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Wednesday, August 20, 2014

Day 1: Sooooo.... school.

Day 1 was not so bad. I like to minimally wah wah wah about the syllabus, because they won't remember anything until the information matters, so we all did some math today.

Algebra 2
Unit 1 is series and sequences so we went in hot with Eating Grapes.
On Monday Angela ate some grapes. On Tuesday she
was hungrier and ate six more grapes than she ate on
Monday. Each day that week she ate six more grapes
than the day before. After she had eaten her grapes on
Friday she had eaten 100 grapes in all.
I read the problem as a story out loud, and asked them to tell me a few things they heard. Then I displayed the text and asked them to read silently, looking for anything that was different from what they remembered. I only showed the scenario, not the question, so next they independently wrote down anything they wondered. We got some fun wonderings like, "Does Angela have an official diagnosis of OCD, or... I mean, Miss Nowak, who counts grapes?" but focusing on questions we have the power to explore mathematically, quickly settled on "How many grapes did she eat on Monday?" They had five minutes of silent individual think time, though some couldn't help themselves from discussing with their groups and I didn't really enforce silence. Then their groups (of 3 or 4) were charged with reaching consensus on a solution and writing it on chart paper so everyone could see. (Not just the answer! Make your thinking visible! I want to see how you arrived at your answer! What was the thought process? No more than half your solution should be numbers! etc etc).

The two approaches I saw were guess and check, and writing algebraic expressions to make sense of the pattern, and then undoing. Making lists or tables were common strategies. Nobody drew diagrams. Here are two samples:



Where I struggle is, books like 5 Practices, and the anticipated answers given by Math Forum, kind of assume every group is going to do it correctly, just in a different way, and the teacher's job is to sequence the different solutions appropriately. I have seen little guidance on how to provide minimally-invasive help to students who have misunderstood something about the problem, but don't realize that their answer is wrong.

Here is the work of a group I failed spectacularly today:


They were sure they were right because 16 works in their equation, but they weren't checking if 16+22+28+34+40 added up to 100. What I did was, encourage them to use common sense and see if starting with 16 would get her to 100 grapes by the end of the week. What I should have done, I think, was interrogate them about where the 4 and the 6 came from. As a result, they fell back to guess and check, but for some reason only added up four days, because "we don't know how many she ate on Monday." They said the answer was 10 grapes on Monday, because 16+22+28+34 = 100. Interesting, right?

Here was another group I couldn't make see the light:


If you can read it, their answer was 76. They may have had two misunderstandings about the problem: that Angela ate 100 grapes on Friday instead of 100 grapes in all, or that Angela only ate 6 grapes every day Tuesday through Friday, instead of six more than the previous day. I think it was the second one. We went around and around. I tried saying "Well look. If she ate 76 grapes Monday, and 82 grapes Tuesday, she's already eaten 158 grapes. But we know that she only ate a total of 100 in the whole week" but it was like we were not speaking the same language.

Two things I have to work on:
What to do about kids who are trying to do nothing, and hope that if they are quiet, I don't notice? Group work enables this behavior, because their group can still produce something without them. I don't think "roles" is the answer, because you can be "resource manager" or whatever and still not do any math. I don't think "everyone turns in their own work" is the answer, because then there's no compelling reason to talk to each other.

How to present work so that everyone learns something about why the correct solution is correct, and ideally, learns some math I am trying to teach them? Again, 5 Practices acts as if the four anticipated solutions will show up in your classroom and it's just a matter of choosing what order to talk about them in. What about groups that do not reach a correct solution? How do you discuss their work without embarrassing them? (I think the answer is lots of deliberate growth mindset interventions, but man, it's the first day of school and the last thing I want to do is make a kid feel bad for trying.) What if (like today) no one makes a diagram? Do you generate your own teacher diagram on the fly, for illustrative purposes? What about students who are super-reluctant to speak to the whole group? Is it okay if the teacher explains their work, and maybe asks them some specific clarifying questions along the way?

Help me, people who know what you are doing. I need you.

I also taught a Geometry class! I think I will blog about that tomorrow. (Two days of block scheduling, so same lessons tomorrow.)