What has happened here? Two things. The teacher erred by placing the lesson's emphasis on the result of the problem solving, rather than on the exploration of how the problem should be attacked. Second, language, figurative language, is used to great effect: students remember comparisons almost as well as they remember stories. We have then, a mislearned result - and an absence of true learning.
It is contention of Howard C. McAllister, the author of the site (called 21st Century Problem Solving), that students can improve their matematical problem-solving by following the steps:
- understand the problem - what are you being asked for?
- analyze the problem - what do you know? not know? what resources do you have? what models do you have?
- make a plan - respond to your understanding of and knowledge about the problem
- carry out the plan
- look back - evaluate your solution
Hey - it's a plot diagram on its side! (Or part of the face of a loud talking, large-nosed man. Or a zoom in on a cut made by a pair of my crazy scissors). McAllister asserts that from the mathematical POV, it is important to view a problem as a pattern of related and sequential substeps, which taken together lead to a solution. From a mathematical literacy POV, communicating these steps, their patterns, their connections, and the solution is an essential element of learning. There has been a considerable amount of research done to support these claims since 1996, when the site was posted. In fact, I have taken a graduate course in Writing in Mathematics.
From an ELA/reading POV, recognizing and communicating the patterns and connections found within and between texts is an essential element of learning to read better. Thus we use the plot diagram and other organizers to help students frame their reading.
From an ELA/writing POV, recognizing that there are patterns is helpful, but recognizing that there must be connections is an essential element in learning to write well. The problem with outlines and lists and most organizers is that a single iteration of the pattern is made clear, but the concept of connection (and hence completion) is not graphically clear to the student. Result-oriented writing, which tends to be test-response writing (and therefore hugely important to teachers), is flat. What the student produces is the equivalent of an 8 on its side.
So, how does this play out instructionally? The Better Answers or Gist paragraph, as generally taught now-a-days, produces an OK (2 of 4) standardized test response:
We find here:
- introduction and conclusion [state the problem - return to the problem to evaluate your solution]
- stays on topic [sign of good understanding]
- uses 3 facts from the [imaginary] text to support assertion / answer question [analyze the problem: what resources are available to you? what do you know?]
But what happens when the student is alerted that all writing is a series of connected problems to be explained and solved?
- Understanding the prompt
- Making an good plan of attack: 3 points or arguments, each point explained or expanded in a different way, all points clearly connected to topic and to each other, the end adding to the beginning.
- Make each paragraph a narrative, not a list.
First, the mosquito larvae was studied. Scientists found that as many as a thousand larvae are eaten by a single water flea during a day. Water fleas, in turn, are eaten by dragonflies, again in startling numbers - as many as 1,000 a day. Dragonflies also feast on surviving, adult mosquitoes. Together these small animals are the dragonfly's main diet. The final step in this cycle is the predators that feeds on the dragonfly and the mosquito - birds like the purple martin and the barn swallow, and bats. Each of these predators must eat at least its body weight in insects daily in order to reproduce. In coastal Maine, the majority of this diet is dragonflies and mosquitoes. At each step of this food chain, it is essential that at least 51% of the mosquitoes survive to carry the chain forward and to day the eggs that begin the cycle again. Should this not happen, the population of at least three other species will be negatively impacted. [also made up]
What fun! This is a paragraph that I call Beyond Better Answers, or the 7th Grade Paragraph of at least 11 sentences. The problems (addressing the topic with good points/evidence, expanding evidence statements with lively sentences, linking these sentences, and creating a conclusion that continues the introduction) are solved. If the student does not see this problem set, she will generate no more than a simple list paragraph, like the first example above.
I think I have just invented an organizer better than the outline. Most paragraphs (Describe, Trace, Illustrate, Analyze) will follow the basic Plot Diagram form. A Compare-Contrast will have three peaks (A, B, A&B). But there are a few exceptions. Below is a Slippery Slope, followed by a Golden Ladder:
What student would not want to zig-zag through the planning phase of a writing assignment, seeing clearly where extensions and connections are necessary. Just the shape of the planner is energizing. Outlines and lists are not. Webs are too chaotic for most students and counter-intuitive when it is clear that an organized response is required. For quick, timed responses (on-demand writing), I am going to go with a problem-solving model this year.
So the problem-solving method for writing becomes:
- Understand the prompt.
- Draw possible zig-zags for your response [if there is a model, draw that one first]
- Select the best shape.
- Plan your response.
- Reread and revise.