Conference PaperPDF Available

It depends on the students: Influencing teachers' beliefs about the ends and means of numeracy teaching'

Authors:

Abstract and Figures

This paper reports on the impact of a brief professional learning program for K-8 teachers of mathematics, on teachers' beliefs about effective numeracy teaching strategies and appropriate goals of numeracy teaching, for students with mathematics learning difficulties and for students generally. Evaluation data indicated that the teachers finished the program less inclined to espouse differing beliefs in relation to the two types of students, and that their final beliefs were more in line with the aims of the program. Numeracy is accepted as having its foundations in mathematics (Australian Education Council, 1990) and is typically defined as also involving affective dimensions and the ability to use mathematics in everyday life (Australian Association of Mathematics Teachers, 1997). More recently, the Tasmanian Department of Education (DoET) (2002) stated that numeracy; requires the knowledge and disposition to think and act mathematically and the confidence and intuition to apply mathematical principles to everyday problems. … it also involves the critical and life-related aspects of being able to interpret information thoughtfully and accurately when it is presented in numerical and graphic form (p. 21).
Content may be subject to copyright.
137
It Depends on the Students: Influencing Teachers’ Beliefs About the
Ends and Means of Numeracy Teaching
Kim Beswick
University of Tasmania
<Kim.Beswick@utas.edu.au>
This paper reports on the impact of a brief professional learning program for K-8 teachers
of mathematics, on teachers’ beliefs about effective numeracy teaching strategies and
appropriate goals of numeracy teaching, for students with mathematics learning difficulties
and for students generally. Evaluation data indicated that the teachers finished the program
less inclined to espouse differing beliefs in relation to the two types of students, and that
their final beliefs were more in line with the aims of the program.
Numeracy is accepted as having its foundations in mathematics (Australian Education
Council, 1990) and is typically defined as also involving affective dimensions and the
ability to use mathematics in everyday life (Australian Association of Mathematics
Teachers, 1997). More recently, the Tasmanian Department of Education (DoET) (2002)
stated that numeracy;
requires the knowledge and disposition to think and act mathematically and the confidence and
intuition to apply mathematical principles to everyday problems. … it also involves the critical and
life-related aspects of being able to interpret information thoughtfully and accurately when it is
presented in numerical and graphic form (p. 21).
The Australian Government remains committed to the goal of ensuring that all students
achieve acceptable levels of numeracy (Department of Education, Science and Training,
2004), and national numeracy policies have acknowledged that some students require, and
should receive, additional support for this goal to be realised (Department of Education,
Training and Youth Affairs, 2000). Similarly, in Tasmania, it has been recognised that
recent and ongoing curriculum reform must include access to a broad, rich and challenging
curriculum for students with special and/or additional needs (Atelier Learning Solutions,
2004). The Atelier Report (2004) noted that, for a variety of reasons, commitments to
equity and inclusion at a policy level are not necessarily translated into practice in
classrooms. This study represents an initial step towards elucidating the role in this of
teachers’ beliefs about students experiencing difficulty learning mathematics compared to
other students.
Mathematics Learning Difficulties
In 2002 Baker commented on the relative dearth of research on effective mathematics
teaching for low achieving students. Nevertheless there is a body of literature, grounded in
a psychological perspective, that details efforts to identify the causes of learning difficulties
in mathematics. Prominent in this arena is Geary (2004) who defined the term
“mathematical learning disabilities” as applying to students whose mathematics
achievement over successive years is substantially lower than expected on the basis of IQ.
Other researchers have used terms including “developmental dyscalculia” (Kaufman,
Handl, & Thony, 2003), “mathematical disabilities” (Keeler & Swanson, 2001), and
“arithmetic learning difficulties” (Micallef & Prior, 2004) for similarly defined constructs.
Estimates of the prevalence of these conditions fall between 3% and 8% of school children
138
(Geary, 2004; Kaufman et al., 2003) and the students affected tend to use immature
calculation strategies for longer than other children (Geary, 2004; Torbeyns, Verschaffel, &
Ghesquiere, 2004), take longer to learn mathematical procedures, and have chronic
difficulties retrieving basic facts (Micallef & Prior, 2004).
In addition to procedural and memory difficulties, Geary (2004) described a third
subtype of mathematical learning disability characterised by difficulty in using and
interpreting spatial representations of mathematical material. He acknowledged that
relatively little is known about this subtype and it is unclear to what extent difficulties with
tasks with a significant visual component are in fact due to other procedural and/or memory
deficits. Geary (2004) observed that students with other types of mathematics learning
disabilities appear to have spatial abilities comparable to those of other children. Evidence
from other sources (e.g., Bobis, 1996) suggests that visualisation has an important role to
play in the development of children’s number sense and fact retrieval.
Studies of efforts to ameliorate memory difficulties include that of Tournaki (2003)
who compared the effectiveness of drill and practice with that of strategy instruction in
relation to basic addition facts, for students generally and with learning disabilities. She
found that both interventions were effective for general students, but that only strategy
instruction led to improvements for students with learning disabilities, and that only
strategy instruction led to improvements for both groups of students in relation to tasks that
required the use of basic facts in more complex calculations. Keeler and Swanson (2001)
suggested that strategy instruction in relation to remembering may also be helpful for
struggling students. These findings lend weight to assertion of Aubrey (1993, cited in
Robbins, 2000) that “the majority of children identified as having special needs require not
specialist teaching but good, high quality and effective teaching.” (p. 55).
In this study the term “mathematics learning difficulties” (MLD) was used and its
meaning negotiated with participating teachers to apply to the 10% or less of students who
experience greatest difficulties with mathematics. Since the study was concerned with
teachers’ beliefs (defined as anything they held to be true) about students, the IQ of
students was not considered. It is likely, therefore, that the students the teachers had in
mind as they participated included some with low IQ. Despite this difference from most of
the studies cited above, discussions with teachers throughout the project indicated that the
students of concern to them exhibited the kinds of difficulties described in the literature.
Specifically these students were substantially behind other students in their mathematical
development, had difficulty learning procedures and were still struggling with recall of
basic facts at the end of primary school and beyond.
Teachers’ Beliefs
Teachers’ beliefs have long been regarded as critical to the reform of mathematics
education (Cooney & Shealy, 1997) and the ineffectiveness of reform efforts has been
attributed to failure adequately to address them (Battista, 1994). The gap between policy
and practice identified by the Atelier Report (2004) may well be attributable, at least in
part, to a disjunction between policy and the beliefs of teachers that underpin their practice.
Very little has been written about the teachers’ beliefs in relation to students who
experience difficulties in learning mathematics, but teachers’ disagreement with inclusion
policies generally have been documented (e.g., Coates, 1989, cited in Shade & Stewart,
2001) There is also evidence that high teachers’ expectations of students in relation to
academic tasks are associated with improved achievement (Schoen, Cebulla, Finn, & Fi,
139
2003). Such expectations are likely to be underpinned by positive beliefs about student
capabilities which Beswick (2004) found, even for a teacher with beliefs broadly consistent
with a constructivist view of mathematics learning, to be associated with classroom
practice more aligned with recent and ongoing mathematics education reform efforts. This
study sought to measure the extent to which the participating teachers held differing views
concerning appropriate teaching approaches and goals of numeracy teaching for students
whom they perceived as having difficulty with mathematics and students more generally.
One element of Green’s (1971) description of belief systems related to the observation
that beliefs may be held either on the basis of evidence or for non-evidential reasons such
as the perceived authority of the source of information, or the fact that a particular belief
fits with other centrally held beliefs. Evidentially held beliefs are, by definition, susceptible
to change in the light of contrary evidence. The professional learning program that formed
the context of this study aimed to present teachers with evidence likely to challenge
negative beliefs about the capabilities of students with MLD and the appropriateness for
these students of innovative curricula, such as that being implemented in Tasmania (DoET,
2002), that emphasise the importance of deep understanding. Consistent with the literature,
visualisation and strategy instruction aimed at conceptual understanding were emphasised.
The Study
The study comprised part of the evaluation of a professional learning program aimed at
improving the numeracy education of students with learning difficulties. The program
consisted of three spaced half days of interactive workshops and was based upon the
following beliefs concerning mathematics/numeracy:
1. All students are entitled to a rich, broad and challenging mathematics curriculum
(Atelier Learning Solutions, 2004).
2. All students are able to learn mathematics (Ollerton, 2001).
3. A belief that mathematics makes sense is an essential part of being numerate (Van
de Walle, 2004).
4. All students should experience mathematics teaching aimed at the development of
deep conceptual understanding.
In addition the program was designed, to the fullest extent possible, to embody
characteristics of effective professional learning including: meeting the immediate
perceived needs of participants (Atelier Learning Solutions, 2004); addressing both
teachers’ practice and beliefs (Wilson & Cooney, 2002); relating theory and practice, and
including an expectation that participants would trial new ideas in their classrooms
(Guskey, 1995); and providing opportunities for teachers to share ideas and experiences
(Franke, Carpenter, Levi, & Fennema, 2001).
The program began with discussions aimed at eliciting the teachers’ beliefs about the
needs and capacities of students with MLD in relation to numeracy, and about the
appropriateness of innovative curricula for various students. The responses of the
participants to questions relating to these issues determined the specific content of the
program. The program thus provided participants with specific ideas relating to teaching
mathematics topics that they considered problematic yet crucial to the development of
numeracy, as well as opportunities to discuss a range of issues related to the program’s
aims. The topics and issues nominated by the participants and addressed (however briefly)
by the program are shown in Table 1. Asterisked items were treated in somewhat more
140
detail than the others and many of the issues raised were recurrent themes in the teachers’
discussions as various topics were addressed.
Table 1
Topics and Issues Addressed by the Program
Topics Issues
Place value*
Mental
computation*
Visualisation*
Time
Money
Fractions*
Estimation
Decimals*
Rational
numbers
What survival life skills do students need?
When to use which concrete materials
Mathematical thinking and reasoning*
Developing metacognition*
Meaning of numeracy – application
Affective responses of students
Time implications of teaching for understanding/with concrete materials*
Retention of knowledge*
Repetition of content
Engaging older students with concrete materials
Promoting student autonomy
Role of language in mathematics
Encouraging students to verbalise their thinking*
Moving from concrete to abstract reasoning
The place of calculators/spreadsheets
In each of the second and third sessions several teachers brought along examples of
relevant materials and programs that they had found useful in the past. Text-based
resources were made available for all to inspect and discuss at specific times during the
workshops, while a set of Linear Attribute Blocks (Stacey, Helme, Archer & Condon,
2001), useful in facilitating the development of students’ understandings of decimals, that
one participant had made were demonstrated and discussed in some detail. Many teachers
implemented ideas from the earlier sessions with their classes and reported on these
experiences in subsequent sessions. A set of six readings was also provided as a further
stimulus to discussion.
The evaluation of the program included an examination of the extent to which teachers’
beliefs about appropriate goals of mathematics teaching and approaches to teaching
mathematics, differed according to their perceptions of the students’ mathematics learning
abilities at both the beginning and end of the professional learning program.
Subjects
The 22 teachers who participated in the professional learning program were the
subjects of the study. Five identified as early childhood teachers, eight as primary teachers
and nine indicated that they taught middle school grades, meaning the lower grades of
secondary school in this context.
Instrument
The survey, Numeracy for Students with Mathematics Learning Difficulties (NSMLD),
comprised three sections, the last of which is reported on in this paper. This section
141
comprised 22 items, many of which had been used in earlier work on teachers’ beliefs (e.g.,
Beswick, 2003) concerning approaches to teaching mathematics and the goals of
mathematics instruction. Each item required responses on two five-point Likert scales, one
relating to students generally (labelled, ‘All students’) and the other relating to students
with mathematics learning difficulties (labelled ‘Students with MLD’). Responses were
scored from one for “strongly disagree” to five for “strongly agree”.
Respondents were asked to use a code name in order to allow the initial and final
surveys to be matched while preserving the respondents’ anonymity.
Procedure
Subjects completed the NSMLD at the beginning of the first professional learning
session and again at the end of the last. Ideally two versions, dealing respectively with
beliefs about students with MLD and students generally, would have been administered on
separate occasions but time did not allow this. As it was, teachers were fully aware of the
extent to which they were distinguishing between all students and those with MLD and this
may have reduced the differences reported.
Results and Discussion
There were statistically significant differences in relation to the two groups of students
for the items shown in Table 2.
Table 2
Items Eliciting Significantly Different Responses for All Students and Students with MLD
Mean
(all
students)
n=22
Mean
(Students
with MLD)
n=22
Mean
diff.
(All-
MLD)
Std
Dev. Sig. (2-
tailed) Effect
size
3. Conceptual understanding is an
appropriate goal of mathematics
students.
4.09 3.81 0.29 0.56 0.030* 0.51
3. Conceptual understanding is an
appropriate goal of mathematics
students.
4.43 4.24 0.19 0.40 0.042* 0.47
8. Students should not rely on concrete
material rather than thinking, for solving
mathematics problems.
2.05 1.64 0.41 0.67 0.009
**
0.61
11. Providing students with ‘survival’
mathematical skills is an appropriate
goal of mathematics instruction.
3.27 4.18 -0.48 1.15 0.001
**
0.79
*p<0.05.
**
p<0.01.
Higher mean scores indicate greater agreement with statement, and italics indicate
differences that were obtained on the second administration of the survey. The effect sizes
were calculated by dividing the mean difference by the standard deviation of the
differences to provide an indication of the relative size of the difference in means in
relation to the general variability of responses (Burns, 2000). The effect sizes obtained
142
were medium in the case of Item 3 at both administrations of the survey and medium and
large for Items 8 and 11 respectively.
The participants began the program significantly less inclined to see conceptual
understanding as an appropriate goal for students with MLD compared to students
generally. Rather, they regarded survival skills as more appropriate for these students and
were more inclined to see concrete materials as supporting answer getting, rather than the
development of understanding for these students. This is consistent with their
conversations in the first professional learning session about the problems such students
tend to have with retaining facts. There was still a statistically significant difference
between participants’ beliefs about the two groups of students in relation to conceptual
understanding as a goal, at the end of the program but Table 2 shows that the means had
increased for both groups and come closer together. The difference was slightly less
significant at the end of the program (p=0.42) than at the beginning (p=0.03). Both the
direction of the change and the convergence of the means are in accordance with the
principles upon which the program was designed. The very significant difference in
relation to Item 11 at the start of the program did not exist at the end, suggesting that
participants finished the program less inclined to believe that ‘survival’ mathematics was
the province of students experiencing difficulty learning mathematics.
Table 3 shows items for which there were significant changes from one administration
of the survey to the next, in relation to either all students or to students with MLD. In this
case items relating to students with MLD are italicised. Again effect sizes were calculated.
Table 3
Items Eliciting Statistically Different Responses at the two Administrations of the Survey
Initial
Mean
n=22
Final
Mean
n=22
Mean
diff.
(initial-
final)
Std
Dev. Sig. (2-
tailed) Effect
size
4. Telling children the answer is an
effective way of facilitating their
mathematics learning.
2.82 2.14 0.62 0.92 0.006** 0.67
4. Telling children the answer is an
effective way of facilitating their
mathematics learning.
2.77 2.10 0.62 0.92 0.006** 0.67
8. Students should not rely on concrete
material rather than thinking, for
solving mathematics problems.
1.64 2.10 -0.48 1.03 0.047* 0.46
21. Explicit teaching in mathematics
should focus on task requirements,
strategies, and highlighting significant
mathematical learning.
3.59 3.95 -0.38 0.80 0.0.42* 0.47
*p<0.05. ** p<0.01.
Following the program participants were less likely to believe that telling students
answers was an effective way of teaching them. The change was significant and the effect
size medium to large in relation to both students generally and those with MLD. Consistent
with this was the change in relation to participants’ opinions regarding what should be
made explicit in mathematics teaching for all students. Care was taken in the delivery of
143
the program to define explicit mathematics teaching in terms consistent with item 21 (see
Table 3) and not as prescribing procedures for solving problems or performing
calculations. The participants were also more inclined, after the program, to reject the
notion that students with MLD should use concrete materials as a substitute for thinking to
get answers.
Overall it seems the program had some success in influencing the academic
expectations of teachers in relation to students with MLD in ways likely to contribute to
their improved achievement (Schoen et al., 2003).
Conclusion
The results of this study need to be viewed with some caution due to the small number
of teachers involved and the brevity of the intervention. Nevertheless it provides some
evidence that teachers do hold differing beliefs about appropriate means and ends of
numeracy teaching for students depending upon their perceptions of the students’ ability to
learn mathematics. In particular, they are likely to regard a skills based curriculum focussed
on ‘real world’ survival, rather than one aimed at the development of deep conceptual
understanding to be appropriate for students with MLD.
In addition, they are more likely to approve of the use of concrete material for answer
getting rather than for supporting conceptual development, for students with MLD. This
illustrates the point made by Askew, Brown, Rhodes, Johnson and Wiliam (1997) that
superficially similar practice may in fact have quite different outcomes depending upon the
underlying beliefs of the teacher. It is certainly not sufficient to mandate particular
practices in hope of achieving real change in students’ learning.
It seems that the problem of translating policy concerning equity and inclusion into
classroom practice that was identified by the Atelier Report (2004) is at least partly due, in
the area of mathematics/numeracy, to beliefs that some teachers hold in regard to students
with MLD. Attention will need to be paid to teachers’ relevant beliefs if inclusive policy is
to have a real impact on students with MLD. This study provides some encouragement that
these beliefs are evidentially held (Green, 1971) and hence susceptible to change when they
are made explicit, and evidence to the contrary is presented. In this study some of this
evidence was sourced from research on effective numeracy teaching for students generally
and selected, mindful of what little is known about effective numeracy teaching for
students with MLD, to address the immediate perceived needs of the teachers in relation to
these students. In addition, evidence from the teachers’ own experiences as they trialled
various approaches and activities, albeit briefly seemed, anecdotally at least, to have a
positive impact. These observations are consistent with the notion of a dialectic
relationship between beliefs and practice in which both change together in complex ways.
References
Askew, M., Brown, M., Rhodes, V., Johnson, D., & Wiliam, D. (1997). Effective teachers of numeracy.
London: School of Education, King's College.
Atelier Learning Solutions. (2004). Essential learnings for all: Report of the review of services for students
with special and/or additional educational needs. Hobart: Department of Education, Tasmania.
Australian Association of Mathematics Teachers. (1997). Numeracy = everyone's business. Adelaide:
AAMT.
Australian Education Council. (1990). A national statement on mathematics for Australian schools.
Melbourne: Curriculum Corporation.
144
Baker, S. (2002). A synthesis of empirical research on teaching mathematics to low-achieving students. The
Elementary School Journal, 103(1), 51-74.
Battista, M. T. (1994). Teacher beliefs and the reform movement in mathematics education. Phi Delta
Kappan, 75(6), 462-470.
Beswick, K. (2003). The impact of secondary mathematics teachers' beliefs on their practices and the
classroom environment. Unpublished Doctor of Philosophy, Curtin University of Technology, Perth.
Beswick, K. (2004). The impact of teachers' perceptions of student characterstics on the enactment of their
beliefs. In M. J. Hoines & A. B. Fuglestad (Eds.), Proceedings of the 28th Annual Conference of the
International Group for the Psychology of Mathematics Education (Vol. 2, pp. 111-118). Bergen:
Bergen University College.
Bobis, J. (1996). Visualisation and the development of number sense with kindergarten children. In J.
Mulligan & M. Mitchelmore (Eds.), Children's Number Learning (pp. 17-34). Adelaide: Australian
Association of Mathematics Teachers.
Burns, R. B. (2000). Introduction to Research Methods (4th ed.). French's Forest, NSW: Longman.
Cooney, T. J., & Shealy, B. E. (1997). On understanding the structure of teachers' beliefs and their
relationship to change. In E. Fennema & B. Nelson (Eds.), Mathematics teachers in transmission (pp.
87-109). Mahwah, N.J.: Lawrence Erlbaum.
Department of Education Science and Training. (2004). Researching numeracy teaching approaches in
primary schools. Canberra: DEST.
Department of Education Tasmania. (2002). Essential Learning Framework 1. Hobart: DoET.
Department of Education Training and Youth Affairs. (2000). Numeracy, A Priority for All: Challenges for
Australian Schools (Commonwealth Numeracy Policies for Australian Schools). Canberra: DETYA.
Franke, M. L., Carpenter, T. P., Levi, L., & Fennema, E. (2001). Capturing teachers' generative change: a
follow-up study of professional development in mathematics. American Educational Research Journal,
38(3), 653-689.
Geary, D. C. (2004). Mathematics and learning disabilities. Journal of Learning Disabilities, 37(1), 4-15.
Green, T. F. (1971). The activities of teaching. New York: McGraw-Hill.
Guskey, T. (1995). Results-Oriented Professional Development: In Search of an Optimal Mix of Effective
Practices. Retrieved May 27, 2002, from http://www.ncreal.org/sdrs/areas/rpl_esys/pdlitrev.htm
Kaufman, L., Handl, P., & Thony, B. (2003). Evaluation of a numeracy intervention program focusing on
basic numerical knowledge and conceptual knowledge. Journal of Learning Disabilities, 36(6), 564-573.
Keeler, M. L., & Swanson, H. L. (2001). Does strategy knowledge influence working memory in children
with mathematical disabilities? Journal of Learning Disabilities, 34(5), 418-434.
Micallef, S., & Prior, M. (2004). Arithmetic Learning Difficulties in Children. Educational Psychology,
24(2), 175-200.
Ollerton, M. (2001). Inclusion, learning and teaching mathematics: Beliefs and values. In P. Gates (Ed.),
Issues in Mathematics Teaching (pp. 261-276). London: Routledge Falmer.
Robbins, B. (2000). Inclusive mathematics 5-11. London: Continuum.
Schoen, H. L., Cebulla, K. J., Finn, K. F., & Fi, C. (2003). Teacher variables that relate to student
achievement when using a standards-based curriculum. Journal for Research in Mathematics Education,
34(3), 228-259.
Shade, R. A., & Stewart, R. (2001). General education and special education preservice teachers' attitudes
toward inclusion. Preventing School Failure, 46(1), 37-41.
Stacey, K., Helme, S., Archer, S., & Condon, C. (2001). The effect of epistemic fidelity and accessibility on
teaching with physical materials: A comparison of two models for teaching decimal numeration.
Educational Studies in Mathematics, 47(2), 199-221.
Torbeyns, J., Verschaffel, L., & Ghesquiere, P. (2004). Strategy development in children with mathematical
disabilities: Insights from choice/no choice method and chronological-age/ability-level-match design.
Journal of Learning Disabilities, 37(1), 119-131.
Tournaki, N. (2003). The differential effects of teaching addition through strategy instruction versus drill and
practice to students with and without learning disabilities. Journal of Learning Disabilities, 36(5), 449-
458.
Van de Walle, J. (2004). Elementary and middle school mathematics. Boston: Pearson.
Wilson, M. S., & Cooney, T. J. (2002). Mathematics teacher change and development. In G. C. Leder, E.
Pehkonen & G. Torner (Eds.), Beliefs: A hidden variable in mathematics education (pp. 127-147).
Dordrecht: Kluwer.
... These differences were statistically significant. This is consistent with other research on teachers' beliefs about the capabilities of students with disabilities (Beswick, 2005) and demonstrates teachers need support to provide access to the general education curriculum for CWD (Courtade et al., 2012;Olson et al., 2016). ...
Article
Early mathematics skills are predictive of later achievement, but there is evidence teachers generally provide little mathematics instruction in preschool classrooms. We conducted this survey study to better understand teachers’ reported beliefs about their own mathematics skills, expectations, and practices for children with and without disabilities, and the impact of these reported beliefs and practices on the perceived effectiveness of their instruction. We found teachers had less confidence in their own mathematics skills than their mathematics teaching abilities and had differing expectations for children with and without disabilities. Their beliefs about their own mathematics abilities predicted their perceived effectiveness for typically developing children only, but their beliefs about their teaching abilities predicted their perceived effectiveness for children with and without disabilities. Implications include the need to better prepare and support teachers to teach mathematics to all children and collect data from varied sources on teachers’ practices.
... Teachers' activities in the classrooms are creations of their beliefs (Zakaria & Maat, 2012) with an argument regarding the constancies amid teachers' practices and their beliefs. This issue is somehow complex as some authors have found consistency between teachers' beliefs and classroom practices (Peterson, Fennema, Carpenter & Loef, 1989;Kupari, 2003), while other studies agreed to the contradictions between beliefs and practices (Brown, 1986;Beswick, 2005) in mathematics education. Teachers' mathematics beliefs are affected by prior experiences at school, teachers' present practice and teacher education courses (Raymond, 1993) in which teachers' beliefs influence students' learning of mathematics (Kagan, 1992) and consequently affect students' perceptions of mathematics (Yesil-Dagli, Lake & Jones, 2010). ...
... Swan's diagnostic questionnaire interrogates the idea of three preferred "orientations" for teachingnamely transmission, discovery, and connectionistfirst discussed by Askew et al. (1997), as mentioned in the literature review. Teacher beliefs and attitudes are notoriously difficult to influence in any significant way, as the research has shown (Beswick, 2005(Beswick, , 2006Grootenboer, 2008;Goldin, Rosken, & Torner, 2009;Hurrell, 2013). This was largely seen to be true in this study also. ...
Article
This paper discusses the results of a three-year mixed methods study into the effectiveness of a mathematics education unit. This was written for both pre-service primary education students and re-training in-service teachers, to prepare them for the teaching of pre-algebra and early algebra. The unit was taught rom 2013 to 2015 inclusively in a School of Education setting of a university in an Australian capital city. Focusing on the Number and Algebra strand in the Australian Curriculum, its purpose was to better prepare some novice teachers through modelling a more coherent approach to mathematics teaching. The unit 's genesis lies in the author 's belief that many mathematics teachers conduct their classes in isolated "pockets" of instruction that are not sufficiently informed by a broader, connected understanding of the mathematics. The unit was also prepared as a contribution to the recent call by the Australian Association of Mathematics Teachers for more targeted initiatives to combat the decrease of STEM skills in our schools (AAMT, 2014). Results from the analysis of this study suggest that there might be much to be gained from this new approach.
... The report highlighted shortcomings in the transfer of policy level commitments to equity and inclusion into practice. In the area of mathematics, there is evidence that teachers do not regard curricula that emphasise conceptual understanding as appropriate for students with learning difficulties (Beswick, 2005), however De Geest, Watson & Prestage (2003) demonstrated that low achieving students can engage in the kinds of mathematical thinking usually associated with higher achieving students. Such findings suggest that the goals of equity and inclusion policies are indeed attainable and worth pursuing in the context of ambitious or 'higher order' pedagogies. ...
Conference Paper
Full-text available
This paper reports on the findings of a Tasmanian study for the Department of Education, Science and Training (DEST). The study, Repertoires for Diversity, soon to be published by DEST through the Literacy and Numeracy Clearinghouse, was funded through the Australian Government's Effective Teaching and Learning Practices for Students with Learning Difficulties Initiative. Its purpose was to provide specific support to increase teachers' capacity to enhance the literacy and numeracy development of students with learning difficulties in the early and middle years of schooling. The Tasmanian study was designed to explore connections between school and teacher practices used in inclusive primary grade classes and schools' levels of 'value-adding', determined from national benchmark testing. The results showed that value-adding schools used a range of policies, programs and school-wide processes and professional learning to support literacy and numeracy pedagogies. The study acknowledged the multiple challenges facing teachers who are attempting to balance continuous improvement of students' literacy and numeracy learning with that of increasing social and educational diversity of inclusive school communities.
... Conduct changes the environment conditions and, vice versa, gets formed itself under certain conditions of environment (Bandura, 1989). The correlation of teachers' beliefs, preferences, and practice is not directly proportional (Liljedahl, 2009), but, as revealed by Beswick (Beswick, 2005), beliefs and practice develop together and affect one another, as they are dialectically interrelated. ...
Article
Full-text available
The outcomes of the research on teachers’ beliefs and their relatedness to their work routine are often contradictory, hard to systematize and predict. This may be evidence that research on teachers’ beliefs need a change in research paradigm: it should regard various approaches, in other words, research should be carried out in the Holistic paradigm framework. As a field inquiry concerned with the holistic exploration of phenomena and events, systems theory pertains to both epistemological and ontological situations. The present article suggests to emphasize in inquiries on mathematics teachers’ beliefs the bond of teachers’ beliefs with their practice and analyze the system of mathematics teachers’ beliefs and practice (SMTBP) from the position of Complex Adaptive System (CAS). The aim of the article is to show that the basic features of CAS are present in SMTBP as well and that there exist some features of CAS that have not been sufficiently regarded in previous research on teachers’ beliefs but that could be used to characterize the change of teachers’ beliefs and the processes of practical implementation of teachers’ beliefs in their work. The article is focused on the external factors that may affect these processes, and the principle of Bronfenbrenner ecological system theory has been used for systematizing them. Application of CAS Theory for the studies of SMTBP will provide a unified approach to interpreting the outcomes concerning teachers’ beliefs that, in turn, will make it possible to account for a large number of revelations that were impossible to account for within the framework of non-holistic paradigms.
... A combined effort of lecturers in mathematics and foundation subjects might be beneficial for designing such activities. This level of teacher education will provide a basis for the integration of the concepts across KLAs, which is a key requirement for teaching numeracy (Beswick, 2005). The understanding of the theoretical principles of scaffolding will allow pre-service teachers to anchor their repertoire of scaffolding techniques provided by recent research. ...
Article
Full-text available
Scaffolding has become increasingly popular as it provides teachers with an appealing alternative to traditional classroom techniques of teaching. Recent research identified a number of different ways that scaffolding can be used in the classroom to improve students' numeracy levels in primary schools. However, despite the importance of scaffolding, pre-service teachers experience difficulties in understanding the complex techniques of scaffolding and often fail to make connections between theoretical explanations and their practical use. This paper examines current perceptions of scaffolding by a cohort of pre-service teachers, both in its conceptual framework and its practical implications to teaching in the classroom, and to teaching numeracy in particular. The results indicated that the participants appreciated the importance of scaffolding as an alternative to the traditional forms of educational instruction. However, they continue to demonstrate a limited appreciation of the more complex and theoretical aspects of scaffolding.
Article
In this article, we present findings that examined special education teachers’ perception of students’ with disabilities ability, instructional needs, and difficulties for using visual representations (VRs) as a strategy to solve mathematics problems. In addition, whether these perceptions differed by instructional grade or setting currently teaching was examined. Survey data from 97 in-service teachers revealed, regardless of instructional setting or grade level taught, that they believe students with disabilities have the ability to learn about and use VRs and need to be taught to use VRs. Furthermore, the special education teachers perceived students with disabilities to have difficulty with all aspects related to using VRs in mathematical problem-solving. Implications for teacher training and development are provided.
Article
Full-text available
This research explored opinions of 115 parents who had children in kindergarten and prekindergarten classrooms about the degree of importance of some school readiness skills. The findings showed that for almost all parents “developing self-help/self-need skills” are important. On the other hand, academic skills are rated important by fewer numbers of parents. In general, parental perceptions did not vary by children’s gender and educational level of mothers or fathers. The only exceptions found were that the degree of importance of “child’s ability to seat still for 15-20 minutes” varied by father’s educational level and the degree of importance of “letter recognition” and “counting up to 20” varied by mother’s educational level.
Article
The aim of the research was to compare espoused beliefs about teaching and learning and reported practices for the teachers of mathematics in Latvia. The sample consisted of 390 teachers of mathematics from different regions of Latvia. The present research is a part of an international comparative research within the NorBa project (Nordic–Baltic Comparative Research in Mathematics Education) that makes use of a quantitative questionnaire for mathematics teachers. The results show that the espoused beliefs of Latvian teachers of mathematics on efficient teaching tend more to a constructivist approach, whilst reported practices are more oriented to a traditional approach; yet, there exist statistically significant differences for teachers of different social and demographical groups. The research outcomes may be used for the improvement of teacher further education programmes.
Article
Full-text available
This paper reports on the beliefs of a group of K-8 mathematics teachers about appropriate goals and methods of mathematics teaching for students with mathematics learning difficulties and for students generally. The teachers were involved in a brief professional learning program that aimed to provide them with effective strategies for mathematics teaching for numeracy, and to influence their relevant beliefs towards a more inclusive view of mathematics teaching. The questionnaire used in the study revealed differences between teachers' beliefs in relation to students generally and those with mathematics learning difficulties, and provided evidence that carefully designed professional learning may be able to reduce these differences. Numeracy has its foundations in the discipline of mathematics. Definitions of numeracy typically emphasise the use of mathematics in everyday life and highlight the importance of affect (Australian Association of Mathematics Teachers, 1997). The Department of Education Tasmania (DoET) (2002) refers explicitly to the application of mathematics to everyday life and acknowledges affect in its use of the terms disposition and confidence in its definition of numeracy, which includes the following: Being numerate involves having those concepts and skills of mathematics that are required to meet the demands of everyday life. It includes having the capacity to select and use them appropriately in real settings. Being truly numerate requires the knowledge and disposition to think and act mathematically and the confidence and intuition to apply particular mathematical principles to everyday problems … it also involves the critical and life-related aspects of being able to interpret information thoughtfully and accurately when it is presented in numerical and graphic form. (p. 21) Interpreting the Tasmanian definition necessitates some understanding of what the mathematical demands of everyday life might be and it seems self evident that such demands differ from individual to individual depending upon, among other things, occupation. Being numerate can thus be regarded as requiring different mathematical skills and concepts for different people. Nevertheless, the Australian Government remains committed to ensuring that "all students attain sound foundations in literacy and numeracy" (Department of Education, Training and Youth Affairs (DETYA), 2000, p. 6). DETYA (2000) also made clear that this goal was based upon a firm belief that all students can acquire the mathematical skills necessary for life in modern society and recognised that
Conference Paper
Full-text available
This paper reports on one aspect of a larger study and comprises an analysis of the beliefs concerning mathematics, its teaching and its learning, and the classroom practice of one secondary mathematics teacher. It focuses on the question, "What specific teacher beliefs about students are relevant to teachers' classroom practice in various classroom contexts?" The teacher's practice was examined in relation to several of his mathematics classes and significant differences, consistent with the teacher's beliefs in regard to the various classes, were found. The findings confirm the contextual nature of beliefs and highlight the importance to teachers' practice of specific teacher beliefs about the various students that they teach. BACKGROUND AND THEORETICAL FRAMEWORK
Article
Full-text available
The purpose of this study was to synthesize research on the effects of interventions to improve the mathematics achievement of students considered low achieving or at risk for failure. Meta-analytic techniques were used to calculate mean effect sizes,for 15 studies that met inclusion criteria. Studies were coded according to 5 categories of mathematics interventions, and effect sizes were examined on a study-by-study basis within each of these categories. Results indicated that different types of interventions led to improvements in the mathematics achievement of students experiencing mathematics difficulty, including the following:.(a) providing teachers and students with data on student performance; (b) using peers as tutors or instructional guides; (c) providing clears specific feedback to parents on their children's mathematics success; and (d) using principles of explicit instruction in teaching math concepts and procedures.
Article
We report results from a study of instructional practices that relate to student achievement in high school classrooms in which a standards-based curriculum (Core-Plus) was used. We used regression techniques to identify teachers' background characteristics, behaviors, and concerns that are associated with growth in student achievement and further described these associations via graphical representations and logical analysis. The sample consisted of 40 teachers and their 1,466 students in 26 schools. Findings support the importance of professional development specifically aimed at preparing to teach the curriculum. Generally, teaching behaviors that are consistent with the standards' recommendations and that reflect high mathematical expectations were positively related to growth in student achievement.
Article
This study documents how teachers who participated in a professional development program on understanding the development of students’ mathematical thinking continued to implement the principles of the program 4 years after it ended. Twenty-two teachers participated in follow-up interviews and classroom observations. All 22 teachers maintained some use of children’s thinking and 10 teachers continued learning in noticeable ways. The 10 teachers engaged in generative growth (a) viewed children’s thinking as central, (b)possessed detailed knowledge about children’s thinking, (c) discussed frameworks for characterizing the development of children’s mathematical thinking, (d) perceived themselves as creating and elaborating their own knowledge about children’s thinking, and (e) sought colleagues who also possessed knowledge about children’s thinking for support. The follow-up revealed insights about generative growth, sustainability of changed practice and professional development.
Article
This study explored the arithmetic skills of 39 children with arithmetic learning difficulties (ALD), compared to two control groups, one consisting of normally achieving children matched to the ALD sample for chronological age (n=28), and another comprising younger normally achieving children matched to the ALD sample for arithmetic level (n=28). To determine the relative contribution of reading impairments to arithmetic performance, the ALD group was further classified according to the presence or absence of reading difficulties into a mixed ALD (n=24) and specific ALD (n=15) group, respectively. The ALD group performed well below their chronological age counterparts in terms of the developmental maturity of their arithmetic problem solving. These children demonstrated a reliance on slow, counting-based procedures to solve arithmetic problems and, in this regard, they performed similarly to their younger, arithmetic-matched normal peers. Children with ALD also employed direct fact retrieval reliably less often than their same-age peers, but did not differ from their younger, arithmetic-matched counterparts. There were few significant differences between ALD children with and without accompanying reading problems. The findings suggest that children with ALD appear to show delayed development of arithmetic skills rather than specific processing deficits or abnormalities when comparison is made with both their chronological age and arithmetic-matched normal peers.