel. Many students chose to write formulas and use the fill handle to brute force the time it would take to fill the various bathtubs.
• Students deduced the connection between casework, writing out values, finding a trend, and generalizing through trying to write formulas for the volume of each bathtub as time passes.
• Students discovered finding the "center" of a semicircle is more challenging than they anticipated. (see the half-cylinder tub on the original worksheet) Several used Geogebra to find the center through geometric construction.
• One of the students made some progress on understanding 3-dimensional graphing with Geogebra 5.0 Beta.

Here are some examples from student work on Excel:

This student (below) chose to write the functions for fill and drain as linear functions in Geogebra:

Here is some work by a student trying to find the center of a semicircle:

A student modeled a theoretical room using Geogebra 5.0. (Apologies in advance for the sound quality.)

The students submitted written reports summarizing their thinking in this activity. I wanted to see a progression in their thinking from specific cases - like the percentage of a tub filled after 10 minutes - to generalized cases.

# The Broken Pipe

A teacher from another school across the state emailed me last week. She described a terrible facilities issue she has dealt with for days.

I don't have all the details and may not be able to get them... a pipe in my room burst on Monday and caused water damage in the hallway end and a total of 6 classrooms in full or in part... I have been told varying amounts as to how high the water was in my room and mere estimation by our custodial staff on how much water they themselves eliminated... the plumber said that the little copper pipe was spewing 12 gal per min...

We had been working on a different project during MTPS (Math Theory Problem Solving) class that I didn't want students working on without me being present... I had a sub coming in last Friday and decided this would be a great opportunity for students to do some modeling work in the computer lab.

Take a moment to consider how elaborate this situation is. Water accumulates within a room. Many places exist where the water could escape. Some places are obvious, like underneath the door. Other places are not as obvious, such as through electrical outlets or through the cracks in the drywall where the tile meets the wall. There are many potential sources for error. Modeling simple cases is easy, flow in is positive, flow out negative. But this is definitely a problem from the world outside school (I've never been a fan of the term 'real world'... that would imply high school isn't real... and I remember sitting through some interminable classes with a very real feeling of when will this class ever end...)

I worked to write a worksheet that would help students identify some of the potential complications in modeling how water would accumulate in various rooms. I thought about using theoretical "bathtubs" to simplify the computations and help the kids understand some of the complexities they would encounter in modeling a real room.

The students have spent one 46 minute period and one 90 minute period in the computer lab working on modeling the situation. The source worksheet appears below.

The Broken Pipe Problem-1

Students will have another 90 minute period in the computer lab Wednesday, then 30 minutes on Friday before each student gives a 3 minute presentation on Friday about their lab work and findings. I will share some of the students' work on this dilemma later.

P. S. On the worksheet, I use Google images to retrieve pictures of each of the bathtubs. The copy did not turn out as nicely as I had hoped, so I traced over the images and scanned the resulting worksheet. I told the students the half-cylinder tub should be oriented in a way where the curved side is tangent to the floor. (The 3D image on the worksheet makes it look like the tub is tilted, but I did not intend for the half cylinder tub to be tilted). The values for the time in minutes were arbitrary. All of my students have chosen to set up tables of values in Excel so they can address some of the interesting questions (like how many minutes will it take to fill the rectangular prism bathtub?)

# Are Students Evolving or Changing?

The break gave me a fair amount of time to reflect on teaching and my philosophy of education. I was scrolling through my Twitter feed and came across this tweet from the National Council on Teacher Quality (NCTQ):

While reading this tweet, I was struck by an interesting memory. In my first year of teaching in Omaha, back in 2004, the school I taught at had just implemented 1-to-1 computing with Macbooks. I had a student that year from China. After class ended one day, she and I talked about how interesting it was to watch how students were using the Macbooks in positive ways but also in negative ways. I asked the student what she thought about the computers. She offered a brilliant insight:

I think the computers are making the students impatient. It used to take a lot of time to look something up. All the students want the answer right now.

That conversation has stuck with me. I see this impatience in my classroom and other classrooms on a regular basis, whether it's students moaning and groaning when a problem takes more than thirty seconds or the students' body language revealing frustration or boredom.

Two years later, I was teaching at a different school, a school that has not yet gone to 1-to-1 computing, even in 2014. Even without 1-to-1 computing, the ubiquity of smartphones and tablets has had a profound impact on our young people. Personally, I wonder how different my school would have been with the temptations of Twitter, Facebook, and SnapChat lurking in the background. I am not saying technology is evil; I am saying educators need to be mindful of the impact technology has on students' physiology and psychology.

Our culture in the United States is the embodiment of instant gratification. This has dire consequences for math teachers attempting to help students learn patient problem solving. Joachim de Posada discusses the predictive impact of studies on delayed gratification. Intuitively, it makes sense the students willing to delay gratification - those that push through and past the point of frustration - tend to be the successful students in school.

Back to the earlier question: are students evolving or changing? Technology provides the context for our students to literally stand on the shoulders of giants and answer more interesting questions that can profoundly impact the modern world. But with great power comes great responsibility. In nature, we sometimes see evolutionary dead ends. Think duck-billed platypus. We want our students to be critical thinkers. To be problem solvers. And while it is convenient to rely on Google's search algorithms to find an answer quickly, we need our students to be able to analyze, synthesize, and evaluate the information they encounter in the world. It's not a stretch of the imagination to think certain technological behaviors we allow, like clicking on the first link on a Google search or citing only Wikipedia sources, are the evolutionary equivalent of a duck-billed platypus. I wonder what my students and their children will be doing in the 22nd century and how different the world will be. I want to empower my students with robust strategies to prepare them for this world that does not yet exist. What can I do as a math teacher to maximize these students' potential? Because barring medical miracles, I am undoubtedly preparing my students for a world in a new century I will not likely see. So what should be my function, my purpose for teaching students mathematics?

Awaken raw curiosity. Provide the context and vocabulary to describe the universe and everything in it. It starts with asking interesting questions and finding technological resources to address these questions. Michael Stevens is doing a phenomenal job leveraging curiosity to create teachable moments at Vsauce.

The spirit of the STEM movement is the interconnectedness between disciplines. Why we limit this 'interconnectedness' to only four disciplines causes me to scratch my head a bit. Mr. Stevens' video above references figures that cannot be measured directly with current measurement tools... but they can certainly be calculated. We can even compute how aesthetically pleasing an object is or isn't. What should the role of the math teacher be, then? What does this have to do with our students, and whether they are evolving or changing?

I think my best answer right now is to provide a balanced approach between traditional mathematics and problems from the world outside school. And for those items from the world outside school students may not have the mathematical horsepower to address, there is now a phenomenal resource to address both traditional topics and modeling challenges. Math teachers may experience some nervous excitement on the Wolfram Demonstrations website. I encourage you to check out the resources available on Wolfram Demonstrations.

Regardless what educational approaches we utilize, the world will continue to move forward. Students will continue to move on through the sorting algorithm that is school and the world will continue to realize the potential of students from many different school systems. What do you think? Are students evolving or changing? What can we do differently as teachers to prepare students for the future?

# Procedural Understanding in Teaching Mathematics

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:

Chicken 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?

# Final Exams: You Be the Judge! • 1 Comment

Each semester, I stand in awe of how many students do not understand how to calculate the impact a semester final exam has on their semester grade. Our math department grading policy roughly breaks down in each class to the following:

90% Formative/Summative measures from the semester
10% Cumulative Final Exam

Kids carry many misconceptions about the final exam. A common one is that the final can somehow miraculously overwrite a semester's effort (or lack of effort). Here's a visual representation, to scale, of the 90%/10% model.

Exhibit A: The orange team blows out the red team, 90 to 10.

I have often instructed students to make the following program to help them forecast the impact of the exam on their semester standing.

Exhibit B: Program on the TI-84 for computing a 90%/10% weighted grade.

Here's a numeric example for how this sometimes surprises a student.

Exhibit C: Not much movement of the overall grade, despite a solid final exam score.

This is where the discussion gets interesting. Students will often use trial and error, over and over and over again, with the above program to compute the final exam score they need in order to get an A (which is 90% in our grading scale).

Exhibit D: An A doesn't appear to be in the cards, especially since extra credit does not exist in my class. (I'll share my views on EC on a different day). But I digress.

This is where I like to invoke Ken O'Connor's philosophy from How to Grade for Learning. This is a terrific book for teachers of all disciplines because it carefully examines the potential dangers of deferring to the electronic or paper gradebook to make all the heavy decisions.

Guideline 6 (page 153): Crunch numbers carefully - if at all.

a. Avoid using the mean; consider using the median or mode and weight components to achieve intent in final grades.

b. Think "body of evidence" and professional judgment - determine, don't just calculate grades.

I have been using the following practice for nearly ten years. I have a one-on-one conference with the student. I speak to them about what they need to get on the final to earn a particular grade. Since the breadth and depth of my final exam questions are more considerable than on unit exams, the student with an 86 would earn a 90 if they earn a 90 or better on the final exam, despite the fact the numeric stuff doesn't turn out that way.

This policy works wonders with students at the upper end. Sitting on a 96 for the semester, Johnny? What if you had the chance to up your semester grade? What if you could get a 100 for the semester by acing the final exam? All of a sudden, the student has a carrot to chase. The students appreciate this approach because it is a system that rewards effort and hard work. This approach gives the control back to the student.

Otherwise, isn't it possible that neat-and-tidy 96% in the grade book is simply a graveyard of sign errors? Or computation errors? Philosophically, what do I want as a teacher? Do I want a student's grade to reflect their learning? I have worked with thousands of students, yet only a handful I have encountered get most things on the first attempt. I tell my students every semester I have yet to meet a person that learned something without making mistakes.

I approach this problem reasonably. Don't get me wrong - if a kid is sitting at a 62% for the semester and gets a 95% on the final, that doesn't necessarily mean the kid deserves a 95%. I would need to reflect on the student's formative and summative measures as a body of evidence to help me make an informed professional judgment. However, if that situation happens - and it hasn't happened to me yet - I would need to re-examine my professional practice. That situation would indicate there is a disconnect between the content mastery a student demonstrates and what the grade says the student knows.

Teachers need to think carefully about how their grading practices capture - or don't capture! - student content mastery. Virtually all measurements are imperfect. The burden of evidence to prove or disprove the student knows something should lie with the student. If this is true, then the burden of judging whether the evidence suggests the student is learning lies with the teacher, not the plastic or silicon genie.

Think carefully on your grading this holiday season. Good luck to everybody - teachers and students alike - with semester finals!

If you are a teacher, and these images are foreign, alien, or unfamiliar, it's time to consider how to better leverage all the professional resources available to you.

If you are a teacher at any level, and you're not on Twitter, you're missing out.

Let's look at this empirically. Above are some Google search results from a query I typed a couple minutes ago. Suppose 95% of these results contain redundancies or snippets of the original query. Really, even this blog post is redundant. I'm definitely far from the first teacher to say Twitter is good for teachers. Even if 95% of these web results are redundant, that still leaves approximately 4,480,000 webpages that say something about teachers using Twitter. This may be a bit of hand waving, but I would hope this would cause a teacher or two in the crowd to consider joining Twitter. Really, this is an insidious sales pitch. If you are a teacher, I want you to join Twitter so I can steal your best ideas and make them my own... because it's all about my students and giving them the best education possible.

Trust me, I know about the missing out. I was reluctant to join Twitter myself.  I've had a Facebook account for about ten years. I love the social aspect of Facebook and that it allows me to keep tabs on all the wonderful post-secondary institutions my former students attend, what career paths they have chosen, and how they are doing as a person and as a lifelong learner.

Until this past summer, I avoided Twitter, and the possibility of establishing a Twitter account, like the plague. Why? The root of my avoidance, I think, is much like the avoidance many teachers have to new technology. I am just north of 31 years old, and I am finally starting to notice feeling some reluctance to incorporating new technologies in some of the things I do as a teacher. This is a new sensation to me because I grew up alongside technology. It's not fear of technology. It's the concern that learning a new piece of software, or a new tech trick, may end up being time consuming and not as efficient as some of the practices with which I am familiar and comfortable.

Many schools in the area where I live seem to be the middle of nowhere in the middle of the middle of nowhere. Rural schools sometimes have one math teacher 7-12. ONE. Not just a department, THE department. The advent of Twitter empowers these isolated teachers to establish a Professional Learning Network (PLN) to enhance and refine their instructional practices.

I am still a Twitter novice. I know I could be using it more efficiently, and I continue to learn about it when I have time to do so. Here are some great resources on Twitter use for teachers.

Mission 5: Twitter Chatter, Subject Matter: how to use chats
Mission 2: Twitter Me This - the how-to guide on Twitter Account setup for teachers

ASCD article: Digitally Speaking / Why Teachers Should Try Twitter

# Filming the Classroom to Improve Teaching

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.

Second, 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.