## Mathematical Literacy: Algebra Struggles, Building Algebraic Reasoning

One of my concerns with a traditional curriculum is that we put the content in ‘boxes’ — this week, we combine like terms … next week, we work with graphs … the following week we work with exponents & polynomials.  An average student proceeds through the course with very few opportunities to mis-apply concepts.

Our Math Lit class had a quiz today.  The first two problems are shown below:

1. Simplify the expression  -8x+2y-5x²-6y+2x

2. Simplify the expression (-8x)(2y)(-5x²)(-6y)(2x)

Most students did fine on the first problem, with combining like terms; a couple changed the exponent when adding.  The second problem caused the class to have a 15-minute discussion about what our options are.

To back up a bit, the prior class had worked on like terms (as a counting activity) and some very basic exponent patterns (multiplying with the same base, for example).  We had not formally covered the commutative property (did that today!), nor the distributive property (a start on that today).

The most common misconceptions that students brought to problem 2:

We can only operate on like things.

The numbers are connected only to the variable.

These were often presented as a package of ‘wrongness’, to create a common wrong answer:  -16x(-12y)(-5x²).  That is not a typo — students multiplied coefficients but did not change the variable (did not multiply those).  There was a general resistance to a suggestion that the constant factors could be separated from the variable factors — essentially, an over-generalization of the adding rule that we can only combine like things and the variable part stays the same.

A good outcome of this quiz is that students are more aware of some problems with their algebraic reasoning; every day, we talk about the reasoning being the important goal of this class, more important than ‘correct’ answers by themselves.  Students  partially buy in to this goal of reasoning; we did have a tense period in class when several students said ‘why do you have to make this so complicated!’.  I was honest with them that the second problem is overly complex compared to what we will need in our course.  And honest with them that the goal is knowing what our options are.

In our typical algebra course, these two problems are not addressed on the same day (except on one test day — even then, the problems are separated by space … one early on the test, one later on the test).  In our intermediate algebra course, I see the alumna of our algebra course struggle with basics — adding, multiplying, properties; the Math Lit experience sheds some light on how this might happen.  Students can pass a beginning algebra course and not understand the difference between processes for adding and multiplying.

We are early enough in the semester that I have to be cautious; just because an issue was raised does not mean that the students resolved the problem to get better understanding.  We will continue working on algebraic reasoning, so I will be looking for progress.

One thing I can say: If an issue is not raised for students, there is a very low probability that they will address the underlying problem.

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• By Sue J, February 22, 2013 @ 2:49 pm

How did you show the rightness and wrongness? Numbers, letters, or a combination?

• By Jack Rotman, February 22, 2013 @ 4:00 pm

Sue:
I’m not sure what you meant … we explore the reasoning of problems to figure out the right thing and to show what is wrong; in this case, we compared the adding and multiplying processes and talked about what is required and what is optional. The context here involved variables, so letters were used in all work.
Jack

• By Sue J, February 22, 2013 @ 5:06 pm

Sometimes I’ve found it helpful to show students that when I plug in numbers for the variables that the thing they said it “equaled” … doesn’t. Helps with the student who is sure taht the rules are different for adding and multiplying because… well, because your math teacher said so.

• By Diane W., February 27, 2013 @ 6:01 am

I find that students tend to confuse multiplication and addition even in their math facts. A common mistake even among my Algebra 2 students is to multiply, say, 6 and 3, and get 9.

I have signs up around my room: “Multiplication is repeated addition.”
“Powers are repeated multiplication.”
“Subtraction is addition of the opposite.”
“Division is multiplication by the reciprocal.”

I also have a graphic organizer with columns where the same number combinations are added, multiplied, and used as base and exponent, so students can see how radically different the results of these operations are.