## Reflection: Student Ownership How Can an Abstraction Show Me How Things Work? - Section 5: Homework: The Number Trick Project, Part 1

Take a close look at the first task on the Number Trick Project handout.  Try it for yourself, down through at least step three.  What's going on here?

The first task is difficult, because it includes subtraction before addition.  In full disclosure, this was my own oversight the first time I created this project, but it has gone on to become a very important part of this task.

Based on what I've shown kids so far, we have a problem in representation.  It's easy to subtract four dots when we have dots from which to subtract.  But what should we do in this case?  Now, when I introduce this task, I tell students to expect a little bit of a challenge on this problem.  If it seems like they need a push in that direction, I'll say, "you may have to invent a new symbol to be able to complete this problem.  See if you can figure out what I mean."

It's so valuable to see what kids do with this!  This is the first task of the first project of the year, and there's this challenge, right away!  This is an opportunity to show students that they're going to have to think hard - to be creative, even - in this class.  It's also formative assessment: I'm learning about my students.  Just as important as knowing what math skills they're bringing to my classroom, I want to learn about their dispositions toward open-ended, unexpected tasks.  Following the recipe I've just laid out on the examples is fine, but what will my new students do with something they haven't really seen yet?

Moving forward, the task gets a great conversation going, one that we might not get to until the next class.  What we have is an endorsement for algebra!  If I try to create a new system of symbols, it quickly becomes clear that I'm going to need to keep adding new symbols in order to make it work for more complicated cases.  (If you think subtracting 4 is hard, what about dividing an unknown number by 4, for example?)  But algebra gives a set of symbols that are ready to go out of the box, and that would help us avoid all this chatter about squares and dots and some yet-to-be-named symbol all together.  That's pretty exciting, right?

In an upcoming class, one move that's been very successful has been to take pictures of the work of 5 or 6 students at varying levels of success on the first task, and to spend a few minutes looking at each as a class.  I put the work images on a Powerpoint, and then teach a mini-lesson by critiquing the work of each student.  I make the work anonymous, and I make sure to highlight what works well about each student's work first, before suggesting a few improvements.  I'm modeling the kind of critique that I'll expect students to engage in a few weeks from now.

Student Ownership: What was once a mistake is now indispensable:

# How Can an Abstraction Show Me How Things Work?

Unit 1: Number Tricks, Patterns, and Abstractions
Lesson 3 of 12

## Big Idea: Abstraction is a such an interesting word, because everyone does it, yet few know what the word means.

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Standards:
Subject(s):
Math, order of operations, Algebra, project, formative assessment, Algebraic expressions, number tricks
43 minutes

### James Dunseith

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