Mathematical Literacy: Understanding Algebra

Our mathematical literacy class is taking the 2nd Test today. Over the past few weeks, we have been working on several topics but the most emphasized of these is using basic algebraic expressions. We are seeing some real progress, and also some areas that are resistant to improvement.

A central issue in algebra is: What objects can be combined in adding, as opposed to multiplying? Initially, most students used the same rule for both types — if they knew to add like terms, the thought that they could only multiply like terms. After a few cycles through this territory, most students now see the correct rule for each type.

However, we are still struggling with the details. Even after exploring problems in small groups and in class discussions, several students in class are making these mistakes:

2n³(4n³) = 8n^9

2n³ + 4n³ = 6n^6

I think the problem with correctly learning these ideas is that it involves “2 dimensions” like a graph — students need to hold coefficients in one process and variables (exponents, really) in another process. Humans are not naturally that good with visual learning; interpreting a bivariate graph does not happen spontaneously. In algebra, we ask the same type of 2 dimensional thinking; instead of vertical and horizontal change, we are looking at coefficients and exponents.

The most challenging problem on this test? This one:

Pick a negative integer and perform the following operations.

Add three.

Multiply the result by 4.

Subtract 4 from the result.

Divide the result by 2. Write that answer.

Write the calculations for the generic number x and simplify the result.

A little more than half the students managed the basics of the operations on a constant. Several students came up during the test to ask what the last part was talking about. Sadly, we had spent one entire class day working on numeric patterns (seats around table arrangements) and generalizing the result for n tables. Nobody got the algebraic version of the question correct, and only a couple of students came close. It is true that the class this semester has students with weaker reading skills than we normally require, so it’s not surprising that those students had trouble. Even those students who are ‘well qualified’ for the course and completed homework did not get this one. [The problem is very similar to one in the homework.]

In a way, that problem illustrates the central theme of the material for this test — making the transition from numeric information to algebraic representation. Clearly, work in class needs to require more attention to that transition.

Some areas seem to have worked well; issues that we struggled with earlier came out okay on the test. Order of operations, including the pesky opposite of a square [like -6²] are definitely going better; I’d like to think that this is due to our working on ‘priorities of operations’ in class and de-emphasizing PEMDAS (I actually omitted PEMDAS from their reading).

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