At the risk of not being able to include everything thoughtful (which is indeed lots of them), here are some quotes:
"the purpose of math class is to build a student's capacity to puzzle and unpuzzle herself"
"The “real world”-ness or “job world”-ness of the task is one of its least important features."
"Your classroom culture will beat any curriculum I can recommend."
Summarized below with a few additions for my own thinking (i.e. what I have chosen to take away from all of this... for now):
1. The important goal of a mathematics classroom is to developing students' desire and ability to puzzle and unpuzzle themselves. Not only for individuals developing their own identities and sense of agency, but together as a class. I believe collaboration is also an important key to this. Thinking together, discussing ideas together, failing together, and succeeding together.
2. Connections to the "real world" is by no means a requirement for engagement. It may help, but it depends on many other factors... where the most important factor is likely...
3. Understanding your classroom culture is extremely important. How do your students work together? How do they think? What interests them? If the "real world" is what engages them, then use it -- but build in opportunities for them to appreciate the "fake world." The same works vice versa (not saying that there's only the two different types). Developing the classroom culture is equally important. But how do we do this appropriately?
Ultimately it also comes to the fact that every teacher is different, and teachers should also use materials that they themselves find engaging.
Okay, sounds good. Let's cut into this with some of my own experiences.
I began a draft a while ago about me being weird and needing to filter my own ideas. Time went on and the examples were no longer relevant (happens to plenty of my drafts lol).
But this is the short version of it:
I am pretty odd.
I find almost everything interesting. It's like the only thing I do well. I think and over think and over think. I can imagine and develop mathematics questions out of everything I see. Walking from my car to the school, I can think about the amount of snow fall and how long it takes for the snow removal truck to plow everything. I think about how many steps more I'd need/save if I cross a snow bank versus going around it... for a quick example.
But that doesn't automatically make it engaging for students.
The following example is from a while ago, but is still fresh in my memory (and I am super forgetful)
It was great. My classes were used to me throwing in random things and them asking interesting questions. We explored it, discussed it, worked out some math for it, but it did not have a nice explosion that Dan typically talks about and engineers beautifully.
Regardless, I liked it. I mentioned this to my colleague the next day and he was surprised that I used the picture for what I used it for. At the risk of quoting wrong... he responded with something like "it was probably mostly your enthusiasm, and they were probably not really seeing the math right away."
That hit me.
He's probably right. Just because I find it interesting doesn't mean that others would automatically. Just because I think of all these questions doesn't mean that the students would.
Granted, that's actually what a site like 101questions is very good at identifying.
My impulsive use of fascinating pictures/videos is something that I have been thinking a lot about. Is it a good thing or bad thing? I've gotten into a habit of saving amazing pictures and photos from places like Stunning Pictures. Not to mention snapping pictures of things I find interesting. But what is this good for?
As a testament to these looming thoughts, my wife actually just walked by and saw my screen. She rolled her eyes when I pointed at the picture of the turtle and said that I used this last year. She continued on to tell me that turtles don't actually see their children. Okay, she was half joking with me because she knows how ridiculous I am -- thinking about math all the time. But she has a point.
Being spontaneous and using recent or recently interesting events may be engaging, but it does not work well in a model that moves through units after units that requires specific curriculum goals. I thought about this about two years ago, and decided that interest trumps goals. This was mainly because I wanted to weave through the curriculum purely through projects. The goal was purely to develop puzzles and to encourage students to solve them. No need for specific learning goals like slope or fractions. What is more important is just student questions and student interest.
Fast forward to today.
Due to a variety of reasons beyond my control, I have been unable to do what I wanted. What came out was (to me) an ugly hybrid. Interesting pictures/videos that were semi-forced into the picture frames of specific learning goals. Specific goals that other teachers teaching the same course are also aiming. Students and I still chat about the interesting questions they came up with - but we have no time to explore them.
So I have come to realize that culture matters in more ways than one. Until I can branch out and do whatever I want with a course (no units), I have to stop doing this. Attempting to combine the essence of not having units -- while still trying to fit units in... is not only ridiculous -- it's stupid.
I need to recognize not only the culture of my classroom, but shape it within the larger culture of the school. Which leads me to think that at this point, focusing on the packaging is much more important. Not to mention that the ultimate goal of building and solving puzzlement is still within reach.
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