Department of Teacher Education

Transcription

Enhancing mathematical thinking in
Finnish elementary school mathematics
- pedagogical models (and modelling)
Dr. Heidi Krzywacki
University of Helsinki, Finland
www.helsinki.fi/yliopisto
Where is Finland?
Finland is in the
North Europe, part of
the Nordic countries
Finland is located by the
Arctic Circle…
…and the home of the
REAL Santa Claus is in the
Finnish Lapland!
Finland in facts
• Population: 5.4 million, 17 inhabitants / km2
• Languages: Finnish (spoken by 91%)
Swedish (~ 5%)
Sámi (about 1,800 people)
• Religion: 79.9% Lutheran and about 1.1% Orthodox, other
1.7 % and none 17.7 %
• Independence: Declared on December 6, 1917. Previously
a grand duchy in the Russian empire (1809-1917), and
before that, a part of Sweden for 600 years
• Currency unit: Euro
• Area: 338,424 km²
• Capital: Helsinki (population ~600 000)
Dr. Heidi Krzywacki
Modelling Conference, January 10, 2013
3
Outline of the presentation
1) Mathematics education aiming at development
of mathematical thinking
2) Pedagogical models (and modelling)
A. The use of manipulatives and illustrations
B. Inductive approach to elementary school
mathematics
C. Word problems and problem solving
3) Concluding remarks
Dr. Heidi Krzywacki
4
Aiming at development of
mathematical thinking
Dr. Heidi Krzywacki
5
National core curriculum:
mathematics education (1/2)
• the significance of developing mathematical
thinking skills
... offering opportunities for the development of
mathematical thinking and for the learning of mathematical
concepts and the most widely used problem-solving
methods.
The discipline's concrete nature serves as an important
aid in bringing together the pupil's experiences and
systems of thought with the abstract system of
mathematics.
Modelling is hardly mentioned in the curriculum!
Dr. Heidi Krzywacki
(FNBE, 2004)
6
National core curriculum:
mathematics education (2/2)
• Links with everyday life and practical problems
• The use of concrete materials and illustrations
supporting conceptual and procedual understanding
• Various ways to process mathematical tasks and
give reasons for solutions
• oral and written communication, the use of
manipulatives, illustrations and symbolic
expressions
Dr. Heidi Krzywacki
(FNBE, 2004)
7
Teachers in Finnish elementary
school
• is typically qualified to teach all 13 school subjects at
grades 1 to 6
• has a Master’s level university degree
(3+2 years, 300 credit points in total)
• majoring in educational sciences / psychology
• only 7 cp dedicated to mathematics education
• Teachers are considered as autonomous academic
professionals who
• choose learning materials and take full responsibility
for teaching and learning in the classroom
• design and carry out assessment
Dr. Heidi Krzywacki
8
Characteristics of Finnish school
system (1/2)
• responsibility at the local level: neither school
ranking, national exams nor pre-evaluation of
learning materials
• Compulsory and intended instruction time of an
elementary teacher is 667 lessons per year
NOTE: the number of teaching hours does not
reflect explicitly the teachers’ total workload
Dr. Heidi Krzywacki
(FNBE Kyrö, 2012)
9
Characteristics of Finnish school
system (2/2)
• Total amount of lesson hours in 9-years comprehensive
school is 5,750
(cf. OECD average 6,500; max. Italy 8,300)
• 32 weekly mathematics lessons
(~ 3.5 lessons per week)
• Schools are of various sizes and rather small in average
• About 44% of all schools <100 pupils (12% of all pupils)
• About 22% of all schools >300 pupils (54% of all pupils)
• Average class size at elementary leve is 19.8 (OECD 21.4)
(FNBE Kyrö, 2012)
Heidi Krzywacki
EDUVISIT, November 22, 2012
Pedagogical models
(and modelling) in mathematics
education
Dr. Heidi Krzywacki
11
Learning mathematics:
students have an active role
A child is seen as an active and social learner who is
basically willing and motivated to learn as well as
capable for self-regulated process
• thinking at concrete level (Piaget, Galperin)
• embodied cognition (tactile and kinestetic approach)
• verbalization and oral communication
(also performing initiatives)
• support to building up self-efficacy (Erikson)
• positive learning experiences, assessment and
feedback
Dr. Heidi Krzywacki
12
Learning arithmetic mathematical
concepts
V
E
R
B
A
L
I
Z
A
T
I
O
N
1. Concrete strategies
• Conceptual understanding based on concrete models
and illustration
• manipulatives
• illustrations and figures
• images of the use of the concrete
2. Mental strategies
• conceptual understanding and processes without
images of the concrete
• One or more steps are embodied in the arithmetic
processes
3. Automatizised conceptual understanding
Dr. Heidi Krzywacki
Learning arithmetic mathematical
concepts
V
E
R
B
A
L
I
Z
A
T
I
O
N
1. Concrete strategies
• Conceptual understanding based on concrete models
and illustration •demonstration in frontal teaching
• manipulatives
• manipulatives used by individual
• illustrations and figures
(or by pairs)
• images of the usestudents
of the concrete
•together with teacher guidance
2. Mental strategies •students in their own pace
• conceptual understanding and processes without
images of8the
+ 5concrete
= 8 + 2 + 3 = 13
• One or more steps are embodied in the arithmetic
processes127 + 38 = 127 + 30 + 8 = 157 + 8 = 165
or
= 127 + 3 + 35 = 130 + 35 = 165
3. Automatizised conceptual understanding
etc.
©
Hellevi Putkonen
Pedagogical models
education
A. The use of manipulatives and
illustrations
B. Inductive approach to elementary
school mathematics
C. Word problems and problem
solving
Dr. Heidi Krzywacki
15
A . The use of manipulatives and
illustrations
• manipulatives and illustrations play a helpful role in
learning mathematics, especially in enhancing
conceptual understanding and also in problem
solving
e.g., attribute blocks, geometric shapes of different colours
and sizes; counting cubes, base ten blocks, fraction
pieces, number line, surface model etc.
• BUT these are potentially confusing if their
presentation is haphazard, disorganized, or lacking
appropriate guidance and instruction from the
teacher
teacher’s
training and pedagogical thinking (!)
Dr. Heidi Krzywacki
16
illustrations
Getting to know different kinds of concrete materials
(manipulatives)...
0,03
17
illustrations
... and learning how to use materials and models
18
illustrations
In teacher education student teachers
• acquire knowledge and skills (cognitive potential) to
use manipulatives and illustrations
• learn how to make (abstract) and formal mathematics
approachable
• get to understand what school mathematics is about
and the meaning of well-structured and meaningful
instruction
EXPERIENCE makes a difference!
19
Pedagogical models
education
illustrations
school mathematics
solving
Dr. Heidi Krzywacki
20
B. Inductive approach to mathematics
• Individual cases as a basis for understanding
mathematical rules and facts
• investigating phenomena through student work
e.g., commutativeness (a + b = b + a), the sum of
angles of a triangle, calculating the area of
parallelogram etc.
A teacher needs to guide students to find general
notions based on their investigations
students’ previous knowledge and skills form a
starting-point for investigations
Dr. Heidi Krzywacki
21
B. Inductive approach:
The sum of angles of a triangle
1. Introductory
phase
2. Examining and
finding the
mathematical
rule
3. Rehearsing
4. Limitations of
the rule
• Previous knowledge about triangles and
angles
each student draws a triangle
(or a teacher gives triangles of different
shapes and size)
setting the aim for investigation
Dr. Heidi Krzywacki
22
1. Introductory
phase
2. Examining and
finding the
mathematical
rule
3. Rehearsing
4. Limitations of
the rule
• Examining the triangles: measuring angles
and guiding students to make notes about
their measurement
listing outcomes of student work and
letting students to figure out that
the overall sum of angles is 180°
Dr. Heidi Krzywacki
23
• Applying new knowledge and rehearsing
1. Introductory
phase
2. Examining and
finding the
mathematical
rule
3. Rehearsing
4. Limitations of
the rule
various tasks
making sure that all students understand
Dr. Heidi Krzywacki
24
1. Introductory
phase
2. Examining and
finding the
mathematical
rule
3. Rehearsing
4. Limitations of
the rule
• Summing up the investigative session and its
outcome
the sum of angles of a triangle applies with
all triangles but not with other shapes
Dr. Heidi Krzywacki
25
B. Inductive approach to mathematics
• get familiar with inductive approach as a method in
mathematics classroom
discuss which contents can be approached
especially through inductive approach and student
investigations
a special challenge is to have an impact on
student teachers’ views of school mathematics
Dr. Heidi Krzywacki
26
Pedagogical models
education
illustrations
school mathematics
solving
Dr. Heidi Krzywacki
27
• Writing down and drawing
1. All information given in the initial task
2. Drawing a picture or other illustration (using
manipulatives)
3. All steps of the solving process
4. Finding the solution(s) and reasons behind it
• Making the problem solving process visible
support to the process and understanding
communicating mathematics with others
becoming aware of personal problem solving
process and reflecting on the process
Dr. Heidi Krzywacki
28
C. Word problems and problem solving:
Word problem ’Flying witches’
Altogether 20 witches are to travel to magic
mountains by 8 brooms. Each broom can fit 4
witches at the most but each broom must have
at least 2 witches. Find out how all witches can
travel to the mountain.
Solve the problem above and reflect on the
problem solving process and the strategies you
used in finding the solution.
NOTE: illustrate and communicate your
problem solving process as accurately
as possible.
Dr. Heidi Krzywacki
29
Word problem ’Flying witches’
Illustration
Describing
the process
Mathematical
formulation
Dr. Heidi Krzywacki
Reasoning
30
Word problem ’Rolls for sale’
Class 4A sells packages of rolls at the school ‘s
Xmas market. A package of rolls includes 4
whole wheat rolls and 3 small breads made out
of organic rye. Altogether 4 packages are sold
before noon. How many rolls are then sold?
Start solving the problem by drawing a picture
of the situation and then continue with
formulating an equation describing the situation
based on your picture.
4+3
Dr. Heidi Krzywacki
31
Word problem ’Rolls for sale’
Class 4A sells packages of rolls at the school ‘s
Xmas market. A package of rolls includes 4
whole wheat rolls and 3 small breads made out
of organic rye. Altogether 4 packages are sold
before noon. How many rolls are then sold?
4 · (4 + 3)
Start solving the problem by drawing a picture
of the situation and then continue with
formulating an equation describing the situation
based on your picture.
Dr. Heidi Krzywacki
32
• …learn the basics about problem solving
• … have an experience of solving problems and
analysing process
a special challenge is to help improving teachers’
self-confidence and willingness to carry out problem
solving activities in their classroom
subject matter knowledge
posing problems and guidance
Dr. Heidi Krzywacki
33
Concluding remarks
Dr. Heidi Krzywacki
34
1. Reachable aims and
differentiation
• Developing activities suitable for different kind of
learners
• What is(are) the objective(s) of the activity?
• Reflecting on how well the chosen tasks worked in
the classroom and how to modify them if needed?
• Assessment procedures should reflect the nature of
teaching and learning
Dr. Heidi Krzywacki
2. Paying attention to affective
aspects
• becoming aware of personal strategies and
qualities as a mathematics learner
• commucating (sharing) with others
verbalization
• positive experiences and encouragement
avoiding categorisation in the classroom based on
mathematical performance
’everyone can do mathematics’
routine tasks are also needed
Dr. Heidi Krzywacki
3. Approaching mathematics through
pupils’ eyes
• Concrete
abstract
• The contextualisation of mathematical tasks should
be close to children’s everyday life or something they
find easy to approach
• Mathematical performance (thinking) is not only
performed through correct symbolic expressions
• Manipulatives and illustrations
• sensitiveness for understanding pupils’ ways of
thinking
Dr. Heidi Krzywacki
4. Resources and materials
supporting school teaching
• Teachers need support in enhancing their practice in
mathematics classroom, for example, suitable tasks
and pedagogical ideas
learning materials (textbooks and worksheets)
teacher guide books (pedagogical support for
teachers)
computer-aided teaching and learning
Not only materials but also education that
supports teachers (and schools) to enhance their
practice
Dr. Heidi Krzywacki
4. Resources and materials...
39
Kiitos!
Further information
Finnish National Board of Education
FNBE, (2004). National Core Curriculum for Basic Education.
www.oph.fi/english
Kyrö, M. (2012). International comparisons of some features of
Finnish education and training. FNBE.
http://www.oph.fi/english/publications/2012/international_com
parisons_of_some_features_of_finnish_education_and_traini
ng_2011
Ministry of Education and Culture
http://www.minedu.fi/OPM/?lang=en
Department of Teacher Education (UH)
http://www.helsinki.fi/teachereducation/
E-mail: Heidi.Krzywacki@helsinki.fi
Dr. Heidi Krzywacki

Department of Teacher Education

Transcription

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