There’s so much to cover that it was hard to pick a topic for this week’s longer post.
It’s the end of Catholic Schools Week. Our schools have been actively emphasizing the Catholic identity of their communities this week, following from the NCEA theme of “Learn. Serve. Lead. Succeed.” Throughout the Archdiocese, as you’ll see in the Superintendent’s Memo this week and on Twitter, students, parents, teachers, and leaders have come together to celebrate what makes our schools great.
And it’s Groundhog Day. Punxsutawney Phil saw his shadow (even though here in New England, we’re always prepared for six more weeks of winter). Science teachers across the Archdiocese are most likely leading conversations about what meteorology is, why rodents aren’t typically known for their weather prognostications, and trying to discuss with students whether to observe Astronomical Winter or Meteorological Winter (I’m on Team Meteorological Winter). Meanwhile, Social Studies teachers are engaging in fascinating conversations about the fact that there are actually MULTIPLE groundhogs that AREN’T Punxsutawney Phil, each with their own cultural background, ethnic heritages, and community customs (Staten Island Chuck is one of my favorites, mainly because he frequently disagrees with Punxsutawney Phil).
But, to get the focus back to teaching and learning, I couldn’t resist writing about a news item I read earlier this week. Recently, in one district’s schools in China, students were confronted with the following math question on an exam:
If a ship had 26 sheep and 10 goats on board, how old is the ship’s captain?
As you’ll see in the articles linked here and here, some students declared this an impossible, ridiculous question. Some students, on the other hand, tried their hardest to come up with a solution to this very difficult problem. Commentary about this question has ranged from the shocked to the intrigued. But I think this is exactly the type of math question we should be asking our students more frequently in our schools’ math classrooms. And here’s why:
1) Multi-step, Complex Thinking and Reasoning
In order to solve this problem, the students cannot just plug and chug. They cannot simply adapt an already learned, discrete mathematical skill and come up with an easy solution. This question requires a complicated multi-step solution that works slowly from the inputs and givens (a ship that can carry at least 36 animals and a captain) to an answer involving the age of the captain.
With so little given, students have to think critically to fill in missing information, to infer connections between and among the givens and the desired solution. This means that students have to ask questions like, “How old are ship captains normally? How big is the ship? How much do the animals weigh? Do size and weight effect who can captain this ship?” Then students would have to carefully organize these complicated pieces into a complex, abstract whole, using the basic arithmetic operations to construct a response.
That’s a level of reasoning we simply don’t offer our students in math classes frequently enough. But it’s exactly the kind of abstract and complex reasoning that we need more of if our students are going to become the kinds of critical thinkers they need to be in our complex world.
2) Deep, Real World Knowledge
In order to solve this problem, students also need to have deep knowledge (or be able to ask deep questions about) the real world applications of the problem’s pieces. This means that a problem like this not only requires mathematical knowledge and reasoning. It also requires interdisciplinary reasoning. One solution to this problem for the Chinese students who received it involved knowledge of ship weight regulations and an awareness that ship captains require licensure with age restrictions. That real world, authentic knowledge is a design feature of this problem, not a flaw. The whole point of the problem is that this real world, deep knowledge is required. Otherwise, you could just say, “This is a made up math problem; it doesn’t matter how old the ship captain is.” No. The problem is written in a way to make you solve for the ship captain’s age, because it is inherently valuable to struggle with math concepts incorporating deep, real world knowledge.
The teacher’s task, then, is to guide students through the construction of the solution and help them find the right questions to ask so they can eventually access this deep knowledge. Students may not have much or any knowledge about ship captain regulations and policies. But a well-designed in-class mathematical investigation with this problem at its core can be designed to help you guide your students’ critical thinking. Which brings me to my last point…
3) No Single Solution
There’s no “correct” or single answer to this problem. Students are forced in this problem to articulate not only mathematical reasoning but a broader chain of logic and argumentation leading to their solution. The fun of the problem would be in probing multiple students’ reasoning to see which one is most plausible. Instead of having all students arrive at a predetermined solution (“The ship's captain is 50 years old”) you can create a classroom discussion about how and why different students arrived at different ages based on their mathematical and abstract logic. As you guide your students through questioning and constructive criticism of other students’ responses, you can build consensus about which chains of logic make the most sense given the known information in the problem. This not only gives students an opportunity to think critically, it gives them an opportunity to engage in a level of metacognition often missing from math classrooms focused on skill development.
So I challenge all math teachers: between now and February break, give your students a problem like this (or this very problem). I would love to hear how students in the Archdiocese are processing these types of math problems. Devote an entire math class to it if necessary. Allow your students to engage in complex and complicated reasoning, push them to incorporate deep real world knowledge, and guide them in the pursuit of multiple solutions. These are the types of problems that will truly start to engage students in math. They’re fun, they’re quirky, and they really make students (and teachers) authentically think.
PS: for those who demand an answer, I’ll just say that I was persuaded by the Chinese student mentioned in the articles who determined the captain had to be at least 28 years old.
CSO Academics Blog
Director of Academics, Andrew Miller, will post regular commentaries in this space about teaching and learning throughout the Archdiocese.