Notational Obstructions to Learning
in Calculus and
Precalculus Mathematics
WORKING DRAFT -- JUNE 1999
Feedback welcome
Harel Barzilai
Department of Mathematics and CS
Salisbury Univesity
Salisbury, MD 21801
harel[at]barzilai[period]org
A class of commonly recurring student mistakes in pre-calculus and calculus is examined, and evidence is presented that such notationally related student mistakes are fairly common. A subset of such mistakes is further analyzed, with the data suggesting that certain conceptual student misunderstandings, relating to the dual nature of functions (operators versus objects), and other dualities (e.g. constants versus variables) underlie many of these notational mistakes.
Quantitative and qualitative analysis of a sample of calculus final exams is used along with entries from a journal in which commonly observed student mistakes were recorded, and evidence from these sources is used to support the above conjectures. The concluding discussion includes possible implications for teaching, notational reform, and avenues for further investigation.
Overview
Starting in 1997 a journal was kept at several institutions of some of the more common mistakes made by students in several calculus and pre-calculus classes. The most common mistakes seem to be somewhat independent of institution, that is, similar mistakes were recorded at a large public research oriented university, at a medium sized private research university, and at a small private liberal arts college1.
More recently (spring '99) a sample of final exams from a first-semester calculus course was analyzed. It appears that many of the most common, recurring calculus and pre-calculus mistakes are notationally related, and some measures of the frequency of such mistakes are given. In addition, deeper conceptual roots appear to underlie many of these mistakes, and data from the journal and the finals was examined in light of these two conjectures, for which evidence was found.
Although this suggests that certain underlying mathematical concepts need to be reinforced for students, the cited data also suggests that some current mathematical notation may be the source of notational or visual "banana peels" for students, cause such mistakes, although based on the deeper conceptual misunderstandings, to be committed by students with higher likelihood.
Recurring Calculus and Pre-calculus Mistakes
It is commonly known and voiced by college mathematics teachers of calculus and pre-calculus courses that as a new semester approaches, they can expect to find that many of the mistakes their students are likely to make will probably be the very same mistakes they have seen made by students semester after semester in such courses.
It seems worthy of the attention of and research by mathematics educators to better understand such recurring calculus and pre-calculus mistakes (RCMs), gain some insight into their deeper roots, and make some tentative suggestions on how to reduce the incidence of these RCMs. Ideally, such improvements should come not merely from better "training" of students ("memorize these common mistakes, and don't ever make any of them"), but from addressing the conceptual difficulties (and perhaps notational contributing factors) which underlie and give rise to RCMs as their symptoms. These are the aims of this investigation.
Evidence presented here suggests that many RCMs are in a sense notationally related, and that such mistakes are common. From the evidence and subsequent analysis tentative suggestions are made about what the deeper roots behind certain RCMs might be, that is, suggestions about their origins in students' conceptual difficulties. Finally, an attempt is made to address more broadly how RCMs can be analyzed, what might be done to reduce their incidence, and possible future avenues of research on these issues.
We begin with some examples of RCMs.
Types of Mistakes
Consider the following familiar student mistakes:
_______ __ __
(1.1) \/ a + b = \/a + \/b
(1.2) cos(x+y) = cos(x) + cos(y)
(1.3) eab = (ea)·(eb)
1 1 1
(1.4) ----- = --- + ---
a+b a b
These mistakes were taken from the journal of student errors, and may
serve as a set of examples of notationally related
RCMs. Although these examples are in a sense not representative of all
types of notationally related mistakes, which one might try to
classify into subtypes2, a more immediate question is: which
kinds of mistakes might not
be considered notationally related?
Other types of mistakes include arithmetic mistakes, pressing the wrong key on the calculator, wrong memorization (e.g. "d/dx(x2)=x", and perhaps "d/dx(cos(x)=sin(x)"), false generalization (e.g. "d/dx(tan(x))=cot(x)" or "d/dx(xx)=x·xx-1"), and various other logical errors (for example, in solving two equations, E1 and E2, each in x and y, the student, possibly after making an arithmetic mistake, gets y=1, then uses E2 to get that y=1 implies x=2. The student then checks the point (2,1) by plugging into E2 rather than into E1).
To understand the origins of RCMs, one would like to identify students' underlying mathematical misconceptions, or their mis-conceptualizations, that is, cases in which the students have incomplete or erroneous mental models, rather than false assumptions about correct mental models.
In the case of notationally related RCMs, such as (1.1)-(1.4), one may pose the question of whether they are related to deeper misconceptions beyond or in addition to the aesthetically appealing2, enticing appearance of these false equations. In other words, is it just "wishful thinking" which leads students to write equations which "look nice" but are false? Are there other, deeper origins?
At first glance, the obvious commonality among (1.1), (1.2), and (1.4) would appear to be an implicit assumption that a particular function is linear. Indeed, if we generalize "is linear" to "is a homomorphism", all four cited RCMs fall under this category. Were this the only source of these and related RCM's, pedagogical implications might include suggesting increased stress by teachers that not all functions are linear, along with counterexamples.
However, upon closer examination, evidence can be found which suggests that deeper mathematical misunderstandings underly at least one of these, namely (1.2). Evidence from homeworks, papers, informal interaction with students, and from the sample of final exams analyzed suggests that a collection of RCMs including (1.2) may often be due to the misuse of the Distributive Property rather than to false assumptions about functional linearity. Such false applications of the Distributive Property, in turn, suggest directly that such mistakes arise from deep misconceptualizations of the notion of a function.
Consider the following entry from the journal of student mistakes:
sin(5x) 5x
(2.1) ------- = ----
sin(4x) 4x
This equality would certainly hold were sin(x) a linear function, were
the student perhaps using not the assumption that for L(x)=sin(x), (i)
L(x+y)=L(x)+L(y), but instead property that (ii) L(cx)=cL(x) holds for
all c, where c is any constant3. However
the following observation suggests that this mistake is not based on a
false assumption of linearity.
Had the student used "false linearity" the calculation would have been:
sin(5x) 5sin(x) 5 / `` sin(5)sin(x) sin(5)'' \ ------- = ------- = --- | or conceivably, = ------------ = ------ | sin(4x) 4sin(x) 4 \ (`homomorphism') sin(4)sin(x) sin(4) /in which 5x/4x does not arise as a final (or middle) step in the calculation.
Thus there are RCMs which at first glance appear to arise from a false assumption of linearity but which may in fact have different roots. In particular, example (2.1) provides at least prima facie evidence that the error may have been due to an attempt to 'cancel' the sines as if they were factors, that is, as if sin(4x) were a product of a factor "sin" times a factor "4x", since (2.1) is consistent with such a mistake. However since as we have seen appearances can be misleading, we turn to an example where there is more direct evidence. Another entry in the journal begins as follows:
h 1 h
(2.2a) ------ = - · -----
sin 3h 3 sin h
which at first appears to be a mistake due to a false assumption of
linearity (sin(ax)=a·sin(x)), however the student continues:
1 h 1 1 h
(2.2b) - · ----- = - · --- · -
3 sin h 3 sin h
before replacing that last "h/h" factor by 1. In this case the
conceptualization and treatment of sin(h) as a product of two
factors is explicit. Thus, as in (2.1), what appeared at first glance
to be a case of a false linearity assumption in fact had roots in
different mathematical misconceptualizations. In the case of (2.2) in
particular, there is an explicit instance of "sin" being
treated as a number or object which can be isolated or canceled. Indeed,
the dual nature of function is such that it can be
canceled in the proper context, namely against other functions, such
as
(cos(x)) (sin(x))
(2.3a) (cot(x))·(tan(x)) = -------- · -------- = 1
(sin(x)) (cos(x))
Since it is common notation to abbreviate f(x),
f'(x), and d/dx[f(x)] as f, f' and
df/dx, it does not seem unreasonable that such notation might
encourage some students to remember (2.3a) as
(cos) (sin)
(2.3b) (cot)·(tan) = ----- · ----- = 1
(sin) (cos)
Whether students in fact move from (2.3a) to (2.3b) is not clear, and
judging whether
abbreviations such as f' contribute to
students arriving at (2.3b) would require further investigation;
minimally however,
it seems that such
abbreviations do not help students avoid a mental
transition from (2.3a) to (2.3b)4.
Notation aside, it seems that fundamentally, the subtlety many students appear be falling prey to is rooted at least in part in the dual nature of functions as both operators and as objects in their own right; in the latter case, functions may indeed be canceled in division (against other functions, and provided fine print about domain of definition is included5).
Analysis of Errors on Final
To examine notationally related mistakes more systematically, a sample of 13 students' final exams from a first-semester calculus course was collected and analyzed. The following mistakes were found in four of the exams:
(3.1) The first student, R., was skillful enough on the final to correctly find the derivative of x2+2x+1 by hand (that is, using limits and the definition of the derivative), while in another problem R. wrote:
(3.2) The second student, B., in trying to calculate the derivative of x2+2x+1 by hand, wrote:f(x) = sin(3x + x2) f'(x) = (sin)'(3x + x2) + (sin)(3x + x2)' f'(x) = (cos)(3x + x2) + (sin)(2x + 3) f'(x) = cos 3x+x2 + sin 2x+3
(i) f(x+h) - f(x) (ii) f(x2 + 2x + 1 + h) - f(x2 + 2x + 1) (iii) fx2 + 2xf + f +hf - fx2 + 2xf - f
(3.3) Among the stronger students in class, K., who in this first-semester calculus class handed in a nearly perfect paper on the use of partial derivatives in chemistry, nonetheless included the following notation in the margin of her test:
xn=n·xn-1
(3.4) The fourth student, J., explicitly wrote:
and continued with:(i) f(x) = sin((cos(x)+3)9) (ii) = sin · (cos(x)+3)9
(iii) f'(x) = (sin)' (cos(x)+3)9 + (sin) (cos(x)+3)9'
Mistakes (3.1), (3.2), and (3.4) are notationally related in that parentheses, as in A(B+C) are indeed proper notation for denoting implicit multiplication, which is the meaning mistakenly being used, in addition to being used for denoting functional composition, while (3.3) may be considered notationally related in that it suggests a lack of fully internalized understanding of what the notation "=" signifies.
There is a second notationally related error in (3.2), for a total of five such errors: in going from (i) to (ii) the student initially interprets the symbols as representing a double composition, with f(x+h)-f(x) interpreted as f(f(x)+h)-f(f(x)) before ignoring the compositional interpretation of the parentheses and lapsing into interpreting them as signifying multiplication in going from (ii) to (iii). There is also a minor sign error. As before this seems to entail a simultaneous confusion about the role of "f" as operator versus as object.
These five mistakes out of thirteen exams (six if the parentheses realted sign error is counted) give an empirical frequency (about 38% or 46% respectively) for notationally related mistakes. In all cases possibly excepting (3.3) the examples suggest that the ambiguity of current mathematical notation might have contributed to the likelihood of the error being made, giving a frequency of about 31% (4 out of 13) for what might be notationally based mistakes.
Just how common are notationally related errors, and just how common is the subset which is suspected as being notationally based? One way to answer this question is to attempt to measure what percentage of "all" errors made by students fall into these categories. Taken literally, this would certainly be a monumental if not impossible task.
As a first approximation, I have used the May 13, 1999 version of Eric Schechter's The Most Common Errors in Undergraduate Mathematics [ES]. Although the counting process is somewhat subjective (readers are invited to carry out this exercise themselves), I counted 23 types of errors listed, with 5 (22%) not notationally related, 6 (26%) ambiguous, and 12 (52%) notationally related.
Individual notationally related mistakes may or may not be considered
aesthetically or visually appealing equation (VA), such as our
examples (1.1)-(1.4); or they may be considered notationally based (NB)
mistakes in which the notation provides a "banana peal" upon which the
student slips, such as "-x" being suggestive of a negative number,
lost invisible parentheses, or the use of parentheses to denote
implicit multiplication in one context, and functional composition in
another; or both, or neither. In the 12 notationally related mistakes I
counted 7 (58% of the 12, and 30% of all 23) in the (VA) category, 7 in
the (NB) category, and 2 (16.6% of the 12, and 8.7% of all 23) in both
subcategories.
The following have all been observed (recorded in the journal):
One journal entry involved a student confusing three-and-one-eighth
with the product (3)·(1/8) rather than the sum 3+(1/8).
Indeed the notation xy signifies the product x·y,
and 5x signifies the product 5·x while
2¾ signifies the sum
2 + ¾, not the product 2·¾.
In fact, as noted in [RN], historically, some mathematicians wanted to
use 3x to denote the sum 3 plus x, rather than the product.
"Mixed numbers" such as
2¾ are the only survivors of this otherwise
(???) convention
Interestingly, [RN] lists in passing an example similar to (?.?) here,
namely x sin x "in which the first x is a multiplier and the
second x is input to the since function" -- an ambiguity we agree is
potentially confusing to students, and about which we would add that
(making student think of "sin" as an object one multiplies by rather
than an operator, again the duality...)
Notationally Related: Other Examples (DRAFT OF
A SHORT SECTION)
/ f(a+h) - f(a) \
(I) f'(a) = lim | ------------- |
h®0 \ h /
/ f(x) - f(a) \
(II) f'(a) = lim | ----------- |
x®a \ x - a /
Some students successfully derive f'(x) form (I) by substituting "x" for
"a" on both sides, getting
/ f(x+h) - f(x) \
(I') f'(x) = lim | ------------- |
h®0 \ h /
but others, trying to make the same substitution by using (II), get stuck.
Discussion
As a graduate student in mathematics during an oral exam, I remember being momentarily stuck on a particular question relating to a certain space (a K(G,1) space), then realizing that if another "name", that is another notation for the same space is used (a projective space) the answer is almost immediate. Upon answering the question, and adding "it's embarrassing how much the notation I use affects my ability to see what's going on!" I was somewhat surprised to hear one of the two examiners, among the leading researchers in his field, reply "the same thing happens to us too!"
Notation does matter. It affects our ability to understand mathematics and to think mathematically. That our students are not immune to having difficulties which arise at least in part from difficult, ambiguous, or misleading notation should come as no surprise in retrospect. How common "notational banana peels" are in calculus and pre-calculus, how much damage they facilitate, and what can be done are some questions which seem worthy of further investigation.
On a second level of inquiry, we've looked into conceptual roots which may lie behind RCMs, including many notational ones. It's also not uncommon to hear mathematicians say that "I didn't really understand calculus until I was in graduate school and took my first course in Differential Topology". Indeed, the set of real numbers is a complicated concept, and there is much "between the lines" (and perhaps "under the rug") in a calculus course, including the concept of a function, with many layers of meaning and context. Perhaps we should not be discouraged therefore if we discover that students' attaining a firm understanding of calculus or even pre-calculus may turn out not one but many levels more difficult than obtaining a purely mechanical pseudo-understanding of (or what might be termed manipulative dexterity in) these subjects.
What might be accomplished by classifying RCMs? On the most basic level, such a classification process will produce as a by-product a "canonical list" of mathematical errors, as found in [ES]. Such a list could be used to help teachers prepare students to avoid making these most common mistakes. Even if this is done merely at the level of "training" students to avoid many RCMs the outcome could be positive. As noted in the introduction however, a higher aim would be to reduce the incidence of such RCMs not via training but by addressing the conceptual deficiencies which plague students, which research on RCMs suggests underly the most common mistakes.
It is also hoped that fellow mathematics educators who are well versed with and knowledgeable about existing and emerging theoretical frameworks will either in parallel or in collaboration with the author broaden the present analysis so that it may both inform and be informed by theoretical perspectives and models of student thinking learning.
Further evidence suggesting we may wish to consider notational reform includes the observation that when we work late at night, mistakes such as switching "its" with "it's" are more likely to happen even when the author is perfectly well aware of the distinction and the meaning of each; the visual pun alone serves as banana peel, that is, makes an error more likely to occur even when the underlying concept is known/understood.
Certainly then, notational pitfalls of this type in mathematics may be particularly hazardous to students who in addition to being subject to such visual banana peels, might not understand the underlying concepts. Schechter, speaking on another topic (functions of two variables), suggests much the same, stating that "our beginning math students have difficulty enough with abstraction even when they are provided with decent notation; how can we expect them to think abstractly without [decent] notation?" This raises several questions, such as : Should such notation be replaced? Or else, how can mathematics educators and teachers best "make peace" with such notation?
Though we might suggest an alternative system of notation for calculus to be provocative, that is, to provoke general thought and discussion about the notational system we use, it seems that no simple and easy solution exists to the notational banana peel problems discussed here. This may be due in large part to deeper, underlying mathematical conceptual complexities underlying RCMs and which are facilitated by notationally based mistakes. This triad -- RCMs, deeper misunderstandings underlying them, and notationally contributing factors -- perhaps deserves regular discussion and analysis in both pragmatic-pedagogical and more theoretical studies.
Questions for Further Investigation
2. All the mistakes (1.1)-(1.4) are aesthetically enticing, that is, one might conjecture that they are often caused by the student wanting these equations to be true by virtue of their being visually/symmetrically pleasing. However, there are mistakes which are notationally related but which do not involve visually aesthetic equations, for example the the assumption that "-x" must be a negative number. Here, one might investigate the conjecture that a false generalization is at play, since -N is indeed negative if N is a positive number (in particular, in earlier experience, if N is a counting number).
3. Of course (ii) follows from (i) if L(x) is known to be continuous, however it is probably safe to assume that the freshmen making these mistakes have not made this deduction
4. For example, it would seem that commonly used shorthand notations such as sin2x+cos2x and tan2x + 1 can only contribute to students' misconceptualizations of "sin" and "cos" by themselves as factors, and thus sin(x) to be seen as a product. Similarly, we often say "remember, the derivative of sine is cosine, while the derivative of cosine is minus sine" rather than "the derivative of sine-of-x is..."
5. The mistake, "(sin(ax))/(sin(bx)) = (a/b)" was recorded
several times in the journal and it seems that students are
particularly susceptible to making this mistake [(f(c))/(f(d))=c/d]
with trigonometric functions. Perhaps this is partly due to the
"black box" nature and lack of an explicit formula for trigonometric
functions, such as students have upon which to perform memorized
operations in the case of polynomials, in which case it may seem only
natural to try to cancel the "sines" in (sin(Ax))/(sin(Bx)) or to try
to cancel "Ax" in (sin(Ax))/(Ax). A further complication is that,
"unfortunately" the limit as x approaches zero of (sin(Ax))/(sin(Bx))
happens to equal A/B.
References
[ES]: Eric Schechter, The Most Common Errors in Undergraduate Mathematics,
[GS]: Gries and Schneider, A New Approach to Teaching Discrete Mathematics,
[RG]: Robert Gardner, A Useful Notation for Rules of Differentiation,
[RN]: Rick Norwood, A Star to Guide Us, Mathematics Teacher 92 (2)