It was fun to share the textbook scans with Dan. Christopher Danielson jumped in with some thoughts about elementary curriculum. He works with future teachers and is a very insightful fellow! Check out his Tootsie Roll and the Ootsie.

As a middle school math teacher, this elementary stuff is all new to me, but I'm open to the challenge. Especially since I'll be helping coach a handful of teachers next year with 3 Act lessons. I'm using the summer to investigate some ideas and build up a small catalog of lessons. My working goals are to:

- Work on turning the abstract to concrete.
- Look for overlapping concepts throughout K-8 grades.
- Further investigate the importance & relevance of estimation.

Lastly, it's great to know there are teachers helping students with life skills and not workbook skills. Sadie Estrella shares an experience she had this year by embracing those teaching moments we frequently have with our students. Should we eliminate 're-teach' from our vocabulary and change it to re-explore? Your thoughts on all this?

Future,

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I’m looking forward to seeing more of your 3 Act lessons for young students. If I were doing the Popcorn Picker task in a Grade 2 classroom, I’d recreate at the rainbow table what Dan does in the video. Whether teachers show these videos or are inspired to replicate them with their students, it’s all good.

ReplyDeleteI’m confused by one thing– your first goal. Why “turn the abstract into the concrete”? Students best learn math by moving from the concrete to the abstract.

Grade 2 students will solve Dan’s Popcorn Picker problem concretely (i.e., using popcorn or some other manipulative found in the classroom). Only concretely. That’s okay. They’re seven. What does the abstract look like here for a seven-year-old?

I’ve given this problem to elementary school teachers. The most common initial guess is “they hold the same because the two pieces of paper are the same size”. This is a wonderful (albeit unfortunate) misconception to have available as a workshop facilitator. Much like second graders, most teachers will fill each cylinder with cm cubes (or small plastic teddy bears, colour tiles, etc.) and compare the results. There are variations on this. My favourite: place the tall cylinder inside the short one, fill the tall one with concrete materials, pull the tall cylinder away. Your move from concrete to abstract is why I’m such a fan of your File Cabinet task. Students move from the concrete (post-it notes) to the abstract (SA = 2*l*w + …). Brilliant.

My initial approach to the cylinder problem was much different than this. I reasoned r had a greater effect than h in the formula V = pi*r^2*h. My background teaching secondary math at work here.

Maybe I’m misreading your goal. It happens. Maybe we have different definitions of ‘concrete’. If the goal is about you, as a math teacher, taking your abstract understanding of math and figuring out how it plays out concretely in the classroom, I’m right there with you (see above). If it’s about turning abstract to concrete for students, I’m a bit lost.

Thanks for the contribution Chris. Your last paragraph nailed it. My working goal as a teacher, curriculum designer, and coach will be to take those abstract ideas (in my head, from a textbook, or somewhere else) and make them concrete lessons/tasks. My working goal is for the teacher to have the concrete example and understanding in order to deploy to their students. I want the 3 Act lesson to be something they can reproduce in the classroom. Teachers can have students measure the paper, fold the paper, measure the folded sections, and begin piecing together the relationship between the tall, skinny piece of paper and the shorter, wider piece. I envision this as a prequel to the Popcorn Picker. I hope that helps.

Delete@reflectionsinthewhy: Your comment "I reasoned r had a greater effect than h..." reminded me of my student's comment of the same Popcorn Picker lesson (but ours were prisms instead of cylinders http://fawnnguyen.com/2012/06/14/20120613.aspx)"The tallness didn't make up for the fatness." :)

ReplyDeleteAndrew,your working goal #2 is also mine. When kids struggle in middle school, I'm so befuddled as to where/when they had missed the key concepts. Re-teach sounds very teacher-centered, while re-explore is student-centered, so yes, I'd vote for the latter.

Keep up the great work, Andrew. We can't get enough of them 3Act lessons!

A time machine would be nice to go find out when students missed key concepts. Ha! It sure would be nice to help instill a sense of desire with students to go back and re-learn or re-explore those missed concepts. Hmmm...

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