Developing a different view on looking at student work through Baldermath
This blog post is about the class I teach at MIT called Understanding and Evaluating Education.
Student work is evidence of how they are thinking through material, which is a starting point for helping students get to more accurate (and eventually more efficient) ways of thinking mathematically and scientifically. However, in the quest for the Getting Of the Right Answer, it can be tempting to look at student work from an evaluative (right or wrong) rather than an interpretive perspective. Evaluating students’ ideas immediately leapfrogs right over the information students offer us when they show their original problem solving strategies. With that in mind, we spent a session in the Understanding and Evaluating Education course I teach at MIT exploring how to look at student work by playing a game called Baldermath.
A key skill in understanding student thinking is being able to listen effectively as they discuss their problem solving. Before class, we read an article Warning Signs by Victoria R. Jacobs, Heather A. Martin, Rebecca C. Ambrose, and Randolph A. Philipp. In this article, the authors discuss three ways that effective listening can be short circuited: interrupting the student, manipulating the materials, and asking closed ended questions. While interrupting students is probably never a good idea, manipulating the materials could potentially be helpful once the student has shown his or her thinking, and you can then use that to say “how about we think of this in a different way? This could be more efficient”. Asking some closed ended questions can be a way to check for student understanding quickly, as a summative assessment, but yes or no questions often gloss over the reasoning behind them, so use them carefully.
During class, we played the game Baldermath. Baldermath was developed with a math teacher named Michael Pershan. For many years, Michael maintained a blog called mathmistakes.org. Baldermath is a game that’s modeled after Balderdash, where a group of people all get a word and one person gets the correct definition and everyone else makes a definition up. Each player shares their definition and the others guess who is bluffing. If you guess who has the correct definition, you get points. Successful bluffers also get points.
In Baldermath, we use this game mechanic, but we use it to explore how people view student work in mathematics. Many individuals immediately look to see whether a math problem is “right” or “wrong”, and this can snowball into making assumptions about students, what they know, and can influence how a teacher interacts with his or her students. In Baldermath, we have an opportunity to imagine what a 4th grader might be thinking as they solve a problem, both from the perspective of a student (when playing a student) and from the perspective of a teacher (when playing the judge). It can be difficult to remember how confusing fractions were at some point, or imagine why a student might get carried away drawing butterflies. Looking at student work is one activity that teachers do in Professional Learning Communities (PLCs).
What we did during class
We started by having students look at actual student work on a Desmos activity with the question “what do you notice about this problem”? Then I had students divide into five groups of five, and passed out the cards for the game. Everyone got a math problem to look at. They chose a judge, the judge left the room. One student got actual work from a student, and copied it into his/her own handwriting. The others made up answers as if they were fourth graders. The judge returned, and everyone shared their problem. “I am the real student because…….”. The judge chose who they thought were bluffers.
After the game, we had a discussion of what was challenging about this activity. One student mentioned that it was difficult to remember what it was like to not understand fractions. Another mentioned that they had actually miscounted on a number line (so simple, yet easy to make mistakes sometimes!). We discussed how to ask students about explaining their work, and strategies for how to scale this to a whole class activity by gathering ideas from a student or students, asking other students for their input, and then synthesizing a whole class conclusion.
I asked three questions on the exit ticket, and here is a summary of some students’ responses to those questions.
What did you learn about student mathematical thinking as a result of playing Baldermath?
Nine students mentioned how the game gave them insight into student thinking. This included “remembering the mistakes students can make” and that students make silly mistakes that “can seem ridiculous to us”. Five students mentioned that fourth grade students took a visual approach to problem solving, which can be “good for understanding but bad for making careless mistakes”. Three students mentioned the challenge in moving students from a concrete representational approach to a more efficient symbolic approach to math. Finally, two students mentioned that there were many ways to do a problem wrong — and right.
Which role (judge or bluffer) did you think was most helpful to you in learning about student math thinking and why? What did you find challenging?
Seventeen of the nineteen students mentioned that being the bluffer or student was more helpful than being the judge. Being the bluffer allowed them to “think in the student’s shoes” and “explain student logic”. Students also though being the bluffer was difficult. Using visuals rather than symbols was a challenge, as was coming up with a believable rationale. Fewer students were able to be judges, but found value (and challenge) in that role. The judges had to think about misconceptions as they selected who they should was “the real student”, and one student noted that even just reading through the cards, which had several examples of 4th grade work, was helpful in considering how students may approach the problem.
Things I’d do differently
In the future, I think I will use a consolidated one page instruction sheet for the game. The cards are useful for self directed play (without facilitation), but a whole class experience that was synchronous would have given me a better sense of how many rounds students had completed, and allowed more time for discussion. I’d like to check in after round 1, and be sure to rotate the judge role so more than one student had that chance. I’d also make it clear that the judge can also look at the problem (but not the student work) while they are sitting out, and thus stay engaged in the game.
One student suggested modeling a student/ teacher conversation about a problem, which would have been a nice connection to the reading, and a way to consolidate what we had learned through the game. I think I would do that next time I teach with this game.
Final thoughts
Playing the game in class was a way of bringing everyone together in an activity that was engaging and helped approximate some key skills of teaching, such as thinking about student viewpoints and looking at student work. Playing a game also provides a new way to interact with others in a way that’s not just listening or small group or full group discussion.
The students seemed engaged with the game, and appeared to have fun while playing it. In their final comments, a few students found the game both fun and a positive learning experience. One student commented “The game was interesting! It honestly sounded strange at first, but while playing it I realized it accomplished its objectives quite well.” Another student noted “I was amazed by the creativity of the real student work — their methods (although less “efficient”) were sometimes really complex and outside of the box! — it’s a shame adults lose some of that type of thinking.”
Remembering that students can be creative and sophisticated problem solvers and thinkers is exactly what I had hoped students would gain from looking at student work .
