Could Elementary Mathematics Textbooks Help Give Attention to Reasons in the Classroom?

DOUGLAS P. NEWTON and LYNN D. NEWTON

COULD ELEMENTARY MATHEMATICS TEXTBOOKS HELP GIVEATTENTION TO REASONS IN THE CLASSROOM?

ABSTRACT. Trainee teachers, new and non-specialist teachers of elementary mathematicshave a tendency to avoid thought about reasons in mathematics. Instead, they tend to favourthe development of computational skill through the rote application of procedures, routinesand algorithms. Could elementary mathematics textbooks serve as models of practice andsupport the professional development of these teachers? Eighteen such textbooks for use by7 to 11-year-old children in England were examined for their potential to help teachers attendto reasons. It was found that they were unlikely to point teachers towards reasons for patternsand procedures. Although some had the potential to help in introducing and structuring alesson, they were unlikely to induce the teacher to address matters of understanding intheir discourse. Their implicit message tended to be that mathematics education is aboutcomputational skill development through routines, algorithms and practice.

KEY WORDS: models of practice, textbooks, teacher knowledge, understanding reasons,elementary mathematics

1. INTRODUCTION

Cognitive models of thinking and learning are generally based on the viewthat learning is an active process that tries to makes sense of information.We attempt to make connections and construct a coherent and meaningfulwhole of them (Borich and Tombari, 1997). On this basis, Nickerson (1985,201) has described understanding as, ‘the connecting of facts, the relatingof newly acquired information to what is already known, the weaving ofbits of knowledge into an integrated whole’. Such cohesive wholes havea great variety. Concerning number and space, for example, they includepatterns, concepts, procedures, equivalences, relationships, reasonings, andproblem situations (Sierpinska, 1994, 2). Each has its place but understand-ings that enable explanation and application of thought are highly valued.Piaget (1978), for instance, felt that only mental constructions that answerthe question ‘why?’ deserve to be called understandings. These include thereasons that underpin mathematics and its processes. Knowing why haspowerful advantages. It adds a satisfying flexibility to thought, facilitatesfurther learning and enables it to be successful in new situations (New-ton, 2000). Hiebert and Wearne (1996) carried out a longitudinal study inthe USA of children learning mathematics as they moved through grades

Educational Studies in Mathematics (2006) 64: 69–84DOI: 10.1007/s10649-005-9015-z C© Springer 2006

70 DOUGLAS P. NEWTON AND LYNN D. NEWTON

1 to 4 (6 to 9 years). They found that children who had constructed anunderstanding were better at inventing and modifying ways of solvingproblems than those taught only algorithms. Further, they were better atmaking sense of later instruction, were more efficient in their mathemat-ical work, they retained their learning longer and could make more rapidprogress.

The value of verbal interaction with children in mathematics lessonsto scaffold their thinking, examine its quality, ask and answer questions iswell known (Steinberg et al., 2004). But, as Lampert (1985) has pointedout, an understanding of mathematics is essential to teach children how tothink mathematically. In the UK, as in many other countries, teachers ofyoung children often teach a range of subjects. Mathematics may not havebeen strong in many of these teachers’ own education. Teachers who lackknowledge or confidence tend to avoid ‘conversational risk’ and focus onfacts, routines and right answers at the expense of the reasons that underpinthese (Carlsen, 1991; Grossman et al., 1989; Bennett, 1993; Bennett andTurner-Bissett, 1993). Even teachers with a strong background in a subjectdo not always press for understanding (Schoenfeld, 1983). Hiebert et al.(2003) argue that training which puts all its efforts into making good suchdeficits in a teacher’s knowledge will generally be found wanting. In theUK, for instance, most teachers of younger children are trained to teach awide range of subjects in one academic year, after a degree course, seldom inmathematics. This period of training has a significant school-based elementwhich leaves little time to develop mathematical knowledge. Hiebert et al.(2003) suggest that training to teach mathematics is more effective if itconcentrates on ‘helping students acquire tools they will need to teachrather than on the finished competencies of effective teaching’ (Hiebertet al., 2003, 202).

Textbooks are frequently used to support teaching and learning in manycountries (see, for example, Teters and Gabel, 1984; Sheldon, 1988; Sharp,1999; Pepin and Haggarty, 2001). In the USA, Sheldon (1999) identifiedthe main reasons: books are often the basis of a course and preparingclassroom materials is difficult and time-consuming. Skierso (1991, 432)adds that ‘most teachers tend to follow the text . . . to the letter’. While thismay be an over-statement, teachers often use textbooks to help them planlessons and as sources of in-class activities for the children (Tolman et al.,1998; Newton, et al. 2002). Following a text to the letter, however, canhave benefits. In Australia, for instance, Rymartz and Engebretson (2005)found that a textbook made a big difference to the quality of teaching. Mostteachers and particularly new teachers and those teaching outside their areaof expertise found that they taught better, that they fostered better qualitythinking, and assessed more purposefully and to better ends by using a

COULD ELEMENTARY MATHEMATICS TEXTBOOKS HELP GIVE ATTENTION 71

textbook. They also reported better attitudes and motivation amongst theirstudents.

Practices regarding textbook selection and use, however, vary. In theUSA, for instance, there is a long history of guidance on textbook selectionby state departments of education (Orlich, 1983). Nevertheless, teachersin the USA are also often free to choose the book they will use them-selves (Tolman et al. 1998). In the relatively small markets of Cyprus andHong Kong, on the other hand, the educational authorities have funded theproduction of textual materials in order to support educational initiatives(Christou et al., 2004; Lo, 2005). In the UK, it is the schools that generallychoose and buy books for use in the classroom. In some countries, and inmany UK private schools, the schools choose the books and the parentsbuy them for their children (Newton, 1989). Major teaching schemes mayhave a Teacher’s Guide to explain their use but many books offer guidancein a few, introductory pages for the teacher. How books present the sametopic varies between books and educational contexts (Howson et al, 1999,269; 2001). For instance, Howson et al. (1999, 269; 2001) found that booksvaried from country to country in ‘the extent to which they draw upon aclear understanding of the links between different aspects of mathematicsand [how] they may be represented in text or illustration’.

In some countries, however, there can be a mistrust of a reliance ontextbooks. Teachers who rely on textbooks may be described as lazy or asencouraging passive learning (see, for instance, Tolman et al., 1998; Issett,2004). In the context of mathematics, Pepin and Haggarty (2001), foundteachers widely felt they should present a topic and explain it themselvesbut can textbooks help them do it? Brown (2001), for instance, has shownhow a teacher could acquire the tools to use textbooks well in mathematics.But, are the books up to it? In particular, have mathematics textbooks thecontent to help a teacher address reasons and, through them, understanding?As Lipman (1991) has observed, to foster reasoning, we need books thatembody and model it.

This is not to say that the textbook is a panacea that overcomes thedifficulties of mathematics teaching nor is it to say that practical activitiesare unimportant (Newton, 2000). Being for everyone, textbooks generallycannot meet the precise needs of the individual, they are blind to off-taskbehaviour and poor reasoning, and they can lend themselves to passivelearning. But, as Brown has pointed out, ‘no teaching material is completewithout a teacher’ (Brown, 2001, 245) and a deficient book may still beof use to a knowledgeable teacher. A textbook can offer well-structuredmaterial that keeps the teacher’s attention on what matters, allows childrento work at their own pace, and provide for their mental engagement with thesubject. The focus of this study is on what children’s textbooks might do for


the new teacher and those teaching outside their main area of expertise. Inparticular, do mathematics textbooks for young children point such teach-ers in the direction of reasoned explanations? Or do such textbooks, likesome teachers, avoid reasoned explanations? What message do these booksconvey about the nature of mathematics by virtue of what they include andomit? Is it one that portrays mathematics as a collection of routines andprocedures that are to be rehearsed until acquired? If so, such books mayoffer little to alleviate the problem. Or do they concern themselves withunderstanding? In particular, do they deal with the reasons underpinningmathematical routines, procedures and phenomena? If they do, they couldhelp a teacher redirect attention, overcome a reluctance to engage in dis-course involving reasons, and give confidence to press for understanding(Kazemi, 1998). In other words, although such books are not written forteachers, they may be a ready means of self-help and professional devel-opment. This study seeks to answer these questions and comment on thepotential of textbooks for enhancing the understanding of mathematics inthe primary school through the support they offer a teacher.

2. THE STUDY

2.1. Materials

In this study, a textbook was taken to be ‘a focused educational programmein text that relates to a scheme of work’ intended for use by the learner(Issitt, 2004: 65). Eighteen textbooks for teaching mathematics in KeyStage 2 (7 to 11-year-olds of the primary school in England) were exam-ined for the attention they gave to reasons. The books were an opportunecollection from the mathematics education sections of teacher training li-braries and resource centres and were for use in teaching National Cur-riculum mathematics, introduced in England and Wales in 1989. Thesebooks were for use in the classroom by the children, they had been writtenfor the children and were not teachers’ manuals. There were no separateteachers’ guides although the books usually included brief introductionsfor the teacher. These described broad aims, the children the books wouldsuit, and how the content related to the National Curriculum. They werenot explicit about particular learning outcomes, such as the potential valueof knowing reasons. As was mentioned earlier, the adoption of a book inthe UK is a matter for each school. It is not possible to point to a collectiveagreement or directive as evidence of book use. Nevertheless, these bookswere published by major UK publishers and most of the books had beenreprinted several times. This was taken to be an indication that they werenot languishing in publishers’ warehouses but were in use.


2.2. Sampling

Ignoring the title, contents and index pages and any introduction for theteacher, the remainder of each book was for the children. This was dividedinto quarters. The beginnings and ends of the topics under discussion atthese quarter points were located and the text between was parsed intoclauses. On average, this amounted to about 300 clauses per book. (Fora 32 page book, for example, these would typically be drawn from someeight pages.) This process allows for the possibility that the expositionalstyle may vary throughout a book. All the books included illustrations. Textin illustrations that occurred in the samples was analysed similarly.

2.3. Procedure

There is evidence that the strength and number of cause and purpose con-nections determine the probability of comprehension and the recall of in-formation read (Britton and Graesser, 1996) and can indicate a teacher’sor writer’s concern for reasons (Newton and Newton, 2000). Even whenwriters withhold reasons and provide activities to help children constructthem, they cannot assume that this will happen. In books, a concern for rea-sons, therefore, is often indicated by their presence. Clauses of cause andpurpose can, within limits, serve as indicators of this concern (Britton andGraesser, 1996; Newton and Newton, 2000). On this basis, the procedure isone developed for analysing discourse in the classroom and text (Newton,et al., 2002) and derives from Marton’s method of phemonographic analy-sis (Marton, 1981). There are two steps. In the first, the text in each sampleis divided into clauses (a combination of words in which something is saidabout something else using a finite verb). Clauses are commonly used asunits of textual analysis (Weber, 1990). Amongst these clauses, clauses ofcause (typically signalled by words like as, because, since) and purpose(typically signalled by in order to, to, so that) were noted. Clauses of causeand purpose provide reasons and are intended to help a reader know ‘why’.To illustrate, consider, ‘In order to find 19×17, work out 20×17 then takeoff 17. This works because 20 seventeens are one 17 too many so we takeone 17 away at the end.’ This contains both purpose (‘In order to . . .’)and causal clauses (‘because . . .’). The proportions of cause and purposeclauses for each book are recorded. The set of all clause and purpose clausesfrom all the books forms the data pool. It includes questions directed at thechild that ask for a cause or purpose.

The clause analysis was done by someone trained for the purpose andexperienced in teaching and using books of this kind with 7–11 year olds.The training comprised a review of the nature of a clause and the parsing


of text into clauses, a joint analysis with the authors of samples of textto illustrate the identification of clauses of cause and purpose and an ac-count of procedures for dealing with non-grammatical text and instancesof ambiguity. Text of this kind usually has a simple structure to increase thelikelihood that children will grasp its meaning. This tends to make most ofthe analysis unproblematic. There was sometimes doubt in the parsing ofcomplex sentences and from the omission of signals, such as, in order to orbecause. When this happened, complex sentences were parsed jointly withthe authors into agreed constituent clauses that maintained the meaning ofthe text. When signals are omitted, supplying them can test the function ofthe clause (Greenbaum, 1996). For example, ‘In order to find 19×34 and21×34. . .’ was considered to be, ‘In order to find 19×34 and in order tofind 21×34. . .’ and so would be recorded as two clauses of purpose. Inaddition, ten percent of each sample from each book was also analysed bythe authors and compared with the analyst’s record. There was agreementof 99% in the identification of the clauses.

The proportions of clauses alone are not sufficient to understand howexplanation is used in a book. A direct inspection of the text is also neededto ensure that these clauses reasonably reflect the writer’s concern for rea-sons. The second step, therefore, subjects the data pool to a kind of phe-nomenographic analysis in accordance with Marton’s procedure (Marton,1981). In essence, this involves an iterative sort of the clauses into groupsor categories on the basis of their similarities. Here, the process began byplacing together clauses used for the same kind of explanatory purpose.So, for instance, one clause might tell the child the purpose of a practi-cal activity while another might give a reason for a mathematical assertion.These groups are tentative at this stage. They are now considered separatelyfor homogeneity and together for overlap and duplication. In the process,they may be abandoned, replaced, merged or divided. This continues untilstability is achieved. (This stage corresponds, to some extent, to the datareduction stages of grounded theory and three-part analysis when they areused to produce descriptive categories (Strauss and Corbin, 1990; Milesand Huberman, 1984). In these, however, feedback from participants mayalso inform the process.) At this point, the sorters (the authors, in this study)label, select a representative example, and compile a concise description ofeach group. The product in this study was a set of categories that describeand illustrate the ways the clauses of cause and purpose were used in thedata set.

It is probably self-evident but should be said that this analysis looksonly at the potential of certain textbooks and not at the extent to which thatpotential is realised in the classroom. (For useful reference to the limitationsof various approaches to textbook study, see Pepin and Haggart’s paper


(2001).) In this study, it is the potential that is of interest but it has to beremembered that potential to be a model of teaching may be unrealised. Astrength of this approach is in the very high level of agreement that can beachieved in the clause analysis of relatively simple text and, hence, in theconstituents of the data pool. This is not to say that writers explain onlythrough clauses of cause and purpose. They may use other devices to thesame end and this analysis does not detect them. There is also what theteacher and the child do with the textbook to support understanding, perhapsthrough practical activity (Entwistle and Smith, 2003). This approach doesnot detect these directly. The aim of the study, however, is to consider thepotential of the children’s text to direct a teacher’s attention to reasons. Abook that does not address reasons explicitly is likely to have less potentialin the hands of a teacher lacking in confidence, one who does not knowthose reasons at the outset, or one who does not know that a grasp of reasonsis a valued goal. Such a book, by omission, could encourage a teacher todo the same or leave them unsure of the language they might use or theanswers they might expect.

Regarding the phenomenographic process for identifying categories ap-plied in the second step, two important cautions are needed. First, the pri-mary purpose of the approach is to identify, label and describe categoriespresent in the samples of text. Additional samples or additional bookscould, however, add new categories. In practice, a point of diminishingreturns is reached where successive books simply add examples to exist-ing categories. That point was reached in this study but we acknowledgethat other categories may exist. Second, it is not the purpose of the phe-nomenographic approach to provide quantitative data about the categories.The most common type of clause may not be the most common in thepopulation of mathematics books at large or in every book examined. Forthat reason, the figures we have attached to each category should be treatedwith due caution.

3. RESULTS

The results have two parts. In the first, we first present some statistics to setthe scene. In the second, we state the outcome of the phenomenographicanalysis. After that, we present an overview.

3.1. Incidence of cause and purpose clauses in each text

Clauses of cause ranged from nil to 3.96% of text (using clauses as theunit) with a mean of 0.68% (s.d. 1.08). Clauses of purpose ranged from nil


Figure 1. The percentages of clauses of cause and purpose in the maths books

to 8.03% of text with a mean of 4.77% (s.d. 2.08). On average, there werethree times as many clauses of purpose as clauses of cause, a difference thatwas statistically significant (t-test, p < 0.000). Figure 1 shows the profileof each book. For instance, there were no clauses of cause or purpose inBook 18. A direct examination of the content of this book produced noevidence of other ways of supporting thought about reasons, explanationsand similar understandings. Book 1, on the other hand, contained clausesof purpose. Book 5 contained a mixture of cause and purpose clauses. Howthese were used was the object of the phenomenographic analysis.

3.2. Categories of cause and purpose clause use

The phenomenographic analysis produced seven categories for the use ofcause and purpose clauses. Categories 1 to 5 relate to the use of purposeclauses. Categories 6 and 7 are for the use of clauses of cause. Note, how-ever, that some uses do not support a child’s understanding of mathematicsdirectly. Category 1, for instance, contains clauses used to tell the chil-dren the purpose of games and activities. While games and activities mayenhance their grasp of the mathematics, the clauses cannot be said to doso themselves. This could be described as an ‘extra-mathematical’ useof reasons. On the other hand, Category 6 clauses provided reasons formathematical assertions explicitly and might support mathematical under-standing directly (although this is not to say that they necessarily are goodreasons or are certain to produce an understanding). This could be describedas a ‘mathematical’ use of reasons. When describing the categories, it isusual to provide typical examples. Those provided here are taken from thedata pool. Note that the percentages have been rounded.


Category 1: Providing the purposes of games and other activities forthe child.

This category of use was extra-mathematical. The clauses providedthe purpose of and instructions for games and other activities intended toprovide experience of a topic. It was the largest category, comprising 60clauses and amounting to 40.0% of the cause and purpose data set.

For example:‘Both players work out how many steps it takes to reach number 1’Category 2: Providing purposes in contexts and examples for the child.This category of use was also extra-mathematical. The clauses provided

reasons in stories, real-world examples and applications and in descriptionsof the basis of analogies. They provided contexts or supported situationalunderstanding to facilitate the introduction of a topic or explanation of themathematics involved but did not explain the mathematics directly. It wasthe second largest category, comprising 33 clauses and amounting to 22.0%of the data set.

For example:‘Gruesome Gertie is buying some new ingredients to use in her spells.’‘[Bank] Notes are made of paper and have complicated designs on them

to make them hard to copy.’Category 3: Providing the purposes of mathematical procedures, oper-

ations and algorithms for the child.This category of use could be described as mathematical as the clauses

provided the intentions of procedures, operations and algorithms for pro-ducing a particular mathematical end. The clauses do not, however, tell thechild why these work (although associated clauses in another category maydo so). The category comprised 20 clauses and amounted to 13.3% of thedata set.

For example:‘[In order] To find a quarter of a number, find half of half.’Category 4: Providing the purpose of words, units, signs, abbreviations,

conventions and non-verbal representations for the child.This category was used for clauses providing the purpose of certain

words, units, signs, abbreviations, conventions and non-verbal represen-tations. It also contained instances indicating the purpose of mathemat-ics in general. Included in this category were mathematical and extra-mathematical uses. Examples of the former were:

‘We often use the word sum [in order] to describe any arithmetic prob-lem.’

‘Block graphs are often used [in order] to show differences betweenthings.’


An example of the latter was:‘We use am [in order] to show 12 midnight until 12 noon’.No attempt was made to separate mathematical and extra-mathematical

uses as some could, arguably, be either. For example:‘A pictograph is a graph which uses pictures [in order] to show infor-

mation.’The category comprised 19 clauses and amounted to 12.6% of the data

set.Category 5: Providing the purpose or intention of the text for the child.This category of use was extra-mathematical. The clauses provided the

purpose of text in terms of its learning aims and objectives and could bedescribed as metadiscourse. They were not common. Two clauses werenoted, amounting to 1.3% of the data set.

For example:‘On these two pages are tips in order to help you multiply.’Clauses of cause were much less common than clauses of purpose and

were mainly used for:Category 6: Providing reasons underpinning mathematical assertions

for the child.This category of use was mathematical. These clauses attempted to

justify assertions and comprised the largest category of causal clauses (al-though it was small when compared with the major categories above). Itcomprised 14 clauses and amounted to 9.3% of the data set.

For example:‘The number will be 2 because 2×1 = 2.’‘It is a square number because 5×5 = 25’.Category 7: Providing reasons underpinning extra-mathematical asser-

tions for the child.This category of use was extra-mathematical. The clauses justified asser-

tions of a non-mathematical nature although these were allied to the topicunder study as in, for example, an analogy. This small category comprisedtwo clauses and amounted to 1.3% of the data set.

For example, in an account of an analogy was:‘You can see how the machine below works because the machine’s

functions are clearly labelled on the outside’.

3.3. Overview

To summarise, the results indicate little explicit concern for purpose orcause in some of the books. Of those that offered reasons, more attention


was given to purpose than to cause. Where reasons were provided, theywere mainly extra-mathematical and related to the purpose of games andactivities and to contexts and real-world examples to support the teachingof the subject. In addition, there was a small amount of metadiscourse de-scribing the purpose of the text and some causal justification of assertionsin accounts of a contextual nature. These categories together amountedto 64.6% of the purpose and cause clause data. Of text which expoundedmathematics more directly, the purpose of procedures, operations and al-gorithms, and reasons underpinning the use of language, symbols and rep-resentations received some attention. In total, these amounted to 25.9% ofthe data set. Clauses providing reasons for assertions of a mathematicalnature (Category 6) amounted to 9.3% of the data set.

The categories derive from the full data set but, of course, categorypresence varies from book to book. For individual books, that presencewas generally like the pattern indicated above. For example, the majorityof cause and purpose clauses in Books 1, 2 and 14 provided the purposeof games, activities, contexts and examples (68.2%, 66.8%, and 72.3%,respectively). These books also included clauses providing the purpose ofprocedures, language, symbols and representations (27.3%, 16.7%, and18.2%, respectively). They included relatively small numbers of clausesproviding reasons for mathematical assertions (4.5%, 11.1%, and 9.1%,respectively). There were, however, exceptions. For instance, no clausesof cause or purpose were found in the samples from Book 18. In Book 5,on the other hand, 39.6% of cause and purpose clauses gave the purposeof mathematical procedures, language, symbols and representations, and31.2% of the cause and purpose clauses provided reasons for mathematicalassertions.

DISCUSSION

We do not claim understanding reasons is all that matters in mathematics.There is more to mathematics than that. Nor do we say that understandingreasons rests entirely on the explicit provision of clauses of cause or pur-pose or on asking for them. Counts of these clauses can, however, serve asgeneral indicators of that support (Graesser and Hemphill, 1991). Further,the explicit provision of reasons is potentially useful for a teacher wholacks confidence in his or her own grasp of mathematics or ability to teachit. It both reminds the teacher to give them attention and it facilitates dis-course with the children with less conversational risk. From the children’spoint of view, if their teacher does not know the reasons for what theydo and all else fails, what can they do? Again, their textbook could help.


Nevertheless, choosing a book that could help is not simple. Two bookswith the same numerical scores (like 3 and 4, or 10 and 11 in Figure 1)may, in fact, concern themselves with different kinds of reasons. Directexamination of the books is essential if the figures are to be interpretedmeaningfully.

Could these textbooks help teachers talk about and press for reasons inmathematics in the primary school? As far as the explicit provision of rea-sons is concerned some books, like 18 in figure 1, were likely to be of littlehelp. In many of the other books, there was evidence that some attentionwas given to reasons but this was largely in the provision of purpose inlearning activities, of purpose and cause to make contexts meaningful andof the intention of the text. This attention is extra-mathematical but thatis not to say it is worthless. When children understand the aims and thevehicles used to teach them, they are more likely to be motivated and tolearn (Newton, 1990). New teachers and those without a sound backgroundin mathematics are often weak at this (Bennett, 1993) so this could be ofconsiderable help to them (Newton, 1990). Many of the books also of-fered explanations of words, symbols and representations by stating theirpurpose. Similarly, some explained algorithmic procedures in this way.These could be described as mathematical explanation and again have thepotential to support understanding. The provision of extra-mathematicaland mathematical reasons of these kinds could remind a teacher to sup-port understanding and, being explicit, remind them what to say. But, forinstance, they do not help a teacher explain why an answer is correct ora mathematical assertion is sound. There was evidence that some books(like Book 5) attempted to be explicit about this although, overall, theattempt was uncommon. Nevertheless, it has to be remembered that, asstated above, explicit statements of cause and purpose are indicators ofa concern for reasons, not absolute measures of it. Reasons may be con-structed by a reader in the absence of such statements. However, it wouldbe easy to pick up a book which offered no or few explicit reasons thatunderpin the mathematics being taught. We see such an absence as a signalto new or non-specialist teachers that knowing reasons is not a matter forconcern.

These books, however, were not written for teachers, but for children.Activities for the children were intended to prepare them for the mathemati-cal objectives. Although this study did not consider the quality of expositionand its likely effect, some of these activities could help the children graspwhy mathematical operations and procedures work as they do. A teacherwho knows the mathematics that is targeted by these activities could makea lot of them. Beyond those activities, however, the books would do little


to point a new teacher or one without a sound foundation in mathematicstowards addressing reasons for patterns and procedures, even when Cate-gory 6 clauses are taken into account. An experienced and knowledgeableteacher probably could use such books to good effect; others are unlikelyto go beyond the book and put themselves at conversational risk (Carlsen,1991; Bennet, 1995). In general, the implicit message in the books is thatmastery of computation is what counts. Of course, it does count but thatmessage is likely to reduce any inclination to ask for explanations of, forexample, children’s methods and reasoning and for mathematical patterns(DfEE/QCA, 1999, pp. 60–75).

These books were generally not accompanied by guides for the teacher.Such guides could, of course, be useful but they remain a step away fromthe classroom. A textbook for the children can take the extra step thatis needed to express the lesson in learner-friendly terms in a teacher’svoice (Pepin and Haggarty, 2001; 160). This step bridges the gap betweenmathematics (the discipline or subject) and mathematics as it is taught,taking the words straight to the point of use. Lucas (1996) suggested thatauthors should be seen as colleagues. But, like all colleagues, some arebetter models than others and some are better at some things than others.Some books made provision in some aspects of learning but not in others.A teacher may need to use more than one book to build up a picture ofthe kinds of understanding expected and how they might be addressed.This is where the ‘tool’ they need is the skill to choose and use a bookto support particular kinds of understanding. That skill could be devel-oped through exercises such as those described by Brown (2001). The firststep, however, is to ensure that teachers know what counts as understand-ing in elementary mathematics and what is being targetted and supportedat any one time. New teacher, non-specialists and experienced specialistsmay not work equally well with the same book. The practice in someschools is to use a series of books throughout the school. This uniformitylends itself to progression and coverage but it does not allow for differ-ent teacher needs and preferred ways of working. One size may not fitall.

CONCLUSION

Knowing why in mathematics is valuable. With it, knowledge can be moredurable, more powerful and easier to acquire. It is what lifts mathemat-ics from facts, rules and recipes for successful computation to a body ofknowledge to think about and think with. Of this knowledge, knowing why


an assertion is sound, why a procedure works, why an answer is correct,for example, can be particularly valuable. We conclude that, as modelsof practice, most of these books are unlikely to help new teachers, oneswithout a sound background in mathematics, or those lacking confidencein their subject knowledge or teaching skills to give attention to these kindsof understanding. Publishers and writers need to give more attention to thiskind of understanding and take into account the potential of a textbookas a model for teaching, at least in the UK. Teacher trainers should helpteachers acquire conceptions of mathematics that encompass both causeand purpose and give them the tools to exploit classroom resources withthe potential to support understandings based on these.

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DOUGLAS P. NEWTON AND LYNN D. NEWTONSchool of EducationUniversity of DurhamLeazes Road, Durham, DH1 1TAEngland, U.K.Telephone: +441913348379E-mail: [email protected]

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Could Elementary Mathematics Textbooks Help Give Attention to Reasons in the Classroom?