Let’s talk variables. Do we want students to develop an understanding of variable concepts (whether in a developmental math course or not)? Is accurate use of variable notation enough? If students can model applications and accurately determine solutions … is that enough? Is there a role for linguistic literacy in mathematics?
A few years ago, I was able to use a sabbatical leave to explore a number of issues related to learning in developmental mathematics; the primary product of this leave was a series of short reports intended for my department though appropriate for faculty at other colleges; one of the reports dealt with variable concepts — see http://jackrotman.devmathrevival.net/sabbatical2006/1%20Variable%20Understanding%20and%20Procedural%20Skills.pdf. The other mini reports from that sabbatical are available at http://jackrotman.devmathrevival.net/sabbatical2006/index.htm
One of the issues we face in college is dealing (or not) with prior learning. Without intervention, prior learning (even when inaccurate) survives — often surviving in the face of conflicting information in the current learning environment. Visualize the prior learning as being as a stable mass of ‘knowledge’ (even though it has gaps and errors); as students go through a class as adults, information that connects positively with the old reinforces the old. When new information does not connect or conflicts with the old, the low-energy (natural) response is to build new storage … resulting in that solid core being supplemented by weak veneers of new knowledge. This, of course, is an incomplete visualization for the actual processes in the human brain. The suggestion is that students approach a math class with an attitude that supports old information and minimizes cognitive effort for dealing with new or incompatible information.
In my beginning algebra class this week, we did the test on exponents and polynomials. Although the test includes some artificially difficult problems with negative exponents, most of the items deal with important ideas. One of the most basic items on the test was this:
Evaluate a² + (3b)² for a = -3 and b = 2
Several students made this mistake with the first term:
-3² = -9
A smaller group of students made this mistake with the second term:
Now, this is a good class — all students are actually doing homework and attending class almost every day. We had dealt with the first situation at the start of the semester. How could these errors survive to this point?
Both errors are based on variable as a symbol to be replaced by a number, which is not complete. They might represent a visual approach, not verbal. Variables represent quantities involved in sums and products, where products with variables are implied … and more than this. Simplifying expressions might — or might not — uncover the incomplete understanding. What can I do to help students with this?
I am planning on incorporating some linguistic activities around variables in the first week of the semester. Some of the ideas are from a old book called “English Skills for Algebra” from the Center for Applied Linguistics (Joann Crandall, et al); I believe this book is out of print. The authors wrote this book from the viewpoint of helping students with ‘limited English proficiency’, which might just apply to many of our developmental students. Some of their activities involve listening to somebody read mathematical statements and the student writing them down. I think I will mostly activities that deal with written statements — identifying translations and paraphrasing (both to algebra and from algebra).
I do know that just saying “that was wrong … this is right” will not help these students develop a more complete understanding. I need to create situations where they get uncomfortable and really dig into the concepts related to variables. Some energy needs to be created so that we don’t just place a veneer on top of that mass of prior knowledge; parts of that prior knowledge need to be broken up and put back together. Without that process, many of these students will be limited in their mathematics and blocked from many occupations.
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