Reading Mathematics Text: a Study of Two Empirical Readings
Margot Berger
International Journal of Science and Mathematics Education ISSN 1571-0068 Int J of Sci and Math Educ DOI 10.1007/s10763-017-9867-6
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Author's personal copy Int J of Sci and Math Educ https://doi.org/10.1007/s10763-017-9867-6 O R I G I N A L PA P E R
Reading Mathematics Text: a Study of Two Empirical Readings Margot Berger 1
Received: 18 May 2017 / Accepted: 3 November 2017 # Ministry of Science and Technology, Taiwan 2017
Abstract This paper explores different ways in which mathematics students read a section of a prescribed mathematics textbook. In this case, the students are mathematics teachers learning new mathematics in a self-study calculus course. Two students were video-taped, while they studied a prescribed portion of the textbook. The reading styles of these two empirical readers are analysed according to how they exploit opportunities for learning from the textbook, how they inject productive knowledge such as prior knowledge, from outside the textbook, into their reading and how they make connections between different objects in the textbook. Their reading styles are compared to that of a hypothetical ‘implied’ reader. This implied reader engages with the text in a fruitful manner. The analyses are used as a platform from which to highlight productive and less productive ways for reading to learn mathematics. Keywords Empirical reader . Implied reader . Reading mathematics . Self-study . Use of mathematics textbooks
Introduction How students read mathematics textbooks, if at all, is a very under-researched area in mathematics education (Osterholm & Bergqvist, 2013; Rezat & Straesser, 2014; Shepherd, Selden, & Selden, 2012). Similarly, I suggest that reading of appropriate mathematics textbooks by students is an under-appreciated pedagogic activity: reading, if done with particular skills, may provide access, initial or extended, to mathematical concepts; to applications of these concepts; to particular mathematical examples and so on. Indeed, reading text is an indispensable skill in the modern world (be it reading of
* Margot Berger
[email protected]
1
Marang Centre, School of Education, University of Witwatersrand, Private Bag 3, WITS 2050, Johannesburg, South Africa
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internet materials, literature, textbooks and so on). The probability, though, is that many students are unable to read mathematics textbook for understanding. But this is not researched. Appreciating the different ways in which various students do read mathematics texts when required to do so is an important first step in understanding this potentially educative activity and in devising ways to help students read mathematics text. To gain some insight into the ways in which students read mathematics text, I embarked on a research project in which a sample of students who were enrolled in two self-study mathematics modules at university were video-taped while they read out loud from a prescribed mathematics textbook. In this article, I examine the ways of two of these students and I show how different protocols of reading afford very different ways of exploiting or not the written text for the learning of mathematics.
Literature Review Although there is growing interest and research into teachers’ use of textbooks (for example, Gueudet, Pepin, & Trouche, 2012; Leshota, 2015; Remillard, 2005), there is a very little research on learners’ use of mathematics textbooks, let alone their reading styles. Rezat (2013) argues that this may relate to methodological problems in researching an area involving activities which often take place in silence outside the classroom or it may derive from a view that students’ use of textbooks is always teacher-mediated. Related to the latter point, it may be the result of an underappreciation of the opportunities for learning that many contemporary mathematics textbooks do provide to students. Fan, Zhu, and Miao (2013, p. 774) in their review of research on mathematics textbooks acknowledge that ‘studies addressing the issues about the use of textbooks with data collected through empirically based methods such as classroom observation, interviews etc.’ are fraught with difficulties. The study reported here uses empirically collected data (as opposed to self-reports or surveys) and hopefully is a step to redressing what Fan et al., (2013) regard as a gap in mathematics textbook research. With regard to research specifically directed at learners’ use of mathematics textbooks, the work of Rezat (for example, Rezat, 2008, 2013) stands out for its theoretical rigour and methodological care. Rezat (2008) uses Activity Theory to develop a typology of learners’ uses of mathematics textbooks. In a later paper, Rezat (2013) uses an instrumental genesis framework for analysing 74 learners’ utilisation schemes when involved in self-regulated practice. Self-regulated practice involves learners choosing and doing mathematics exercises from a mathematics textbook to ‘improve’ their mathematical knowledge; it is self-regulated in that the teacher does not explicitly tell students which parts of the textbook to use. Rezat identifies three different utilisation schemes within self-regulated practice. Sierpinska’s (1997) study on the interaction between students, textbooks and tutors in a linear algebra course is also directed to student learning as mediated by a tutor, from a textbook. Sierpinska uses the construct ‘format of interaction’ to unravel the mechanisms whereby mathematical meanings are made in these interactions. There is also some research based on literary theory which looks at students’ use of textbooks. Shepherd et al. (2012) adapted the Constructively Responsive Reading
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
Framework from literary theory to explore why many mathematics undergraduate students do not read their mathematics texts in a useful way. Weinberg and Wiesner (2011) adapted reader-oriented theory (deriving from a branch of literary criticism) to illuminate the relationship between the reader and the mathematical text. Weinberg and Wiesner’s (2011) reader-oriented theory distinguishes between the attributes of an actual reader (the empirical reader) and the expected attributes of a reader who is able to make meaning from the text as she reads (the implied reader). As such, it is a very useful framework for my comparison of two readers (the empirical readers) and the reading of the same section of a mathematics textbook by an implied reader. I expand on this point in the Theoretical Framework.
Theoretical Framework Textbooks Within a sociocultural context, a mathematics textbook is theorised as a cultural artefact. It contains mathematical knowledge developed over hundreds of years, carefully selected and sequenced from both a mathematical and a pedagogical perspective. Kilpatrick (2014) broadens the notion of a textbook to include material presented in electronic format. Although I do not doubt the value or the potential use of electronic resources, the focus (both theoretical and empirical) in my research is on the written textbook, still a very common resource in most classrooms of the world. Within a socio-cultural perspective, knowledge is not constructed anew by each generation; rather each generation builds on the knowledge of its predecessors (much of which is contained in textbooks) to construct its own mathematical knowledge and possibly to create new knowledge. Analytic Framework The analytic framework is based on two pillars: the discourse of the mathematics textbook (the written discourse) and the ways in which the student interacts with this discourse (the enacted discourse). The theorised sorts of interaction between these two forms of discourse derive largely from Vygotsky’s theory of internalisation: ‘Every function in the child’s cultural development appears twice: first, on the social level, and later, on the individual level; first, between people (interpsychological) and then inside the child (intrapsychological). This applies equally to voluntary attention, to logical memory, and to the formation of concepts’ (Vygotsky, 1978, p. 57). In this research, the social level takes the form of the written discourse in the mathematics textbook. The individual level is made manifest in the enacted discourse through which the particular student interacts with the written discourse. Learning takes place through the individual’s thoughtful transactions with the written mathematical discourse. The distinction between written and enacted discourse also corresponds to the distinction between the implied reader and the empirical reader in reader-oriented theory (Weinberg & Wiesner, 2011) and so these constructs are useful in my research. According to reader-oriented theory, there are three types of readers: the empirical
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reader, the intended reader and the implied reader. As previously stated, the empirical reader is the actual reader of the text and is a focus of my research. The intended reader is the reader that the author imagines is reading the text. The intended reader is not examined in this article. The implied reader is the ‘embodiment of the behaviours, codes and competencies that are required for an empirical reader to respond to the text in a way that is both meaningful and accurate’ (ibid., p. 52). These qualities of the implied reader are determined by the text itself (as opposed to by the author) and are decided on by expert readers. In general terms, ‘behaviour’ refers to the reader’s activities with the text; for example, in undergraduate textbooks, worked examples (WE) are frequently accompanied by diagrams with which the self-study reader is expected to engage while doing the WE. Another aspect of the implied reader is her ability to appropriately interpret codes, often implicit, in the textbook. These codes include aspects of formatting, language and symbols which have particular meanings for the textbook writer. For example, correct reading of diagrams may require knowledge of certain codes; research related to this comprises a small but important part of the literature in reading textbooks (for example, Dietiker & Brakoniecki, 2014; Yang & Li, 2016). A third aspect of the implied reader are her competencies. These refer largely to the prior knowledge of a reader which are deemed necessary to understand the text. In this paper, I elaborate the construct of the implied reader with regard to the context of self-study where a prescribed mathematics textbook is the primary resource. Written Discourse The different components of many traditional undergraduate mathematics textbooks (such as the American undergraduate textbooks used in my research) comprise theoretical components such as theorems (statements and proofs), definitions and properties of mathematics objects; procedural components such as worked examples, exercises with full, partial or no given solutions; multiple representations of mathematics objects such as graphs and diagrams and explanatory text. Worksheets and notes written by the teacher and given to the students are also part of the written discourse. This discourse also includes more subtle, explicit or implicit aspects, such as expected behaviours, codes and competencies, which delineate the implied reader (Weinberg & Wiesner, 2011) as discussed above. Enacted Discourse The enacted discourse describes ways in which the empirical reader interacts with the written discourse. The empirical reader brings particular beliefs and experiences to the reading process (Weinberg & Wiesner, 2011). In addition, each individual brings his or her own prior mathematics knowledge. These characteristics all shape the ways in which the empirical reader exploits or not the learning opportunities of the text. Given that my research is concerned with the ways in which students read the textbook to promote learning, categories for analysing how the student reads the textbook are related to the ways in which the student engages with or ignores different components in the textbook. Broadly speaking, these categories are ways of using learning opportunities of the textbook, injection of knowledge from sources other than the textbook and making connections within the given mathematics textbook. I also draw attention to both congruencies and disparities between the self-studying implied reader and the empirical reader with regard to these categories.
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
The analytic framework for the enacted discourse was refined through my observation of five individual students as they studied a particular section of a mathematics textbook (the written discourse) using a talk-aloud protocol (the enacted discourse).
Research Focus In this paper, I focus on the reading styles of two contrasting students, Sol and Tom, as they read from the prescribed calculus textbook. I ask: What are some of the possible ways in which a student (the empirical reader) reads and engages with a mathematics textbook? How do these ways of reading align with the behaviour of the implied reader?
The Context The research takes place within the context of a precalculus and a calculus module which are part of a 1-year postgraduate programme (Honours level) in Mathematics Education for high-school teachers at a South African University. Each module consists of 11 3-h sessions, one session per week. I was the lecturer in charge of the modules. South African high school mathematics teachers often have weak mathematics specific content knowledge. This is partly a result of teachers often having a degree in education and an absence of traditional university-level mathematics courses (a legacy of apartheid). Accordingly, the focus of these mathematics modules is on deepening and broadening mathematics content knowledge and so the teachers in the course are conceptualised as mathematics learners in the design of the modules, and in this research paper. Another important aspect of the modules is an emphasis on self-learning. This emphasis derives from the idea that learning to use texts and other resources to learn or re-learn mathematical topics is necessary for a teacher who needs to keep on expanding and deepening her mathematical knowledge. In order to foreground the practice of self-learning, at the beginning of each module, all students (in this case, the teachers who are conceptualised as learners) are given a hand-out in which they are told exactly which part of the prescribed textbook to study for each weekly session. For the calculus module, the textbook is Thomas’ Calculus: Early Transcendentals (Thomas, Weir, & Hass, 2014). As students are repeatedly told, studying involves carefully working through definitions, theorems and proofs and worked examples, referring to other resources or other sections of the textbook if required, and doing a set of prescribed exercises (taken from the back of the chapter). Thus, ‘new’ mathematical knowledge or revisited mathematical knowledge is accessed by the students through the prescribed textbook prior to the lecture. Throughout the modules, the lecturer discusses ways of reading mathematics text, for example, ways in which to deconstruct a definition. Due to the scarcity of research on how students should or do read mathematics textbooks, these ways are based on the lecturer’s own experiences.
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Collection of Data Six students from the precalculus and calculus module (two low-performing, two medium-performing and two high-performing students) were asked if they were willing to be video-taped while studying a specific section of a chapter, the sub-chapter, from each prescribed textbook. Five agreed. For the video-taped session presented in this paper, students were asked to study the section ‘Intermediate Value Theorem for Continuous Functions’ (Thomas et al., 2014, pp. 99–101) as they would in preparation for class and to talk out loud as they did so. They were also given a set of exercises at the back of the sub-chapter, as was the case for their weekly sessions. Each video-taped session, which took place in my office, was at most 1 h long. This sub-chapter was chosen for the reading task because it involved content which none of the students would have previously encountered. Aside from the transcripts of each video session, the interviewer (myself) wrote a set of notes during each interview, noting points of interest. Also, the writings and solutions to exercises of each student were photocopied after the interview. These field notes and student solutions were used in the interpretation of the transcripts. Despite making efforts to put the student at ease during the video session and to encourage the student to study or read as he or she would in preparation for our weekly sessions, the students were clearly not in their natural setting. In addition, the presence of a researcher/teacher most likely effected their reading practice. However, all students seemed to be fairly relaxed and their readings were not surprising in terms of my prior experience of these students interacting with text in their weekly sessions. This gives a measure of confidence that even if the student did depart from his or her home reading style, the reading sessions in and off themselves produced a valid set of transcripts of different ways in which an empirical reader may interact with a written mathematical discourse.
Analysis and Coding Broadly speaking, the transcripts of all five learners were analysed in terms of the ways in which students read the text, made connections within the text or inserted other resources (such as prior learning) to enable or enhance their learning experience. I used the constant comparative method as described by Merriam (1998) as my method of analysis (cf. Glaser and Strauss, 1967). More specifically, the analysis of data consisted of several iterative steps. First, there was the descriptive level: I read through the transcript of each interview, together with field notes and the student’s photocopied solutions. This allowed me to make notes on the transcripts describing what was happening. (For example, ‘explains solution to WE as he reads it’; ‘points to diagram as he explains the theory’; ‘inserts inappropriate phrases into what she is reading’.) During the descriptive level, I generated three broad analytic categories relating to how the reader was using the different components of the textbook (Textbook Opportunities), whether the reader was injecting knowledge from beyond the textbook (Injections) to support their learning and how the reader was making connections within the textbook (Making Connections). The broad categories were further refined (see Table 1) so as to provide codes for a more in- depth analysis. I then re-read the
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings Table 1 Summary of codes and indicators for an empirical reader and an implied reader Code
Description and indicator
Textbook opportunities AOPTP
Uses textbook to seemingly enrich understanding of mathematics. For example, to help solve problem, to make connections between theory and processes etc. This is also indicated by accurate explanations or paraphrasing of the text.
OPT
Looks at or reads a component in the textbook, for example, explanatory text, proof of theorem, diagram. No indication as to whether this engagement is productive of meaning or not.
OPTM
Misses opportunity to use textbook for help. There is content in textbook which could potentially enrich understanding of mathematics but student does not attempt to find or read this content.
OPTF
Attempts unsuccessfully to use textbook to enrich understanding of mathematics. Looks at theorems, worked examples etc. which are not relevant to the issue at hand.
OPTX
Looks at appropriate objects in textbook but seems to be unable to make appropriate connections.
XT
Does exercises, or worked example (WE) without explicitly using textbook.
Injection AINIR
Explicitly uses intervention of interviewer or peer or technology in a productive way.
INID
Explicitly uses intervention of interviewer or peer or technology in an unproductive way.
AINPR
Explicitly uses prior knowledge in a productive way.
INPD
Explicitly uses prior knowledge in an unproductive way.
AINR
Uses knowledge from a non-explicit resource, in a productive way.
IND
Uses knowledge from a non-explicit resource, in an unproductive way.
INWD
Inserts distracting or incorrect words or phrases into the reading; copies down mathematics or text incorrectly.
Making connections AMCT
Makes explicit connections within text.
AMCV
Makes explicit connections between text and visual component.
transcripts, applying the analytic categories via their codes to interpret the data, for example, ‘makes connection between theory and diagram’ (AMCV); ‘misses opportunity to use textbook’ (OPTM). After this, I wrote a short narrative description of each student’s activities while reading the text. Finally, I counted the instances of each code appearing in each student’s transcript and generated a bar chart for each student. All these steps were repeated several times, so as to refine the descriptions, the analytic categories, the summary accounts and the counts. I also constructed an implied reader (see below) and compared the student’s reading style with that of the implied reader. Coding the Data Construction of Implied Reader In this research, the implied reader of the textbook serves as a backdrop to the empirical reader. I am the ‘expert’ who constructs the implied reader, that is, who judges what the text requires or expects the reader to do in the context of the activity of self-study. As indicated earlier, the implied reader behaves in a way that supports the reading of the text (for example, attempting worked examples
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on her own before reading the solutions); is able to interpret codes in the text (for example, to realise, implicitly or explicitly that a box around text signifies the importance of the content in the box); has sufficient competencies such as background knowledge of the subject matter. In this research, knowledge of functions and of continuity are necessary pre-requisites to making meaning from the text. The context for the reading is self-study in an environment which permits access to peers, teachers or technology. Since this is empirical research in which the activities (behaviour) of the empirical reader are compared to that of the implied reader, I do not examine the interpretation of codes and competencies as such. Rather, these are regarded as being implicit in the activities of an empirical reader. The attributes of the implied reader are encapsulated in the analytic categories, Textbook Opportunities, Injections and Connections. All activities which use the textbook directly to develop or maintain understanding are coded AOPTP. The injection of accurate and appropriate prior knowledge into the implied reader’s reading of the text is coded AINPR. The implied reader is studying text in the context of a video session with an interviewer; in this regard, the productive use by the implied reader of interventions by the interviewer is labelled AINIR. Sometimes, the source of the injection of knowledge is not explicit. In this case, if the injection is used productively, it is labelled AINR. Making connections is another very important and expected activity of the implied reader, hence, the codes AMCT (explicit connections within text) or AMCV (explicit connections between text and diagrams). Empirical Reader As with the implied reader, the main analytic categories for the enacted discourse by the empirical readers comprise Textbook Opportunities, Injections and Connections. In order to highlight the congruencies and discrepancies between the empirical reader and implied reader, the categories of activity of the empirical reader, which are considered commensurate with the activities of the implied reader, are given the first letter A (as are all the codes of the implied reader). If a code does not start with the letter A, it designates an activity which may not be (depending on context) commensurate with the expected activities of an implied reader. As stated previously, I focus on the activities or behaviours of the students. Textbook Opportunities Episodes in which the student explicitly uses the textbook to look at specific components (theoretical, procedural, multiple representations or text) but with no indication as to whether this looking is productive of learning or not are coded OPT. Episodes in which the student does not use the textbook to clarify, illuminate or enrich their understanding of the mathematics discourse, despite such content being available in the textbook, are coded as ‘missed opportunities’ (OPTM). If the student uses the textbook productively, for example, to make meaning (via paraphrasing), to make explicit connections, to generalise, to exemplify etc., this is coded as AOPTP (as with the implied reader). If the student attempts to use the textbook to access or enrich understanding of an aspect of math discourse but looks at a component (e.g. theorems, worked examples etc.) unsuited to the issue at hand, this is coded as OPTF. Finally, if the student looks at appropriate theorems, worked examples and so on, but seems unable to usefully interpret or apply the reading to their particular conundrum, this is coded as OPTX. XT is the code used when the
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student appropriately does not access the textbook; this often occurs when a reader is successfully doing an exercise. Injections If the student injects discourse from a source other than the textbook into the reading of the text, this is coded as AINR or IND depending on whether the injection is ‘robust’ or ‘distractive’. These terms (injection, robust and distraction) were introduced by Leshota (2015) in her research on school mathematics teachers’ use of textbooks as a resource for teaching. Robust refers to productive insertions of discourse; distractive refers to unproductive or confusing insertions. If it is clear that the injection is that of prior knowledge, the code AINPR or INPD (robust or distractive respectively) is used; if the injected discourse derives explicitly from the interviewer or a computer or resources other than the textbook, the code is AINIR or INID (depending on whether it is robust or distractive). The code INWD is used if the reader inserts distracting or inappropriate words or phrases into the reading; it is also used when copying down maths or text incorrectly (for example, when doing an exercise). Connections Since this framework is concerned with how students engage with the discourse of a mathematics textbook, it is important to look at what connections (MC) are made within the textbook. Of course, many connections may be apparent to the reader but not visible to the researcher interpreting the reader’s activities. Hence the use of the term ‘explicit’ in the description of the MC codes. Specifically, the code for making explicit connections between different parts of the text is AMCT; for making explicit connections between two different representations of the same object (e.g. a diagram and text), the code is AMCV. AMCV is a specific form of AMCT and so is not used together with AMCT. In contrast, all other codes (for implied or empirical reader) are not mutually exclusive. For example, the code AOPTP (using the textbook productively) may be accompanied by the code AMCT if the nature of the productive use is making connections within the text. This can be seen in Table 4, where Sol’s utterances on line 13 of transcript has both these codes.
Summary of Codes Table 1 presents a summary of all codes with their indicators for implied and/or empirical readers. Some of the codes are discussed further in the Analysis and Discussion section. The Data Written Discourse The written discourse comprises a worksheet handed out at the beginning of the interview (Table 2), a section from the textbook (Thomas et al., 2014, pp. 99–101) on the Intermediate Value Theorem for Continuous Functions (Figs. 1, 2, 3 and 4) and Table 3. In the textbook, the statement of the Intermediate Value Theorem (IVT) for continuous functions is preceded by a short explanatory paragraph (Fig. 1). IVT is a theorem which is concerned with the existence of a point on any continuous function between
Author's personal copy M. Berger Table 2 The interview handout Chapter 5.5 Study chapter 2.5, p. 99–102: The Intermediate Value Theorem for Continuous Functions Do the following exercises: Exercise 2.5: No 54. At end of question, insert 'between x = 0 and x = π.' Exercise 2.5: No 71. At end of question, insert 'between x = − 1 and x = 1.'
any two other points on the function. The analytic proof for this theorem is not given in the text. However, a geometric interpretation and explanation is provided below the statement of the theorem. See Fig. 1.
Intermediate Value Theorem for Continuous Functions Functions that are continuous on intervals have properties that make them particularly useful in mathematics and its applications. One of these is the Intermediate Value Property. A function is said to have the Intermediate Value Property if whenever it talks on two values, it takes on all the values in between. THEOREM 11 - The Intermediate Value Theorem for Continuous Functions If f is a continuous function on a closed interval a, b , and if y0 is any value between f ( a ) and f (b ) then y0
f (c) for some c in a, b .
Theorem 11 says that continuous functions over finite closed intervals have the Intermediate Value Peoperty. Geometrically, the Intermediate Value Peoperty says that any horizontal line y y0 crossing the y- axis between the numbers f ( a ) and f (b ) will cross the curve y f ( x ) at least once over the interval a, b . The proof of the Intermediate Value Theorem depends on the completeness property of the real number system (appendix 6) and can be found in more advanced texts. Fig. 1 Text and graph from Thomas et al., 2014, p. 99. (Format and layout differ to that of textbook)
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
Figure 2.46
The function
The continuity of f on the interval is essential to Theorem 11. If f is discontinuous at even one point of the interval, the theorem’s conclusions may fail as it does for the function graphed in Fig. 2.46 (choose y0 as any number between 2 and 3). A Consequence for Graphing Connectedness Theorem 11 implies that the graph of `a function continuous on an interval cannot have any breaks over the interval. It will be connected - a single, unbroken curve. It will not have jumps like the graph of the greater integer function (Figure 2.39) or separate branches like the graph of 1/ x (Figure 2.41).
A Consequence for Root Finding We call a solution of the equation f ( x ) 0 a root of the equation or a zero of the function f. The does not take on all values between f (1) 0 and f (4) 3; it misses all the values Intermediate Value Theorem tells us that if f is continuous, then any interval on which f between 2 and 3. changes sign contains a zero of the function. In practical terms, when we see the graph of a continuous function cross the horizontal axis on a computer screen, we know it is not stepping across. There really is a point where the function’s value is zero. f x
2 x 2, 1 3
2
x
2
x
4
Fig. 2 Text and graph from Thomas et al., 2014, p. 100. Format and layout differ to that of textbook
Further explanation of why continuity is deemed essential for this theorem is given in explanatory text with a counter-example in the form of a diagram. See Fig. 2, first paragraph plus diagram. This is followed by two paragraphs of explanatory text in which the consequences of IVT for Connectedness and Root Finding are discussed (Fig. 2). Worked examples (WE) 11 and 12 (Figs. 3 and 4, respectively) follow, accompanied by diagrams to illustrate. Finally, exercises are given. See Table 3. As indicated in Table 2 the students in the video-tape session were expected to do these two exercises with the added constraints given in the handout in Table 2. Enacted Discourse: Transcripts For this paper, the enacted discourse comprises the transcripts of Sol’s and Tom’s video-recorded sessions. Two excerpts of analysed transcripts from these sessions are presented (Tables 4 and 5) so that the reader (of this article) can see how I coded these excerpts. The actual analyses of these lines of transcript are discussed in the Analysis and Discussion section later.
Analysis and Discussion A summary of the analyses of two students’ (empirical readers) approaches to reading the designated textbook section is given here. For each student, due to space constraints, a summary rather than a comprehensive account of the video-taped session is
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EXAMPLE 11 Show that there is a root of the equation x 3 x 1 0 between 1 and 2. Solution Let f ( x) x 3 x 1. Since f (1) 1 1 1 1 0 and f (2) 23 2 1 5 0 , we see that y0 = 0 is a value between f (1) and f (2) . Since f is continuous, the Intermediate Value Theorem says there is a zero of f between 1 and 2. Figure 2.47 show the result of zooming in to locate the root near x = 1.32.
(a)
(b)
(c) Figure 2.47 Zooming in on a zero of the function f ( x)
(d) x
3
x 1 . The zero is near x = 1.3247.
(Example 11) Fig. 3 Worked example 11 from Thomas et al., 2014, p. 100. Format and layout differ to that of textbook
provided. The episodes that I discuss are selected for one of two reasons: either they are necessary for a coherent story of that empirical reader’s engagement with the textbook or they describe a typical activity of that student. The two students were chosen because of their contrasting reading profiles. Sol Sol is a strong mathematics student. His end-of-year marks for both precalculus and calculus are among the best in the class. Sol has a B. Ed in Maths Education degree from a university in a country neighbouring South Africa. Summarised Description Sol reads and adequately paraphrases the paragraph above the statement of the theorem (Fig. 1). This is coded as AOPTP since the paraphrase indicates that Sol is making meaning from what he is reading. He then reads the IVT theorem and relates it to the given diagram (AOPTP and AMCV) (Fig. 1). He proceeds
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
2x 5
4 x2
EXAMPLE 12 Use the Intermediate Value Theorem to prove that the equation 2 x 5 4 x 2 has a solution (Figure 2.48). Solution We rewrite the equation as 2 x 5 x2 4 and set f ( x )
FIGURE 2.48 The curves y 2 x 5 and y 4 x 2 have the same value at x c where 2 x 5 4 x 2 (Example 12).
2x 5
x 2 . Now
g( x ) 2 x 5 is continuous on the interval [ 5 / 2, ) since it is the composite of the square root function with the nonnegative linear y 2 x 5 . Then f is the sum of function the function g and the quadratic function y x 2 , and the quadratic function is continuous for all 2 x 5 x2 values of x. It follows that f ( x ) is continuous on the interval [ 5 / 2, ) . By trial and error, we find the function values f (0) 5 2.24 and f (2) 9 4 7 , and note that f is also continuous on the finite closed interval [0,2] ⊂ [ 5 / 2, ) . Since the value y0 4 is between the numbers 2.24 and 7, by the Intermediate Value Theorem there is a number c [0,2] such that f (c) 4. That is, the number c solves the original equation. ■
Fig. 4 Worked Example 11 from Thomas et al., 2014, pp. 100-101. Format and layout differ to that of textbook
to explore why the function needs to be continuous using the given counter-example with its graph (Fig. 2). This is again coded AOPTP and AMCV. See transcript in Table 4 (lines 11–13 and 21–24). Sol then reads and paraphrases the paragraphs about the implications of IVT for Connectedness and Root Finding (Fig. 2). Given the accurate paraphrasing, these activities are both coded AOPTP. After this, Sol reads out the task in WE 11 (Fig. 3). He indicates that he first wants to try this WE on his own: ‘And before I look at what’s happening, I just want to try and think on my own’. He then correctly solves the problem out loud. Since Sol is not using the textbook directly while solving WE, this is coded XT. He then checks the given worked solution, explaining as he goes (coded AOPTP) (Fig. 3). He also examines the given diagrams in which graphs of different scales, which zoom in on the root, are presented (Fig. 3). This is coded AMCV. Sol then moves on to the next WE, example 12 (Fig. 4) but indicates that he does not know how to approach the given WE: ‘Okay, I think I’ll just read through that example. I don’t seem to have any or some inclination as to how I can approach that’. He first reads silently to himself. This is coded as OPTX since Sol is at first unable to make sense of the question. This silent reading is followed by Sol’s Table 3 Two exercises Exercise 54: Explain why the equation cos x = x has at least one solution. Exercise 71: Use the Intermediate Value Theorem to prove that given equation has a solution: x3 – 3x – 1 = 0
Author's personal copy M. Berger Table 4 Excerpt from Sol’s transcript Line Who What is said
Textbook focus
Activity & comment
Code
11
S
Okay, then they emphasise the P.100 reads top of Paraphrases appropriately importance of the continuity page about of the function f for the necessity for theorem to be valid. If f is not continuity continuous at any point then (Fig. 1). the theorem now becomes invalid.
12
Int
Yes. What were you looking at? You can talk aloud, if you can, when you’re thinking. If possible, yes.
13
S
Oh, okay. Alright, I’m trying to Pg 100. Looking actually see why it’s not at graph of going to work for this discontinuous function, which is function discontinuous. I’m trying to (Fig. 1). follow their clue that if we choose any function value between two and three … I want to see what that function value will do for our graph.
AOPTP Using given graph of discontinuous function to try AMCV to understand why continuity essential for theorem to hold
21
S
Mm, okay, two comma five. Two comma five.
22
S
AOPTP Chooses an appropriate value for y0 to show that IVT does AMCV not hold if f is not continuous
p. 100. Graph of discontinuous function Two comma five, so if I draw a (Fig. 1). horizontal line then it passes through that gap so it won’t intersect with our function.
23
Int
Okay.
24
S
And then it goes against the Intermediate Value Theorem.
AOPTP
reading the solution out loud and explaining it as he goes (AOPTP). He seems to ignore the given diagram (Fig. 4) in which the intersection of the two given functions is pffiffiffiffiffiffiffiffiffiffiffiffiffi shown. Instead, Sol sketches a rough diagram off ðxÞ ¼ 2x þ 5 þ x2 . Since this rough graph of f is appropriate to his reasoning, the ignoring of the given diagrams is not marked OPTM. Sol frequently refers to the statement of IVT (AMCT) and to his own rough graph of f (AMCV) while doing WE 12. Although Sol is able to explain WE 12 to himself, he indicates that he has a problem with the language used in the WE solution: ‘I can picture what they are talking about. It’s only the wording that has taken me off a bit’. This problem with the use of rigorous language when working with IVT relates to Sol’s difficulties with using the discourse of the textbook; it is coded as OPTX. Sol now addresses himself to the first exercise, exercise 54 (Tables 2 and 3). When starting off, he indicates that he will ‘try to relate [the solution to] what I met in …the first example that I did’. He does the exercise adequately, explaining his solution adequately as he writes. Since Sol relates the solution of the exercise directly to the
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings Table 5 Excerpt from Tom’s transcript Line
Who
What is said
Textbook focus
Activity & comment
Code
46
T
And they say, explain why the equation cos x is equals to x has at least one solution.
No 54. (Table 3)
Reads question correctly.
OPT
47
Int
So you’re doing number 54?
No. 54 (Table 3)
T
Yes, ma’am. Okay. they say, explain why the function cos x is equal to x has only one solution.
Inappropriately changes wording from ‘at least one’ to ‘only one’.
INWD
48
49
T
Um, well, from the sketch I’ve drawn, the function of y is equals to x and the function of y is equals to x. y is equals to cos x and y is equals to x. And I can see from the graph that surely there’s no other way in which these graphs can have more than one solution. They only intersect at one point.
No. 54 (Table 3)
Draws graph of y = x and y = cos x. Argues that they intersect at one point only. Has lost intent of question: supposed to show ‘at least one solution’; arguing for only one solution. Does not use IVT.
OPTM IND
WE in the text, his solution is coded AOPTP and AMCT. Sol then starts the second exercise, exercise 71 (Tables 2 and 3) which he solves successfully. His mathematical language is more precise than in the first exercise and he concludes by stating that ‘From Intermediate Value, if y0 is equal to zero, it’s implied that by the IVT there exists a value c, which is contained within my interval a, b which is one, two. There exists a value c, contained in one, two, such that f(c) will be equal to zero’. As with the previous exercise, this is coded AOPTP and AMCT.
Sol - Reading Activities (Calculus) 90 80 70 60 50 40 30 20 10 0
Fig. 5 A bar chart showing level of alignment of reading style of empirical reader, Sol, with the implied reader
Author's personal copy M. Berger
Comparing the Empirical Reader (Sol) with the Implied Reader Sol’s activities are generally aligned with those of the implied reader. As can be seen from Fig. 5, 80% of Sol’s responses align with those of the implied reader. Sol reads attentively, accurately paraphrasing and explaining the text as he goes (about 42% of his responses are coded AOPTP). He also refers to and interprets most diagrams specified in the explanatory text or WEs making frequent connections between the text and the given diagrams (about 34% of responses are coded AMCT and AMCV). An exception to this is the diagram in WE 12 which Sol seems to ignore. Sol’s reading style is flexible. For example, he tries the WE 11 on his own before looking at the given solution. However, given that he is initially unable to do WE 12, he first reads the given solution, explaining to himself as he goes. Although Sol does not explicitly use prior knowledge, he must be using such knowledge implicitly since he is able to engage adequately with various mathematics objects and notions in the text such as functions, graphs and continuity. There are zero instances of distractive use of prior knowledge (INPD). Distractive injections are negatively aligned with those of an implied reader. Tom Tom is a weak but passing, mathematics student. His marks for precalculus and calculus are tenth and eleventh respectively in the class of 14 students. Tom has a B. Ed Degree majoring in Mathematics Education from a South African University. Summarised Description Tom starts off by reading the explanatory text above the IVT, the explanatory text below IVT and then the IVT given in the box (see Fig. 1). He paraphrases parts of what he reads inaccurately, sometimes inserting superfluous or incorrect words or ignoring important information. For example, when paraphrasing the geometrical explanation of IVT given below the statement of IVT (Fig. 1), he writes ‘If f is a continuous function on a finite interval the horizontal line crossing the y-axis will cut the graph at least once between f(a) and f(b)’ (my italics). This should be ‘…any horizontal line …’ not ‘the horizontal line…’. Much of this inaccurate paraphrasing is coded OPT since it is not clear whether Tom is making personal meaning from the rewording or not. When reading the top three paragraphs of page 100 (Fig. 2), Tom reads quickly from the text without paraphrasing. This is coded OPT since it is impossible to say whether Tom is making meaning from what he is reading or not. After reading the explanatory text and statement of IVT several times (Fig. 1), Tom reads the question and solution of WE 11 (Fig. 3). On reading the question and the solution, Tom tells the interviewer: ‘I’m not quite clear with the use of the Intermediate Value Theorem to determine the roots of the function’. Tom’s idea that he needs to determine the roots is confusing and confused. IVT is an existence theorem: it is not used to find the roots of a function; rather, it is used to prove the existence of roots if they exist. This confusion carries through to later in the reading when Tom tries to solve the two given exercises (this is discussed below). The insertion of the phrase ‘determine the roots of the function’ is coded as INWD since it is a distractive insertion. Tom now moves on to the second WE, example 12 (Fig. 4), which he reads out loud. He immediately reveals the same confusion as he did in WE 11 by stating ‘Okay, basically I don’t know if they’re trying to solve the x here …’. This is coded INWD because it involves the insertion of a distractive idea (finding the value of an intersection). Tom
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
reads the given solution silently and states ‘Here they were trying to prove the Intermediate Value Theorem but they were actually using the real equations. I think I can see but I’m not quite happy with my understanding of this’. In this utterance, Tom indicates that he does not understand how IVT relates to the given question and that he knows that he does not understand. This utterance is coded OPTX since Tom, despite reading the appropriate text, is unable to interpret it usefully. Soon after, Tom indicates that he would like to engage with some of the exercises around continuity at the back of the chapter. Since this activity does not directly relate to IVT, it is coded OPTF. After a little engagement with some exercises, the interviewer advises Tom to move on to the given exercises. He starts on exercise 54. See transcript in Table 5. He immediately rephrases the given question ‘Explain why the function cos x = x has at least one solution’ (my italics) to ‘Explain why the function cos x is equal to x has only one solution’ (line 48, my italics). This is coded INWD, since the change of wording obfuscates the purpose of the question. Initially, Tom hand sketches the graphs of y = cos x and y = x to demonstrate that there is only one point of intersection, i.e. one solution. This is coded OPTM, since Tom misses an opportunity to use the textbook, possibly in the form of a WE, to help him solve the problem. The interviewer asks Tom if he has used IVT in his solution. This prompts Tom to return to WE 12 (Fig. 4) for assistance. However, Tom struggles to apply the method of WE 12 to his exercise. Much of the activity around Tom trying to use WE 12 to help solve exercise 54 is coded as OPTX since even though Tom accesses appropriate resources in the textbook, he is unable to use them in a positive way. Since Tom is not making much headway in his solution to exercise 54, the interviewer reminds him to focus on the interval between nought and pi (Table 2). This helps in that Tom evaluates f(0) and f(π) (where he has previously defined f(x) asf(x) = cos x − x). This is coded AINIR in that Tom is able to productively take up the interviewer’s suggestion. However Tom again uses a distractive term, ‘median’ (coded INWD), which he soon changes to ‘the middle’ (also coded INWD) when indicating that the line y = 0 will cut the function f (x) between 0 and π. Thereafter, he writes ‘The equation cos x = x has at least one solution y = 0 between 0 and π because y = 0 is a function value between y = − 4.14 and y = 1, by the IVT (line 96)’. This is a reasonably accurate application of IVT although Tom fails to mention the continuity of f on the relevant interval. It is coded AMCT, since Tom is making a connection to the statement of IVT, and OPTM, since Tom misses the opportunity to use previous WEs or the text to refine his response. Tom then moves onto the next exercise, exercise 71 (Table 3). As with the previous exercise, he starts off by trying to hand-sketch the functionf(x) = x3 − 3x − 1 using calculus. He tells the interviewer that he ‘wants to have the picture of the graph’ because he wants to ‘get all the solutions where I see the graph changing sign’. Since the purpose of the exercise is to use IVT to show that the equation has a solution, Tom’s time-consuming efforts in drawing the function are coded as INPD. This code is used since Tom is using prior knowledge (of curvesketching) in a distractive way. When Tom variously talks about finding the value of a root, this is coded as INWD since Tom is inserting activities not relevant to IVT into the given task (the given task involves showing a root or a solution exists, not its actual value). On the suggestion of the interviewer, Tom rereads WE 11 (coded AINIR). This helps a little in that he starts off by finding the y values of the two given endpoints of the interval, − 1 and 1. This is coded as AOPTP and AMCT. However, Tom’s reasoning soon becomes incoherent: x- and y- axes are interchanged and Tom again tries to find
Author's personal copy M. Berger
the actual value of the root. Most of this unproductive activity is coded as OPTX and INWD. Soon after this, the interview ends due to time constraints. Comparing the Empirical Reader (Tom) with the Implied Reader Tom’s reading activities do not align well with those of an implied reader. Just less than 40% of his responses are commensurate with those of an implied reader. See Fig. 6. The productive responses constitute mainly AOPTP responses (about 14%) and AMCV responses (about 11%). The AOPTP responses involve Tom using the text or WEs to help him solve the WEs or exercises, respectively. Tom’s most frequent code is OPT (about 18%); this is allocated when Tom reads the text without paraphrasing or any other explicit productive outcome. His second most frequent activity (about 15%) is the insertion of distractive or incorrect words into what he has read (INWD). This unproductive activity does not align with that of the implied reader. Similarly, OPTX activities (over 12%) in which Tom seems unable to use the text or diagram he is accessing in the textbook are important parts of Tom’s profile. Distancing Tom’s activities further from that of the implied reader are his scores on the codes of OPTM (just over 5%) and IND (just over 5%). OPTM refers to Tom’s missing of opportunities to use the textbook helpfully; IND refers to a distractive injection (where the source of the injection is not known).
Brief Summary In this research, the reading activities of two empirical readers have been analysed as they read their mathematics textbook for self-study in an interview setting. The focus has been on how they used learning opportunities presented in the textbook, the extent to which the readers injected prior knowledge (robust or distractive) into their reading and the types of connections that the readers made within the text. The activities of each reader were compared to the expected activities of an implied reader. This implied reader was constructed by the researcher in terms of both the actual text and the context of the reading (self-study).
Tom - Reading Activities (Calculus) 100.0 80.0 60.0 40.0 20.0 0.0
Fig. 6 A bar chart showing level of alignment of reading style of empirical reader, Tom, with the implied reader
Author's personal copy Reading Mathematics Text: a Study of Two Empirical Readings
The analyses show how one reader, Sol, reads in ways that are reasonably commensurate with those of the implied reader (about 80% alignment). For example, Sol accurately paraphrases most of what he reads. Also, Sol only once inserts a distractive injection into his reading. See Fig. 5. The other reader, Tom, reads in ways that are often not aligned to those of an intended reader (less than 40% of his reading activities are congruent with those of the intended reader). Indeed, Tom, frequently inserts distractive comments and words or inappropriate prior knowledge into his reading (over 22% of his responses) and he often paraphrases inaccurately.
Conclusion This research takes place in an environment in which mathematics textbooks suffer a double, possibly linked, ignominy: they are under-used as a learning resource and they are under-researched in terms of how they can be read productively by learners. There has been very little research illustrating either an adequate or a less-than-adequate reading of mathematics text by students. Such research is useful in two broad ways: to describe the lived-in reality of many students who are expected to read their textbooks and to give the researcher, teacher or textbook writer, pointers as to what strategies contribute to a productive reading of text. This paper contributes to the meagre mathematics education literature on reading mathematics text. By applying and extending Weinberg and Wiesner’s (2011) theoretical distinction between an empirical reader and an implied reader, an original framework for comparing the reading profiles of an actual reader (empirical reader) and an imagined and productive reader (implied reader) is developed and presented. Underlying this framework is the assumption that an alignment of the reading ways of an empirical reader with an implied reader is desirable and that the teacher or researcher can construct the implied reader appropriately. This article does not examine whether or how it is possible for a teacher to teach students how to read text productively. Such research could examine how different interventions, which are set in place to improve alignment between the empirical reader and the implied reader, affect the profile of the empirical reader. Another related area of research is the investigation of ways in which different texts support different ways of reading. Such research could help textbook writers in determining which features they should incorporate into their texts in order to promote productive reading. Meantime, it is worth noting certain practices (of both Sol and the implied reader) which contribute to a productive reading of text. Accurate paraphrasing of text (as frequently done by Sol) seems to be a very beneficial activity. In contrast, not paying attention to the exact words (for example, Tom) in the text and not paraphrasing accurately or at all may not lead to successful outcomes. Other strategies such as consciously connecting WEs and exercises to the relevant theory (IVT in this case) seem to bring meaning to Sol and so contribute to a positive reading experience. Another productive strategy, evident in Sol’s reading, is the connecting of text to a given diagram. The importance of prior learning, well recognised in the mathematics education community, is also strongly supported by this research. A final word: Most students have or should have access to some form of a mathematics textbook, a relatively cheap learning resource. Given the textbook’s
Author's personal copy M. Berger
potential as an extremely rich and useful learning resource, I suggest that more research and more pedagogic attention be paid to its usage by learners, and to possible ways of teaching the productive reading of mathematics text. Such productive ways should promote the exploitation of the many opportunities for learning that most reasonably designed mathematics textbooks have to offer. Acknowledgements This work is based on research supported by the National Research Foundation of South Africa: UID Number 85685.
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