## Reflection: Continuous Assessment Words and Abstractions - Section 3: Words and Abstractions #1

I've gone back and forth over the years on the role that a pre-test or diagnostic assessment can play within the first few days of an Algebra 1 course.  My current thinking is that I prefer to spend class time setting up class routines, nurturing classroom culture, and demonstrating to students the role of group work, both formally and informally.

Then, assignments like the Number Trick Project and this Words and Abstractions homework assignment play the role of a diagnostic, helping me to see how much each student knows about algebraic representations and vocabulary, as well gauging their dispositions toward the use of the abstract symbols that I've introduced.

For an example of what I can learn from this assignment, take a look at the back of this student's Words and Abstractions assignment.

First, let's think about the "Algebra" column.  I notice that this student has a lot of things right.  The only outright operation errors are on #'s 20, 27, and 28, where she defaulted to addition rather than using multiplication for the word product (#20) and division for the word quotient (#27 & 28).  Many of my students need some clarification that the word "and" doesn't always mean "addition" - that we have to notice words like difference, product, and quotient before we can interpret what the "and" means.  In particular, a lot of kids need clarification on the definition of quotient.

Next, exercise #'s 23 and 24 reveal some common errors with the order of operations.  Many students can recite the fact that multiplication comes before addition and subtraction in the order of operations, but they're not sure how to show this in an algebraic expression.  So on #23, for example, I know I'll need to show students how to use parentheses to indicate that the addition happens first, then we multiply by 3.  A quick glance at a stack of student work tells me which particular kids will need to get this message, and I'll pay close attention to who gets it over the next few days.  The other common error on #24 is the order of 15-x.  We will have to note that "subtract 15 from a number" means that we start with a number, x, then we subtract 15.  Our early dive into the Delta Math assignment (see my other reflection for this lesson) gives kids a chance to see for themselves that the order of terms matters in a subtraction statement, but not so for addition.  Eventually, we're able to generalize that addition and multiplication are commutative, but subtraction and division are not.  Most students are aware of this; this assignment shows me which of my students can apply that awareness and who will need my help.

Finally, there is the idea of what I describe to my students are "algebraic traditions".  Look at this student's solutions to #21 and #22, for example.  It is not wrong to write this expression the way it's written here, but it's not the way that we're going to see it in this class.  Kids have to be trained away from writing "x x 3" and toward writing "3x".  The same goes for that division sign on #'s 17 and 18: it's a fine and useful symbol, but not one you're often going to see in this course.  Here's my chance to re-train students to think of any division expression as a fraction.

Ok, so now that we've noticed all of that, what about the symbols?  It's hard work coming up with novel ways to represent some of these expressions, particularly when we're asked to represent two operations in a particular order.  My big takeaway from looking at this student's work is that she certainly brings a lot of mathematical knowledge to the table, but that I'll want to work with her to think about what the operations really mean, rather than just relying on what she's been taught.  Look at #29 and #30, for example.  That actually seems like a mathematically reasonable way to represent the square of a number!  But what does it assume?  Is the more important part of squaring a number the use of superscript to write exponent?  She's got that part down.  But I don't see an indication that she really knows what it means to "square" something.  In general, this student doesn't yet understand the different between multiplication and addition, and how each are represented with the squares and dots.  There's plenty more to get out of everything she's tried: what else do you notice?

Continuous Assessment: Student Work: This Assignment as a Pre-Test

# Words and Abstractions

Unit 1: Number Tricks, Patterns, and Abstractions
Lesson 4 of 12

## Big Idea: In order to figure out how to represent number tricks, we'll have to build up a language for doing so.

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Standards:
Subject(s):
Math, Algebra, Need to Know, symbolic abstraction, number tricks, translating expressions, Algebraic expressions
45 minutes

### James Dunseith

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