The importance of mathematics teachers' beliefs.It is widely acknowledged that what teachers believe influences their teaching, yet the focus of much professional learning remains on influencing the specific practices and tools that teachers employ in their classrooms. In this article it is argued that a greater and more explicit focus on teachers' beliefs would be beneficial. To this end an overview of aspects of our understandings of the nature of beliefs is presented followed by findings from a recent study that examined mathematics teachers' beliefs and their impact on classroom practice. Finally, implications for mathematics teachers and those involved in designing and implementing professional learning for both teachers and pre-service teachers are suggested.
The idea of belief systems recognises that beliefs are not held in isolation from one another but are in fact inter-related in complex ways. Green (1971) provided a description of belief systems that is still very useful. He described several dimensions of beliefs systems, three of which are of relevance here. The first is the idea of centrality. The centrality of a belief is a function of the strength and number of its connections with other beliefs. Other beliefs may be held because they are consequences of a central belief and any change in a central belief would have important ramifications ramifications npl → Auswirkungen pl for the individual's belief system and could be experienced as quite unsettling un·set·tle
v. un·set·tled, un·set·tling, un·set·tles
1. To displace from a settled condition; disrupt.
2. To make uneasy; disturb.
v.intr. . Centrally-held beliefs are thus relatively difficult to change.
A second aspect of Green's description of belief systems is the phenomenon of clustering. This means that beliefs with a system can be held in groups that are isolated from other beliefs. A consequence of this is that a person may hold beliefs that contradict con·tra·dict
v. con·tra·dict·ed, con·tra·dict·ing, con·tra·dicts
1. To assert or express the opposite of (a statement).
2. To deny the statement of. See Synonyms at deny. one another without being aware of the contradiction CONTRADICTION. The incompatibility, contrariety, and evident opposition of two ideas, which are the subject of one and the same proposition.
2. In general, when a party accused of a crime contradicts himself, it is presumed he does so because he is guilty for . According to according to
1. As stated or indicated by; on the authority of: according to historians.
2. In keeping with: according to instructions.
3. Green (1971) such clusters are likely to develop when beliefs are formed in disparate contexts. An example might be a student's belief that he is a poor mathematics student, formed perhaps on the basis of negative experiences of school mathematics, held at the same time as a belief in himself as mathematically competent formed as a result of experiences of part time work in a retail context. The student may not be consciously aware of one or other or both of these beliefs and may continue to believe both in the absence of any experience that makes them explicit and stimulates reflection on their contradictory elements.
The third aspect of beliefs relates to the basis on which they held. The basis of a belief may be evidence, in which case the belief is said to be evidentially held, or it may be held for other reasons such as the perceived authority of its source, or because it is regarded as a consequence of a another belief which may or may not be evidentially held. Evidentially held beliefs are by definition susceptible to change on the basis of evidence to the contrary, while non-evidentially held beliefs are impervious im·per·vi·ous
1. Incapable of being penetrated: a material impervious to water.
2. Incapable of being affected: impervious to fear. to evidence and hence very resistant to change.
Implicit in Adj. 1. implicit in - in the nature of something though not readily apparent; "shortcomings inherent in our approach"; "an underlying meaning"
underlying, inherent both the centrality and clustering of beliefs is the importance of context. The relative centrality of beliefs varies according to the context. For example, in the context of a professional learning session, a teacher might express a belief in the importance of providing students with ready access to manipulatives as they engage with mathematics, but in the context of his grade 8 classroom his belief that the teacher must maintain control of classroom activity and the related belief that this particular class would not use manipulatives in the intended way could be more central. The result might be that manipulatives are nowhere to be seen in that classroom. It is important to recognise that this would in no way mean that there was any lack of sincerity associated with the teacher's statement during the professional learning session.
The notion of clustering provides an alternative explanation for apparent contradictions between stated beliefs and practices like that described above. It allows the possibility that a teacher might simultaneously hold contradictory beliefs that have developed in different contexts. Beliefs formed as result of his/her own experiences of learning mathematics, those formed during teacher education, and others that have developed as result of classroom experience may contain contradictory elements that the teacher is unaware of.
The study aimed to examine the connection between secondary mathematics teachers' beliefs and their mathematics classroom environments and was described in detail by Beswick Beswick could be
CLES Constraint-Limited Exhaustive Search ) (Taylor Taylor, city (1990 pop. 70,811), Wayne co., SE Mich., a suburb of Detroit adjacent to Dearborn; founded 1847 as a township, inc. as a city 1968. A small rural village until World War II, it developed significantly in the second half of the 20th cent. , Fraser Fraser, river, Canada
Fraser, chief river of British Columbia, Canada, c.850 mi (1,370 km) long. It rises in the Rocky Mts., at Yellowhead Pass, near the British Columbia–Alta. line and flows northwest through the Rocky Mt. & Fisher, 1993) sought their perceptions of their classroom environments and asked them to rate the frequency of occurrence of various classroom events. The teachers were asked to complete this survey twice with a particular mathematics class in mind on each occasion. Several of the teachers completed just one survey either because they believed that the classroom environments of both classes were the same or because of time constraints In law, time constraints are placed on certain actions and filings in the interest of speedy justice, and additionally to prevent the evasion of the ends of justice by waiting until a matter is moot. . The teachers administered a student version of the CLES to the students in the two classes with respect to which they had completed the teacher version.
The vast majority ([greater than or equal to] 88%) of the teachers agreed or strongly agreed with statements such as the following:
1. A vital task for the teacher is motivating children to solve their own mathematical problems Mathematical problem may mean two slightly different things, both closely related to mathematical games:
2. Ignoring the mathematical ideas that children generate themselves can seriously limit their learning.
3. It is important for children to be given opportunities to reflect on and evaluate their own mathematical understanding.
4. It is important for teachers to understand the structured way in which mathematics concepts and skills relate to each other.
5. Effective mathematics teachers enjoy learning and "doing" mathematics themselves.
6. Knowing how to solve a mathematics problem is as important as getting the correct solution.
7. Teachers of mathematics should be fascinated with how children think and intrigued by alternative ideas.
8. Providing children with interesting problems to investigate in small groups is an effective way to teach mathematics
It is important to note that, with the exception of number 8, none of these statements prescribe pre·scribe
To give directions, either orally or in writing, for the preparation and administration of a remedy to be used in the treatment of a disease. any particular teaching strategy or classroom arrangement. The teachers were less inclined to agree with statements that did. For example, less than two-thirds of the teachers agreed or strongly agreed with the following items:
9. It is the teacher's responsibility to provide children with clear and concise solution methods for mathematical problems.
10. There is an established amount of mathematical content that should be covered at each grade level.
11. It is important that mathematics content be presented to children in the correct sequence.
12. Mathematical material is best presented in an expository ex·po·si·tion
1. A setting forth of meaning or intent.
a. A statement or rhetorical discourse intended to give information about or an explanation of difficult material.
b. style: demonstrating, explaining and describing concepts and skills.
Number 12 more evenly divided the teachers than any other (32% agreed or strongly agreed, 40 % undecided, 28% disagreed or strongly disagreed) indicating a diversity of opinion, as well as considerable uncertainty, regarding how beliefs such as those expressed in statements 1-8 should be enacted.
Cluster analysis Cluster analysis
A statistical technique that identifies clusters of stocks whose returns are highly correlated within each cluster and relatively uncorrelated across clusters. Cluster analysis has identified groupings such as growth, cyclical, stable, and energy stocks. (Hair, Anderson Anderson, river, Canada
Anderson, river, c.465 mi (750 km) long, rising in several lakes in N central Northwest Territories, Canada. It meanders north and west before receiving the Carnwath River and flowing north to Liverpool Bay, an arm of the Arctic , Tatham Tatham may mean: Places
1. Content and clarity
These teachers believed that they had a responsibility clearly to explain mathematical content and that it may be necessary to tell students the answers. They believed that they must cover the prescribed pre·scribe
v. pre·scribed, pre·scrib·ing, pre·scribes
1. To set down as a rule or guide; enjoin. See Synonyms at dictate.
2. To order the use of (a medicine or other treatment). content in the correct sequence. They also regarded computation Computation is a general term for any type of information processing that can be represented mathematically. This includes phenomena ranging from simple calculations to human thinking. is a major part of mathematics and believed that effective mathematics teachers enjoyed the discipline.
2. Relaxed problem solvers
Teachers in this cluster viewed mathematics as more than computation and were the least inclined to believe that it was their role to provide answers or even clear solution methods. They were also less concerned than other teachers about either content coverage or sequencing.
3. Content and understanding
These teachers could be described as the most concerned about the coverage and sequencing of the content, but the least likely to seek guidance regarding sequencing from a textbook textbook Informatics A treatise on a particular subject. See Bible. . They were focussed on students' understanding of the content, but not comfortable with students suggesting alternative solutions.
The CLES (student version) resulted in five clusters of classes based on the classes' average perceptions of the extent to which they were responsible for their learning and were engaged with the mathematics and connecting their learning with their existing knowledge (Beswick, 2005). The more these elements were in evidence the more consistent with constructivist con·struc·tiv·ism
A movement in modern art originating in Moscow in 1920 and characterized by the use of industrial materials such as glass, sheet metal, and plastic to create nonrepresentational, often geometric objects. principles the classrooms were deemed to be.
Subtle but important relationships were found between the teacher's beliefs and their students' average perceptions of their classroom environments (Beswick, 2005). Classes in clusters characterised by classroom environments most consistent with constructivist principles were more likely than others to be taught by teachers whose belief survey responses placed them in the Relaxed Problem Solvers cluster. It is important to remember, however, that teachers in this cluster (and each of the others) did not achieve these classroom environments by implementing identical, or even superficially su·per·fi·cial
1. Of, affecting, or being on or near the surface: a superficial wound.
2. Concerned with or comprehending only what is apparent or obvious; shallow.
3. similar, practices, but in spite of in opposition to all efforts of; in defiance or contempt of; notwithstanding.
See also: Spite the variety of ways in which they were implemented, their beliefs impacted their classrooms in ways that their students could discern dis·cern
v. dis·cerned, dis·cern·ing, dis·cerns
1. To perceive with the eyes or intellect; detect.
2. To recognize or comprehend mentally.
3. . This fact is illustrated by two of the teachers in the Relaxed problem solvers cluster, Jim and Andrew (pseudonyms This article gives a list of pseudonyms, in various categories. Pseudonyms are similar to, but distinct from, secret identities. Artists, sculptors, architects
The following quotations, some of which also appear in Beswick (in press), are taken from the interviews with Jim and Andrew and provide an indication of their beliefs about the discipline of mathematics and mathematics teaching and learning.
Jim I read about it, and I enjoy it and I sit here folding bits of paper in times when I could be doing something adults think might be more important ... and I'm constantly excited by it, and I do a fair bit of personal professional development and every time I go somewhere, I find extra little things ... (Beswick, in press) [I]f we're investigating some aspect of that, and the kids come up with a "what if" idea or, "I wonder what happens if we do this", then I'd absolutely grab it all the time ... If you think you've planned this lesson, and it's beautiful and linear and, it's going to work ... but I think the kids sometimes, won't believe they've got anything to offer ... and if we're going to keep inventing this stuff called mathematics, or discovering it, or making sense of it, we've got to believe some of our kids are going to go a lot further than we did, and if they don't think they can actually offer anything they won't. Sometimes when kids have suggestions, they're incomprehensible, if you just listen to their words because they haven't got the language and they haven't got the background ... and it's easy to dismiss stuff as being ludicrous, but if you then have got a culture where they can sit and try and tease it out and explain it, often they come up with amazing sorts of things ... Andrew I'm very teacher directed but at the same time what I like to do is not to give the kids the answers, but what I try to do is to make them think ... Getting them to come up and put on the board their ideas, what they think might be, what they should be doing or their way of doing something is a struggle ... Yes, they should guess. They should conjecture, but at some stage the teacher's going to have to call a halt and just say well, what about trying this? ... You're not just a supporting role, you are a facilitator, but you're also more than that. You're someone who hopefully understands the clear path that might be needed and can also see different paths to get to the end point and send the kids off on appropriate paths, not just let them wander through the minefield.
Observations of Jim's classes (grades 9 and 10) and Andrew's grade 7 classes confirmed their interview responses and revealed very different teaching approaches at least superficially. For example, Jim's students almost always worked in small groups and his interactions with them were primarily at the individual or small group level. In contrast with this, Andrew's students sat in rows of twos or threes facing the front of the room and most of the interactions were at the level of whole class discussions facilitated by Andrew. Nevertheless, the students perceptions of their classroom environments indicated that there were similarities in the extent to which they were responsible for their learning and were engaged with the mathematics and connecting their learning with their existing knowledge.
The beliefs that emerged as underpinning un·der·pin·ning
1. Material or masonry used to support a structure, such as a wall.
2. A support or foundation. Often used in the plural.
3. Informal The human legs. Often used in the plural. the practice of Jim and Andrew related to the nature of mathematics, their students and their capabilities, the teacher's role in the classroom and professional learning. Beliefs about mathematics, students, and the importance of professional learning were most central in Jim's case, whereas beliefs about the teacher's role were most central for Andrew. The particular beliefs that emerged as most central to one or other of Jim and Andrew were:
1. Mathematics is about connecting ideas and sense-making.
2. Mathematics is fun (in the sense of playful play·ful
1. Full of fun and high spirits; frolicsome or sportive: a playful kitten.
2. confidence with and enjoyment of mathematics).
3. Students' learning is unpredictable.
4. All students can learn mathematics.
5. The teacher has a responsibility to maintain ultimate control of the classroom discourse.
6. The teacher has a responsibility actively to facilitate and guide students' construction of mathematical knowledge.
7. The teacher has a responsibility to induct in·duct
To produce an electric current or a magnetic charge by induction. students into widely accepted ways of thinking and communicating in mathematics.
8. The teacher is the authority with respect to the social norms that operate in the classroom.
9. Teachers have a professional responsibility to engage in ongoing learning.
Beswick (in press) argued that this set of beliefs seems to be related to teachers' ability to create classroom environments that can be described as constructivist and that it is such beliefs, rather than particular teaching methods or materials, that matter in terms of students' perceptions of their classroom environments. This is consistent with the findings of Watson and De Geest n. 1. Alluvial matter on the surface of land, not of recent origin. (2005) and Askew a·skew
adv. & adj.
To one side; awry: rugs lying askew.
[Probably a-2 + skew. , Brown, Rhodes Rhodes (rōdz) or Ródhos (rô`thôs), island (1990 est. pop. 90,000), c.540 sq mi (1,400 sq km), SE Greece, in the Aegean Sea; largest of the Dodecanese, near Turkey. , Johnson and Wiliam (1997) concerning the importance of teachers' beliefs in shaping their practices.
The literature on teacher change is replete re·plete
1. Abundantly supplied; abounding: a stream replete with trout; an apartment replete with Empire furniture.
2. Filled to satiation; gorged.
3. with evidence that real and lasting change is achieved only if teachers' belief systems support the underlying premises of the changes they are asked to implement (e.g., Chapman, 2002). Little is achieved by getting teachers (or students) to mouth "suitable" views or perform certain actions if they are not convinced of their value. It is, therefore, not enough to provide teachers with resources, curriculum materials and ideas without attending to their relevant beliefs. The point here is analogous analogous /anal·o·gous/ (ah-nal´ah-gus) resembling or similar in some respects, as in function or appearance, but not in origin or development.
adj. to the more widely espoused view that it is not enough to get students to recite facts or perform procedures if they are not meaningful to them--i.e., if they do not really believe the procedures or their results.
Findings concerning the importance of teachers' beliefs to the kinds of classrooms that they create highlight the importance of individual mathematics teachers, and providers of professional learning or pre-service teacher education This article or section is written like an .
Please help [ rewrite this article] from a neutral point of view.
Mark blatant advertising for , using . related to mathematics, reflecting carefully on the beliefs that they hold about the nature of mathematics and about mathematics teaching and learning. The following is a list of questions that may be helpful in stimulating such reflection:
With respect to each of the nine beliefs listed above:
1. To what extent do I hold this belief?
2. Why do I believe this? What hard evidence underpins my belief? Is this evidence more than anecdote anecdote (ăn`ĭkdōt'), brief narrative of a particular incident. An anecdote differs from a short story in that it is unified in time and space, is uncomplicated, and deals with a single episode. ?
3. How/in what way(s) does this belief shape my practice?
4. How would my practice be different if I believed this?
5. Would an observer in my class (including my students) be surprised if I told them I believed this? Why?
6. What other beliefs about mathematics or mathematics teaching and learning, influence my practice? Why do I believe these things "These Things" is an EP by She Wants Revenge, released in 2005 by Perfect Kiss, a subsidiary of Geffen Records. Music Video
The music video stars Shirley Manson, lead singer of the band Garbage. Track Listing
1. "These Things [Radio Edit]" - 3:17
2. ? Is there hard evidence for their veracity veracity (vras´itē),
When considering new practices, ideas, or materials:
7. What beliefs about mathematics and about mathematics teaching and learning does the author/creator of these materials hold?
8. What does this professional learning provider believe about the nature of mathematics and mathematics learning and teaching?
9. To what extent do I share these beliefs? Why?
10. What beliefs underpin my negative/positive reaction to this idea? Are these beliefs reasonable?
In relation to students' perceptions of your beliefs:
11. What might my students think I believe about:
a. their capacity to learn mathematics?
b. how they learn mathematics?
c. what it means to "do mathematics"?
d. my role as a mathematics teacher?
12. How might these perceptions vary from student to student or from class to class? Would there be differences according to mathematical ability or grade level? How might students notice these differences?
Opportunities to talk with trusted colleagues about responses to these questions would likely be helpful. It would also seem sensible for professional learning providers to be explicit about their own beliefs and those that underpin their own practices and recommendations. Providing time, opportunities and stimuli for teachers' reflection on their beliefs is also important and certainly consistent with a social constructivist view of learning that recognises that teacher change is learning. Similarly teachers should make their own relevant beliefs explicit for their students. Perhaps teachers and teacher educators alike could benefit from asking their "students" what they think their "teachers" believe. All of this has the potential to be quite confronting and uncomfortable but I believe that such unsettling is fundamental to learning.
Askew, M., Brown, M., Rhodes, V., Johnson, D. & Wiliam, D. (1997). Effective Teachers of Numeracy numeracy Mathematical literacy Neurology The ability to understand mathematical concepts, perform calculations and interpret and use statistical information. Cf Acalculia. . London London, city, Canada
London, city (1991 pop. 303,165), SE Ont., Canada, on the Thames River. The site was chosen in 1792 by Governor Simcoe to be the capital of Upper Canada, but York was made capital instead. London was settled in 1826. : School of Education, King's College King's College, former name of Columbia Univ. .
Beswick, K. (2005). The beliefs/practice connection in broadly defined contexts. Mathematics Education Research Journal, 17(2), 39-68.
Beswick, K. (in press). Teachers' beliefs that matter in secondary mathematics classrooms. (accepted February 2006 for publication in Educational Studies in Mathematics).
Chapman, O. (2002). Beliefs structure and inservice high school mathematics teacher growth. In G. C. Leder, E. Pehkonen & G. Torner (Eds), Beliefs: A Hidden Variable in Mathematics Education? (pp. 177-193). Dordrecht: Kluwer.
Green, T. F. (1971). The Activities of Teaching. New York New York, state, United States
New York, Middle Atlantic state of the United States. It is bordered by Vermont, Massachusetts, Connecticut, and the Atlantic Ocean (E), New Jersey and Pennsylvania (S), Lakes Erie and Ontario and the Canadian province of : McGraw-Hill.
Hair, J. F., Anderson, R. E., Tatham, R. L. & Black, W. C. (1998). Multivariate The use of multiple variables in a forecasting model. Data Analysis (5th ed.). Upper Saddle River Saddle River may refer to:
Taylor, P., Fraser, B. J. & Fisher, D. L. (1993, April). Monitoring the development of constructivist learning environments. Paper presented at the Annual Convention of the National Science Teachers Association, Kansas City Kansas City, two adjacent cities of the same name, one (1990 pop. 149,767), seat of Wyandotte co., NE Kansas (inc. 1859), the other (1990 pop. 435,146), Clay, Jackson, and Platte counties, NW Mo. (inc. 1850). .
Watson, A., & De Geest, E. (2005). Principled prin·ci·pled
Based on, marked by, or manifesting principle: a principled decision; a highly principled person. teaching for deep progress: Improving mathematical learning beyond methods and materials. Educational Studies in Mathematics, 58(2), 209-234.
University of Tasmania (body, education) University of Tasmania -