University Students Reading of Their Fir

  University Students’ Reading of Their First-Year Mathematics Textbooks

  

Draft

Mary Shepherd, Annie Selden, and John Selden

ABSTRACT

  This paper reports the observed behaviors and difficulties that eleven precalculus and calculus students exhibited in reading new passages from their mathematics textbooks. To gauge the effectiveness of these students’ reading, we asked them to attempt straightforward mathematical tasks, based directly on what they had just read. The students had high ACT mathematics and high ACT reading comprehension test scores and exhibited much of the Constructively Responsive Reading of good readers in reading comprehension research,as described in the reading comprehension research literature. However, they were not effective readers of their mathematics textbooks. We discuss some reasons for this that might not be easily discernable among the variety of task-working difficulties of less able students. Finally, we pose questions for future research and suggest some implications for teaching.

  

INTRODUCTION

  From our own teaching experience and in talking with colleagues, it appears to be common knowledge that many, perhaps most, beginning university students do not read large parts of their mathematics textbooks in a way that is very useful in their learning. Whether this is because they cannot read in such a way, or choose not to do so, seems not to have been established. However, there have been a number of calls for mathematics teachers to instruct their students on how to read mathematics (Bratina & Lipkin, 2003; Cowen, 1991; Datta, 1993; DeLong & Winter, 2002; Draper, 2002; Fuentes, 1998; Pimm, 1987; Shuard & Rothery, 1988). In addition, textbooks for many beginning university courses, such as college algebra, precalculus, and calculus seem to be written with the assumption that they will be read thoroughly, precisely, and with considerable benefit.

  For example, the preface of the precalculus book used by the students in this study asserts: The following suggestions are made to help you get the most out of this book and your efforts. As you study the text we suggest a five-step approach. For each section, 1. Read the mathematical development.

  2. Work through the illustrative examples.

  3. Work the matched problem.

  4. Review the main ideas in the section.

  5. Work the assigned exercises at the end of the section. All of this should be done with a graphing utility, paper, and pencil at hand. In fact, no mathematics text should be read without pencil and paper in hand; mathematics is not a spectator sport. Just as you cannot learn to swim by watching someone else swim, you cannot learn mathematics by simply reading worked examples—you must work problems, lots of them. (Barnett, Ziegler, & Byleen, 2000, p. xxxi). Most teachers of beginning undergraduate mathematics would probably agree that the above is good advice. But it is unclear whether it is realistic to assume that students will, or even can, adequately carry it out.

  In this exploratory study we examined the mathematics textbook reading of eleven beginning undergraduate students from the perspectives of the students’ mathematical difficulties in working tasks in the passages read, the writing style of beginning mathematics textbooks, and the reading comprehension literature. We also attended to whether these students could reasonably be seen as good at, or promising in, both mathematics and general reading comprehension.

  In the first section, we briefly describe reading comprehension research including the Constructively Responsive Reading (CRR) perspective (Pressley & Afflerbach, 1995), discuss how beginning mathematics textbooks differ from other books, note the limited amount of research that has been done on how students read their mathematics textbooks, describe the ACT reading comprehension and mathematics tests, and mention some relevant psychological research. In the next section, we describe the goal of the study, indicate what we mean by effective reading, and mention four questions to be answered. After that, in the following section, we describe the students, their courses, and our research methodology. Next there follows a section in which we describe our data, including observations concerning students’ use of CRR, and their difficulties in working straightforward tasks from their mathematics textbooks. Finally, we discuss our findings, propose some directions for future research, and suggest some implications for teaching.

  Reading Comprehension Research

  During the past fifty years, conceptual shifts have led reading researchers to view reading as an active process of meaning-making in which readers use their knowledge of language and the world to construct and negotiate interpretations of texts in light of the particular situations within which they are read. (Borasi, Seigel, Fonzi, & Smith, 1998; Brown, Pressley, Van Meter, & Shuder, 1996; Dewitz & Dewitz, 2003; Flood & Lapp, 1990; Kintsch, 1998; McNamara, 2004; Palincsar & Brown; 1984; Pressley & Afflerbach, 1995; Rosenblatt, 1994; Schuder, 1993; Siegel, Borasi, Fonzi, Sanridge, & Smith, 1996). These conceptual shifts have expanded the notion of reading from that of simply moving one’s eyes across a page of written symbols and translating these symbols into verbalized words, into the idea of reading as a mode of thinking and learning (Draper, 2002). From this perspective, the reader integrates new information with preexisting knowledge structures to create meaning (Flood & Lapp, 1990; Rosenblatt, 1994), for example, through assimilation and accommodation (Pressley & Afflerbach, 1995, p. 103).

  Reading and literacy researchers agree that reading includes both decoding and comprehension. Research on comprehension, often based on think aloud protocols, indicates that there are many responses and strategies that good readers employ before, during, and after they read. These responses and strategies seem to vary from reader to reader and to depend on the material being read and the goals of the reader (Borasi et al., 1998; Brown et al., 1996; Flood & Lapp, 1990; Fuentes, 1998; Palincsar & Brown; 1984; Pressley & Afflerbach, 1995; Siegel et al., 1996). One of the most comprehensive metastudies of reading research was conducted by Pressley and Afflerbach (1995), combining the results of many previous reading researchers (e.g., Brown, Kintsch, Rosenblatt, and many others). They developed the idea “that reading is constructively responsive – that is, good readers are always changing their processing in response to the text they are reading” (p. 2). Pressley and Afflerbach synthesized some of their findings into a “Thumbnail Sketch” of the Constructively Responsive Reading (CRR) responses of good readers. These CRR responses are used later in the results section in the construction of Table 1. Because this Thumbnail Sketch came from many studies of differing kinds of readers and texts, and because good readers vary their responses according to their own knowledge and the kind of text read, one cannot expect to measure reading quality based on the number of CRR responses a reader makes to a text. However, the existence of such responses in a student’s reading can support the idea that the student is a good reader.

  None of the studies used by Pressley and Afflerbach involved reading of mathematical text. What makes mathematical text different? The Writing Style of Mathematics Textbooks

  Mathematicians appear to prize brevity, conciseness, and precision of meaning in mathematical writing. Further, there is often little room for an acceptable interpretation of a passage that is different from the one intended by an author.

  Some special features of the style of mathematical writing that can sometimes lead to student difficulties, as indicated by Barton and Heidema (2002) and Shuard and Rothery (1988), include:

  1. Reading mathematics often requires reading from right to left, top to bottom, bottom to top, or diagonally.

  2. The writing in mathematics textbooks has more concepts per sentence, per word, and per paragraph than other textbooks.

  3. Mathematical concepts are often abstract and require effort to visualize.

  4. The writing in mathematics textbooks is terse and compact—that is, there is little redundancy to help readers with the meaning.

  5. Words have precise meanings which students often do not fully understand.

  Students’ concept images of them -- mental links to such things as relevant examples, nonexamples, theorems, and diagrams of them -- may be “thin.”

  6. Formal logic connects sentences so the ability to understand implications and make inferences across sentences is essential.

  7. In addition to words, mathematics textbooks contain numeric and non- numeric symbols.

  8. The layout of many mathematics textbooks can make it easy to find and read worked examples while skipping crucial explanatory passages.

  9. Mathematics textbooks often contain complex sentences which can be difficult to parse and understand.

  Most first-year university mathematics textbooks currently published in the U.S. contain exposition, definitions, theorems and less formal mathematical assertions, as well as graphs, figures, tables, examples (i.e., tasks, some with solutions), and end-of-section exercises. Theorems may be accompanied by proofs, but students are not normally required to produce or reproduce them. Typically there is a repeated pattern consisting of first presenting a bit of conceptual knowledge, such as a definition or theorem and perhaps some less formal mathematical assertions, followed by closely related procedural knowledge in the form of a few worked examples (tasks), and finally, as a self test, students are invited to work very similar tasks themselves. Both the conceptual knowledge and the related tasks are subsequently useful in working the end-of-section exercises.

  In mathematical writing of this kind, it is intended that everyone who reads the definition of a concept with comprehension will have essentially the same basic understanding of the definition. Different individuals’ concept images are likely to differ, but everyone should be able to agree on whether or not an example satisfies the concept’s definition. In the same way, it is expected that everyone who reads an explanation of how to work a task will be able to work a very similar task in essentially the same way as described in the explanation. This expectation departs from the modern view of reading as interpretation of text, individually constructed and negotiated, that was mentioned in the previous section. It appears to be possible only due to the use of analytic (stipulated) definitions (Edwards & Ward, 2004; Selden, 2005) and classical logic in modern mathematics (Stenning & van Lambolgen, 2008, p.27). This contrasts with some earlier mathematics, for example, that underlying Proofs and Refutations (Lakatos, 1976), in which a purported counterexample might be seen as not in the “domain” of a statement,

  th

  rather than as indicating the statement’s falseness. By the beginning of the 20 century, the mathematics community abandoned this idea of the domain of a statement, and treated definitions as analytic. More important here, the inclusion of self-test tasks in the students’ mathematics textbooks appears to reflect a recognition that the above kind of precise communication may not always be achieved on a first reading.

  In all of these respects, the textbook passages (Barnett, Ziegler, & Byleen, 2000; Larson, Hostetler, & Edwards, 2002) read by the students in this study appear to us to be typical. (See Appendices A and B.)

  Previous Research on Reading Mathematics Textbooks

  Only a little research seems to have been done on how students read their mathematics textbooks. Osterholm (2008) surveyed 199 articles having to do with the reading of word problems, but found little about reading comprehension of more general mathematical text. He has done several studies on secondary and university students’ reading of mathematical text (Osterholm, 2005, 2008) using passages written especially for that research. In contrast, the students described here read passages from their own textbooks.

  There has been an interest in, and some research on, how students read science textbooks in order to learn science. A recent issue of Science had a special section devoted to research on, and to the challenges of, reading the academic language of science. It was noted that, while students have mastered the reading of various kinds of English texts (mostly narratives), this does not suffice for science texts that are precise and concise, avoid redundancy, use sophisticated words and complex grammatical constructions, and have a high density of information-bearing words (Snow, 2010, p. 450). These are some of the same features of mathematics textbook writing noted above.

  Finally, Weinberg and Weisner (2010) have recently introduced a framework for examining students’ reading of their mathematics textbooks. A major part of their perspective is an emphasis on the richness of personal meanings that readers construct, as opposed to the proximity of those meanings to the author’s meaning or the meaning in the text (as interpreted by the mathematical community). While we see this as an interesting and useful perspective agreeing with current reading comprehension research, in this paper we take a different perspective and consider whether, and how, students construct meanings very close to those of the author and mathematical community. This seems appropriate when dealing with passages introducing new mathematical ideas that readers could not intuit, reason out, or even alter much, based on prior knowledge alone.

  The ACT Tests Most students in the U.S. are required to have at least minimum scores (set by each

university) on a national reading comprehension and a national mathematics test, either

  

the ACT or the SAT, as well as other qualifying materials, in order to be admitted to the

university. At the university where this study took place, the ACT tests (routinely provided by American College Testing, Inc.) were required.

  

The ACT reading comprehension test calls on students’ to read and interpret

passages of text. For each passage, students are to determine the main ideas, find significant details, understand sequences, make comparisons, understand cause-effect

relationships, understand context-dependent words, draw generalizations, and understand

an author’s voice or method (ACT Reading Test Description, 2010). The ACT

mathematics test is a 60-question, 60-minute test designed to measure the mathematical

skills students have typically acquired in courses taken by the end of 11th grade (ACT Mathematics Test Description, 2010).

  Psychological Research

  Some of the difficulties students have in working straightforward, very recently explained, tasks appear to be due, not to their mathematical deficiencies, but to misapprehending explanations. The literature on mind wandering (a.k.a., zoning out) suggests an explanation for this. Mind wandering refers to a person having task-unrelated images and thoughts while carrying out a task (Smallwood & Schooler, 2006). In an experiment in which participants read War and Peace for 45 minutes, on average participants zoned out more than five times and were often unaware of doing so (Schooler, Reichle, & Halpern, 2004). Also, Schooler et al. obtained behavioral evidence that “zoning-out episodes are associated with particularly low levels of attention to the [meaning of the corresponding part of the] text” (p. 212). That is, readers may have unnoticed “cognitive gaps” in what they read.

  With this study, we hope to begin to understand why many beginning university students, including many who are neither unmotivated, nor ill-equipped to learn mathematics, apparently do not read their mathematics textbooks in a very useful way. We investigated the following four questions.

  1. Did the students exhibit some of the characteristic responses of good readers from the CRR perspective, and canare some of their difficulties be explained by a due

  to lack ing of such responses?

  2. Could our students read their mathematics textbooks effectively? That is, could they carry out straightforward tasks associated with the reading soon, often immediately, after reading passages explaining, or illustrating, how the tasks should be carried out, and with those passages still available to them?

  3. Were some student difficulties in working tasks traceable to the writing style of their mathematics textbooks?

  4. What kinds of mathematical difficulties did the students encounter in working the tasks in, or arising from, their reading?

  

METHODOLOGY The ACT Tests Most students in the U.S. are required to have at least minimum scores (set by each

university) on a national reading comprehension and a national mathematics test, either

the ACT or the SAT, as well as other qualifying materials, in order to be admitted to the university. At the university where this study took place, the ACT tests (routinely provided by American College Testing, Inc.) were required.

  

The ACT reading comprehension test calls on students to read and interpret

passages of text. For each passage, students are to determine the main ideas, find significant details, understand sequences, make comparisons, understand cause-effect

relationships, understand context-dependent words, draw generalizations, and understand

an author’s voice or method (ACT Reading Test Description, 2010). The ACT

mathematics test is a 60-question, 60-minute test designed to measure the mathematical

skills students have typically acquired in courses taken by the end of 11th grade (ACT Mathematics Test Description, 2010).

  The Students

  The students in this study attended a U.S. mid-western comprehensive state university. They were selected from a precalculus class of 17 and from two sections of Calculus I with 41 students total. In the fourth week of the first semester, we first identified 33 good readers (12 precalculus, 21 calculus) with national reading

  th th comprehension test (ACT) scores ranging from the 70 percentile to the 99 percentile.

  Based on the instructor’s judgment, nine students (4 precalculus, 5 calculus) were eliminated because it appeared they had no problems reading mathematics because of

  

previous independent work in mathematics or and may have seen the material in previous

  courses. This was done to maximize the chance of identifying reading problems. Of the remaining twenty-four students, eleven (5 precalculus, 6 calculus) volunteered to participate in the study. Eight of the students were female, none came from minority ethnic groups.were minorities.

  The university has a moderately selective admissions standard and a student body of 6,500 students of which 5,500 are undergraduates. Five of the eleven students, all precalculus students, were university students also enrolled in a very selective two-year science/mathematics “early-entrance-to-college” (after grade 10) program. These five took all of their courses together with regular university students, and had been ranked in the top 10% of their first two years of secondary school. In general, they were better academically than the university’s normal intake of first-year students.

  The average reading ACT score for all eleven students in the study was 28.6

  

th

  (where the median, 28, corresponds to the 87 percentile). This compares favorably with the average ACT reading score of 22.3 for all incoming first-year students at this university. All but two volunteers, both calculus students, had ACT mathematics scores

  st th

  ranging from the 71 percentile to the 96 percentile. The remaining two students, Tara and Vannie, were not first-year students. They had originally been weaker than the other students in the study, but had completed university courses in preparation for studying calculus. They were not atypical among calculus students at the university.

  Thus, according to their incoming reading comprehension and mathematics test scores, as measured by the ACT tests, all of the students were good at reading and nine of the students were good students generally and good at mathematics.

  The university mathematics courses taken by the eleven students were taught by the first author, a member of the university’s mathematics department. Ten of the students received a small amount of extra credit for participating in the study. The amount of extra credit received did not change any final grades. One calculus student dropped the class before the fourth week of the semester, but agreed to participate anyway. That student was grouped with the precalculus students since it was the precalculus passage the student read for the study. The first letters of pseudonyms for the names of precalculus students came from the beginning of the alphabet and those for the calculus students came from the end of the alphabet. Also in the excerpts where students are speaking, omissions are indicated by … and pauses are indicated by […].

  The Courses

  The courses were standard U.S. university mathematics courses taught by the first author. The precalculus course met four hours per week and the content included functions and graphs, equations and inequalities, analytic geometry, and trigonometry. The calculus course also met four hours per week and the content included limits and continuity, differentiation of elementary functions, curve sketching, extreme values, areas, rates of change, and the Fundamental Theorem of Calculus. Both courses include d theorems, with proofs or explanations. However, students were not asked to reconstruct proofs or construct new ones.

  From the beginning, both the precalculus and calculus courses from which the students in this study were chosen had a strong emphasis placed on students reading their mathematics textbooks. The students were given handouts about reading mathematics on the first day of class, and beginning the second class period, students were given reading guides for use with the first several sections of their mathematics textbooks. An example of a reading guide and additional information about the teaching practices of this instructor appeared in Author (2005).

  During the first two weeks of the courses, all 58 students from the pre-calculus and calculus classes participated in a diagnostic interview as part of the instructor’s normal teaching practice. This consisted of reading one of four short (one-half to two page) passages on partial fractions, algebraic vectors, absolute value, or symmetry.

  Students at this level are unlikely to be familiar with readings on these topics, but will normally find them accessible. After reading the short passage, each student was asked to complete a task, based on that passage. In addition to being used diagnostically in teaching, these interviews served to familiarize the 11 subsequent volunteers with the interview procedures that they would experience later.

  The Conduct of the Study

  During the sixth and seventh weeks of the courses, the volunteers each selected a 90-minute time slot during which they were asked to read aloud a new section of their respective textbooks to the interviewer, who was also the instructor and first author. In addition, they were asked to think aloud during periods between active reading. The students did not appear to be inhibited by the interviewer being their teacher, and indeed, volunteered unsolicited information about how they would, or would not, read alone.

  The passages read were selected so that the students would be as familiar as possible with the notations and prior definitions used in them and so that they would have the necessary mathematical prerequisites for reading and understanding them. However, this does not mean the students’ relevant general prior knowledge from earlier courses was far beyond what one would normally expect. That is, in working tasks a student’s prior knowledge might occasionally be flawed. The five pre-calculus students, and the one calculus student who had dropped the calculus course, read the section entitled “The Wrapping Function” in Barnett, et al. (2000, pp. 336-343). The five calculus students read the section entitled “Extrema on an Interval” in Larson et al. (2002, pp. 160-164). Along with definitions, theorems, examples, figures, and discussions, the precalculus book has “Explore/Discuss” tasks, and the calculus book has “Exploration” tasks to encourage students to become active as they read. The precalculus section had one definition, three examples, one theorem with an informal proof, several figures, exposition and two Explore/Discuss activities. The calculus section had three definitions, four examples, two theorems (one with a formal proof, one without proof), several figures, exposition, and one Exploration activity. (See Appendices A and B for some of this.)

  The interviewer remained generally silent except to stop the students at intervals during their reading when they were asked to try a task based on what they had just read, or asked to try to work a textbook example (task) without first looking at the provided solution. These were the places that the textbook authors would probably have assumed readers would independently pause for such activities. The precalculus students were stopped an average of three times (a maximum of four times, a minimum of three times).

  The calculus students were stopped an average of eight times (a maximum of nine times, a minimum of seven times). The tasks were straightforward ones based directly on the reading with its examples and explanations, and required little or nothing of the kind of sophisticated problem solving discussed by Schoenfeld (1985) or Polya (1957). For instance, after the pre-calculus textbook had defined, and diagrammed, the Wrapping Function, W, and had explained the calculation of the coordinates of

  W (0), W 2 , W ( ), W 3 

  2

    and   , the task given was: Find the coordinates of the

      circular point W  2  . Parts of the selected reading passages along with the

  interruptions and requested tasks appear in Appendix A for precalculus and in Appendix B for calculus. After the entire section had been read and a few final tasks had been attempted, the students were questioned about how reading during the interview differed from their normal reading of their mathematics textbooks (Appendix C).

  All interviews were audio-recorded and transcribed. The interviewer also made notes during the interviews. The written work produced by the students during the interview was collected. The first author listened to the recordings carefully at least three times, noting responses associated with good reading according to the CRR Thumbnail Sketch. These additional notes, along with the notes taken during the interviews and the students’ written work, were compared with the transcripts to create Tables 1, 2, and 3 below. The overall number and kind of CRR responses observed (see Tables 1 and 2) and the kinds of mathematical difficulties the students incurred were noted.Also, for each student, we noted the number of tasks attempted, done correctly, done incorrectly, not done (skipped or given up), left incomplete, read but not worked, read as worked, and “correct” with a wrong reason (Table 3).

  

OBSERVATIONS

Activities Associated with Good Reading

  The students all exhibited a wide range of reading responses from the CRR perspective, and the number of such observed responses per student ranged from 24 to 128 observations.

  In Table 1, for each of the fifteen kinds of activities described in the CRR Thumbnail Sketch, we provide examples of observed behaviors, together with the numbers of students exhibiting those behaviors. For example, six students read the title of the section, the introduction, or the caption at the start of their reading and were judged to have employed the CRR-based response “Preview the text to be read” (CRR response 1). Also, Christie made the following comment after reading the definition of the Wrapping Function. “So, to me it sounds like that they have a circle at […] and it has to have [a] start at this point and there’s going to be a line going around it and we’re going to find the points on that line.” This was coded as a paraphrase, a strategy to better remember the text (CRR response 10, example).

  Table 1 indicates that the eleven students, as a group, were exhibiting the kinds of responses characteristic of good readers from the CRR perspective, and Table 2 indicates a similar result for each individual student. These support that the students were, in general, good readers, as indicated by their ACT reading scores.

  None of the student difficulties seemed to be attributable to their lack of a specific characteristic of good readers from the CRR perspective.

  INSERT TABLES 1 AND 2 ABOUT HERE

  Reading Effectiveness

  It is perhaps one of the main goals of beginning undergraduate mathematics textbooks in the U.S. that readers should understand the content well enough to be able to work the provided tasks, or similar tasks, shortly after being shown how to do so. It is important that this can be done reliably before readers go on to later passages. However, all of the students in our study had considerable difficulty correctly completing straightforward tasks based on their reading. For example, five of the six students who read the precalculus passage did not find correct values of the Wrapping Function, W, in two or more instances. Also, four of the five students who read the calculus passage containing the definition of extrema of a function on an interval could not determine from its graph whether it had a minimum. Only three of our eleven students (Bryan, Ellis, and Vannie) could independently work at least half of the tasks, and only one of these, Ellis, could independently work three-fourths of them. Thus, by our measure none of our students read effectively. The number of tasks each student attempted, worked correctly, worked incorrectly, gave up or skipped over (after starting), left incomplete, read but did not work, worked while reading the book’s solution, and worked “correctly,” giving a wrong reason, is given in Table 3.

  INSERT TABLE 3 ABOUT HERE

  The Writing Style of Mathematics Textbooks

  Previously, we pointed out a number of ways that mathematical writing can differ from that of other text. Such differences can in some situations interfere with comprehension or effective reading. However, most of these differences did not occur in the passages our students read, and what differences there were did not cause the students to stumble in reading. For example, they could easily read equations and the notations for functions, intervals, and points. We could not trace any student difficulties to the writing style of their textbooks.

  Kinds of Difficulties in Working Tasks

  The students’ difficulties working tasks all ar oi se from, or depend ed largely on, at least one of three main kinds of difficulty: (1) insufficient sensitivity to, and inappropriate response to, their own confusion or error; (2) inadequate or incorrect prior knowledge, and (3) insufficient attention to the detailed content of the textbook. The difficulties working tasks and their origins occurred throughout the passages read and were associated with exposition, definitions, theorems, worked examples, and explorations.

  Responses to confusion or error. Mathematics (as it is presented in universities)

  “builds on itself,” as opposed to being built on descriptive information from the external world. This is reflected in the textbooks, in that earlier concepts and tasks are often components of later ones. A reader’s confusion or error that is not attended to will likely cause a later difficulty that is harder to sort out than the original one. Thus, a reader (who is reading to learn mathematics) should be sensitive to his or her own confusions and errors. Upon noticing a confusion or error, a reader should respond by reworking tasks and rereading parts of the textbook until the difficulty is removed. Properly prepared students, such as those in this study, should be able to do this, even though they might not correctly work all, or even most, tasks on the first attempt. But being able to do something, in the sense of having sufficient knowledge or skill, is not the same as actually doing it at the appropriate time.

  Most of our students appeared to be strikingly unconcerned about their errors or confusion and did not seem to believe they could have independently done anything about it. Ten of the students stated at some point that they did not understand something, but made no attempt to understand whatever was causing confusion. Five students, three precalculus and two calculus, gave up at some point. They stated that they had no idea what to do, either while trying to work a task or when reading through a worked example.

  When questioned, one calculus student stated she would just move on, the other four stated they would quit and ask for help before continuing. However, they continued to read at the request of the interviewer. Examples of inappropriate responses to confusion or error tend to be associated with other difficulties, and some will be pointed out below.

  Prior knowledge. For the calculus students, one difficulty appeared to come from

  an inadequate concept image of the word “function.” After correctly reading the definition of extrema (Appendix B), Vannie was asked to look at the graphs of eight functions and to determine whether they had minimum values. As she looked at the graph of 51a, which was a function with a jump discontinuity, she went back to the definition and tried to compare it with the graph.

  “You’re on the interval I as they designate. You’re supposed to look at […] Is it c

  f (c ) or x they use? … For all the x’s, is supposed to be your minimum point. f (c )

  Well, on this portion is your minimum point, is a real number, but on this one it is not [i.e., there is no minimum] because it [i.e., the interval] is open. So, if you look at it from [...] since it’s totally two different things coming in. I don’t know if you say, well this one does have a minimum and this one doesn’t, or if they go together, then they don’t. I don’t [...] that part I [...] I’m not clear on.”

  Vannie came to no resolution, and did not persist in attempting to find the origin of her confusion and reworking the task. Although she clearly tried to use the definition of extrema, she did not appear to recognize the graph of a function with a jump discontinuity as representing a single function. She had the book available but did not

  attempt to review her conception of function by looking up a definition or reviewing examples in the textbook. Thus Vannie had inadequate/incorrect prior knowledge, and

  although she recognized her confusion, she did not respond to it appropriately in a manner that was helpful to her understanding .

  None of the calculus readers completed the final three worked examples (tasks) in their textbook passage without looking at the solutions provided or comparing their work with that of the book. These worked examples concerned estimating extrema for graphically presented functions and finding the extrema of a trigonometric function. The calculus textbook provides procedural knowledge in the form of a list of steps to follow to find extrema on a closed interval. Although the calculus students tried to follow these steps, three of them had difficulty with algebraic concepts (negative exponents, factoring, trigonometric identities), and all of them gave up trying to find the extrema of the

  f x x cos 2 x trigonometric example, ( ) 2sin on [0, 2 ]  .

    Although they continued to read for the interview, two calculus students stated they would normally give up before reaching this final example. Vannie indicated she would ask her group for help before continuing, and Tara indicated she would ask the teacher about the trigonometric example in the next class period.

  In particular, Vannie’s incomplete prior knowledge of negative fractional exponents caused her to become frustrated and give up attempting to understand a calculation. She tried to work Example 3 that asked the reader to find the extrema of

  2

  3

  on the interval [-1,3]. Vannie first attempted to take the derivative and

  f ( x )  2 x  3 x 1 

  3

  set it equal to zero. She incorrectly wrote . At this point she

  f (' x )  2 x  2 x 

  checked the solution to confirm her derivative and said, “They did something crazy. Ok. What did they do? […] I’m confused. . . . I don’t understand their math or their […] what they did. … I figured it was just a basic [...] you did the derivatives in the subtraction.”

  She eventually fixed her derivative but still could not get the form of the _{1 3}

    2 x 

  1 f ' ( x )  2  

  2

  derivative shown in the textbook, which was . The negative _{1 3 1 3}

    x x

   

  exponent confused her, even though she had tutored college algebra in the past. Her final comment after reading through the entire solution was, “At this point, if I was really reading this I would be frustrated and quit and then I would go ask somebody.” This may

  

have been efficient from Vannie’s point of view, knowing there were people who would

help her, she did little to help herself work through the frustration of the not well understood algebra presented.

  These students’ difficulties appeared to be due to inadequate/incorrect prior knowledge. Also they did not respond to their errors appropriately in a manner that would

  have directly addressed their inadequate/incorrect prior knowledge , that is, by more

  carefully reworking tasks and by searching for errors in their knowledge base or looking

  up relevant definitions or algebra rules . Such an effort might well have removed some

  errors and, even if it did not, would have been a useful exercise before ultimately asking for help.

  In sufficient attention to details . We noted two kinds of errors, or misreadings, apparently duewhich we believe can be at least partially attributed to an apparent to

  insufficient attention to details in the textbook passages. The less frequently observed of these two was what we will call enunciated misreading, that is, a student enunciated the words or symbols incorrectly, yielding an incorrect meaning. The more frequently observed kind of misreading, that we will call interpretive misreading, occurred when a student, who might have enunciated the words and symbols in a passage correctly, soon thereafter used the passage as if it had said something else, or even asserted that it had said something else.

  Here in referring to “ insufficient in attention” we are not claiming that we can directly observe attention. Rather we are using this terminology because the errors we can see seem most likely to arise from reduced attention.

  The less frequent kind of error, enunciated misreading, occurred when Tara read the guidelines on how to find extrema on a closed interval and attempted to work a related task. She (and other students in the study) did not carefully distinguish between f and f ‘ and appeared to be rereading them more or less interchangeably. For example, on rereading the definition of critical number, Tara said, “Let f be defined at c, if f of c [not

  f ‘(c)] equals zero, then c is a critical number.” Somewhat later, Tara also misread the

  symbol “>” as “less than”. Here we cannot be certain whether Tara simply misspoke or whether she did not know the meaning of “>”, but most U.S. calculus students are quite familiar with the meaning of “>”, having used it for several years.

  Turning now to the more frequent, interpretive misreading, we have an example that combines misreading with an inadequate response to error and confusion. Christie correctly read the passage about the Wrapping Function, which includes diagrams, and

  

  how to calculate its values for integer multiples of /

  2 , orally answered two worked

  examples incorrectly (with the work hidden), and then read their solutions. Next she tried

  W (   )

  to answer the first matched problem, Find the coordinates of . She said, “It’s

   going to be (1,0) because you’re going . . . up every time, every quarter of a circle….

  So if we just start at the top [i.e., (0,1)] and then go down [i.e., clockwise] one  , I think we’d be at (1,0).” Not only did Christie start wrapping at the wrong point, (0, 1), but she

  2 

  also did not appear to know that the measure of a quarter circle is and that positive angles are measured in the counterclockwise direction. All of these were in the passage, but she had not gone back to it to check them. Thus this is an example of interpretive misreading, and inappropriate response to error.

  Somewhat later in her reading, Christie did discover that the starting point was (1,0) instead of (0,1). However, at the end of the interview, when asked if there was any notation that had bothered her, she said, “And I still don’t [...] I mean they still start you at the v-axis sometimes [i.e., at the top, (0, 1)] , and they start you at the u-axis sometimes [i.e., at (1, 0)], I think. So, I’m not real sure on that aspect of it.” Thus this is an example of interpretive misreading, and inappropriate response to error.

  Another result of in sufficient attention to detail is difficulty distinguishing between definitions with similar wording, such as relative extrema versus absolute extrema. One of the tasks given the calculus readers included the directions, Determine

  whether the function has a relative maximum, relative minimum, absolute maximum, absolute minimum, or none of these at each critical number on the interval shown

  (Larson, et al., 2002, p. 165). Zoe worked through the exercise, looked up the definition of extrema on an interval which included absolute extrema, but not relative extrema (Appendix B). In the debriefing (Appendix C), she was asked if there were any words that had bothered her. From her comments, one can see that Zoe had not distinguished between definitions of related concepts.

  “It said to find any relative minimum, relative maximum, absolute minimum, and absolute maximum. But in the first of it [definition of extrema], they said that those are the same things. So I wasn’t quite sure why they were asking me to find possibly four different things if they’re supposed to be just the same thing, but synonyms. … Since I didn’t know, I just went under the assumption that they’re the same thing.”

  

Clearly, she could have gone back in the textbook and looked up the definitions, but she

didn’t. This is then another example of interpretive misreading and inappropriate

  response to confusion and error.

  DISCUSSION

  Our students were all good readers according to the ACT reading comprehension test. This is also supported by our observation that the students exhibited the characteristic responses of good readers from the CRR perspective, both as a group (Table 1) and individually (Table 2). Furthermore, we found no student difficulties traceable to lacking such responses, and the students’ mathematics courses provided some training in reading mathematics textbooks, which is not a common practice in the United States.

  Our students also showed considerable ability and promise mathematically. Nine of the eleven students had done very well on the ACT mathematics test and five of those were members of a very selective science and mathematics program at the university. The remaining two students, Tara and Vannie, were not first-year students and had taken university courses that supported their current calculus course.

  Although the symbols and writing style of mathematics textbooks can make them hard to read, that was not the case for the passages read by our students, and none of their difficulties could be traced to the symbols or writing style of those passages.

  All of the above, taken together, might suggest that the students should have been effective readers, but our observations show that none of them were. At first this might be surprising, but we can suggest a likely explanation. Effective reading of mathematical text appears to call on different abilities from those required by the two ACT tests.