In a traditional beginning algebra course (like one I am teaching), we spend much of the time working with variables in expressions and equations … and functions. The course first shows variables within simple expressions with a progression towards complexity and equations. One problem (and the student errors on it) really caught my attention this semester; it’s actually a simple problem, not requiring any procedural steps.
Here it is:
Solve -6k + 3 = 3 – 6k
I have to admit that I did not emphasize ‘reading the equation to see what general statement it is making’, though we did actually talk about equations where the variable term was equal on both sides. One of the common errors is shown below:
Every student making this error could some an equation like ’2y – 5 = 4 – y’. What was causing the error?
Sadly, the problem was that many students are learning the ‘algebra dance’: Duplicate these steps, record the result. Part of the dance is to write the opposite of the variable ‘thing’ on the other side to get one variable in the problem. Students used this dance to solve a number of equations to produce quite a few correct answers. For this problem, part of the dance was the ‘get one variable’ — the student knew that -6 + 6 was zero, so we just have the letter. The variable was less understood than we thought, based on the consistent correct answers to other problems.
It’s very likely that you can list some mistakes that are similar in showing a less understood variable concept. One of the errors I am seeing is “5 + 2x” becoming “7x”; the numbers and letters become the whole story … the operations are not even being read.
If you are curious, there is a wide body of research on learning variable concepts; for one summary, see http://www.nctm.org/news/content.aspx?id=12332 (an NCTM item). Some particular research items: http://www.emis.de/proceedings/PME29/PME29RRPapers/PME29Vol3KoiralaEtAl.pdf and http://www.merga.net.au/documents/Steinle_RP09.pdf and http://elib.mi.sanu.ac.rs/files/journals/tm/16/tm915.pdf.
What I am focused on, however, is not the research nor the particular misunderstandings — rather, I am thinking about WHY this happens. It seems the problem is most likely when students have a higher motivation for ‘correct answers’ compared to their valuing of understanding (which is a combination of desire to understand and confidence in being able to). In my classes, I often say that I am not that interested in the answer they get; I am more interested in the knowledge you have about that type of problem (the understanding). Obviously, this statement from me does not change the drivers of student motivation (answers or understanding); I need to create instructional spaces where the understanding is the result being assessed directly. I suspect that I will be using some type of writing for this purpose; this will be a challenge, given the range of writing abilities in the class. For one reference on writing in math, see http://www2.ups.edu/community/tofu/lev2/journaling/writemath.htm.
However, I can count on a basic human trait: We (meaning our brains) naturally prefer to understand the world around us. Knowledge organized by understandings is easier to maintain and use, compared to knowledge that is random memories. The problem with ‘variables less understood’ is that this natural desire to understand has been subverted, perhaps caused by messages about ability … perhaps reinforced by social messages that math is about formulas and answers.
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