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
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;
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).
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:
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
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, rather than as indicating the statement’s falseness. By the beginning of the 20th 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
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
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
RESEARCH QUESTIONS
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 lacking 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
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
comprehension test (ACT) scores ranging from the 70th percentile to the 99th percentile.
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 (where the median, 28, corresponds to the 87th 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 ranging from the 71st percentile to the 96th 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.
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 included 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).
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.
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
(0), 2 , ( ),
W W W and W
3 2
, the task given was: Find the coordinates of the circular point W
2
. Parts of the selected reading passages along with theattempted, 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
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
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
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.
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
or x they use? … For all the x’s, f(c)is supposed to be your minimum point. Well, f(c) 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.
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
trigonometric example, ( ) 2sinf x xcos 2x 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
3 2
3
2
)
(
x
x
x
f
on the interval [-1,3]. Vannie first attempted to take the derivative andset it equal to zero. She incorrectly wrote
f
('
x
)
2
x
2
x
13
0
. At this point she checked the solution to confirm her derivative and said,She eventually fixed her derivative but still could not get the form of the
derivative shown in the textbook, which was
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.
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 inattention” 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.
also did not appear to know that the measure of a quarter circle is 2 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 insufficient 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.
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.
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. Effective reading from a mathematics textbook depends on the ability to use recently acquired information in working tasks that have just been explained or illustrated. In contrast, the ACT reading comprehension test depends on the ability to understand the content of the passage read and infer information from it, rather than on using that information in working new tasks. The ACT mathematics test does depend on working tasks, but on tasks similar to well-practiced ones, not tasks that have just recently been explained or illustrated. Unfortunately this explanation indicates little, or nothing, about how students might avoid errors and come to read effectively. To begin to analyze that, we turn to the kinds of difficulties our students had in working tasks associated with the passages read.
One main source of our students’ difficulties was inadequate or incorrect prior knowledge. But at the time of working a task, an error due to prior knowledge cannot be avoided. Another main source of students’ difficulties and error was insufficient attention to the content of the passages read. While the reasons for this and how it might be
unconsciously but plausibly by the similarity of the unit circle to a clock face, starting at the top (the twelve o’clock position) and progressing clockwise. Christie read the passage correctly and remembered it, but remembered that it said something else – that wrapping started at the top and that the positive direction was clockwise. Like errors due to
incorrect prior knowledge, errors due to the unnoticed results of inattention while reading cannot be avoided. This brings us to the third source of our students’ difficulties.
Although errors due to incorrect prior knowledge or unnoticed inattention may be
unavoidable, they are often correctable – given an appropriate response to the resulting confusion or errors. That is, a passage can be reread more carefully or a task can be reworked. However, our students were remarkably unconcerned about confusion or errors, and did not seem to think they could help themselves. They could have taken the opportunity to respond to such difficulty in an appropriate way, but very rarely did. Only Ellis occasionally responded appropriately to confusion or errors, and he eventually correctly worked 76% of the tasks he attempted, considerably more than any other student. For example, in working with the Wrapping Function, when Ellis found incorrect
coordinates for W(76), he created his own example to better understand the calculation.
Thus it appears that for most students a major factor in effective reading is sensitivity to their own confusion and errors and appropriate response to them. It seems likely that this factor might not have been easily discernable had we been dealing with less able students with a greater variety of task-working difficulties.
effectively. For example, in many U.S. mathematics departments, it is not seen as remarkable to be assigned to teach an undergraduate course that one has not previously taught or to teach from a new textbook. Indeed, it is common knowledge that
occasionally U.S. mathematicians successfully teach topics – even graduate courses – they have never before studied in order to learn them. All of this depends on effective reading, as does reading in support of mathematical research. However few
mathematicians have received any instruction in reading mathematics and seem to have tacitly learned effective reading.
This brings us to a final question. What might explain why mathematicians appear to be effective readers and our eleven students appeared not to be? Our data do not speak to this question and it seems not to be answered in the mathematics education research literature. However, we can offer a tentative suggestion from our own experiences as mathematicians. For sufficiently important reading, we will work tasks to check the correctness of our understanding and tend to look for errors. On finding an error, we rework the task or reread the appropriate passage. We feel that we can benefit from such a process, and suspect that this feeling of self-efficacy arose from our past positive
experiences with reworking tasks and rereading associated passages. We suggest our students lacked the feeling that one can independently rework a task or reread a passage until one ultimately “gets it right”.
FUTURE REASEARCH
It would be interesting to know (based on observations and interviews) how mathematicians react when they find themselves in a situation similar to a common one for our students. That is, when they are trying to learn new mathematics through reading, have read instructions on how to work a task, have worked it, and suspect their answer might be incorrect.
It would also be good to know more about which beginning undergraduates read or do not read, their mathematics textbooks, as well as which parts they read, and why. Our relatively advantaged students volunteered that they would not read major parts of their textbooks, perhaps because they did not feel they could do so usefully, and indeed they were not effective readers of their textbooks.
It would be useful to further explore errors in working tasks due to inattention during reading mathematics, including inattention that the reader is unaware of or soon forgets. Here the methods of the psychological literature on mind wandering during reading (one of several sources of inattention) might be helpful in examining the reading of mathematics texts. It would be especially interesting to understand more about how cognitive gaps due to inattention are sometimes filled with plausible, but incorrect, information as happened with Christie after reading about the Wrapping Function. This phenomenon resembles the way the mind allows one to experience “seeing” a large scene in high resolution, even though one’s eyes have focused on only a few parts of it.
feeling of the value of persistence be engendered – perhaps by providing supporting experiences?
IMPLICATIONS FOR TEACHING
The results of this study suggest that good conceptual reading, as indicated by high ACT reading scores and the use of many CRR responses, is not sufficient, by itself, for students to become what we have called effective readers of mathematics.
Nevertheless, some instruction in reading mathematics, perhaps based on the CRR responses and similar to the instruction described in Author (2005), seems likely to be beneficial. In this connection, the everyday interpretation of some CRR responses (Table 1) may need alteration for reading mathematics textbooks. For example, CRR response 9 in the Thumbnail Sketch (i.e., Determine meanings of new/unfamiliar words) can be accomplished with much non-mathematical text by inducting from, or reflecting on, a variety of uses of the unfamiliar words. In reading mathematical text however, such reflection is usually inadequate for establishing the meaning of technical terms that, unlike many words, have analytic or stipulated definitions. For such terms, nothing short of carefully reading, and reflecting on, the definitions themselves will suffice. We suspect that many students do not fully appreciate this fact (see for example, Edwards & Ward, 2004). Thus, an explanation of the nature of mathematical definitions should be included with any instruction involving CRR response 9.
need to learn to recognize errors, to see them less negatively and as a normal part of doing mathematics, and to respond appropriately. We suspect that most professional mathematicians have tacitly learned this over considerable time. However, it may be possible to explicitly teach students to look for their own errors and to appreciate that careful rereading of an explanation or reworking of a task can be very productive. One might ask students in class to work a task after having read how to work it. Should a mistake occur, one might lead students to reread the explanation or rework the task in an effort to give them the opportunity to experience success with such rereading or
reworking. It may be possible to “educate” students’ feelings so that they can sense when an error may have occurred and come to view success as often following from rereading or reworking tasks.
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