Transcript of Slagjana Jakimovik University “Ss. Cyril and Methodius” – Skopje Faculty of Pedagogy “St....
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- Slagjana Jakimovik University Ss. Cyril and Methodius Skopje
Faculty of Pedagogy St. Kliment Ohridski Republic of Macedonia
IMI-BAS, Sofia, R. Bulgaria, February 14, 2015
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- Teacher competencies Cognitive abilities: Professional
knowledge Mathematics content knowledge (MCK) Pedagogical content
knowledge (PCK) General pedagogical knowledge (GPK) Affective
motivational characteristics: Professional beliefs, motivation and
self- regulation Beliefs about: mathematics, teaching and learning
mathematics Professional motivation and self - regulation 2 Figure
1. Conceptual framework of teacher competencies (Blmeke, S. &
Delaney, S. (2012). Assessment of teacher knowledge across
countries: A review of the state of research. ZDM 44 (3), 223-247.)
(Shulman, 1985) (Richardson, 1996; Thompson et al., 1992)
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- Affect (Beliefs, Attitudes, Emotions) Phillip, R. A. (2007).
Mathematics teachers beliefs and affect (Ch.7 in F. K. Lester
(ed.), Second Handbook of Research in Mathematics Teaching and
Learning, 257-315) Affect a disposition or tendency or an emotion
or feeling attached to an idea or object; comprised of Beliefs
psychologically held understandings, premises, or propositions
about the world that are thought to be true Attitudes manners of
acting, feeling or thinking that show ones disposition or opinion
Emotions feelings or states of consciousness, distinguished from
cognition. BeliefsAttitudesEmotions more cognitiveless cognitivenot
cognitive harder to changeeasier to changechange rapidly not felt
intenselymoderate + or - feelingsintense + or feelings 3
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- Beliefs lenses which filter a complexity of situation to make
it comprehensible, shaping individuals interpretations of events
Beliefs influence perception; predispose toward action Develop
gradually; cultural factors play a key role in their development
Some beliefs (primary) serve as the foundation of other beliefs
(derivative) in a quasi-logical structure Central beliefs are held
strongly, peripheral beliefs are more susceptible to change Beliefs
are held in clusters relatively isolated from other clusters
Beliefs systems may appear contradictory or inconsistent to an
observer Beliefs are context specific and situated. 4
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- Why? Teachers beliefs and knowledge shape teachers actions
Research studies demonstrate that teachers practices are consistent
with teachers beliefs about mathematics to a higher degree than
with teachers beliefs about teaching and learning (Phillip, 2007)
Teachers practices impact students development of mathematics
proficiency Mathematics proficiency comprised of five interrelated
strands: Conceptual understanding Procedural fluency Strategic
competence - the ability to formulate, represent, and solve
mathematical problems Adaptive reasoning - the capacity to think
logically and to justify ones reasoning Productive disposition
perceiving mathematics as useful and worthwhile, seeing oneself as
an effective learner and doer of mathematics (Adding It Up, NRC,
2001) 5 Studying Teachers Beliefs (and Knowledge)
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- An IEA cross-national study to provide data on the knowledge
that future primary and lower-secondary school teachers acquire
during their mathematics teacher education including their beliefs.
Likert-type scales development informed by several studies:
Teaching and Learning to Teach Study, at Michigan State University
(Deng, 1995; Tatto, 1996, 1998, 2003) MT21 Report (a feasibility
study for TEDS-M) (Schmidt et al., 2007) I. Beliefs about the
nature of mathematics II. Beliefs about learning mathematics III.
Beliefs about mathematics achievement Questions represent two views
consistent with: A. Conceptual and cognitive-constructivist
orientations B. Calculational and direct transmission orientations.
6 TEDS-M (Teacher Education & Development Study in
Mathematics)
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- I-A. Mathematics as a Process of Inquiry 1. Mathematics
involves creativity and new ideas. 2. In mathematics many things
can be discovered and tried out by oneself. 3. If you engage in
mathematical tasks, you can discover new things (e.g., connections,
rules, concepts). 4. Mathematical problems can be solved correctly
in many ways. 5. Many aspects of mathematics have practical
relevance. 6. Mathematics helps solve everyday problems and tasks.
7 I.Beliefs About the Nature of Mathematics
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- I-B. Mathematics as a Set of Rules and Procedures 1.
Mathematics is a collection of rules and procedures that prescribe
how to solve a problem. 2. Mathematics involves the remembering and
application of definitions, formulas, mathematical facts, and
procedures. 3. When solving mathematical tasks, you need to know
the correct procedure else you would be lost. 4. Fundamental to
mathematics is its logical rigor and precision. 5. To do
mathematics requires much practice, correct application of
routines, and problem solving strategies. 6. Mathematics means
learning, remembering, and applying. 8
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- 9 II-A. Learning mathematics through Active Involvement 1.In
addition to getting a right answer in mathematics, it is important
to understand why the answer is correct. 2.Teachers should allow
pupils to figure out their own ways to solve mathematical problems.
3.Time used to investigate why a solution to a mathematical problem
works is time well spent. 4.Pupils can figure out a way to solve
mathematical problems without a teachers help. 5.Teachers should
encourage pupils to find their own solutions to mathematical
problems even if they are inefficient. 6.It is helpful for pupils
to discuss different ways to solve particular problems. II.Beliefs
About Learning Mathematics
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- 1. The best way to do well in mathematics is to memorize all
the formulas. 2. Pupils need to be taught exact procedures for
solving mathematical problems. 3. It doesnt really matter if you
understand a mathematical problem, if you can get the right answer.
4. To be good in mathematics you must be able to solve problems
quickly. 5. Pupils learn mathematics best by attending to the
teachers explanations. 6. When pupils are working on mathematical
problems, more emphasis should be put on getting the correct answer
than on the process followed. 7. Non-standard procedures should be
discouraged because they can interfere with learning the correct
procedure. 8. Hands-on mathematics experiences arent worth the time
and expense. 10 II-B.Learning Mathematics by Following Teacher
Direction
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- III.Beliefs About Mathematics Achievement Mathematics as a
Fixed Ability 1. Since older pupils can reason abstractly, the use
of hands-on models and other visual aids becomes less necessary. 2.
To be good at mathematics, you need to have a kind of mathematical
mind. 3. Mathematics is a subject in which natural ability matters
a lot more than effort. 4. Only the more able pupils can
participate in multi-step problem- solving activities. 5. In
general, boys tend to be naturally better at mathematics than
girls. 6. Mathematical ability is something that remains relatively
fixed throughout a persons life. 7. Some people are good at
mathematics and some arent. 8. Some ethnic groups are better at
mathematics than others. 11
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- Pilot Study for a Larger National Study The beliefs of
prospective primary school teachers and pre-school teachers about
the nature of language and of mathematics and about teaching and
learning language and mathematics national project at Ss Cyril
& Methodius University Skopje respondents prospective teachers
in their final semester at each teacher education department in
Macedonia Pilot survey administered at the beginning of the 2 nd
semester (before taking the first university course in mathematics)
Sample 102 first year students (87 % female) 71 primary scho0l
teacher education students 31 pre-school teacher education students
Sub-sample (survey & mathematics items respondents) 71 12
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- Survey results in % (102 respondents) Disagree- Strongly
disagree Slightly disagree- Slightly agree Agree - Strongly Agree
Mathematics as a Process of Inquiry/25.874.2 Mathematics as Set of
Rules and Procedures/22.477.6 Learning Math. through Active
Involvement/24.875.2 Learning Math. through Teacher
Direction/78.521.5 Mathematics as a Fixed Ability10.581.08.5 13
What is the nature of these beliefs? Which beliefs are
teacher-students inclined to act upon?
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- Mathematics Survey Items: Descriptions and Results Respondents
were asked to: 1) Analyze 4 pictorial representations of
multiplication of two fractions (3 were correct models, two of
which very similar) and choose if only one of them (which one),
more than one (which ones), or none correctly modeled the
operation. More than 50 % of the respondents chose only one! 2)
Find the solution to a given quadratic equation using one of two
methods (an example was presented using each of them): finding the
roots as two whole numbers whose sum is the negative of the linear
coefficient (7) and whose product is the free term (12) finding the
roots using the quadratic formula More than 85 % of the respondents
chose to use the formula! 14
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- 3) Choose the method expected to be used by grade 4 pupils in
finding the area of a triangle with vertices on 3 different sides
of a grid composed of unit squares: A. Directly using the formula:
length times height over two (impossible) B. Using Pythagoras's
Theorem to calculate the lengths of the sides and then Herons
Formula for the area of a triangle (both given). C. Using the
Distance formula to calculate the lengths of the sides and then
Herons Formula for the area of a triangle (both given). D. Finding
the area of the 3 outer rectangular triangles by halving the number
of unit squares in 3 rectangles and subtracting their sum from the
area of the grid. E. Other
_________________________________________________ Responses:A 34 %B
31 %C 9 % D or E (offering an appropriate method) only 21 % 15
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- Conclusions At the beginning of initial teacher education: i. 3
out 4 students endorse statements supporting a view of mathematics
as a process of enquiry and of mathematics learning through active
involvement (conceptual and cognitive-constructionist orientation)
ii. 3 out 4 students endorse statements supporting the view of
mathematics as a set of rules and procedures 1 out 4 students
supports mathematics teaching through direct transmission; 3 out 4
slightly agree/slightly disagree with it (calculational and direct
transmission orientation) Yet, when approaching mathematical
problems, high percentage of prospective teachers actions are
aligned with the second view. 17
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- Possible Explanations Teachers MCK and PCK are positively
related to teachers conceptual and cognitive-constructionist
orientation, negatively related to teachers calculational and
direct transmission orientation (TEDS-M, 2008) The pattern of
beliefs held by future teachers matches the pattern of beliefs of
teacher educators (TEDS-M, 2008) Teachers beliefs (and knowledge)
develop as a result of their learning experiences (years of
apprenticeship) The process of mathematics learning (and of
becoming a mathematics teacher) is a process of enculturation, of
becoming a member of a community of mathematics learners. 18
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- Changing Beliefs If beliefs serve as lenses which filter how an
individual sees the world, how can they be changed? Many
researchers attempt to develop mechanisms for influencing teacher
beliefs: Teachers beliefs change if teachers evidence positive
changes in student learning outcomes. Providing (prospective)
teachers with opportunities to learn about students mathematical
thinking and reflect upon the experiences. Immersing (prospective)
teachers in a community so that they become enculturated with
beliefs through cultural transmissions. Changing teacher education
in line with these beliefs. (For a review of research literature on
teachers beliefs, see Phillip, 2007) 19
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- 20 Stairs to the Castle b, bs, bl, bsl, bssl, bsssl (b-bridge,
s-short, l-long) Magdalena, independent constructor (3,5 years old,
August 12, 2013)