Primary teachers’ epistemological beliefs: some perceived barriers to investigative teaching in primary mathematics

Educational StudiesVol. 34, No. 5, December 2008, 419–432

ISSN 0305-5698 print/ISSN 1465-3400 online© 2008 Taylor & FrancisDOI: 10.1080/03055690802287595http://www.informaworld.com

Primary teachers’ epistemological beliefs: some perceived barriers to investigative teaching in primary mathematics

David S. Bolden* and Lynn D. Newton

School of Education, University of Durham, UKTaylor and FrancisCEDS_A_328926.sgm10.1080/03055690802287595Educational Studies0305-5698 (print)/1465-4300 (online)Original Article2008Taylor & Francis0000000002008Mr. [email protected]

A recent investigation of primary teachers’ epistemological beliefs concerning theteaching and learning of mathematics discovered that teachers’ beliefs cannot be said toform neat world views. Teachers’ hybrid world views often included epistemologicalbeliefs that supported teaching approaches which evidence suggests leads to greaterconceptual understanding of mathematics. Classroom observations and semi-structuredinterviews with primary teachers suggested that although there is a desire to adopt aninvestigative approach, this is perceived to be largely incompatible with some of therequirements of the UK National Curriculum. Common potential barriers identified byteachers included: the volume of curriculum content they are required to cover, thelimited time available to cover it, some working practices perceived to be associated withthe current emphasis on teacher accountability and the current method of assessment byStandard Assessment Task tests (SATs). The findings are discussed in relation tochallenges facing UK policy-makers if an approach to teaching primary mathematics,which is known to support conceptual understanding, is to flourish.

Keywords: teacher epistemologies; epistemological beliefs; mathematics; barriers toeffective teaching; investigative approach; primary mathematics

Introduction

Epistemological beliefs

Epistemology is defined by Hofer and Pintrich (1997, 88) as: “an area of philosophyconcerned with the nature and justification of human knowledge.” An epistemologicalbelief is what an individual holds to be true about such knowledge. Schommer (1990) hassuggested that individuals may hold several independent epistemological beliefs about anaspect of knowledge. Schraw and Olafson (2002) have coined the term epistemologicalworld view to refer to the personal construct that is constituted by a wider set of individualepistemological beliefs, held by someone concerning the nature and acquisition of knowl-edge. They define an epistemological world view as: “a broad intellectual perspective thatserves as a lens to see the world that transcends individual beliefs about knowledge”(Schraw and Olafson, 2002, 104). Such beliefs and world views have been shown toimpact on teaching practices in mathematics (Lerman 1983; Thompson 1984; Marks 1987;Dougherty 1990; Askew et al. 1997).

Research concerning teachers’ epistemological beliefs lends some support to the exist-ence of three broadly different epistemological world views; the realist, the contextualistand the relativist world views (Kuhn 1991; Prawat and Floden 1994; Cunningham andFitzgerald 1996; Fielstein and Phelps 2001; Kincheloe, Slattery and Sterberg 2001; Schraw

*Corresponding author. Email: [email protected]

420 D.S. Bolden and L.D. Newton

and Olafson 2002).1 The realist world view assumes that knowledge is absolutist. Itassumes that there is a direct, one-to-one relationship between epistemology and ontology,that is there exists independently of the knower an objective and unchanging body of knowl-edge. This epistemological view can be linked to a behavioural model of the teaching andlearning of mathematics (Burton 1994) as when it is assumed that knowledge is bestacquired via a teacher-centred approach using the “transmission” of facts from the expertteacher to the passive learner. Within such a model teachers might focus on proceduralrather than conceptual understanding and employ drill and practice as ways of transmittingknowledge. For example, teaching pupils to calculate the area of a triangle could involvethe memorisation and application of the formula: area = 1/2 base × perpendicular height. Wemight also expect these teachers to prefer observable behaviours as ways to assess whetherlearning has taken place, through, for instance, norm-referenced and externally produced,standardised tests.

The contextualist world view assumes that there is no direct, one-to-one relationshipbetween epistemology and ontology and that knowledge is consensually agreed and sharedwithin communities and so is changeable over time. This epistemological world view canbe linked to a social-constructivist model of the teaching and learning of mathematics.Within such a model teachers might place more emphasis on conceptual understanding andact as facilitators in the classroom. For example, teaching pupils the area of a triangle heremay involve allowing the pupils the freedom to discuss and explore relationships betweenthis and, say, the area of a rectangle, as a way of better understanding the 1/2 in the aboveformula. Teachers might also be more concerned with the less obvious processes with whichpupils construct knowledge. Consequently, we would also expect them to use scaffoldingtechniques and to prefer an investigative approach to teaching and criterion-referencedassessment over the use of standardised assessment.

The relativist world view is very similar to the contextualist world view in its epistemol-ogy–ontology relationship. However, it differs from the contextualist world view, in that itassumes that each learner constructs a unique representation of that knowledge base. Thisworld view can be linked to a radical constructivist model of the teaching and learning ofmathematics. For the purposes of this research, however, the contextualist and relativistworld views were conflated because it was uncertain what difference holding a contextualistor relativist world view would have for a teacher teaching in the classroom (Prawat 2002).The conflated world view will be referred to as the relativist world view. It is consideredthat at any point in time a teacher’s world view is largely consistent with one or other of thedefinitions above and that hybrid positions are rare (Prawat and Floden 1994).

Understanding in mathematics

It would seem obvious to state that understanding in mathematics is important. Newton (2000)lists a number of reasons why understanding is important but perhaps chief among these arethat understanding can facilitate learning and produce flexibility of thought and action innovel situations. There is a growing body of research in mathematics to support these benefitsof understanding (Schoenfeld 1985; Silver 1985; Winkles 1986; Hiebert and Wearne 1996;Bransford, Brown and Cocking 1999). For instance, research by Winkles (1986) and Hiebertand Wearne (1996) showed that children exposed to teaching which emphasised understand-ing in mathematics were able to retain information for longer and be more flexible in theirsubsequent approaches when attempting to solve new mathematical problems.

Understanding can take many forms (Newton 2000). For instance, Ajdukiewicz (citedin Sierpinska 1994) defined understanding as an act of mentally relating the object of the

Educational Studies 421

understanding to another object. Sierpinska (1994) also views understanding as complex andmultifaceted, dependent on context and what it is that is to be understood. However, sheplaces at the heart of understanding the idea of mental order and harmony. Understandingin mathematics is often discussed in terms of conceptual and/or procedural understanding.Procedural understanding (computational understanding in the USA) is concerned with thelearner’s grasp of mathematical calculations, whereas conceptual understanding is concernedwith the grasp of the ideas underpinning those procedures. Understanding cannot be trans-mitted, but it requires the learner to make the necessary mental connections themselves.

An investigative approach

Space precludes an exhaustive account of an investigative approach to mathematics teach-ing, but for the purposes of this study, it is consistent with a social-constructivist model ofteaching (e.g., see Greeno, Collins and Resnick 1996). Social constructivism’s central claimis that human knowledge and learning is not passively received, but acquired through theprocess of active construction via experience (von Glasersfeld 1987). As such, a teacherusing an investigative approach allows pupils far greater freedom to develop and constructtheir own understandings of a concept or problem. The teacher negotiates rather thanimposes meanings and takes into account pupils’ previous knowledge and understanding,and presents learning activities which will allow pupils to reorganise their cognitive struc-tures which, in turn, extends and develops their knowledge and understanding. Teacherswill also tend to emphasise differentiation by outcome over differentiation by task as thisallows pupils a more open-ended activity where they take more control for their own learn-ing (Davis 1994). In essence, investigative approaches based on social-constructivist prin-ciples view learning as a personal and private activity but teaching as a public and socialone (Watts and Jofili 1998).

This approach is important because it can be a way of helping children make mentalconnections. There is now increasing evidence to suggest that an investigative stance inclassrooms by teachers can help their pupils gain a deeper understanding of mathematics(e.g., Woods and Sellers 1996; Woods 1996a, 1996b, 1999; Wilson and Cooney 2002) andthat allowing teachers to teach for understanding results in a variety of learning gains(Carpenter et al. 1989). For instance, some of the elements described above are present inthe effective teachers of numeracy identified by Askew et al. (1997). They identified effec-tive teachers of numeracy and described their approach to mathematics as connectionist,that is such teachers were aware that pupils arrive in the classroom already in possession ofmental strategies for calculation. They also emphasised the links between different aspectsof mathematics and believed that effective teaching was based on dialogue between teacherand pupil. Wilson and Cooney (2002, 134) suggest that: “evidence that understanding bystudents, defined as the ability to construct rational, meaningful solutions to meaningfulproblems, is enhanced by such an inquiry approach by mathematics teachers.”

Aims of the study

The aims of the study were to:

(1) Identify the nature of primary teachers’ epistemological beliefs about the teachingand learning of mathematics.

(2) Investigate the assumption that teachers’ epistemological world views are consistentwith only one world view as described by past research.


(3) Consider theoretical and practical implications of teachers’ epistemological beliefsabout the teaching and learning of mathematics.

Method

Data collection

Collecting data concerning teacher epistemologies can be difficult as teacher behaviour isnot always consistent with their stated beliefs (e.g., Cohen 1990; Schoenfeld 2002). Thisnecessitates a way of verifying the relevance of teachers’ expressed beliefs. One way ofdoing this is to use both in-depth semi-structured interviews and classroom observations.

Observations and semi-structured interviews

Lesson observations and semi-structured interviews with teachers were conducted concur-rently over six months between October and March of the school year. Approximately fourobservations and interviews were conducted with each teacher during this period (Wragg1997). In observations the researcher took a position at the rear of the classroom, behindthe children. During the lesson, notes were made concerning aspects of the teachingconsidered to be revealing about that teacher’s epistemological beliefs to be probed later ininterview (for instance, a particular approach to teaching or questioning). The authorsacknowledge the personal nature of what was deemed revealing. The interviews followedthe lesson observations and allowed teachers to reflect upon the rationale for particularexamples of their teaching while it was fresh in the memory. The questions asked weresimilar in the four interviews although they depended to some extent on what was observedin the lesson.

Questions were asked informally and probed teachers’ epistemological beliefs in twoways: by directly asking questions about specific teaching and learning issues, and byasking questions concerning specific instances of teaching observed. For example, teacherswere asked their opinion on current assessment procedures and about their rationale forusing particular teaching strategies – e.g. “talking partners”. A triangulation process on themeaning and interpretation of words in these sessions was used in an attempt to avoidpreconceived meanings and develop a shared vocabulary with teachers (Schoenfeld 2002;Speer 2005).

Sample

Three teachers volunteered to take part, so the sample was self-selected. Nevertheless, theteachers were considered to be broadly representative of Years 5–6 primary teachers (teach-ing children aged 9–11 years) in many English schools (see below). These teachers will becalled Lisa, John and Mary.

Lisa was 29 years old, had a Pure Mathematics degree and had gained all her six years’teaching experience in the same Roman Catholic primary school. Lisa’s school wasdescribed in its most recent Office for Standards in Education (OFSTED) report as smallerthan average with considerable deprivation in its catchment area. The school had a higherthan average number of pupils with learning difficulties and/or disabilities or who wereeligible for free school meals (a proxy indicator for low socio-economic status). Pupilmobility was higher than average. Comparatively few pupils were from minority ethnicbackgrounds or spoke English as an additional language. Children entered the school with


attainments which were considerably below what would be expected for their age but by thetime they left in Year 6, standards were above the national average.2

John was 32 years old and had been a teacher for eleven years. His current school wasdescribed by its most recent OFSTED report as an average-sized primary school. Mostpupils were of White-British origin and none had English as an additional language. Thesocio-economic indicators for the area were said to be broadly average but the percentageof children eligible for free school meals was described as below the national average.The attainment on entry was broadly average but at the end of Year 6 the standards werevery high.

Mary had been a teacher for 31 years. She had a teaching certificate but had no qualifi-cations in mathematics. Mary’s school was described by its most recent OFSTED report asa small primary school standing in an area of social and economic disadvantage. Theproportion of children eligible for free school meals was broadly average and none of itspupils had English as an additional language. Children’s attainment on entry was just belowaverage but, by the end of Year 6, children were achieving results above the national aver-age in all subjects.

Data analysis

Analysis of data from observations and interviews took the form of identifying emergingthemes from fieldnotes and interview transcripts using the constant comparison method ofGrounded Theory (Glaser and Strauss 1967). The approach is underpinned by two commit-ments: the constant comparison method, and theoretical sampling. The constant comparisonmethod leads to a “fracturing” and rearranging of the data into categories that facilitatecomparison between elements in the same category or between categories (Maxwell 1998).Pidgeon (1997, 78) suggests that: “By making such comparisons the researcher is sensitisedto similarities and differences as part of the exploration of the full range … of a corpus ofdata, and these are used to promote conceptual and theoretical development.” The categoriesthat emerged from the initial data inform the decisions made in the next phase. For example,analysis of fieldnotes from early observations suggested areas and aspects of teaching offurther interest, which then informed subsequent observations. This is an iterative processwhich blurs the traditional boundaries between data gathering and data analysis.

Theoretical sampling is the process of actively sampling new cases of interest in subse-quent phases of data gathering on the basis of what has been learned from earlier data anal-ysis. It is not driven by the requirement for representativeness like more traditionalquantitative research because such an approach would not be resource-efficient. Instead, thistype of sampling selects cases for their potential to extend or develop the emerging under-standing of the phenomenon under study (Pidgeon and Henwood 1997). Consequently, thecoding process involved identifying pieces of written text from fieldnotes or transcripts andnoting the potential theme (or code) in the margin. Fieldnotes from other observations ortranscripts were then studied to ascertain whether similar pieces of text could be assignedthe same code. Codes which repeated themselves were applied until “saturation” occurred– i.e. no further incidents were identified that could “fit” into previously identified codesand no further codes emerged. When codes were found not to repeat, they were discardedand other codes sought. Sometimes codes initially identified did not exactly fit the variouspieces of texts which fell within that code. When this occurred, the code was refined slightlyso as to better fit the instances of text.

Tentative analysis of the interview transcripts started as soon as possible after the inter-views had been transcribed, but only after receiving transcripts from the teachers with their


respondent validation. This allowed the early identification and coding of potential catego-ries while the interviews were still fresh in the memory. These categories subsequentlyprovided a working conceptual framework for the identification of new incidents. A numberof main codes (and sub-codes) emerged: content to be covered; time available; assessmentby SATs; and some current working practices.

Results

Epistemological world views: hybrids

Interviews with teachers showed that they could not be characterised by only one of theworld views described above. That is, teachers appeared to hold hybrid epistemologicalworld views. This was evidenced by teachers expressing views that previous research hadshown to be characteristic of opposing epistemological world views. For example, John saidthat he disliked the publication of league tables as they were reported in the local andnational press. However, he agreed with the use of Standard Assessment Task (SATs) testsat the end of the key stage 2 education in Year 6 and the optional tests that almost allprimary schools use in the years preceding it James 2000). He said he liked the SATsbecause of the data they provided him concerning the learning of his pupils. Early on in thediscussions, John stated his thoughts on the current use of SATs:

J: I agree with SAT testing within class because I think it does give us an idea of where thechildren are at and it gives us an awful lot of data to look at where we are going wrongas teachers or where we are going right. But it tells us which area of our curriculum isweak, and which area we need to strengthen. But in terms of the publication of SATs,that’s where I disagree, because we then have a problem that all schools aim high or aimto achieve, but when they don’t achieve the teachers feel bad, the kids feel bad, theschools drop in the league tables, and when they drop in the league tables, parents don’twant to send their children there. I’ve been in two schools that have been high in theleague table and you have parents phoning when they come out to try and register theirchildren in that school because it is seen as a good school. So there is too much emphasisplaced on SATs.

DB: But you also said that they give you a lot of valuable information about where the chil-dren are at?

J: Yes, they do. The optional testing we do gives us a lot of information.

John’s agreement with and acceptance of standardised testing in class was interestingbecause he had previously espoused a largely relativist world view concerning the teachingand learning of mathematics. John was unaware of the inconsistencies in his words here.That is, on the one hand he argued that he tried his very best to adopt a social-constructivistapproach to his teaching of mathematics, and that was evidenced first hand, but on the otherhand he seemed to place great store in the veracity of the information gleaned about hispupils’ learning from standardised tests that require standardised answers. This latter viewsits more easily with the absolutist epistemology of the realist world view and with the asso-ciated transmission approach to teaching and learning, discussed earlier. However, all teach-ers involved in this research had more elements to their world views that were relativist thanrealist.

Epistemological beliefs: perceived barriers to an investigative approach

Despite the hybrid nature of teachers’ epistemological world views, all teachers expressedthe belief that an investigative approach in mathematics can lead to greater understanding


in their pupils. During interviews, all teachers also expressed an enjoyment of teaching inan investigative way and most expressed a wish to teach this way more often. All teachersalso said that using this approach more often than they did was not possible given therequirements of the National Curriculum. These perceived barriers to adopting an investi-gative approach are discussed below.

Content and time available

An evaluation of the early implementation of the National Curriculum in England and Walesconducted as long ago as 1993 by Askew et al. identified the volume of content teacherswere expected to cover and the time they had to cover it as a major concern for teachers:“Comments from some interviewees indicated that they were doing less practical and inves-tigative work than they used to [before the introduction of the National Curriculum] becauseof the pressures of what had to be covered in the mathematics curriculum” (Askew et al.1993, 127).

More recent research continues to suggest that “coverage” takes precedence over“understanding” (Dadds 1994, 2001; Pollard and Triggs 2000). It seems that little haschanged in the intervening period. During the interview with John, he said how much hehad enjoyed teaching a Year 5 class (aged 9–10 years) earlier in his career because he hadfreedom to adopt a more investigative approach to his teaching:

J: My favourite year in teaching, which was a while ago now, when I look back that was awonderful year, there was lots and lots of practical stuff and I didn’t care about whetherit was recorded or … and the kids learned a massive amount, not just in terms of singularsubjects, but in terms of the interconnectedness of subjects as well because it could betruly cross-curricular. You know, we would look at one thing in science that would thenreflect in their English, which would then reflect in their history, which would thenreflect in their maths. I think they really got the interconnectedness, now because it is socompartmentalised, we are losing that interconnectedness.

And later in the same interview he said:

DB: Would you like to teach more like that? From what you said earlier about your reallygood teaching year?

J: Yes, I would prefer to have more … Given that we had more space, more time, giventhat there wasn’t an expectation every day to have something in the book which resem-bled work and which was monitorable, an actual answer, something to tick or somethingto mark wrong, something that that child has achieved or that child hasn’t achieved, yesI would love to but all that would have to disappear first.

This notion that some of the requirements of the National Curriculum were restricting theway in which teachers wanted to teach mathematics was pursued by asking Mary how manytimes in the year she was able to teach in an investigative way:

M: About five, but I would like to do more but time does not allow it.DB: Is that because of the requirements of the National Curriculum?M: Yes, it means we have to get through the material in a given amount of time.DB: Would you like to teach that way more often? If all the constraints were taken away,

would you?M: Yes, I would. It’s a wonderful thought that, taking all the constraints away and teaching

the way you think teaching should be done.

The same notion was pursued with Lisa:


DB: How many times per year are you able to do that type of investigative teaching?L: I tend to do a lot more in the summer because then SATs are finished. We do transition

units to go to secondary school and a lot of the work we do for them is investigative work.

There is an implication in Lisa’s words that she did not have the freedom to teach in aninvestigative way more often before the SATs because the pressure of time was too great.This led to the following exchange:

DB: You said you do a lot more in the summer because then the SATs are out of the way.L: Yeah, and you can take up more of the timetable with investigations.DB: So, before the SATs, there is less time to do that type of investigative teaching? That

open-ended type of teaching?L: Yeah, and especially because it is open-ended and from one lesson to the next you have

to see which way the children are going before you do the next and it could go on for awhile.

Some aspects of accountability

The curriculum in the UK during the past 30 years has become more centrally prescribedand there is now a much greater emphasis on accountability of schools and individual teach-ers (e.g., Elliott and Elkins 2004). Recent evidence suggests that accountability systems canlead to the deskilling of teachers and a narrowing of the curriculum as delivered in the class-room (McNeil 2005).

Teachers involved in this study said the view that teacher accountability and the way inwhich teachers’ work is monitored restricts their opportunities to adopt an investigativeapproach to the teaching and learning of mathematics. Lisa described it thus:

L: Yeah, because you have work scrutinies and things where they get books and say, “Hasthis topic been taught? Has this topic been taught? Has this topic been taught?” And thenthey can come and say, “This topic hasn’t been taught”….

John, too, alluded to this problem:

J: Erm, through them, yes. In terms of what they have actually produced, no … if we hadhad time at the end, which we didn’t have, then we could’ve got them to write downsomething, something they had discovered from today and that would’ve then been suffi-cient for me to be happy with what they produced. At the moment what we’ve got is acollection of sheets that, some of them have got a rule written on it which is accurate,others are still in the midst of things.

DB: Why is it important that they write something down? Is it so they can go back and lookat it?

J: No, to satisfy the “powers that be”.DB: Because they come and look at the books?J: Yes, they would be asking questions like, “What did the children do on this day?” Why

have you got a day missing?” If I say, “we did a practical activity that day”, they ask,“where’s the evidence for that practical activity?” It’s wrong. It is wrong.

The degree to which this particular manifestation of accountability was perceived as abarrier varied from teacher to teacher, and it is likely to be linked to years of experience.For instance, Mary attributed her laissez-faire attitude regarding strict record keeping to hermany years of teaching experience:

DB: Does the fact that there is nothing much in their books at the end of it all make youanxious or uncomfortable?


M: No, because I am capable of justifying my reasons for doing it this way. It might havedone in the past when I was younger and less experienced, but I’m now experiencedenough to not to let that bother me. I could justify it and set out my objectives for thelesson, etc.

Assessment by Standard Assessment Task tests (SATs)

Assessment within the National Curriculum amounts to a quantitative measurementapproach to learning within which assessment simply estimates the degree to which astudent has acquired the knowledge from a teacher by taking a sample of that knowledge.This seems at odds with the idea that children learn by constructing their own understand-ings. Shepard (1991, 2002) argues that this measurement approach to classroom assessmentreflected by standardised tests and teacher-made emulations of those tests is essentiallyincompatible with the central ideas of constructivism and only serves as a barrier to theimplementation of the ideas of constructivism, of which an investigative approach can beviewed as central. As long ago as 1990, Lerman (1990) was arguing:

Clearly, new ways of learning call on new ways of assessment and interpretation of “ability”.Such very different directions as those … focusing on the child’s constructions, which ofnecessity originate in the understanding that children bring into the classroom, cannot beassessed by the usual traditional methods which, in general, examine children’s grasp of thingstaught by the teacher rather than the children’s understanding … Yet we adhere to this modeof assessment of children’s mathematical ability. (Lerman 1990, 60)

This scientific measurement approach is more consistent with earlier forms of the curriculaand their associated beliefs about learning – e.g. rationalism and behaviourism.

From interviews a conclusion was forming that some of the requirements of the NationalCurriculum were incompatible with an investigative approach to the teaching and learningof mathematics and that assessment could be legitimately added to the list of barriers:

DB: If we agree that children learn best by discovering things for themselves, then doesn’t itfollow that assessment needs to be tailored to each individual child to see what they havelearned?

L: Yes, but you can’t do that for every single topic or for every single….DB: Why not?L: Because you would spend your entire time assessing each child, and you would never be

able to move anybody else on.DB: So you haven’t got time to do that?L: Not on every single topic….

John too suggested that assessment by SATs was inconsistent with an investigativeapproach to the teaching and learning of mathematics. This was surprising because he hadpreviously said he liked the SATs for the information they provided him concerning hispupils’ learning. In discussing SATs, he wondered whether it was important that the testsshould be standard, in the sense that each child receives the same test:

DB: I’m just trying to figure out how SATs as a way of assessment fit in with this idea ofconstructivist teaching and learning, given that ….

J: I don’t think it does [laughs].DB: Why?J: … Because children aren’t really guiding their own learning, we are guiding their learn-

ing for them. If, once they had done the SATs test, the children could identify the areasfor improvement themselves.


DB: What type of assessment would be consistent with it, then? What could we do? If weagree that children learn best by constructing their own conceptions of things …

J: Well, ideally, it would be like a portfolio-based assessment, that’s the standard answer.

Discussion and conclusion

Earlier, reference was made to research which described pure epistemologies, but this studyhas shown that contrary to these findings, these teachers did not hold pure, self-consistentepistemologies. Instead, they were found to hold hybrid epistemologies, that is contradictorybeliefs, which contradict Prawat and Floden’s (1994) rarity view. It is also clear that theseteachers valued and enjoyed the opportunities to teach primary mathematics using an inves-tigative approach, expressing a desire to teach like this more often. This value and enjoymentstem, at least in part, from their capacity to allow children the freedom to explore mathe-matical ideas and the perceived potential for increasing understanding in mathematics, asdefined above, in the children that experience them. Teachers’ comments also hinted at theintrinsic motivation they feel in adopting an investigative approach to teaching mathematics.

It is also clear that these teachers perceive some of the requirements of the currentNational Curriculum to be largely incompatible with an investigative approach to teachingmathematics. Given the research evidence suggesting that such an approach to teachingleads to greater mathematical understanding in children (Carpenter et al. 1989; Woods andSellers 1996; Woods 1996, 1999; Wilson and Cooney 2002), this represents a seriousconcern for anyone interested in mathematics education. Problem-solving approaches asencouraged in the Primary National Strategy (OfSTed 2003) and personalised learning asunderpins the Every child matters agenda (DfES 2004) would both require the considerationof such broader investigative approaches.

Yet, time, the amount of content to be covered, current working practices and SATs werecited as barriers. Past research suggests “coverage” is taking precedence over “understand-ing” (Pollard, Muschamp and Sharpe 1992; Dadds 1994, 2001; Pollard and Triggs 2000).The evaluation of the early implementation of the National Curriculum in England andWales by Askew et al. (1993) also suggests these are long-standing concerns for teachers.Teachers perceived some current working practices to be a barrier to adopting an investiga-tive approach to teaching mathematics. One working practice, in particular, the auditing ofwhat teachers have taught over the academic year by their local education authority wasviewed as restrictive. This particular form of accountability places emphasis on correctanswers at the expense of understanding and therefore an investigative approach to teaching.

This sample is small and self-selected, and it is possible that there are teachers with simi-lar world views but who do not feel constrained by the same requirements of the NationalCurriculum. While it is also acknowledged that there are likely to be some teachers whohold a predominantly realist epistemological world view concerning the teaching and learn-ing of mathematics, it is thought unlikely that such teachers would place the same value onan investigative approach to its teaching. Nevertheless, the teachers described here are typi-cal of many teaching mathematics in primary schools in England. Consequently, someteachers will be able to relate the findings described here to their own background and expe-rience and will find a resonance with the views expressed (Bassey 1981).

Implications of the research

What, then, are the major challenges to policy-makers if an investigative approach to theteaching of mathematics is to be allowed to flourish? Essentially, this research suggests the


need for changes to the National Curriculum which remove the barriers discussed and freeteachers to teach more often in a way that they enjoy and prefer, and which have also beenshown to produce greater conceptual understanding in pupils. For instance, this researchreinforces the belief that the current form of assessment in the UK’s National Curriculumraises serious problems and that there is a need for a revision of the way in which we assesschildren within it (Shepard 2002). Schools and teachers are acutely aware of the high stakescurrently attached to SATs, so it is not surprising that teachers place greater emphasis onchildren’s procedural rather than conceptual understanding since this is exactly what SATscurrently require of children. The assessment system needs to change to recognise thepotential benefits of an investigative approach, that is a system that looks for understandingto a greater degree (Newton 2000). Some alternative assessments which have been devisedin the USA may have something to offer here. These include larger projects that may takeseveral days or weeks to complete (Greeno, Collins and Resnick 1996).

Having such a revised form of assessment would have other benefits for teaching forunderstanding using an investigative approach. Such assessment does not require that teach-ers teach quite so much content. It allows more time to adopt an investigative approach.Teachers’ awareness of their own epistemologies and the potential inconsistencies withinthem are likely to result in a greater reflection on their teaching of mathematics. This canbe achieved through discussion with other teachers but the size of many primary schoolsmeans that this is likely to require the governing education authority to take a leading andorganising role in establishing teacher forums. It is perhaps best viewed within the frame-work of continuing professional development (CPD) courses offered to teachers by theirgoverning authorities.

However, implementing such changes will not always be quick and easy and a greaterevidence base is required to persuade those in positions of authority to make the changesnecessary. On a positive note, it is pleasing to see that this evidence base is alreadybeginning to grow. Recent research from the USA investigated teachers’ commonlyquoted barriers to the use of teaching approaches that emphasise conceptual understand-ing. Desimone et al. (2005) reviewed the evidence underpinning the barriers to the adop-tion of such an approach, including its perceived incompatibility with large class sizes,less able pupils and more algorithmic strategies, as well as the resulting requirement ofhigher teacher mathematical qualifications. In each case, they presented evidence tosuggest that these are misconceptions and that teaching for understanding in mathematicseducation can be achieved.

Notes1. Some of these researchers use different terminology but the central ideas are broadly the same.

For instance, Prawat and Floden (1994) distinguish between the mechanistic, organismic, andcontextualist world views but these match the assumptions of the realist, relativist, and contextu-alist world views respectively.

2. This and later references to OFSTED may be found at http:\\www.osfted.gov.uk/reports/ butspecific details are omitted to protect anonymity.

Notes on contributorsDavid Bolden joined Durham University’s School of Education in December 2006 as a researchassociate. Previously, he worked as a lecturer and researcher at Northumbria University for six years.He has recently completed his doctoral thesis focusing on primary teachers’ epistemological views ofmathematics. His research interests include teacher epistemologies, the use of performance data inschools and teacher recruitment and retention.


Lynn Newton is a specialist in primary education, and primary science in particular. Having workedin schools in north-east England as a teacher and as an advisory teacher for science, she moved to theUniversity of Newcastle School of Education in 1987. While there she completed her PhD, focusingon mental model theory and teaching for understanding in primary science. She moved to DurhamUniversity in 1997 where she is professor of Primary Education. She has published widely on primaryscience, teaching and learning in the primary school and primary initial teacher training. She iscurrently Head of the School of Education at Durham. Her research interests include teaching forunderstanding and effective communication strategies.

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Documents

Primary teachers’ epistemological beliefs: some perceived barriers to investigative teaching in primary mathematics