johan häggström elisabeth rystedt - ncm:s och nämnarens...
TRANSCRIPT
A workshop at
Homi Bhabha Centre for Science Education,
Mumbai, India
22–23 February 2011
Elisabeth Rystedt and Johan Häggström
National Center for Mathematics Education, NCM,
Gothenburg, Sweden
Teacher
Pupils: 10–16 years old
Mathematics and the Swedish language
Educational inspector in mathematics at The National
Agency for Education
National Center for Mathematics Education, NCM
Project leader of a national network
Math lab at NCM
The book Matematikverkstad
In-service projects
Lectures
Conferences
Courses: all over Sweden, in some places in Norway + London, Brussels, Mexico City …
Advisors
Research overview
Teacher in mathematics and science
Mathematics teacher educator
Research in mathematics education
Editor of Nordic Studies in Mathematics Education
Learning study – professional development
How can I take the ideas and change them, so it will be possible toconduct them together with my pupils?
Gothenburg
Stockholm
Population 9 000 000 1 161 000 000
Population
density, inh/km
22 353
Area
km
450 000 3 290 000
”Big cities” Stockholm
1,2–2 milj
Mumbai
14–20 milj
The only things required for
this game are pens and paper
and also an ordinary dice
Each participant will draw a
square which is in turn
divided into 3 3 squares
In other words: a lot of
people and only one dice for
all!
+
+ + =mental math
Why will it be a favourite game?
How can you vary the game?
Think of these two questions while
we play the game …
Why will it be a favourite game?
It has to be quiet (and it will be!)
It´s easy to understand the rules and it´s
easy to carry it through
It will reveal which pupils have a good
number sense
How can you vary the game?
Make smaller grids, e.g. 2 2 squares
Make grids for e.g. subtraction
Appoint different masters by playing in different languages
Use a ten-sided dice instead (change the total, e.g. 5000)
If you are going to learn to count in a new language it´s more natural to count to ten.
http://www.zompist.com/numbers.shtml
1 3 5 4
2 8
7
6 9 10
How can you vary the game?
Put a decimal point in the grid
Change the total, e.g. 5
+
.
.
.
.
4-sided 6-sided 8-sided 10-sided 12-sided 20-sided 30-sided 100-sided
… and a lot more of different kinds
What?
An inspiring furnished and
with hands-on materials
well-filled place
A way to work hands-on
A way to relate to
mathematical
education
A lot of photos from Swedish
schools, showing WHAT a math
lab can look like.
Make use of the walls, windows, floors
and ceilings
Different kinds of shelves: some old, some
new, some big, some small, some open,
some covered
Boxes, tins and drawers
Water
Interiors
Make use of the walls,
windows, floors and ceilings …
Pink horses 8 %
Grey horses 28 %
cloth pegs
A half square metre …
Different kinds of shelves:
some old, some new
some big, some small
some open, some covered
Boxes, tins and drawers …
Cardboard boxes wrapped in wallpaper
Water …
Interiors …
Probably the
smallest math lab
in Stockholm …
Probably the
second smallest
math lab in
Stockholm …
How tall are you?
Some Swedish words:
Cirkel
Rektangel
Triangel
Kvadrat
More Swedish words:
Addition
Subtraktion
Multiplikation
Division
What number is missing?
1, 2, ?, 4, …
What is the next number?
1, 1, 2, 3, 5, 8, 13, 21, ?
Box files
The maths grocery/shopExtremely popular!
Use empty packages
Let the pupils produce groceries
Use money
To …
reach educational goals
increase the interest in – and knowledge in – mathematics
create variation
make the subject mathematics visible
broaden the approach to mathematics
draw out curiosity and creativity
individualize – both extra support and extra challenges
support language development
develop different competences
…
matches
straws bamboo sticks
Trunks of
trees –
not too big :-)
You may use …
All you need is 12 sticks of the same
length
= a length unit = l.u.
= an area unit = a.u.
perimeter
area
= 12 l.u.
= 9 a.u.
perimeter = 12 l.u.
area = 8 a.u.
Instructions:
1. It always has to be the perimeter 12 l.u.
Challenges:
Try to find polygons with 7, 6, 5, 4 and 3 a.u.
2. It has to be one polygon every time
Definition of a
polygon:
plane figure
bounded by a
closed path
a finite sequence of
strait line segments
area = 7 a.u.
area = 6 a.u.
area = 5 a.u.
area = 4 a.u. ?
A common question: Is it ok to use triangles?
Yes, it is ok. But what triangle to use?
A right-angled triangle? (half a square)
The hypotenuse is longer than one stick
2 1,414
A common question: Is it ok to use triangles?
An equilateral triangle?
Yes, it is ok. But what triangle to use?
Is the area 1/2 a.u.?
No, the area is less than 1/2 a.u. because the altitude is less than 1 l.u.
The area 0,43 a.u.
Area: = =base · altitude 2
1 3
2·
2 3
4 0,43 a.u.
1 1
1
x
60°
60°
60°
Area of an equilateral triangle?
b
a c
a2 + b2 = c2
Altitude: x2 = 12 – (1/2)2 x = 32
a
b
c
sin( ) = a = c · sin( )ac
1 1
1
x
60°
60°
60°
32
x =1 · sin(60o) = l.u.
Area of an equilateral triangle?
Area: = =base · altitude 2
1 3
2·
2 3
4 0,43 a.u.
area = 4 a.u.
area = 4 a.u.
area = 6 a.u.
3 – 4 – 5
Egyptian triangle
Proof: Pythagorean theorem
32 + 42 = 52
area = 5 a.u.
area = 4 a.u.
area = 3 a.u.
Summary:
Easy to find material (sticks)
Easy to introduce the activity
The material is necessary
More mathematical content than you perhaps first realized
It can be a challenge even for gifted pupils
One perimeter – several areas
…
12 sticks to each pupil or pair of pupils.
Basic understanding of the concepts of perimeter and area
measuring and logical reasoning
length and area units, polygons
Ask the pupils what they know about perimeter and area. What do they think about the connection between perimeter and area? If one is altered, what happens with the other? Hypotheses?
I have to talk more about area units before we start next time I use this activity.
In figures with the same perimeter the areas can differ.
To avoid the common mistake when pupils think that if you alter the area the perimeter (always) will alter at the same time.
We will use the activity ”Area with sticks”
Were your hypotheses correct? Why or why not?
How can you express the connection between perimeter and area in general? Give some examples.
They can use the maths hands-on workshop journal.
Ask them to give an example of their own.
Cheap material
Little storage space
A lot of different mathematical contents:
Perimeter
Area
Pythagoras
Geometrical figures
Patterns
Algebra
Logical thinking
Spatial reasoning
…
A tessellation is created when a shape is repeated over and over
again covering a plane without any gaps or overlaps
Maurits Cornelis Escher, a Dutch artist, 1898–1972
1 = ett
2 = två
3 = tre
4 = fyra
5 = fem
6 = sex