johan häggström elisabeth rystedt - ncm:s och nämnarens...

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