By Amanda Jansen, author of *Rough Draft Math*, along with Megan Wickstrom and Derek Williams

Whenever we learn anything new, bringing a spirit of rough drafting to the process can open us up to be more free to try on new ideas. We can ask students to share their first drafts, then we can discuss the ideas as a class to provide new insights on the drafts. During these discussions, it’s helpful to maintain a stance of curiosity and seeking to understand rather than judgment while we help each other’s ideas evolve. Students can then revise their initial thinking and document how their thinking shifted.

Now that so many of us are shifting our mathematics instruction online, I consulted with two colleagues who have been using rough drafts in their online mathematics courses. Megan Wickstrom and Derek Williams are professors in the Department of Mathematical Sciences at Montana State University. They both teach courses about mathematics online; these courses are professional learning experiences for future and current teachers. They have had success integrating rough drafts into their online mathematics courses. In this post, I share what I learned from them in hopes that K-12 mathematics teachers would find their teaching practices to be useful in an online setting.

**Why Use Rough Draft Thinking in an Online Course?**

Rough draft thinking is just as useful online as it is in face-to-face mathematics learning environments. According to Megan, “In my experience, posting online can feel pretty final and students feel pressure not to share ideas because they don’t think they are worthy of posting. Although students still begin the course with some hesitations, within a couple weeks of the course, students are posting ideas, ponderings, confusions, and extensions. It is very exciting to see!”

**Inviting Students to Draft and Revise**

Inviting students to draft and revise helps students overcome possible reluctance to share their work-in-progress. We can let students know rough drafts are welcome in this class and that they will have chances to revise their work.

Derek explicitly states to students that the purpose for discussion forums is to share initial thoughts about weekly assignments, interact with each other’s thinking, and to be able to revise before submitting the assignment at the end of the week. He posts the following to his students:

Each week you will be prompted to share your initial (rough draft) thinking about certain problems from the weekly assignment. You are not expected to have these problems complete or completely correct when sharing your first post. The purpose of these posts is to promote discussions based on the different ways of thinking you all bring to the problems, and to interact with each other’s thinking. You will have time to revise your original work before submitting.

Students are also told in the beginning of the course that all submitted assignments may be revised again based on feedback.

Megan shares the following with her students, after they read and discuss an article about rough draft talk (Jansen, Cooper, Vascellaro, & Wandless, 2016):

What does this mean for us and our classroom? The online learning environment can be a little challenging for learners to take risks. In our classroom, I want you to think of each assignment as an evolution over time. Do not feel like a proof or task has to be perfected before you post. I want you to feel comfortable posting your initial hunches, ideas, and ponderings along the way to your final, polished product. I also want to acknowledge that it often takes working through an idea in a rudimentary way for us to realize better, more succinct methods. If we never engaged in the initial work, we wouldn't have found a cleaner path forward. When working with Geogebra, remember this! Your first sketches might be cumbersome and maybe a little ugly, but they build to more beautiful and functional constructions.

It might also be possible that your work is not done when you hit submit. After each assignment, we will be in conversation on ways your assignment might be improved, if any. I will also do my best to draw connections and highlight student responses that might be helpful to everyone in moving forward.

**An Example from Megan’s Math Class**

Megan teaches her course asynchronously. Most of the class assignments are done through writing, short video presentations, and discussion boards.

For a typical mathematics task in her online classroom, she usually starts with an intriguing problem and gives students space to wonder, observe, and make a plan. They post their initial ideas in a discussion board, and they then read and respond to other classmates’ posts. One key component of her course is the practice of devising a plan and providing feedback on each other’s plans. Students brainstorm ideas without being forced to commit to one plan or another. This allows students to construct a draft.

We begin Unit 1 with an interesting dilemma (some of you may have seen this before). We have triangle A composed of 4 shapes. When we rearrange the shapes to make triangle B, a square goes missing. What has happened?

I would like you to use this as a pondering moment to do the following:

- Wonder: What do you want to know about these two pictures?
- Observe: What are attributes that you observe that might help you in reasoning about this dilemma?
Devise a plan: What would be your plan of action if you had to determine what happened? (Hypothetical...don't worry about a formal solution, yet).

After these initial ponderings, the students tackle the problem. Sometimes this involves posting solutions in progress and giving feedback to other solutions. While, other times it involves submitting the assignment directly to the instructor for feedback.

**Online Course Structure**

According to Derek, discussion boards allow students to try on ideas and peers’ responses help them refine their work. Then he provides feedback on the assignments, and students are given another opportunity for revision.

Derek thinks about online courses in one-week chunks of time. A parallel model could be a project-based math class or a problem-of-the week for a K-12 course. At or before the beginning of the week, he informs students about the upcoming weekly assignments and gives the students a few days to begin working through the problems and activities. Most students use this time to work individually, but he also puts students into small groups so they can work collaboratively if they choose.

Then, in the middle of the week, he creates prompts in discussion forums so that students can post their initial thinking as an entry post. Normally, he requires students to make a post before they are able to see other’s posts.

For the rest of the week, students spend time providing feedback on their classmates’ posts and discussing student-generated ideas. Students can use any of the discussion threads to revise their work on the problems before they submit their work for the first time.

Finally, he provides feedback, and students are given an additional opportunity for revision. Integrated with his feedback, he provides encouragement to students. Derek anticipates students to be a little apprehensive about sharing their incomplete or potentially “wrong” work during the mid-week posts, so he replies to all mid-week posts during the first few weeks with statements like, “Great thinking! I cannot wait to see the discussion your work generates.”

**Debriefing with Students**

At the end of each week, Derek makes closing announcements where he makes sure to feature the revisions that took place, and he shares the authors’ submitted assignment (blinded and with permission) as examples of the benefits of interacting with each other’s rough draft ideas and revisions.

Megan debriefs with students before they revise. After she receives initial solutions from students, she then crafts a video or document that summarizes the class’s consensus. In the summary, she reflects upon what they, as a collective class, know for sure about the problem. She highlights interesting approaches (making sure to credit different students), lingering wonderments or sticking points that might need more work, and interesting extensions that classmates proposed or considered. Then, students go back to their work and revise and resubmit a final product.

**Impact of Rough Drafting in Online Math Classes**

Megan and Derek both report being excited when they logs online and see students responding back and forth describing their mathematical thinking, supporting one another, and referencing revisions. It is remarkable to see students interact with each other’s ideas in non-evaluative ways. Megan’s students have made the following sorts of statements:

- Here are my thoughts. I might be missing something but I decided I needed to post anyway. Let me know what you think.
- I loved your logic about this topic. I didn’t think of it in that way! Something to chew on before we have to submit our final assignment in the upcoming week. Thanks for making me rethink my reasoning!
- This made me curious about another case and decided to investigate if it will work.
- Right after I submitted my revision, I thought of something else I wanted to add.
- After looking at another student’s approach, I started thinking about another solution and added additional thoughts.
- I had some of the same thoughts as you but hit a roadblock with this concept. I appreciate your diagrams, as they are giving me something more concrete to think about.

Megan believes that rough draft thinking has helped her students in at least three ways. First, the fear and self-imposed restrictions of mathematics were lifted. Students saw that it could be okay to post ideas in progress and continue to grow them over time. One of her students, when reflecting on the last assignment stated,

The fact that I wasn't being graded for my initial work on those problems gave me a lot of freedom to try new ideas and put my work out there without having to be anxious about it being 100% correct. I knew I could share my ideas, read ideas from other people, and then refine my own ideas.

Second, rough-draft thinking helped her classroom become a community. The students no longer saw problems as something they were required to know and do individually or in isolation, but as a group endeavor that we tackle together. Conversations and feedback became vitally important in the problem-solving process. One student reflected,

For all of these activities we were asked the basic questions of “why” or “how”. We were tasked with exploring using Geogebra to make claims about what the answers could be. We collaborated and bounced ideas off of each other to further our exploration. The problems were intriguing and engaging resulting in some really cool conversations. We were all working towards a common goal…solve the problem!

Finally, math has become less about the answer and more about the process. In describing proof, one of the students reflected that sometimes we ignore the process and focus solely on the solution when, in reality, the process is just as important as the solution.

We often have to play with different ideas to arrive at a proof. Usually that play is very results oriented -- we just want the proof to be done, and we tend to consider the time we spend thinking about different approaches as wasted time. When we get the result we are often critical of ourselves and think “why didn’t I think of that sooner?” I now realize that play time and exploration time is not wasted time, and perhaps it is even the most important part of the process.

Derek also experienced success with incorporating rough draft math online. He provided some quick data from his online math class: There were more than 2500 student-authored posts to discussion forums prompting rough draft thinking in a class of 18 students. Students were only required to make an initial thread and respond to two other threads for 14 weeks. If students completed only the minimum, then there would have been closer to 18 x 3 x 14 = 756 posts. The overall involvement and amount of interaction between students is evidence that rough drafting is effective for online teaching because it promotes discussion and participation in ways that center on students’ mathematical thinking.

According to Derek, this was a beneficial practice for online learning because he never felt compelled to post his own thinking or solutions to assignments. Regardless of how students were doing (from an evaluative perspective) on their initial posts, the collaboration on discussion threads and revised assignments were almost always at or above his objectives. The course was completely run and taught through student-generated ideas, which was really powerful for him as a teacher.

**How have you tried incorporating rough draft math online? Share your thoughts on Twitter, using the hashtag #RoughDraftMath. We hope to hear from you.**