And math teachers aren't wrong- a poor quality task is not worth the time it takes to implement it. That's what I really appreciated about the treatment of tasks in Necessary Conditions. Quality is king.
Ideas I’m Implementing/Saving for Later:
- Tasks don't just allow for creativity- they should rely on it
- It's so easy for kids to hide behind rote, low level learning in many traditional classrooms. If they can regurgitate it back to you, GREAT! They can even do it with the numbers changed?! AMAZING! But those same students often cannot think around a problem posed differently or see when their knowledge might be necessary. They think math relies on memorizing processes and then implementing those "when the question looks like this" and who in their right mind would want to study THAT? Real world problem solving involves creativity that is intrinsic to true mathematics. I want to prioritize that in my classroom more.
- I loved this quote from the book: "If you deny students the opportunity to engage in this activity- to pose their own problems, make their own conjecture and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs- you deny them mathematics itself" (p.59)
- Two solutions from the book for this:
- tasks that are primarily student generated
- tasks that have a singular solution that can be arrived at through a multitude of solution methods
- Curiosity is key
- Tasks should naturally spark curiosity- building engagement and buy in. I had 2 favorite suggestions from this section of the book:
- Solicit Predicitions! Start with one that is obviously too low and one that is obviously too high, then a best guess. These get students to think about what their results should and should not be- sense-making and allowing them to justify what they think.
- "Steeped in Dissonance"- cognitive dissonance is such magic in the classroom. It gets kids thinking, wondering, trying to explain (or argue against!). This phrase gave me that warm, fuzzy feeling all over. You have to let kids sit in their dissonance a little to let them process when something is counterintuitive to them.
- Access is non-negotiable
- We talk a lot about "low floor, high ceiling" tasks in math ed and for good reason. If every single student in your class can't access the meaning of the task in some way, they have no way to be curious about it, much less get started.
- One suggestion from the book for sussing out the level of access is asking students to restate the task on their own. This is so simple, but brought me back to my time at the NYC Math Lab. Kids will explain to you what THEY understand, whether that's what you want them to or not. It's my job to find where that knowledge is anchored- what they DO know, instead of what they don't. Again, let them talk. Strategies after empathy.
- Simplifying tasks can lead you down a more complex road in the end
- I loved the discussion of simplifying what we ask- not only to extend access to more students, but also to open up the doors of creativity. It set off a lightbulb in me....I have been in a graduate cohort for the last 1.5 years that partnered with NASA and I couldn't figure out why so many of the NASA resources felt "off" to me. Here's an example about radiation in astronauts.
- It's got good bones- I like the real world applications and the use of data. But I've often felt like these feel "forced" to me- like a person said "HEY! There's math in this! I bet we can make this seem interdisciplinary!" Last year I started trying to open up my tasks more and it made a huge difference. This was my favorite- discovering the derivative. I asked me kids "What's the actual instantaneous velocity of a marble at this point?" and made them figure it out. It being open ended allowed them to ideate, test ideas, fail, and learn. It wasn't just a bunch of steps they were following because I asked them to do so. They were answering a big question in a way they saw fit. My goal this year is to simplify more.
- There is task inspiration everywhere!
- I loved the practical suggestions the book gave for adapting existing tasks. Textbook writers take a lot of time to write application problems, but they're often too scaffolded or too contrived. These tasks- which often rely on simplifying, as discussed above- help transform a meh question into a great task:
- Start with the answer
- Remove the steps and sub-problems
- Make the problem one of optimization (hellooooo, calculus)
- Choose your own problem
- Blur or withhold critical information
- Move from application problems to authentic experiences
- Keep fighting the good fight, even when kids fight you back!
- Sometimes you just need to hear this. A student that I taught 2 years ago and who graduated this year wrote me a card at the end of the year thanking me for not giving up on teaching the way I do, even though he fought me on it daily. He said it took him being years removed from it to realize the impact it had had on his ability to think. So for any of you fighting the good fight, keep fighting!
Questions I’m Pondering:
- What are good strategies for a student who really struggles with accessing at-level tasks? Especially with open enrollment in classes?
- What are more resources for upper level tasks? Who wants to share some with me (hint, hint ;o) )?
- What are the best ways to balance tasks with timing in an AP course, where you are already missing a month of instruction?
Last day tomorrow- effective facilitation & wrap up!