When teaching algebra, or courses which depend on algebra, we as teachers aim to provide our students with rich experiences and a deep understanding which they will be able to retain and apply. Yet algebraic thinking is one of the areas of mathematics where students are most susceptible to rote learning, unthinking manipulation, and lack of long term retention. When we encounter these things, how can we respond?
In this article, the author describes the need for a constructivist approach grounded in the meanings students experience while learning. This entails an examination of the nature of the subject itself -- that is, of what we mean by "algebra" -- and a similar examination of what is meant by the term "learning". The author then offers practical suggestions for how to detect and respond when students' algebraic thinking falls short of our goals.
Specifically, in the sections below, the author will discuss the following: the nature of algebra and of conceptual learning; the use of error analysis and visual representations of algebraic concepts; turning student frustration into teaching opportunities; and helping students see and overcome hidden subtleties.
What is Algebra? What is Conceptual Learning?
We all know what "algebra" means. As teachers of mathematics, we are aware of its multiple strands -- symbolic, graphical, and numerical -- and of its role as a rich language for representing real-world situations and their underlying meanings. Properly understood as a mathematical language for modeling real phenomena, such aspects as patterns, generalizations of arithmetic, variables, equations, and the idea of function emerge within meaningful contexts.
Yet do these understandings inform our working definition of "algebra" in our practice of teaching? The level of success and depth of learning which students enjoy in an algebra class are closely connected to the answer they give to the question, "what is algebra?" The same is true, in fact, for teachers' and students' definition of learning.
When algebra is viewed strictly as a set of "rules" which govern the manipulation of symbols, student motivation suffers, as does retention since a learner's capacity to remember is dependent on having a context within which to place new ideas. Similarly, the type of studying in which our students engage -- for example, whether they seek out relationships and meanings or only look for which facts and formulas to memorize -- is dependent on their internal view and working definition of the term "algebra." Just as inadequate working definitions can hold students back, clearer ideas of what algebra "is" can aid learning. When algebra is placed in concrete settings, its multiple dimensions naturally emerge and foster deeper understanding in students.
The question of how one defines algebra is closely connected to which definition of conceptual learning is used. How do we define conceptual learning ? The answer depends on what we mean by the term "content acquisition." One definition relates to students' ability to correctly carry out a process. Alternatively, one may define and approach content acquisition by focusing on student understanding of the meanings behind the mathematics they are learning -- what Tom Bassarear refers to as "owning" the mathematics (Bassarear, 2001).
There is a very practical reason for using a meanings-based definition of content acquisition: a mostly process-based learning model tends to have little carry-over in terms of student retention of the material, or ability to apply it in a slightly different context either later in the course, or (gasp!) in a future course.
If students view mathematics as consisting of a largely unrelated set of tricks, calculations, formulas, algorithms and rules for manipulating symbols to "get the right answer," they will not grasp the material as deeply, or retain it as well. The psychological learning modes encouraged by such beliefs lead to detrimental study habits (e.g., a primary focus on memorization). In other words, significant obstacles to learning are often due to both student habits, and the beliefs about the subject which underly those habits.
This is not to say that process is not relevant. Ultimately, students do need to be capable of obtaining correct answers and of performing necessary calculations. But as teachers we must help our students avoid missing the forest (i.e., real-world patterns and underlying meanings) for the trees (i.e., the mechanics of calculations). Most of us, regardless of what age level we teach, regularly struggle with these issues. We know that "hands on" and constructivist learning are critical for helping our students understand mathematics in conceptual, meaningful ways. Yet how can we respond as teachers when students do not "get it"?
In what follows, several approaches are suggested for tackling these challenges. These include: (a) error analysis and visual representations of algebra; (b) ways of detecting when content acquisition by students is not conceptual, and of turning frustrations into teachable moments; (c) and ways of reminding ourselves of subtleties in seemingly basic algebraic ideas, and how to help students understand the meanings underlying algebra.
(a) Focus on Error Analysis
Throughout the K-16 curriculum, noticing and analyzing the common mistakes which students make can reveal problems in their mathematical thinking (Ashlock, 2002; Barzilai, 1999; Schechter 2002). Consider the following example from research cited in "Children's Understanding of Equality: A Foundation for Algebra" (Falkner, Levi, Carpenter, 1999):
In this problem, the correct value for □ is 7. Teachers who are able to correctly guess the most frequent student error most likely rely on past experiences and on their knowledge of the types of mistakes students make. To seek out the conceptual root causes underlying student errors, it's important to examine the mental frameworks and paradigms used by students which lead them to these common mistakes. That is, what internal notions about the nature of mathematics, about what constitutes "learning" mathematics, and about how one approaches problem-solving, do such students employ?
The cited study had students of various grade levels fill in the box with a value. The responses the students gave are not encouraging. The percent of students giving the wrong answer "12" initially decreased from 79% in a sample of first-graders, to 54-60% in second and third grade groups, to a low of 9% in fourth graders; the percentage then significantly increased in fifth and sixth graders to 48% and 84% respectively. The frequency of the answer "12" suggests a lack of understanding of mathematical grammar which defines the meaning of the equation as a statement. The equals sign and the box are often seen in isolation, with the former treated as a command to perform an operation, and the rest of the equation is ignored. This example underlines the importance of having students move from a mostly process-based framework (which may be more appropriate for elementary arithmetic problems) to a more conceptual framework which takes context and meaning into account. This context may be based on a word problem, or on the mathematical grammar in an equation, for example.
Another common mistake is the misuse of the equals sign as an all-purpose symbol connecting one expression or statement to the next. Suppose, for example, a student knows that making N gallons of pink lemonade carries a flat cost of $5 plus $2 per gallon. Students who can correctly calculate the total cost of making 9 gallons often use a "nonsensical" statement to arrive at the correct answer, writing:
Although this appears to be a more minor error, it reflects a similar lack of the critical thinking necessary for cognitive ownership of the algebraic procedures being carried out. Even "dumb" mistakes in which students assert, for example, that the value of 4x+5 is 45 when x=0, often reflect reasonable analysis applied to incorrect conceptual frameworks; in this case, failing to understand that x is a variable, not a numeral, in the algebraic expression 4x+5. Since the mistake is a surface manifestation of this deeper mistaken conceptualization of "x", the most effective response is not to apply remedies to the symptom, but to help the student grapple with the underlying conceptual roots of the error: understanding what the notation "ax+b" means.
One of the most common paradoxes of ephemeral learning in algebra is exemplified by the student who correctly performs "FOIL" on general expressions like:
only to later tell his or her teacher that:
Perhaps one reason students are so often "FOILed" in this manner is the very mnemonic ease the acronym provides, which thus allows the mechanics of the computation (I) to be very quickly absorbed -- often without the student also constructing an understanding of the reasoning and meanings behind FOIL.
Figure 1.
A conceptual framework such as an area model for multiplication can help students realize that (II) does not make sense (see Figure 1). This diagram visually demonstrates that the square of (A+B) has four terms: (A+B)2 = A2 + AB + BA+ B2 which simplifies to A2 + 2AB + B2. Students can also be helped to discover that FOIL is the result of two consecutive applications of the distributive property. More basically, the distributive property too may be represented with an area diagram (see Figure 2).
Figure 2 .
In addition to area models, other visual models which represent algebraic meanings and relationships are available to us as teachers (see Figure 3). Mistakes like "2(4+3)=(2·4)+3" are common when students try to manipulate symbols without meaning. We can help address the root causes of such errors, and thus help prevent them, by using visual depictions of key mathematical differences:
Figure 3.
The same can be done for algebraic versions of mistakes related to the distributive law:
Figure 4.
Diagrams similar to these could be drawn to help students meaningfully understand the difference between 6(1+4x) and 6(5x), in addition to methods suggested in (Ashlock, 2002). The key lesson lies beyond the specifics of these visual examples. The common theme is the need to look for the root causes behind common mistakes to truly liberate our students from their grasp.
(b) Turning Frustration into Teaching Opportunities
The pedagogical approach suggested above, essentially, is that we think of ourselves as detectives when we teach. It can be a frustrating experience to observe students making the same mistakes each year. One way to turn frustration into something constructive is to ask ourselves: "What is this mistake trying to tell me? What are its conceptual roots?"
In this role of playing detective, I have been led to discover, for
example, that the perennial mistake
"sin(x+y)=sin(x)+sin(y)" is often
a more simple, algebraic mistake in disguise: the student does not
understand the concept of a function as it applies to
sin(x). By playing Sherlock Holmes, one often finds revealing
evidence elsewhere in the written work of such a student. In one case,
the quotient of h over sin(h) was manipulated by
factoring the numerator as 1 times h, and "factoring" the denominator
as "sin" times h. Thus after "cancellation,"
the term h/sin(h) became "1/sin."
Such students clearly see "sin" as a separate entity which is
multiplied by whatever lies to its right; consequently, "sin(x+y)" is
seen as an opportunity to apply the distributive
property!
Thus the error here was not a false assumption about the
properties of sin(x) (i.e., linearity), but was rooted in
confusions about the function concept itself, and possibly augmented
by difficulties with the notation.
Similar confusions often occur at a less advanced level too, long before sin(x) is introduced. The following took place in a College Algebra course I have taught, and inservice high school teachers in my graduate courses have reported observing very similar student errors in their own classes. The class was looking at the "difference quotient" which represents the slope between two points on the graph of the function p(x): the numerator is p(b)-p(a) and the denominator is b-a. Before writing the formula on the board, I asked for a volunteer to state the expression. A student raised his hand and gave a would-be correct answer, except that he stated the numerator was "p times b, minus p times a."
Recognizing this as a case of confusing functional evaluation with multiplication, I initiated a classroom discussion. "What should we really say?" I asked. Someone else replied: "`p-of-b', not `p-times-b.'" Encouraged, I pressed forward, "Very good, but what exactly is the difference between `of' and `times' in this situation?" This led to a long silence, then a lot of grasping at straws by students. The discussion lasted over 10 minutes. At first the students were frustrated with me and with themselves. Ultimately however, the class was able to carefully reconstruct a correct, precise answer, and the students recognized the value of our discussion. Mistakes like these point to a need to reemphasize that an open-parenthesis does not always mean "multiplied by", that sin(x) is a function of x, and the need to provide meaningful examples and practice with these concepts.
Readers may want to recall and reflect on some frequently recurring algebra mistakes their own students make. Upon closer examination, consistently recurring student mistakes often turn out to have conceptual roots, rather than being due to sloppiness or to student memorizing the wrong facts. Such recurring mistakes are often not just false beliefs about correctly constructed mental models of concepts, but due to a false or incomplete internalization (mental model-building) of the concepts themselves, (e.g., variables or functions). Once such mistakes are traced to specific concepts students are struggling with, teachers can find ways of turning the mistakes into learning opportunities for these students and for the class as a whole through feedback, classroom discussion, or modified assignments.
(c) Helping Students See and Overcome Hidden Subtleties
In addition to actively looking for and diagnosing recurring student mistakes, teachers can directly examine algebraic ideas more deeply. Often, seemingly basic mathematical concepts turn out to be surprisingly subtle. For example, NCTM points out that the terms "variable" and "equation" can have many different meanings [NCTM, 2000]. Consider the following four equations:
(ii) 1 = t (1/t)
(iii) A = LW
(iv) y = 3x
These respectively exhibit:
Thus, the concepts of variable and equation are not monolithic. Although we use one set of terms, the underlying notions are complex and come in many varieties and shades of meaning. How can students be expected to avoid mistakes later on, when later material like higher algebra or trigonometry builds on the notion of a variable, if they remain blind to these subtleties? I still recall my own high school chemistry teacher trying to explain a quantity and stating that "it's a constant which varies." That apparent oxymoron -- which resulted in quiet amusement at the time -- in fact underscores the subtleties hidden away by the neat, cut-and-dry definitions and terms like variable and constant which teachers use everyday.
Similarly, one can ask what a function is. Is it a process? An operator? An object in its own right? Of course it is all of these, and more. To address these difficulties, concrete equations may be given for students to examine. For example, in the equation A=L·W, L can be the constant and W the variable, or the roles of variable and constant can be reversed, depending on which is most appropriate in the given context.
These examples highlight the need for teachers to pay attention to the complexities of "simple" concepts. By doing so, we as teachers can remain sensitive to the struggles our students face despite our own familiarity with the material and our ability to shift easily (and often subconsciously) from one shade of meaning to another.
Concluding Thoughts
How can teachers apply these examples and ideas to their teaching? It is probably best to start with a small number of modest goals at first. For example, by honing our listening and questioning skills, and by discussing common mistakes with each class in order to create a plan for learning which is customized to their individual conceptual and pedagogical needs.
Examples include assignments asking students to "find the error" and side-by-side contrasts of different roles for "x". Multiple representations of variables can be included to help students appreciate the meanings behind algebraic techniques, and to advance beyond unthinking symbol manipulation. Context-rich problem solving can also be very helpful (Driscoll, 1999).
It is also important to remember that there is no contradiction between computational facility and deeper understanding. After all, deeper understanding can provide context, meanings-based relevance, and interest which bolster students' ability to master and retain necessary algebraic manipulation skills.
Finally, as teachers we should remember that playing detective and attempting to diagnose student mistakes can be fun. Our feedback on written work can become more constructive for both the individual student and the entire class if conceptual errors are presented as learning opportunities, and used as launching points for further clarification and for deeper investigations by the class. When respectfully asked to take intellectual responsibility for and ownership of their mistakes, students' attention is drawn into a participatory process which truly can help us transform mutual frustration into teachable moments.