## Monday, July 30, 2012

There is two hour parking all around University of Saskatchewan. I once went to move my car (to avoid a ticket) and found that the parking attendant had marked--in chalk--the top of my tire. I wanted to erase the mark so began driving through as many puddles as possible.

I then convinced myself to find a puddle longer than the circumference of my tire--to guarantee a clean slate and a fresh two hours.

As I walked back to campus, I got thinking about the pattern left behind by my tires. For simplicity, let's take the case of a smaller vehicle--a bike.

If you were to ride a bike through a puddle of a certain width, the trail would look like this:
Is this model correct? Evenly spaced iterations of puddle-width splotches.

Assume that:
width(puddle) < circumference(tire)

and consider the following bike-ish contraptions. Can you predict the pattern? Better yet, can you draw an accurate prediction on graph paper? Assume a six-inch puddle (why not?)

That is the task I present to the students. The emerging patterns are interesting.

Unicycles--one wheel; one pattern.

But now combine them. (Of course, the bike goes in a perfectly straight line...)

A standard bicycle-- two wheels; same size.

Alter it slightly. (You may want to encourage colour coding for overlapping paths...)

Old school--two wheels; different sizes.

Exaggerate the difference.

Crazy old school--two wheels; way different sizes.

How does the pattern change? Is it important to know how far apart the wheels are? (Experiment...)

Just for fun--4 wheels; 3 tracks; 2 sizes.

What do you notice about certain radii? What causes certain patterns to "line-up"?

An interesting task to give a class working on circles, algebraic manipulation, factors, etc.

NatBanting

## Friday, July 27, 2012

### Painting Tape

I came across the following situation while shopping for paint at a local home improvement store:
Admittedly, the three varieties were not positioned like this, but this positioning does raise an interesting question.

"We can see the packages are the same height, what is that height?"

I see this question going one of two ways:
1. The students realize that really any conceivable measurement is possible. (Barring, of course, zero and the negatives) One could make the argument that it also cannot be irrational, but this would be nit-picking. Can a roll of tape have a width of pi/6? Exactly?
2. The students fall prey to their subconscious affinity toward the integers and begin constructing common multiples.
In fairness to the problem, both are very teachable moments, but there is nothing scarier for a teacher--under considerable time constraints--than to see a problem steer students in an alternate, but useful, direction. We know they should explore their curiosity, but can we as teachers shut-up long enough to let them?

Situation (1) leads into an explanation of unit analysis. The height can be any "x" because each roll would simply subsume the thickness of x/6, x/4, x/3 respectfully. This demonstrates great number sense. If the class immediately goes that way, I would show this picture.
Revisit the question:
"What is the height of the packaging?"
"How thick is the individual roll in each package?"

Ideally, students dwell on situation (2) long enough to draw out the ideas of factors, common factors, multiples, divisibility, and lowest common multiples. After which I would drop the unit analysis bomb on them anyway.

Just a thought. Yet another way that mathematics proves to be an inseparable mass despite what neatly organized curricula dictates.

NatBanting

## Tuesday, July 24, 2012

I am frustratingly mathematical. Ask my wife. I see the world as a combination of, in the words of David Berlinski, absolutely elementary mathematics.(AEM). The path of a yo-yo, the tiles in the mall, and the trail of wetness after a bike rides through a puddle are all dissected with simple, mathematical phenomenon. The nice part about AEM is that I can talk about it to almost anyone. People are (vaguely) familiar with graphs, geometric patterns, and circles even if they can't decipher what practical implications they have on their city block. Unfortunately, people (and students) don't often want to hear about them--they need to see them.

I can remember the look on my mother's face when I broke out the silverware to show her that the restaurant table corner was not square. Without a ruler, I showed her that trigonometry allows us to rely on ratio rather than set measurements. As I was in the midst of showing her that the 3-4-5 knife-length rule was breached, the waitress came. Mom was horrified; I was thrilled.

AEM has a visual nature; school mathematics often destroys that nature--reducing it to a simple diagram of a rope hanging from a drainpipe or train chugging its way through the prairie. I am guilty of the same thing in my class. I am a tactile learner so spend the majority of my time teaching with things. Students play with triangles to learn trigonometry and we build models to slope specifications. I often describe problems--good problems--to my students and have them struggle through them. Great learning occurs, but I have robbed them of the ability to find their own problems--the very problems of AEM that exist all around us.

I have been watching the work of many educators for a while now. I love the way they use simple, visual elements to create extremely intriguing problems. This past week I was particularly inspired by Timon Piccini (@MrPicc112) and Andrew Stadel (@mr_stadel). They create video problems that not only test the AEM underpinning, but the curiosity and problem solving of the students. It is under this inspiration that I created my first video-based task. It contains a strong visual component and is based on a natural phenomenon that I observed during housework.

Sprinkler Task (V.2) from Nat Banting on Vimeo.

The video shows two takes of me using my "circular" sprinkler on my oddly-shaped piece of lawn. The natural question that came to my mind was, "Where do I place my sprinkler to minimize the amount of water that is wasted?"

The question is broad. The situation is organic. A simple curiosity can be cultivated in a novel way. That is my favorite part of the problem. I will simply play the video for my students. I will pause after the first take and allow students to absorb the situation. Hopefully cynics will point out that the pattern is not circular; this will lead to a great conversation about perspective and spatial reasoning.

I want students to notice that the first pattern touches a corner of the yard. What if the edge of the circle didn't touch any edge of the yard? Could this possibly be the most efficient watering method?

I will clear up any variables that a good mathematician would. We assume the spray is uniform. We assume that the pressure can be turned up or down to any desired radius. Wasted water is considered water that lands outside of the grass. Wind is not a factor. After our initial conversation, I will re-play the video and see the second case.

Which one wastes more? Have students discuss. Ask the students what information they need to solve the problem. Measurements will undoubtedly come up. If students are done theorizing with the problem, I provide them with a picture of the yard complete with measurements.

The lawn has been modeled as a rectangle and two semi-circles. What error has occurred? Can we refine the model? (Possibly by placing a quarter circle on the bottom right-hand corner). Do the measurements help you calculate how much water is wasted?

To aid in their work, I created a scale diagram of the yard. I then created three different worksheets--each has a different size scale drawing on it. This creates three unique scales within the classroom. As a group, we will try the first placement together. We'll draw the sprinkler and create triangles to calculate the distances to the furthest points. The longest of these must be the radius of the sprinkler circle.

I will then send the students out into groups to discover a more efficient placement. If they are going to communicate with one-another, they will have to convert using their scales. For example, "We placed ours 3 inches down and 2 from the left" won't work if a group has a different scale factor on their diagram.

At the end, I will construct a table of results to see which group indeed maximized the efficiency.

To follow up, I created two other "yards" complete with measurements. I will ask the same question. One involves the possibility of introducing trigonometry and the other has students explore the idea of circumcircles. Both of these diagrams along with three worksheets can be downloaded here.

This is hopefully the first video embodiment of my thinking. Tasks like these not only "real-world", they cater to multiple learning intelligences. The visual, spatial, kinesthetic, and auditory are all engaged. Some of the best lessons in the classroom mirror experiences that students may have outside the classroom.

NatBanting

## Thursday, July 12, 2012

### Creating PBL 3.0

I have been on my project-based learning journey for a while now. This blog has served as the main receptacle for my inspirations, ideas, successes, failures, and reflections. It is now time to document my next step: wide scale revision.

This post will be divided into two main sections:
1. A look back at the posts that brought me to this point. (Reading them may provide some context, but not reading them will provide you with more free time...your call)
2. A look ahead into my revisions and their rationale. I will describe the new administrative and assessment framework around the projects and provide links to the first completed framework online.
Now that we have that out of the way, I guess we should start with section one.

1.      My views on PBL have varied drastically as I have experienced it first hand. My initial vision for the course was one of infinite possibilities. Students would develop their own projects and follow them out to fruition. I would provide the supports for them to do so. Only after I tried to do this myself did I find that good projects are hard to find, and even harder to create. My initial (naive) vision can be read here:

http://musingmathematically.blogspot.ca/2011/10/proper-workspace-for-workplace.html

As I re-shaped this initial vision, I discovered that there was a lot more support for these teaching ideas than I originally thought. Every time I found an excuse not to pursue the goal, it was addressed. My skills at project creation were growing and views of prominent educators worldwide began to solidify my belief that a completely project-based class was possible. My solidified vision can be read here:

http://musingmathematically.blogspot.ca/2011/11/more-inspiration-for-math-projects.html

I gathered support and launched two courses that were project-based. I included good problems and tasks for students to learn the basic skills and they were then solidified and utilized in their projects. During the semester, I had a number of roaring successes--both with problems and projects. Students were buying into the deep learning available to them. My largest project success can be read here:

http://musingmathematically.blogspot.ca/2012/04/soft-drink-project-part-1-framework.html

The semester ended and I had a chance to reflect. I knew the students had learned on a deep level, and I left most days surprised with the complexity of their thought and initiative. Upon reflection, I highlighted areas that the class needed to improve on. These included the technology, group work, and assessment. I wanted the course to get stronger in all three areas. My rationale and reflection can be read here:

http://musingmathematically.blogspot.ca/2012/07/project-based-pitfalls.html

That brings us to the next step along the path.

2.     My solution to the three major issues addressed above was to implement a continuous feedback assessment structure. That would keep groups accountable as well as improve the formative and summative assessments on the projects. I dubbed the framework, "Project Binders".

Each group has a unique project binder for each project. The projects are no longer allotted a clump of time in which groups are required to produce the final product. I found this approach left quite a few students lost along the way. Every group would come up with a product, but many would be missing key developmental stages along the way. The project binder clearly truncates the project time into "stages". The students are responsible for a certain sub-section of the project during that stage. Each stage is discussed orally, worked on within the group, and assessed by a "stage rubric". A binder includes two copies of each stage rubric--one for the group to use and one for the teacher.

Along with the stage rubrics, students fill out a daily log to infuse the process with self-evaluation. Students are asked who was present, what they accomplished, what their next steps are, and if there were any issues they needed to report. Issues could be anything from a lack of white glue to a slacking team member.

Also included in the binder is a cover page, a calendar page, and a group contract. The calendar page will be put into a clear sheet protector. That way students can write deadlines, stage assessment days, and teacher-group meeting days right into their binder. Each binder comes equipped with a fine-tip dry-erase marker.

The group contract outlines the responsibilities of each member for the duration of the project time. If a group takes issue with a member's conduct, they can fill out a "issue" on the daily log form. A meeting with me decides a future course of action. If the problem persists, that student can be found in violation of the contract and will be forced to form their own group. If this occurs, the student and I will negotiate the amount of appropriate overlap between their new project and their previous group's.

I plan on setting this all up with a set of dividers and leaving plenty of room for the students to hole punch work from the stages and place it right in the binder. (Calculations, geometric drawings, brainstorming, etc.)

My goal, for this year, is to take a step back from technology. I want to refocus my efforts; I think it became a distraction at times last year. I also want to make these project binders accessible to a large number of teachers. I have abandoned the class wiki in favour of a paper calendar and physical progress sheets. It is my hope that this method appeals to more educators.

I have developed a set of templates for each page in a "project binder". I have also developed the specific contents of the "Pop Box Project" project binder. I believe that the new structure will not kill the innovation from the original project. (Linked above). I have posted all the files I have to date on my personal wiki page, and will continue to post the binders as I develop them.

All I ask is that you use the material and provide feedback so we can make this process continually better. I am sure that this is not the last chapter of my PBL story.

NatBanting

## Wednesday, July 4, 2012

### Project-Based Pitfalls

Those of you who follow me on twitter or read this blog regularly know I have been struggling to implement wide scale Project-based Learning (PBL) into my Workplace and Apprenticeship mathematics courses. This strand of classes is probably unfamiliar to those outside of Western Canada. I have included a link to our provincial curriculum below. You can skip to the outcomes and indicators to view which topics need to be addressed. (Page 33)

Let me start out by saying that I think this is an excellent direction for high school mathematics. Some powers-that-be in Saskatchewan would like to see this pathway die out or become analogous with a modified course. I disagree strongly on both counts. This course is an exercise in teacher flexibility. (That's probably why it is hated so much).

I designed my class around an infusion of technology, a large amount of responsibility, peer collaboration, and large-scale projects. I was very happy how it went (for the most part) but there were a few glaring problems that need to be addressed.

You may read this post as a warning if you are planning to implement projects into math class. You can also see it as an encouragement. It has been done, students did learn, and the teacher didn't collapse from administrative stresses.

The Three Biggest Struggles: A Rookie's Guide

1)     Using Technology as an Enabler

I was graciously given numerous supports to set-out on my journey. First, and foremost, I was given a schedule where three of my four classes were Workplace and Apprenticeship classes. (Two Gr.10 and one Gr.11). This allowed me to focus on the institution of PBL. My principal bought me new tables to make over the physical appearance of the room. This made peer collaboration more accessible. I was given a document scanner to create digital archives, as well as sixteen laptops. The laptops ran Microsoft Office 2003. That puts them in perspective. Throughout the semester, they became more of a hindrance than a support. I was grateful to have them, but they soon started to crash, lose work, and even lose keys!

After the fact, I reflected upon my use of a wiki-centered class. I wanted the class to seem modern. I wanted the website to be our central hub of communication. The fact was, students rarely accessed the wiki outside of class and the computers acted as a barrier to the central hub. I have been denied new laptops for next year, but am choosing to see it as a blessing. (after an initial period of rage). The re-designed course will be organized in binders with more focus on neat construction. In this case, the technology I needed came in the three-ringed variety.

Make sure to ask yourself, "What does this technology enable my students to do that they could not do before?" Scanners, graphing software, and collaborative structures all proved useful. Archaic laptops became a barrier.

2)      Creating Continuous Assessment

I was in constant communication with my students. The beginning of the year was scaffolded to acclimatize them to a system filled with freedom and creativity. As I weened them off of smaller projects onto ones with larger scope, I noticed a drop in commitment. The class evaluations revealed that numerous students felt lost or confused. They became directionless.

Projects that resulted in a creative product were not the problem. Students knew that the creativity of design unlocked diversity. I found that I lacked assessment (and subsequent guidance) on large projects where the product was designated, but pathway was not. For example, one class was designing a moving out plan complete with budget, housing and employment plans, a expense chart, and tax assessment. They knew what they were to hand in, but didn't grasp the possible pathways to get there.

To combat this, I am going to use a series of project checkpoints--flexible due-dates to keep groups on task. Each one includes a self-assessment, a teacher progress report, and a face-to-face meeting. It will keep both students and teacher accountable. In PBL, confusion is your biggest opponent. Students will shut down if they feel like they are on the wrong path. There needs to be scheduled times of encouragement and, if necessary, re-direction throughout larger project phases.

3)      Group Accountability

This is every teacher's biggest fear with group work. One student does it all while others play 'Draw Something' with each other. Again, I felt this happening with the larger scope projects. The new assessment plan will help, but students will always have a natural tendency to wander intellectually. Not all off-task time has malicious intent.

At the beginning of the semester, I had students keep a log of what they did each day. The technology limitations halted the process. Having students create a group road map (so to speak) will cut down on time off task. One student realized he needed to get to work when his daily log included:

March 15th
-Filled out my bracket for March Madness. It's the winner, I can tell. Banting's has nothing on mine, he's gonna lose!
March 14th
-Sick, couldn't come into work.
March 13th
-Finished comparisons and graphs.
-Started making PowerPoint
March 12th
-Made all of my bar graphs and started putting them into a presentation.
-Put up my Andre Iguodala poster in my cubical.

This student has obvious interests that distract him from his work. A quick review of his log revealed a lot to him. This brings me full-circle to the technology piece I began with. I didn't have adequate means of self-tracking. The teacher needs to provide students the technologies to learn this important skill.

It is worth noting that I placed 3rd in the school pool--handily beating this student.

I have begun to re-work my classes with these three important reflections in mind. I want to develop step-by-step project binders for the students to keep them on pace. As the project descriptions, rubrics, and exemplars are created, I will post them (in full) on my personal wiki page.

It is ironic how much of a project designing a project-based class has become for myself.

NatBanting
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