When I began teaching math, my own pristine view of algebra was suddenly under assault by the incoherent and chaotic view that many of my students had of mathematical ideas. I thought the world (that is, the teachers and adult students I knew from several college careers) shared my understanding of equations, variables, constants, operations, properties and various key symbols. More accurately, I gave no thought to my understanding of the symbols and concepts of algebra (and the higher-math topics which depend on algebra), so consistent, reliable and solid was my understanding: I had nothing to question or explain about how algebra worked since I was as ignorant of inconsistency and incoherence as someone in a two-dimensional world is of a third dimension.
Imagine my astonishment when I first realized that my students had it all wrong. I’m not just talking about forgetting how to distribute, or getting the wrong product from 5×6. They used equal signs in ways I never would have considered: as short-hand for “and then the next step is” and as an indicator for “here comes the answer.” They added an amount twice to the left side of an equation and not once to the right side (apparently not understanding what a “side” was). They substituted a number into a variable with a coefficient and happily turned the whole variable term into a brand-new two-digit number (instead of multiplying the substituted variable by the coefficient).
The “ah-HA” moment for me came one spring afternoon when I was asking my students to read out loud some equations I had written on the board. I was attempting to focus in on and correct some problems with students who didn’t understand what the “sides” of an equation were. One girl read every element correctly on the left side of the equation and eventually read every element correctly on the right side, but instead of saying “equals” (when she saw the equal sign), she said, “…And then….” I heard this several other times during the week. A student would see the equation x + 5 = 2x – 7 and read it as, “X plus five … and then … two X minus seven.” There were usually pauses, which I’ve indicated with ellipses, which I took to mean both that the student was not comfortable with the equal sign and that the student recognized the equal sign as some sort of punctuation, to be carefully set off – like a comma – with a little pause in speech.
Way too late in the year, I’m sorry to say, I had discovered like so many before me that language was important to learning mathematics. This is commonly understood in the math-teaching field, although from outside the field, people commonly assume that a math classroom can be a sort of language-free zone, a place where the poor students of English and the English Learners should be able to flourish, “since math is all just symbols anyway.”
This view is refuted in the literature. MacGregor and Price (1999) conducted tests with over 1,200 students and found a positive correlation between language and algebra scores in each of three different grade levels. Vygotsky (1978) described how social interaction and communication are crucial elements in the process whereby children conceptualize words.
You often hear students express frustration over some other teacher who presents an idea and moves on too quickly, brushing off their complaints with some impatient statement such as, “I explained this already. You need to start paying attention.” The observant teacher recognizes that in many cases, the students are paying attention, but they see a completely different picture in their heads than the one in the instructor’s head, and of course a completely different picture than the one the instructor expected them to see. It’s as if the teacher is lecturing on palm-reading, thinking everyone in the class envisions a human hand, when what the students think he’s talking about is a giant tree.
One of the places where problems begin is that the teacher may think the students enter an algebra class innocent of all the material that the teacher will teach. Kaye Stacey and Mollie MacGregor (2000) discuss this situation and its ramifications in their paper Ideas about Symbolism that Students Bring to Algebra. They claim, “Teachers may think that students come fresh to algebra, not considering that they already have ideas about the uses of letters and other signs in familiar contexts.” In their research on more than 2,000 students, they discovered many misconceptions, which they collect into these four categories: “students’ interpretations of algebraic symbolism are based on other experiences that are not helpful; the use of letters in algebra is not the same as their use in other contexts; the grammatical rules of algebra are not the same as their use in other contexts; and algebra cannot say a lot of the things that students want it to say.”
In their paper, Stacey and MacGregor list the most common misconceptions, along with their explanations for the students’ misconceptions. Some of my favorites include:
- Invented abbreviations which violate algebra customs, such as “Dh” to represent “David’s height”
- Assignment of values to variables based on a variable’s position in the alphabet (e.g., h + 10 = 18, since h is the 8th letter of the alphabet and so must have a value of 8 )
- Interpretation of a letter placed next to a number to mean addition, seemingly a best explanation for some students who are comfortable with the workings of Roman numerals.
- Automatic assignment of 1 to any unspecified variable, apparently related to the idea that a variable without a coefficient means one times that variable.
- Use of the equal sign to mean “makes” or “gives” (a habit from arithmetic), leading to such patterns as “3 + 5 = 8 x 7 = 56 ÷ 2 = 28.”
- Application of natural-language grammar rules inappropriately in algebra, for example, interpreting a = 28 + b to mean “a equals 28, then add b”
- Invention of symbols to mean things where students have not been given a symbol, for example in situations requiring iteration which could not be represented by a simple equation (some students demonstrated creativity with constructions such as “x ↑ 1, y ↑ 1” meaning “as x goes up by 1, y goes up by 1”)
Interestingly, Stacey and MacGregor identified many of these problem misconceptions not only in students who were new to algebra, but also in students who were in their third year of algebra.
These examples are relatively specific. A teacher who successfully identifies one of these misconceptions being used by a student might be relieved to discover a problem as limited in scope as the natural-language grammar item above, since the solution begins with the simple explanation that algebra doesn’t obey the same rules as English. But what about more fundamental misunderstandings? Glenda Lappan (2006) writes about problems with the conception of variable itself. She points out that as in English, mathematical terms can have multiple meanings, and the listener or reader needs to choose the appropriate meaning from the context of the term. “Take, for instance, the concept of variable–something students must understand as they mature mathematically,” says Lappan. “Faced with a mathematics problem, students have to find ways to use mathematics to represent the situation, manipulate the representations to find solutions, interpret the solution in the original context, and look for ways to generalize the solution to a whole class of problems. Variables play a key role in the process of mathematizing a situation. But what meaning of variable for a given situation is appropriate? Is it a placeholder for an unknown? Or is it a domain of possible values for one of the phenomena? Or is it used in yet another way?”
Early in a first-year algebra course, we use letters mathematically a number of ways. We present an equation such as “y = a” and describe it as the equation of a horizontal line, casually pointing to the letters and calling them “variables,” without considering the sloppy definitions in play. In this example, although the letter y is properly called a variable, the letter a is actually what we as teachers know from our broader experience to be a constant, standing in for the domain of all possible values for y, but only for a single value from within this domain at any one time. It’s safer to call all letters “variables”; it would be extremely confusing to attempt to describe these highly abstract notions to a roomful of fidgety 13-year-olds.
In addition to what we present for our students to attempt to understand, we also ask our students to practice using the brand-new language we give them. As with the previously discussed dilemmas our students face using symbols, a student who must form his personal concept of what a variable is will often invent something that doesn’t match our concept, and it’s our job to recognize the flawed invention.
As Lappan accurately points out, as we teach our students the language of mathematics, “The challenge for our students is to learn both to use that language to show their ideas, … and to read that language to understand the meaning of someone else’s mathematical representations….” This requires us as teachers not only to listen to our students’ use of language but also to watch and judge how our students hear us and each other as we talk about mathematics.
1. MacGregor, Mollie, and Elizabeth Price. (1999). An Exploration of Aspects of Language Proficiency and Algebra Learning. Journal for Research in Mathematics Education. 30(4), 449-467.
2. Stacey, Kaye, and Mollie MacGregor. (2000). Ideas About Symbolism that Students Bring to Algebra. Algebraic Thinking, Grades K-12: Readings from NCTM’s School-Based Journals and Other Publications, 308-312.
3. Lappan, Glenda. (2006). The Language of Mathematics: The Meaning and Use of Variable. Retrieved November 27, from http://www.nctm.org.
4. Vygotsky, L.S. (1978). Mind in Society: The Development of Higher Psychological Processes. Cambridge, MA. Harvard University Press.