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.


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.