A Perfect Presentation has Mistakes

Written by: Dr. Christine von Renesse.

It's not how we make mistakes, but how we correct them that defines us.

The Dilemma with Mistakes….

I know that it is important for learning to make mistakes. Brains science tells us that the synapses grow when we make mistakes even when we are not aware of making mistakes. See also Mindset and Brain Science. And I have seen in the classroom how students’ conceptual understanding grows out of getting lost, feeling confused and making mistakes. Yet at the end, I still tend to “tell” students to not make mistakes anymore or at least not to repeat mistakes. How? By assessing their learning with presentations, tests, written homework, and final exams using rubrics that give the highest score to the work that has no mistakes. I know that I am giving my students mixed messages. I can see during presentations that students try to erase mistakes or skip over gaps because they believe that showing them would not "be good". See for instance the following student quote from from a journal in Algebraic Geometry, Spring 2017:

"Additionally, it wouldn’t hurt to voice my concerns in class more as well, as I’m sure many of my fellow students are most likely having the same issues as I am. If we open up discussions about problem areas, we’ll probably all have a much easier time making sense of some of the larger topics in class and in our write-ups. I am so used to being afraid of having incorrect answers that I don’t really participate the ideas I have. I know I need to get over this fear if I want to better my learning and provide areas of growth for me and other students as well."

So how can I avoid sending mixed messages and create better rubrics and assessments??

"Process Presentations" and Mistakes

In most of my classes I use group work. Students share out their ideas and process at a time when they are clearly not done yet. These "process presentations" are not graded and are meant to spread ideas, catch mistakes and make connections. In the following video of a mathematics for liberal arts class you can see a typical process presentation. The task was to find all numbers that can be generated by the expression 3a+5b where a and b are non-negative whole numbers. When you watch the presentation, see which mistakes or gaps you can find.

• How would you address the mistakes/gaps in the video clip? What would your next "teacher move" be?
• Imagine that you want to grade this presentation. Which score is appropriate?

"Final Presentations" and Mistakes

In Spring 2017, Volker and I co-taught Algebraic Geometry and used a new (for us) format for the class, relying heavily on readings outside of class and in-class graded presentation. You can read more about the class in this blog. Being new to a graded presentation-centered format of inquiry-based learning, our initial presentation rubric gave the highest score when the "solution was completely, or mostly, correct with only a few insignificant errors." Yet in practice the richest, most valuable learning opportunities for the class arose when presentations featured interesting mistakes. Similarly, certain aspects of the rubric had unintended consequences on students' level of preparedness. Below you can read about the evolution of our rubric. You can also skip the story and just look at two rubrics in comparison: Rubric Comparison.

The Evolution of a Presentation Rubric

Given our lack of first-hand experience with a presentation-focused class, we borrowed a colleague's rubric for presentations, which looked reasonable:

For each Wednesday, we will present selected problems on the board. Your presentations will be graded using the following 4 point rubric:
4	You are ready to present the problem assigned to you; and your solution was completely, or mostly, correct with only a few insignificant errors.
3	You were not ready to present the problem assigned to you but volunteered to present a different problem not already assigned and your solution was completely, or mostly, correct with only a few errors.
2	You presented a partial solution that was mostly correct and you clearly indicated where you were stuck.
1	You presented a partial solution with significant errors, but did not get very far and it was not clear why you got stuck.
0	You were not able to pass on presenting a problem and were not able to present any partial solutions.
You may not pass more than twice during the semester.

Our goal was to encourage students to be prepared and seek out support ahead of time if needed. We also wanted to give students an occasional "out," either through a limited option to "pass" (at most twice a semester) or by valuing their efforts on a different problem (with a small penalty).

To decide on presenters, we would check in with students in the order of "reverse seniority," i.e. students with few presentations had priority over students who had already presented a lot. We then got a quick idea of how prepared students were to present particular problems or hear whether they would prefer to "pass."

As the semester got under way with student presentations, several observations became clear to us about the various scores:

• "4:" The most learning for the whole class often happened when the presentation was in fact not "completely, or mostly, correct with only a few insignificant errors," as the highest result in the rubric requires. Instead, it was often mistakes in, or disagreements with, presented work that led to the richest discussions and deepest insights for a significant segment of the students.
• "3:" If a student presented the problem they were asked to present but the presentations was not "completely, or mostly, correct with only a few insignificant errors" they could now only get a "2" which seemed like a big drop in score. (We adjusted the rubric so they could also receive a "3.")
• "3:" Presenting an alternate problem: The Focus Questions were quite varied in difficulty, so the option of presenting an alternate problem may have encouraged a strategic avoidance of the harder problems in favor of easier ones. We believe this had an impact on students preparedness, which limited their ability to engage with, and critique, a presentation.

New "Final Presentation" Rubric

Broadly, we realized that our rubric did not communicate clearly enough those aspects of a presentation, or of student preparation, that we actually wanted to value most highly. Here is our revised rubric:

For each Wednesday, we will present selected problems on the board. Your presentations will be graded using the following 3 point rubric:
3	You are willing to present the problem assigned to you. Either: You have made a lot of progress and have tried multiple ideas/ways along the way to solve the problem. You make sense of the bigger picture and the ideas that the problem requires you to think about. You are either stuck or your solution contains an interesting gap or mistake. Or: Your solution is correct and you make sense of the bigger picture and the ideas that the problem requires you to think about.
2	You are willing to present the problem assigned to you. You have made some progress and have tried multiple ideas/ways along the way to solve the problem. You do not yet make sense of the bigger picture and the ideas that the problem requires you to think about. You are stuck or gave up.
1	You are willing to present the problem assigned to you. You have made some progress but have only tried a few ideas/ways to solve the problem. You do not yet make sense of the bigger picture and the ideas that the problem requires you to think about. You are stuck or gave up.
0	You were not prepared at all or not willing to present the problem assigned to you.

Valuing and Encouraging Mistakes in Presentations

A rubric is really only one means to communicate to students what practices we value in the class. The language we use to speak about, and frame, the work in the classroom is another important component that I've become more and more interested in. Our colleague Allison Henrich from Seattle University shares some wonderfully powerful language for "supportive facilitation" in her blog "I am so glad you made that mistake":

“That was perfect! Too bad there were no mistakes in your work for us to learn more from. I’d like to hear from someone who tried a method for solving this problem that didn’t work out so well. Would anyone be willing to share something they tried with the class?”

For a broader view of the important role that addressing mindset and learning about brain science can play in mathematics education, see our blog Mindset and Brain Science.