Mathematics Textbooks and Their Potential Role in Supporting Misconceptions
ANN KAJANDER AND MIROSLAV LOVRIC
Mathematics textbooks are integral parts of our daily lives as mathematicians and mathematics teachers. Students use mathematics textbooks to study and to do homework questions, while professors and teachers may use them to prepare classes and to teach. We also use them to look up a formula or a theorem, and to prepare tests and exams for our students. While at the elementary and secondary levels teachers also have curriculum documents from which to work, the reality is that most teachers still use the textbook as their primary resource [1]. It determines both the material that needs to be covered and the way it is presented. ‘Textbooks form the backbone as well as the Achilles’ heel of the school experience in mathematics. The dominance of the textbook is illustrated by the finding that more than 95% of 12th-grade teachers indicated that the textbook was their most commonly used resource.’ [2]. However, while textbooks support our thinking about teaching, we rarely think about studying the textbooks themselves!
There have been attempts at evaluating textbooks, studies exploring the relationship between textbooks and curriculum (mostly K–12), and also global comparisons (for example [3–5]). Some effort has been put into content analysis and exploring the ways in which textbooks are used in classrooms and beyond (for example [6, 7]). However, very few mathematics education researchers have taken a really close look at what is in the textbooks, with the focus on how the material is presented and what kind of learning may be implied. In many cases, ‘unscientific market research is chiefly used to determine content and approach’ ([8], p. 55). Furthermore, ‘Commercially published, traditional textbooks dominate mathematics curriculum materials in US classrooms and to a great extent determine teaching practices’ ([8], p. 55).
Project 2061, an initiative of the American Association for the Advancement of Science, is a long-term project aimed at evaluating teaching and learning resources in science and mathematics. Founded in 1985, its goal is to ‘help all Americans become literate in science, mathematics, and technology’ (Project 2061 [9]). According to one of their studies, ‘. . . the majority of textbooks used for algebra—considered the gateway to higher mathematics—have some potential to help students learn, but they also have serious weaknesses’ [10]. More than half of the 12 middle school and secondary textbooks evaluated were considered adequate, but none was rated highly. Three textbooks that have been widely used were rated ‘. . . so inadequate that they lack potential for student learning’ [10]. The same report claims that ‘. . . No textbook does a satisfactory job of building on students’ existing ideas about algebra or helping them overcome their misconceptions or missing prerequisite knowledge.’ For details and complete report, see [11]. Project 2061 used the guidelines for how and what students should learn (referred to as the Standards), as defined by the National Council of Teachers of Mathematics [12]. According to the Project 2061 findings, authors of textbooks generally ignore the research on how students acquire ideas and concepts.
The theory of conceptual change [13, 14] provides an ideal framework for the study of potential student misconceptions related to learning from textbooks. The theory describes learning processes of adults as well as children and hence is appropriate in addressing high school as well as university students. Of particular interest are situations in which the new knowledge (be it learned by a child or an older student, or discovered by a research scientist) is incompatible with the prior knowledge, and hence might affect the understanding of the material. Based on a certain amount of information (learned from a book, heard in a class, etc.), a learner uses her/his own ideas and current understanding to create an initial explanatory framework. The presuppositions that inhabit this framework, also known as naïve beliefs or alternative conceptions, will form student’s current beliefs about the material. Faced with having to absorb the material that is in some way incompatible with the prior knowledge, the student will try to assimilate new information into their existing framework, thus creating a so-called synthetic model. As a mixture of beliefs and scientific facts, this synthetic model represents student’s misconceptions about the subject.
Evidence suggests [13, 14] that formed beliefs are very strong and, consequently, synthetic models become quite robust in one’s cognitive environment. It is an important purpose of a course instructor to identify these models and to apply adequate means in teaching and learning environments to help students cope with them.
One method of using the textbook to support the use of classroom time in overcoming misconceptions grounded in prior beliefs might be to require that students prepare material in advance of a class by reading a textbook. The course instructor might then transform the way she/he teaches in many productive ways such as focussing on the material that students are unclear about or indicate as challenging, or emphasizing topics that are of particular importance. Discussion replaces lecturing, and there is ample opportunity to go deeper into material and to make connections with other areas of mathematics that, for various reasons, are not covered in the textbook.
Assuming that an instructor will teach with the textbook (i.e. students will use it to read and study), the kind of research presented here raises awareness of a certain type of problems and issues that need to be addressed in classes. The fact that (mathematically incorrect) synthetic models can be quite robust means that a serious effort needs to be employed to devise strategies aimed at weakening the beliefs that inhabit them.
We have begun to take a closer look at textbooks commonly used in Ontario at the grade 12 and first year university levels to teach calculus, in order to determine to what extent, and how, mathematics textbooks potentially contribute to the creation and strengthening of students’ conceptions and misconceptions about mathematics. More generally, we are investigating to what extent textbooks promote (or not) deep, conceptual understanding of the material that they present. The case study we present here serves to illustrate potential issues, and also suggests a framework for our further analyses.
Examining calculus textbooks, we have identified a number of misconceptions related to the presentation (both narrative and visual) of the concept of the line tangent to the graph of a function y = f(x). For clarity, we have classified issues potentially leading to synthetic models into several categories, and selected typical cases to illustrate each category (in a few cases, we mention misconceptions arising elsewhere in calculus). Naturally, there is a certain amount of overlap between these categories (i.e. a synthetic model could belong to two, or several categories). Furthermore, we do not claim that our list is comprehensive in any way.
PREDOMINANCE OF COLLOQUIAL, ‘READER-FRIENDLY’ LANGUAGE
A fairly common synthetic model many entering first-year university students possess includes ideas described by statements such as ‘the tangent is the line that touches the graph of y = f(x) at only one point,’ or a ‘tangent line crosses the graph at one point.’ When we examined several grade 12 calculus textbooks currently in use in some Ontario schools, we found evidence that could support (or at best fail to correct) such a misconception. For example, while the initial explanation near the beginning of the chapter on tangents states (with accompanying diagrams)
In the graphs of the circle and the parabola, a tangent line touches exactly one point of the graph P. For other curves, such as the one in the third diagram [an example of a tangent which also crosses the curve at two other points] a tangent line touches the graph at the point of tangency, P, but may pass through other points on the graph as well ([15], p. 183).
but later on [in] a summary box highlighted in blue, the same textbook contains the unadorned statement
A tangent is a line that touches exactly one point on the graph of a relation (p. 190).
In a university textbook, we find the statement
If there is a tangent line to a curve at P—a line that just touches the curve like the tangent to a circle—it would be reasonable to identify the slope of the tangent as the slope of the curve at P ([16], p. 57).
Later, the textbook presents the usual limit-of-the-secant-lines approach to defining the tangent, and shows an example where the tangent crosses the curve at the point of tangency. However, the authors do not adequately address the ‘touching’ misconception. Another textbook [17] shows the graph of a function and two lines, both of which intersect the graph at the same point. In answer to ‘Are these both tangent lines?’ it states
Certainly not; only one of these acts like a tangent line to a circle, by hugging the curve near the point of tangency ([17], p. 150).
The book then continues by building correct conceptions (such as discussing direction as what is common to the tangent and the curve, and local linearity), but, again, does not revisit the ‘hugging/touching’ metaphor to correct it. Similarly, we find
The word tangent stems from the Latin word tangens, for ‘touching.’ Thus a line tangent to a curve is one that ‘just touches’ the curve ([18], p. 45).
However, there are textbooks that recognize the issues related to the ‘touching’ cognitive model and address it in an adequate way. For instance, in Larson et al. [19], the authors
• recall the concept of the tangent to the circle, and announce its inadequacy in the general case of the tangent to the graph of a function
• address the ‘tangent touches the curve’ notion directly, by showing examples (with illustrations) where the notion no longer applies
• develop carefully and precisely a correct mathematical definition of the tangent line
• discuss the case of the vertical tangent appropriately (i.e. they say how to define the tangent in the case when the limit of the slopes of secant lines is plus/minus infinity and the general definition for slope of a tangent does not apply)
• do not mention the word ‘touching’ even once after the precise definition is given, thus suggesting that the inappropriate ‘tangent touching’ cognitive model be abandoned.
We suggest that such treatment corrects, refines, and builds upon the earlier-held beliefs, and supports deeper understanding.
The textbook ‘Calculus and Its Applications’ [20], in our opinion, provides the most adequate way of addressing the ‘touching’ cognitive model. In the introduction to a rigorous treatment of the tangent and the derivative, the authors explicitly state four most common misconceptions related to the tangent line. Using graphs, i.e. constructing visual (counter)examples, they carefully analyse all of them. With all misconceptions now fully exposed, readers are invited to reflect, and to construct their own definition of the tangent line. Next, the authors begin to develop the concepts in a precise mathematical framework. Once this is accomplished, there is no going back to the colloquial language of ‘touching’ or ‘crossing.’
In conclusion to this section, let us mention that there is another reason why the term ‘touching’ is not appropriate. Imagine touching a book. Although placed very closely next to each other, our finger and the book do not share any part of space (not a single atom!). However, the graph of a function and its tangent actually do share part of the space—the point of tangency.
INCORRECT GENERALIZATIONS, STATEMENTS TAKEN OUT OF PRECISE MATHEMATICAL CONTEXT
Certain misconceptions arise from correct statements, when one forgets the precise (narrow) context in which they originally appeared. For example, the tangent to a circle is usually defined or described as the line that touches (or crosses) the circle at one point only, and the circle lies entirely on one side of it. The concept of the tangent to conic sections, introduced some time later in secondary school, is relatively easily incorporated into students’ cognitive model of the tangent, since it is a straightforward generalization of the concept of the tangent to a circle. However, the introduction of tangents to general curves produces new knowledge (which might not surface before first-year university) that is incompatible with the previous knowledge, and fosters the creation of (or reinforces already existing) synthetic models of the tangent that are mathematically incorrect. Many secondary as well as first-year university calculus textbooks that we examined contain material that strengthens these models even further, or, at best, do not go far enough to convincingly dispel the naïve beliefs students might possess.
Perhaps the most common misconception of this type is the one that relates to vertical asymptotes. In high school, students study the case of rational functions with numerator 1, and use the correct fact that such functions have vertical asymptotes at all points where they are not defined (i.e. where the denominator is zero). Later, when studying general rational functions (i.e. with an arbitrary polynomial in the numerator), they use the same model, although it no longer holds (for instance, (x2 – 1)/(x – 1) does not have a vertical asymptote at x = 1).
ILLUSTRATIONS AND DIAGRAMS AS SOURCES OF MISCONCEPTIONS
Standard university calculus texts contain numerous illustrations of tangent lines. However, in a vast majority of cases, the tangent is shown in the ‘generic’ position where it ‘touches’ the curve at one point (and does not cross it).
In Ellis and Gulick [21], the authors recall the idea of the tangent to a circle (accompanied by an appropriate illustration), and then also show a tangent to a spiral (page 54). However, the point of tangency on the spiral is chosen in such a way (on the outmost winding) that the tangent does not cross the spiral anywhere else. In fact, an ideal opportunity to show how a tangent can cross a curve at many points has been missed! As a side issue, the choice of the spiral is not appropriate to announce the generalization (the generalization deals with graphs of functions y = f(x) and a spiral (as shown there) is not the graph of a function). Interestingly, the authors do note
Yet the idea of a tangent line ‘touching’ a curve does not lend itself well to drawing a tangent line, nor does it give a procedure for deriving an equation of a tangent line, which is important in calculus ([21], p. 55).
but they do not provide a clear alternative to drawing the tangent. Furthermore, the tangent to a circle is used to show (visually) how secant lines approach the tangent, which could also potentially reinforce the ‘tangent touches the curve’ misconception.
The concept of ‘touching’ might be further supported by examples in which students are given the graph of a function f(x) and are asked to sketch the graph of its derivative f’(x). Although the tangent is defined as limit of secant lines, the illustrations or accompanying text in these examples do not attempt to encourage drawing the tangent or thinking about it as the limiting position of secant lines.
To address this misconception, a textbook (or instructor) could ask students to create illustrations that show relationships between curves and lines and identify which are (or are not) tangents. These illustrations should include cases such as (a) a tangent which does not ‘touch’ on one side but instead crosses the graph at the point of tangency, (b) a tangent which crosses the graph at two (or more) points (one of which is the point of tangency), (c) a tangent line which ‘touches’ the graph at more than one point, (d) a line which ‘touches’ the graph, does not cross it, but is not a tangent (cusp), etc. By analysing such situations, students will potentially rework some of their misconceptions and expand their cognitive model of the tangent.
OVERSIMPLIFICATION (SUMMARY DEFINITION; INTERPRETATION) OR OMISSION OF SPECIAL CASES
Related to the issue of over-use of colloquial language, examples of the situation of over-simplification can be found in attempts to verbalize or rephrase a notion that has already been defined, to make it more ‘clear’ or ‘understandable’. Also, they could be found in attempts at summarizing an idea or concept. For example, consider the following description of the secant line:
In particular, consider the line joining the given point P to the neighboring point Q on the graph of f, as shown in Figure 3.1. This line is called a secant (a line that intersects, but is not tangent to, a curve). ([22], p. 128)
The first sentence is a precise and correct definition of the secant. In an attempt to describe it in words (the statement in the brackets in the second sentence) the authors provided a misleading (and actually wrong) interpretation.
All textbooks we have examined define the tangent line as the limit of secant lines. However, very few mention (as part of the definition, or, say, immediately following the definition), the fact that the tangent line could be vertical (this can be illustrated, for instance, by analysing the tangent to the graph of the cube root function y = x1/3 at x = 0; the textbook we mentioned earlier ([19]) is an exception). From the definition of the tangent, it is not at all clear why this case deserves special attention—that is usually given to it later, in a section on differentiability, where it is named ‘vertical tangent.’ The fact that the statements ‘function f(x) is differentiable at a’ and ‘the graph of f(x) has a tangent at (a, f(a))’ are not equivalent (precisely because of ‘vertical tangents’) is, very often, inadequately addressed in the presentation of the material.
A common strategy that many students employ when finding vertical asymptotes is to identify points where the given function is not defined. Quite often, they identify the statement ‘if f(a) is defined then f cannot have a vertical asymptote at a’ as false. (It is easy to construct a counterexample: the function f(x) = 1 if x = 0, and f(x) = 1/x otherwise is defined at x = 0, and has a vertical asymptote there.) The source of this misconception can be traced, again, to attempts at verbalizing a correctly introduced concept in order to ‘simplify’ it (for instance, because the definition involves numerous mathematical symbols and/or concepts).
DISCUSSING CONCEPTS THAT HAVE NOT BEEN PRECISELY DEFINED
In the introductory part of the section on tangents and derivatives in Bradley and Smith [22], the reader is reminded that
a tangent to a circle is defined as a line in the plane of the circle that intersects the circle in exactly one point (p. 128).
Then, the text states ‘however, this is much too narrow a view for our purposes in calculus’ (p. 128) without any indication of why this is so. Continuing in the same, somewhat vague manner, the textbook does not precisely define the meaning of the word ‘tangent.’ Referring to an illustration that shows four secant lines and a tangent line (with no explanation of how the tangent line was obtained), we read
Notice that a secant is a good approximation to the tangent at point P as long as Q is close to P (p. 128).
with no elaboration on meaning of the phrase ‘good approximation’. Later (p. 129) the text begins the explanation ‘To bring the secant closer to the tangent . . .’, i.e. starts discussing properties of the tangent as if the tangent were already defined. In a similar vein, after reading the sentence
By the definition of the derivative, f’(a) represents the slope of the line tangent to the graph of f at (a, f(a)) ([21], p. 106).
we tried to identify where, previously in the text, the authors defined the tangent. Following through the pages, we found the usual material (reference to the tangent of the circle, the mention of the idea of a tangent line ‘touching’ a curve, etc.) but we could not find a rigorous mathematics statement defining the tangent line.
As an alternative model, visualize the tangent definition (i.e., the limit of secant lines) as a sequence of magnifications, zooming in on the point of tangency (for example see the grade 12 text by Alexander et al., [23], p. 21). Some textbooks do show this, but do not emphasize strongly enough its importance. If these magnifications tend to flatten the graph (i.e. make it look more and more like a line), then the graph (most likely) has a tangent line at the point in question. Using this approach, it becomes obvious, for example, that y = |x| does not have a tangent at x = 0.
The ‘magnification’ approach extends naturally to functions of two variables (i.e. to the concept of the tangent plane to the graph of f(x, y)). Moreover, it reinforces the crucial idea that the tangent (line, or plane) is a local—and also linear—object.
The fact that the ‘magnification’ cognitive model of the tangent is fundamentally different from the usual ‘tangent touches the curve’ model potentially prevents the student from building a synthetic model of the tangent. It becomes difficult to assimilate the notion of ‘zooming in’ on the graph of the curve f(x) to dynamically approximate the tangent more and more precisely, with the notion of a (static) line ‘touching’ the curve. Instead, it might force the student to completely abandon ‘the tangent touches the curve’ model and adopt the new, ‘magnification’ model. Furthermore, the ‘magnification’ model naturally leads into the discussion of the role of tangent as the best approximation of the graph of a function by a straight line.
We were surprised that this preliminary analysis uncovered such a breadth of issues related to basically one topic. Situations leading to potential misconceptions occurred consistently in multiple sources. Acknowledging that textbooks remain a fundamental teaching resource, we suggest that more attention be paid to the presentation of mathematics. Furthermore, analyses of textbooks should include developmental as well as subject matter scrutiny.
In our forthcoming work, using the initial exploratory framework suggested here, we plan to investigate further misconceptions related to tangents and derivatives (local linearity, linear approximation and differentiability), as well as those related to other concepts [24].
We believe that as students move through the secondary curriculum and (potentially) develop deeper and more accurate conceptual understandings of fundamental concepts as they enter university, more attention should be paid to textbooks to continue to support rather than marginalize such growth.
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[9]Project 2061 (2000a). Available at http://www.project2061.org/about/.
[10]Project 2061 (2000b). Algebra for all—not with today’s textbooks, says AAAS. Available at http://www.project2061.org/about/press/pr000426.htm.
[11]Project 2061 (2000c). Available at http://www.project2061.org/publications/textbook/default.htm.
[12]National Council of Teachers of Mathematics. Principles and Standards for School Mathematics, NCTM, Reston, VA, 2000.
[13]J. Davis, (2001) Conceptual change, in Emerging Perspectives on Learning, Teaching, and Technology (e-book), M. Orey, ed., Available at http://www.coe.uga.edu/epltt/ConceptualChange.htm
[14]I. Biza, A. Souyoul, and T. Zachariades, Conceptual Change In Advanced Mathematical Thinking, Discussion paper, Fourth Congress of ERME, Sant Feliu de Guíxols, Spain, 17–21 February 2005.
[15]C. Kirkpatrick, et al., Advanced Functions and Introductory Calculus, Nelson, Scarborough, ON, 2002.
[16]J. Hass, M.D. Weir, and G.B. Thomas Jr, University Calculus, Pearson, Boston, 2007.
[17]T. Smith and R.B. Minton, Calculus, McGraw-Hill, New York, 2002.
[18]C.H. Edwards and D.E. Penney, Calculus with Analytic geometry; Early Transcendentals, 5th ed., Prentice Hall, Upper Saddle River, 1998.
[19]R. Larson, R.P. Hostetler, and B.H. Edwards, Calculus, Early Transcendental Functions, 2nd ed., Houghton Mifflin Company, Boston, New York, 1999.
[20]S.J. Farlow and G.M. Haggard, Calculus and Its Applications, McGraw-Hill, New York, 1990.
[21]R. Ellis and D. Gulick, Calculus with Analytic Geometry, 4th ed., Harcourt Brace Jovanovic, Publishers and its subsidiary, Academic Press, San Diego, 1990.
[22]G.L. Bradley and K.J. Smith, Calculus, 2nd ed., Prentice Hall, Upper Saddle River, New Jersey, 1999.
[23]R. Alexander, et al., Advanced Functions and Introductory Calculus 12, Addison Wesley, Toronto, 2003.
[24]M. Lovric, Mathematics textbooks and promotion of conceptual understanding. Presentation to the Education Forum of the Fields Institute for Mathematical Sciences, University of Toronto, Toronto, 2007.