We asked Khan Academy learners to talk to each other about math
Open response on the internet means breaking away from machine gradable multiple choice or numerical entry as the sole method and representation of student thinking. But if student work is not machine gradable, how can the students receive feedback?
Many modern classroom practices involve facilitated peer interactions: students talking to students in a structured way. What happens when we facilitate peer interactions between learners on Khan Academy?
Our open response project at its heart is:
1. Students respond to a prompt. 2. Students give and receive peer feedback on that work. 3. Students use the peer feedback to revise or extend their original work.
In this trial, we showed students two different methods for solving the area of a trapezoid. (These methods are intentionally set up to provoke comparisons in algebra and geometry. I have done lessons around this since the idea came to me via Dr. Judy Kysh years ago.)
Alma (Sample work)
Beth (Sample Work)
Responding to the Prompt
We asked the trial participants for similarities and differences, and for questions that either they had, or that they could imagine another learner having.
Here are some of the questions they wrote:
I still wonder why they didn't use the formula for the area of a trapezoid: (1/2)(base1+base2)(height).
Can all trapezoids be rotated to form one long parallelogram? Or is it just specific to certain types?
Is one easier to visualize than the other?
A student might ask if we always have to cut the shape in order to find the area.
for alma how you know that yellow triangle is half of (2x3)?
These questions display a variety of mathematical engagements with the prompts. If these were asked in a classroom, they’d be great fodder for a teacher to facilitate class discussions around.
The broad prompts of comparing similarities and differences provide many entry points to students at varying levels. In the questions above, we see a learner approaching from the formula, a student thinking about generalizability of one of the methods, a student wondering about communication (visualization) styles, along with students asking about the underlying pieces of the methods like dissection and area of triangle.
I like this question/comment: I'm not sure that Beth's method makes the problem any simpler, as a parallelogram is just a special trapezoid so she could straightaway use the half-height-times-sum-of-lengths rule.
What fun that could be to bridge this area discussion to the classification of quadrilaterals. The advantage of open response is the ability for many related ideas to be discussed at once.
Peer Feedback
So we took these observations and questions and distributed each to another member of the participant group. For example, this question was seen and answered by a peer:
Question: for alma how you know that yellow triangle is half of (2x3)?
Peer Answer: Alma can know that the yellow triangle is half of 3*2 if she is familiar with the triangle area formula which states that every triangle as an area equal to its height times its base divided by two. This comes out of dividing a parallelogram in 2. To calculate the area of a parallegram you mutiple it width by its heigh. A triangle is just half of parallelogram.
This answer asserts the triangle area formula but goes a little farther: supplying a connection to parallelogram area. Another question and answer pairing shows how a good pairing of peers can help someone appreciate a new point of view.
Question: I wonder why both students did not calculate the area of the trapezoid in one step:Area = 3 x (2+6)/2 Beth's calculation amounts to the same but the rotation etc was not necessary.
Peer Answer: Beth's approach is more suited for those who prefer to visually see their calculations. Whereas Alma's technique breaks the trapezoid into simpler shape which most people would likely know the area formula of. This is useful in situations where one does not know the formula of the area of a certain shape but can discern that it is made up of smaller shapes. Often, these smaller shapes have simpler area formulas.
But a challenge in all peer interactions, online or not, is ensuring a positive productive relationship. Not all questions and answers were helpful:
Question: whether shapes fit when rotated
Peer Answer: Of course they fit when rotated. I did somwhat the same thing and it worked
Allowing a user to request new feedback will be important.
An expert teacher with sufficient time may be able to pair students based on which ideas should encounter one another. Currently our prototype has no expert involved. Research from Elena Glassman has demonstrated some of the potential avenues to involve experts and learners at scale.
While the lack of an expert teacher meant the facilitation was essentially random, it still provided most (in our skewed recruited population) of the participants with a chance to discuss a personally expressed idea.
Synthesizing and Extending
To cap off this activity, we asked participants to synthesize the ideas they’ve written with the work and feedback from others to write about a third method:
Coco (Sample Work)
This work sample is similar to one presented in the current Khan Academy Video on area of trapezoids: (4:00 mark) YouTube, KhanAcademy.org
How might a learner’s mathematical conceptions change when doing open ended response and peer feedback work? How might this supplement or support the “just in time” learner whose google search lands them in Khan Academy’s trapezoid content? How might this supplement or support the “sustained learner” who has progressed through Khan Academy’s geometry course? How about a classroom full of students assigned to participate?
Student response: These ideas helped me see how other people may think about finding the areas of different shapes, and it also showed me how to make it easier on myself when I'm finding the area of different shapes.
I’m excited by the potential that students from around the world can engage in mathematical discourse. Imagine the classroom from the inner city talking with the classroom from the suburbs, or the independent learner from Mumbai talking with the google searcher from Atlanta. Imagine all of those learners discussing mathematics together.
