Article Summary
In this article from Mathematics Teaching in the Middle School Vol. 24, No. 7, May 2019, it describes a summer math camp experience run by Jo Boaler and her Stanford YouCubed team. The camp experience was for middle school students of diverse ethnic backgrounds all of whom did not identify themselves as math people. The students were given a standardized test at the start of the program and at the end and the majority of students showed a marked improvement in their overall score on the test.
Key messages that were delivered to the students were:
- Nobody is born with a math brain.
- Making mistakes and struggling to solve a problem is good for brain growth
- Thinking visually and using different representations of problems is import for brain connections and growth. (Boaler, 2019)
The focus of the program was on teaching problem solving and mathematical reasoning and encouraged the students to reason through their solutions with one another. They also encouraged students to exercise skepticism and to ask their classmates to explain their thinking and to justify the validity of their solution. Referring to Cathy Humphreys's skeptics framework, students were encouraged to think in terms of being able to convince themselves, convince a friend and finally being able to convince a skeptic. In this way, students were encouraged to ask questions of each other such as "How do you know that works?", "Why did you use that method?", and "Can you prove that to us?" (Boaler, 2019).
The nature of the work that the students undertook was group work which can sometimes be unbalanced with the strongest student or students doing all the work while weaker students are unable to participate or excluded. The study required each member of a group to answer the question of "How do you see it" before the group attempted to solve the problem. By using this approach, all students had an opportunity to begin participating in the problem solving from the start.
The ultimate result of the summer camp was that many students exhibited significantly improved results on the standardized test at the end of the test but a more important result was that many of the students changed their perception of their own ability to succeed at mathematics.
Reflecting on my own Practice
I'm always inspired by Jo Boaler's work but also frustrated because I don't feel I have the depth of mathematical insights necessary to properly guide students through exploration to understanding. I learned mathematics in the old-school, chalk-and-talk approach and had some success but there were definitely times when I just accepted that a method worked and tried to learn how to apply it correctly. Generally, mathematics wasn't done collaboratively, so if I didn't understand something I would make the decision to settle for maybe getting part marks or a zero on something I didn't understand.
On reflecting on my own learning style, I've learned that in most cases, I need to discuss problems and solutions to reach a full understanding of the material. This is certainly consistent with Boaler's approach to learning and I encourage students to discuss their problem solving methodology. This does have the drawback that students can often get off-topic/off-task in their discussions.
In my first-year practicum, I brought in whiteboards and facilitated collaborative problem solving for a unit in patterning and algebra. Students did adapt well to the change in approach (it was not how my AT worked) and their test results were good, so from that perspective, the unit was successful. During the Christmas party on the last day of school before holidays, I caught my two strongest math students trying to determine the equation for a more complex numerical series using the visual methods we had employed in class, so I added that to the win column as well. However, I'm not at all confident that if I revisited that unit test with them even two months later, that the majority would have retained the knowledge. I expect my experience would have be the same as Cathy Humphreys related in Connecting Mathematical Ideas (Boaler, J., & Humphreys, C. (2005)). It might have come back to them quickly with a review but I don't know. This is, of course, no worse a result than I would have expected if I had used more traditional teaching methods but we're trying to set the bar higher not just look for a different approach that renders the same results.
Another significant challenge I've encountered with students working collaboratively, playing skeptic and reasoning through problems are the diverse, and often limited language skills that many of our students have and the broad range of mathematical abilities of those students. I'm currently in a grade 4 class and have a few students who are grasping the material effortlessly (likely at above grade capabilities) while the student with the lowest ability is struggling to add single digits even when holding a number line (a grade 1 expectation). Groups that are heterogeneous in composition may not function effectively with such a wide range of abilities and may be further inhibited by limited language skills of some or all of the participants. I experienced similar diverse abilities in my grade 7 practicum last year which limits the success of this approach.
The duration of the math class can also be an issue. By the time students settle into class after transitioning from another subject, the math activity is described, student questions are answered and they begin working on the problem, faster students may get through the activity successfully in the available class time but some students may not have sufficient time to work through the activity and gain the expected understanding from the task.
My final concern with effectively implementing this approach to mathematics in my own practice is that I'm not proficient at finding "creative and rich problems" (Boaler 2019) that are relevant to students for them to engage with and that support the curriculum expectations and are appropriate for the abilities of the students. There are, of course, many resources scattered around the Internet but finding an appropriate resource when you need it can be a time-consuming challenge.
I do have one complaint with the article and it is not with the pedagogical approach but with the reporting that the students experienced as a result of the camp. The article reports an average of a 50% improvement in test scores but this statistic isn’t necessarily representative of the performance improvement experienced by most the students. The amount of improvement varied widely with some students tripling their score while others showed no improvement. A more representative measure would have been the mean improvement which will be much smaller although still very notable. It's disappointing to see that a sensational statistic was chosen rather than a more meaningful one.
References
Boaler, J. (2019). Prove It to Me! Mathematics Teaching in the Middle School, 24(7), 424.
Boaler, J., & Humphreys, C. (2005). Connecting mathematical ideas. Portsmouth, NH: Heinemann.