Thursday, December 27, 2012

Becoming "Unflippable"

This post contains no real lesson or task ideas. That is a rarity for me, but every so often a philosophical battle ignites in my brain. More often than not, the question does not come from an established professional development vessel. Our division provides numerous officially sanctioned "PD" events throughout the year. They serve their purpose, but rarely motivate like those questions that come from within--or, in this case, from a student.

Every teacher is familiar with the following conversation:

Teacher: Can you please pay attention?
Student: I was paying attention.
Teacher: No you weren't. Please put your _____ away.
Student: I was so--I have all the notes.

This infuriates me.

It wasn't until a month ago that it dawned on me:

This is what the public thinks I do for a living...

People think my job is to ensure students get the notes. Why else would parents excuse their children from school to go shopping? or make dentist appointments during school hours? or extend Christmas break by two weeks in the Bahamas? Why else would students gauge their "learning" by the amount of times they visit a pencil sharpener? or the number of pages of scribble they manage? Why do they ask,
"What did I miss yesterday?"
and expect to be immediatley back on the class pace. Students have spent enough time in classrooms to get this notion. Those students become parents... etc.

What do we as (math) teachers do to combat this mentality? Mostly--a whole lot of nothing. Our classes remain predictable in nature. Some students even complain when they should be getting the homework but the activity or lecture goes long. We have literally programmed our students. It is in this light that teachers get so offended when websites like the Khan Academy claim to revolutionize education. Teachers hate to think that they are replaceable by a set of videos when, in actuality, many of our lessons are.

Maybe the videos lack the personal nature and opportunity for diversification. But (school) math is very impersonal, and diversification can be achieved through more videos. They also add convenience to the equation. Anytime, anywhere, and at any pace.

The term "flipped classroom" is slowly percolating into the current educational lexicon. The process involves students accessing video lectures to free up class time for different activities. At first I hated the idea. This wasn't changing teaching; it was switching the medium through which the transmission was performed. I pictured class as a time to test examples and do homework sets. 

My perception changed a month ago when I had the following conversation with a student:

Me: Where were you yesterday? You missed my class.
Student: I was sick; I didn't skip.
Me: The motive was different but the results are the same. You need to catch up.
Student: I'll just get the notes.
Me: It is more than notes. You need to understand why and how we do the math. You need to be in class to learn.
Student: Why? I don't miss anything if I get the notes.

This student is right a lot of the time. I do my best to infuse meaningful mathematical tasks and activities into my room. Many of them are scattered throughout this blog. The burdens of time and curriculum force me into corners, and many classes could be easily captured through a video and a set of notes. I realized that if I wanted students to value the class time, it had to be in a classroom that was "unflippable".

I now gauge my lesson success with a simple question:

Was that lesson unflippable?
 
or
 
Could the students learned the same amount through a video and a set of notes?

These types of questions guide my personal growth as a teacher. They allow me to catch myself when planning gets lazy and when the days get long. Naturally, I have begun to look for ways to free up more time and curricular space for unflippable exploits. Ironically, that has led me toward a flipped classroom model.

Ryan Banow (@rbanow) provides a great starting point here.

Students have responded very well to my Project Based Learning courses over the last couple years. I think the appeal comes in the structure of the class time. Small activities and tasks lead into larger projects; the collegial atmosphere and complexity of the tasks make the process unreplicable through a series of videos and solitary projects. The course is--in essence--unflippable. I am still struggling with the higher-level and increasingly abstract courses. As always, time is at a premium. At least for now it seems like flipping a unit or two may be an effective way to create class time that becomes unflippable.

NatBanting

Saturday, December 8, 2012

My Whiteboarding Framework

This year my department decided to make using whiteboards as formative assessment tools our department focus. This was nice because I had already began to experiment with the process. It just meant that:

  1. I wasn't obligated to try yet another "thing" in my room.
  2. I would be given better materials and funding to work with.
  3. Other math teachers in my building would see the enormous benefits of the technique.
For those of you unfamiliar with the term "whiteboarding" it is very simple. Students are given a miniature whiteboard, a whiteboard marker, and a small eraser. Responses are elicited in various ways from the students--using the board as a medium. There are several ways--and question types--that create different classroom atmospheres. It is my goal, in this much belated post, to detail the three styles of questions that I have used. Each carries with it a specific classroom environment. 

A collection of my boards and erasers
The first style of question I call a Basic Response.
This is a classic "teacher asks, student responds" type of question. Short, sweet, and to the point. The great thing about the whiteboards in this case, is not the fact that they tap deep learning. (we'll see ways for that later). The low floor (everyone can answer), variety of questions (you can ask almost anything), and relative anonymity (takes away from public response) make the response rate close to 100 percent every time.

I have had students place a "?" on their board and hold that up as their response. It provides instant feedback to me about who needs the help--and how badly. I have students turn their boards over and hide their answer. When I see everyone is ready, I ask them to hold up their boards. I collect answers, and we muse on where the mistakes may have creeped in. 

I stress the importance of working individually on these types of questions. Having everyone respond at once cures the "smart kid answers all the questions" syndrome. After an answer is deduced and revealed, I allow the conversation turn to the mathematics. Often times I need to calm down the hum after a question. These types of questions are quick, and easily constructed.

The second style of question I call a Dynamic Response.
This style must include two things:
  1. Student Creation
  2. Student Solution
These questions usually appear in multiple parts. The first step has students create something mathematical according to certain parameters. It could be a set of data with a certain mean, a triangle with a certain property, or a quadratic with certain intercepts. The emphasis is on the creation. The response is dynamic because it involves a passing of the boards. Instead of showing--and getting approval from--the teacher, students swap boards and build, alter, correct, critique, or continue on their classmate's work.

This creation and solution builds deeper thinking models. It also creates a much louder and more active mathematical ecology. Some equate quiet with learning--me? Not so much.

These types of questions are harder to create and should be thought about prior to entering an instructional situation. The tasks need to be small enough to fit on the board, but novel enough to stretch their thinking. Tasks like these might be more typically classified as "new" or "reform" math. (Both phrases appear as pseudo-insults to me). 

It is important to cycle through the groups and see what the conversation is going like. Are there any "rules" we can create? Can we explain our thinking to each other? etc. An example of a dynamic response question is below:

The question was inspired by a previous post, and formatted to fit the whiteboarding structure.

The third style of question I call a Grouping Response.
These questions are designed to have a key divergence or decision. I use this divergence to create groups within the class. Rather than explain the possible schools of thought on the board, I like to discover who thinks in which fashion, and get them to campaign for their thought pattern. 

When the class has been split, it also provides me a chance to spend time with a group of students who could not answer. Essentially the class is split into Type A, Type B, and No solutions. 

Sometimes I ask the students to choose a corner based on their response and explain why their answer is the most efficient. Other times, I ask students to find someone with a different response, and exchange rationale. The questions are designed to get peer interaction kick-started. The whiteboards ensure that everyone labels themselves in a category. 

For example:

Notice that students are still asked to compute or work with some procedural mathematics. The whiteboarding just facilitates an effective mathematical conversation while grouping kids not based on ability, but mathematical preference.

My question categories came from my work on a presentation given in November 2012. The complete presentation slides can be found here. More question examples as well as other classroom benefits are  also included.

Whiteboarding is a great initial change agent. It can be used within a variety of classroom types and adapt to a variety of teaching styles. In the end, teaching mathematics is about creating a vibrant ecology where every student can be reached. Whiteboards have been a huge step toward that direction.

NatBanting

Monday, November 19, 2012

A Discussion on Slope

I have taught Grade 10 math more than any other class. I still have lessons that I created during internship that I use. Other sections of the curriculum I have perfected over the years. Today, I added another lesson to the list of those that I will do for a long time. This is my desperate attempt to describe and catalogue it. If I don't do it now, it will filed as a good, but vague, memory.

My goal was to introduce the idea of slope and be able to get numerical values for slopes from graphs. I also wanted to introduce the four types of slope: positive, negative, zero, and undefined.

The class began with a quick discussion on how rate of change relates to slope. I handed out student whiteboards at the beginning and drew four lines GeoGebra.

I gave each line a label (A-D) and asked students to decide which line had the largest slope. Ninety-five percent of the students answered that Line C had the largest slope. This told me two things:
  1. Students recognize that a line has "more slope" as it approached vertical.
  2. Students did not decipher between positive and negative slope.
I took the second slope and began a discussion. One student said:

"It is either A or C"

When I questioned him why he answered,

"Because I don't know if it matters if we are going up or down."

I took this distinction and introduced the idea of positive and negative slope.

From there, I wanted to show students that it is important to have a strict labeling system for lines. I got the whole class to close their eyes, and one student to describe a line projected on GeoGebra. Every student was to listen to the description, and then draw the line on their whiteboard grid.

The first line looked like this:

The student began explaining the line. She said that is was "pretty flat" and "straight". She also commented that it was "pretty long". At this point, students began to become irritated. One asked if it had a positive or negative slope. Another asked to count the boxes. I was pleased that they realized that counting boxes could give the slope of the line, but asked them to be quiet.

When she was done, I had students graph the line. We discussed how the description could have been more specific, and the tactic of counting boxes came up. I quickly reminded them that we did this before with rate of change and we developed the well known "rise over run" formula.

I then gave this line for the same girl to describe:
The girl immediately commented that the slope was negative. She then said it was fairly steep. She then proceeded to count the boxes and devise the slope in her head. When students drew the line on their boards, almost all of them scribed it correctly.

We used whiteboards to practice calculating slope and ended class with a discussion of division by zero. The lesson became an exercise in mathematical communication. Standardizing how we talk about slope helps us interchange mathematical ideas. Students left the room fully understanding why we had to use a numerical scale when discussing slope. Any time students can experience reason behind their work, it is powerful.

NatBanting

Sunday, October 28, 2012

Webbed Assessment

I have been playing around with several ways to get students to realize why they make mistakes. I am fed up with the traditional grading process where the student completes a task and then is handed dead feedback--stuff to do the next time. In my opinion, the student needs to be the one seeing the diagnosis. 

I guess you could call it "active assessment" or "confidence assessment". My goal is to get students looking into the patterns of their mistakes and isolating skills that they need to practice.

My school has a short 35 minute period every Thursday. I decided to take this period as an assessment focus period and give a series of weekly quizzes. They are not for marks (and I am very clear on this point to my students). I have been 100% transparent with them about the purpose and goal of these formative activities.

I call them "webbed" quizzes because each question is constructed to have something in common with another question on the quiz. That way incorrect answer trends can be attributed (with reasonable accuracy) to certain, isolated skills. 

For example, if question #1 and #6 both require a certain skill or concept, and both are answered incorrectly,  I can then point out that pattern to the student.

I build these quizzes using a specific framework:

  1. Create a list of skills to be covered.
    • Should include 3-6 skills
    • Should be specific, measurable skills
  2. Pair up skills to create a question make-up
    • Questions should cover two skills
    • Every skill should be used at least twice
  3. Create questions that test specific skill combinations
When I build the quiz, I place an "assessment map" on the back of the page. When the quiz is over, I work through the questions, students grade their papers, and then we go over the possible combinations of mistakes. Students are asked to circle which situation they fall into and write out the specific skill they need to practice. 

For example, if a student can simplify square roots and write entire square roots, but cannot simplify cube roots, then there is a specific problem with cube roots. Students can then practice changing the index. Maybe students can reduce both square and cube roots, but cannot write any entire radicals. They are then pointed toward their specific problem. Having students search out these patterns will help them with their self-diagnosis.

The web results in a specific diagnosis; it also saves considerable grading time for me. I am able to communicate specific assessment information to large amounts of students with a single handout. Students begin to partake in the assessment process in a low-pressure environment. 

No quiz is perfect, but so many are slapped together haphazardly through Google searches and exam generators. Allowing a short time for students to see assessment as a means and not an end will keep them looking toward what they can improve and not at what they have done wrong.

NatBanting


Sunday, September 30, 2012

Project Work Scaffold

There are two schools of thought when it comes to PBL:

  1. Start with a large-scale project and fit the specific outcomes within it, or
  2. Build toward a larger project with smaller tasks.
I love the idea of large projects, but also aim to make my work as accessible as possible for those who want to take it and improve on it. I just don't see option one working within my traditional classroom of 35 students for one hour a day. The existence of an overarching curriculum only further decreases its accessibility.

As for option two, there is an art in equipping students for project work. A balance needs to be struck between atomic learnings and creative initiatives. Students need to be given tasks that call for specific skills to be used in ingenious ways. As the unit of study progresses, the degree of freedom acts accordingly. The result is a scaffold toward the ultimate goal--a mathematical ecology where students can use math within relevant schema. 

This year I only have one section of the course I previously planned around the ideas of PBL. The development of the course is littered throughout this blog under the tag "PBL". Most recently, I developed project binders to structure student projects. Each stage is very free-flowing, but students are eased into this structure by a series of tasks. The purpose of this post is to detail the transition from low-level recall and execution tasks to high-level project work.

Surface Area and Volume Unit (Grade 11)
This has been my exemplar for many of my posts. It is the unit I have spent the most time developing. The unit begins with the notion of area and surface area. Estimation tasks for both surface area and volume are completed. Formulas are given and practiced. The focus is always on implementation--not memorization. Some of the low level tasks include basic netting, formula utilization, and composite shape recognition. Structure of response always takes a front row seat. I make it clear that a good project or task not only arrives at a solution, but uses structure to communicate it clearly. The worksheet below is an example of a low-level task. 
Nothing special, but the focus on structure allows them to take the next step.

A mid-level task takes a basic learning or skill and forces the student to use it in a given situation. It may move the learning out of a "textbook context" and into an arena where the student needs to decide when to employ the skills. In the SA&Vol unit, this could take the form of the Silo Task. Students were asked to calculate how much grain could fit in a standard grain silo. Extensions include accounting for a larger opening or removing some given measurements to necessitate trigonometry. 
Mid-level tasks begin to use skills in new settings. Again, structure is emphasized. 

High-level tasks present a complex task to students that needs to be solved using a variety of mathematical skills. High level tasks provide multiple solution paths; students are asked to think critically to navigate their way through. In the SA&Vol unit, I ask students to find, net, and calculate the specifications for all the possible rectangular arrangements of 24 cans of cola. Students need to decide all the combinations using spatial reasoning and pattern recognition. Once they have their organizations, they use the dimensions of the can to find dimensions of the boxes. The task takes several days, but better prepares the students for the project atmosphere. A page from the task is seen below. (This submission was 12 pages).
Again, the focus is on structure and communication. 

Three weeks into the unit, the students are given the pop box project. They have been transitioned through a set of tasks that not only give them the content skills necessary to complete the project, they give them the organizational skills to communicate their findings effectively. 

Creating an effective project-based unit is not a matter of throwing an idea at the students. Effective units scaffold students through increasingly difficult tasks with increasing degrees of freedom. Only then can the students enjoy the dual benefit of motivation and mathematical proficiency.

NatBanting

Saturday, September 8, 2012

On a Smaller Scale

I was watching Saturday morning cartoons when this commercial was aired.


High energy music and neon flashes of light are often used to sell car related toys on these stations, but this commercial caught my eye. Upon first viewing, I thought I saw them advertise speeds of 


1500 mph

I was initially surprised at this huge velocity, but then figured that the station was using a play on units to exaggerate. They never mentioned what the abbreviation "mph" actually stood for--I had assumed miles per hour. I wrote down the name of the toy and made a mental note to write a blog post about how numeracy is necessary because even toy companies prey upon deficiencies.

I assumed that "mph" must really stand for "meters per hour" and converted the 1500 figure into miles per hour. It turns out to be around a single mile per hour. I figured this was reasonable, and left the problem at that.

Later that day, I searched YouTube for a video of the commercial and was surprised with what I missed in my original observation. The actual statistic given on the screen is below:



The 1500 mph was meant to be read in "miles per hour" but the company had scaled the speed. What was that supposed to mean? What was the speed scaled in relation to? How were poor kids supposed to know how fast the toys actually traveled?

My guess is that the makers of the commercial took the scale factor of the dimensions of the toys to an actual race car and applied it to their speeds. 

Our toy is this fast but if it were "x" times larger, it could reach speeds of 1500 mph

Is this even mathematically possible? I struggled with the answer. If we were to take the toy and manufacture it larger, would it travel any faster? What if we made it smaller? Wouldn't that make the "scaled speed" even more impressive?

For that matter, what quantities are "scale-able"? I know I can scale lengths and distances, but what about temperatures? Is a smaller iceberg any less warm? Do we predict the temperature of a piece of ice by its size?

What about force? Don't storms become more violent as they grow?

The whole concept of scaling speed threw me off. The problem had interesting applications to the classroom. If we assume that the unit used was "meters per hour" it is a simple conversion problem. If we allow the scaling of speed, we should be able to predict the toy's size if given specifications on similar, existing racing vehicles. 

Curriculum aside, is scaling speed legal? If it is, why do we care? Is there any quantity that cannot be scaled? Seems like too much thought for morning cartoons.

NatBanting

Thursday, August 23, 2012

Gummy Bear Revisited

The giant gummy bear problem has been floating around the blogosphere for a while. When I first saw it, I knew I wanted to use it. I finally have the perfect opportunity in Foundations of Mathematics 20 this year. (Saskatchewan Curriculum).

History of the Problem (As far as I know)
  • Originally presented by Dan Anderson here. Included original Vat19 video and driving question about scale.
  • Adapted by John Scammell here. Edited video and new driving question.
  • Dan Meyer provided a 3Act framework for the problem here.
  • Blair Miller adapted his own 3Act structure here.
My apologies go out to anyone else who played with or re-posted an original interpretation on the problem.

My vision for the problem (As far as I know)

Students will work on the problem as a task force. Whether or not I choose the groups will depend on the mix of students I get. The task will take place in a unit on rates, unit analysis, and scale factor. Students have been presented with mathematics that will help them through the problem. In that sense, this is not a "discovery" lesson, but I feel that that word attracts so much heat--I dare not use it.

The structure of the unit is highly influenced by Bowman Dickson's whiteboard structure. He uses it to induce wall-to-wall mathematics in a class period. That purpose remains here, but I also feel that a single medium to record their thinking (in a limited space) encourages efficiency and teamwork. With these two purposes in mind, students will enter the room to find tables equipped with the following supplies:
  1. A whiteboard. (Not too big, not too small)
  2. Two fine-tipped whiteboard markers
  3. A ruler
  4. A bag of gummy bears
  5. A folder containing documents
The folder will consist of pictures:


The first three are of the dimensions of the giant gummy bear from the original video.




Each group will also get screenshots from Meyer's weighing of the bears. (Act 3).

When students enter the room, they will take a seat and are free to explore the contents of their folder. After a few days of similar tasks, some will assume their job and get down to work. Others will dwell on the novelty of the situation.

When class officially begins, I will play Scammell's version of the video found here.

From there I pose the task:
How much does the giant gummy bear weigh?

Students are required to show all their work, reasoning, and calculations on the whiteboard. My role becomes one of transient learner. I move from table to table asking questions about their strategy and supplying my own questions for those who are stuck or finished. 

The group whiteboard task uses the mathematical ideas in a realistic setting. There are no prescribed methods or rigorous assessments. Students are simply required to use the mathematical information given to them, mix in a few mathematical tools, and provide me with a reasoned argument why they employed their school math in a correct manner. Kind of liberating--in a sense. 

NatBanting

Saturday, August 18, 2012

The Guess Who Conundrum

Every so often, an idea comes out of left field. I woke up with this on my mind--must have been a dream.

Back in the day, my family had a dilapidated copy of the game "Guess Who?" My siblings and I would take turns playing this game of deduction. You essentially narrowed a search for an opponent's person by picking out characteristics of their appearance.

http://www.flickr.com/photos/unloveable/2398625902/
I vividly remember playing with my younger sister one time at a family cottage. She--foolishly--chose a female person for me to identify. Anyone who has played the game before knows that the males far outnumber the females. I had her narrowed down to six or seven possibilities with one question:

"Is your person a male?"

Move the same principles into the math classroom. Numbers have characteristics much like Robert, Frans, and Bob. Some are large, negative, prime, abundant, triangular, square, etc. Would it be possible to play a game of "Guess Who?" with an array of numbers?

Create a five-by-five game board with numbers from 1-100 dispersed on it. Sample boards are available for download. Both players must have the same numbers on their boards for the game to work. They can be arranged differently to avoid peeking. It would be wise to choose a variety of numbers. Some even, some odd, some prime, some multiples of 3, etc. Try to avoid one dominating characteristic. (Like my sister's gender choice).

Have each player choose a number, and play the game. I would place restrictions on what they can and cannot ask:
  1. Must be a yes-or-no question
  2. Can't pertain to a physical characteristic of the symbol. (Does your number have curves?)
  3. Can't ask if the number contains specific digits (Does your number have an 8?)
  4. Can't ask over-under questions. (Is your number over 12?)
Encourage students to ask if numbers are prime, square, cubic, multiples of specific integers, triangular, odd, even, etc. Be sure they are dealing with the characteristics of the numbers.

After a round or two--depending on the group speed--get down to the real task. Have students pair up with their opponent and answer the following question:

Which number is the best to choose and why?

Students will begin brainstorming the most likely questions, and which characteristics pertain to which numbers. They will be practicing prime factoring within the framework of a larger task.

Students are essentially looking for the number that shares the most characteristics with the other numbers. Any unique characteristics run the risk of being singled out with a single question. This categorization of numbers fits very well in the Grade 10 Pre-calculus factors and products curriculum in my province. These topics are fairly universal across school mathematics.

Great question. I bet characteristic games can be applied to the rational numbers as well. I can see a less than--greater than approach working well for ordering of fractions. 

It might be interesting to have students play in pods of three. Each person has to deduce the other two's numbers by finding common characteristics. Every question is posed to the pair of opponents and each must give a yes-or-no response. Maybe they are both even, but only one is a multiple of 6? Maybe one is even and the other is odd but they are both perfect squares? The game ends when a player correctly guesses both opponent's numbers with a single guess. This throws an interesting wrench into the deduction gears. It also provides a nice extension for those students who are ready to move onward while others need more time with basic factoring. 

This is a great way of throwing mathematical deduction and problem solving alongside factoring. Students should see numbers like they see eccentric faces--as culminations of a series of defining characteristics. 

NatBanting

Monday, July 30, 2012

Bike Trail Task

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?"
adding now, 
"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

Sprinkler Task

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.

Enter the twitterverse.

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.

Download the project binder templates from the heading on the top navigation bar.

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