The 5th Grade Playground Project
Neither writing nor quantitative reasoning are ever "completely learned." Rather, in both cases, students build on previous skills to attain a greater degree of sophistication. We can see this in an assignment designed by Katie Allison and employed by Shelly Bean in her 5/6th grade class at Alcott Elementary School. While the task clearly targets a less-advanced audience than our college-level students, the intellectual moves represented here are the same as those we want for our own students.
Traditional "context-rich" elementary school math problems take the form of story problems. These usually algorithmic, "well-structured" problems are designed to allow a student to show that she can arrive at "the right answer." For example:
Our school wants to build a new grass playground in a vacant lot next to the school. The soil has been prepared and we are now ready to plant the new grass. the instructions call for three pounds of grass seed for 1000 square feet of area to be planted. The dimensions of the vacant lot are 300 feet by 750 feet. How many pounds of grass seed should we buy?
Successful students recognize the need to do a little multiplication followed by division and can accurately carry out the operations to arrive at the single correct response, demonstrating what they have already learned.
By contract, write-to-learn math assignments encourage students to focus on learning underlying concepts or reflect on their process:
- Ask students to explain their method of solving a problem
- Ask students to predict an answer and explain their reasoning
- Ask student to explain their process when given a hard problem and to show where they ran into difficulties
- Ask student to explain what they learned from a particular math unit.
- Is there any value in making mistakes in math? Why or why not?
- What did you learn today that reinforces what you already know or believe?
- Explain to someone who missed class yesterday why dividing by 4 and multiplying by 1/4 are the same thing.
- I went through the following steps in solving this problem, first..., second..., etc.
These kinds of write-to-learn assignments take on a new life when placed in a real context with a specific audience and purpose. As "ill-structured problems" with no single right answer, they require the writer to make an argument for a best solution. And, unlike well-structured problems which are intended to show skill-based competency, rhetorical numeracy assignments often immerse students in data that may be incomplete, inconclusive, and/or irrelevant.
Consider Allison's example problem:
XX Student worker type in the problem XX
While her students worked at a lower level of sophistication, Bean found the kinds of issues her students grappled with to be very similar to those more advanced students face in more complex contexts. First, as with all ill-structured problems, it is not clear how best to proceed. While the students quickly headed toward polling or voting among peers, they weren't sure how many votes each student should be able to cast. This is a very complicated voting theory problem that can be profitably discussed with 5th graders or graduate students!
Second, once data are collected students struggled with how to best display the data for the purpose of exploration. The first attempt, a pie chart, was very difficult to read. A bar chart proved more powerful. Tufte shows that this issue of effective visual representation takes on many nuanced layers that are usefully studied in college and beyond.
Finally, the resulting assignments demonstrated strengths and weaknesses very similar to those found in college-level writing. What distinguished the strong from the weak papers was the ability to integrate the numerical evidence into the context of an argument.
The experience of this 5/6th grade class underscores the need to understand student QR-writing on a continuum of sophistication. These are not skills that can be completed in a single high-school or general education course. Rather, we must expect to repeatedly expose students to these issues, asking them to build on previous accomplishments to reach a new, deeper appreciation for the power of number-based argument.