06/07/2022

Mathematics Education to Develop Students Agency
Part III: Shape and Figure
Plane Figure and Properties

Student Agency:
Learned how to learn

Crucial issue is what content
students learned in relation tohow to learn. Depending on theobjective, learning content
itself changes.
(3)

(2)

On this lecture, we use SEABES: CCRLS to explain
objectives(1)

Review:
Please fill in (1), (2) and (3).
And so on!

ISODA Masami, Prof/PhD
Director of CRICED
University of Tsukuba, Japan

12

Mathematization (Freudenthal, 1973) and van Hiele Levels (van Hiele,1986)This lecture use SEA-BES: CCRLS to explain objectives

Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.

Appreciation(6)

van Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.
These terminology explain the process. Thus, if you can imagine
concrete materials for each class, you are able to develop it inyour classroom. That’s why we have been discussing materials.

Le ObjectvelOperation/
Means

Nature of language:ways of thinking

L1

Shape (形)Concrete object istreated by thehidden attribute ofeach shape

Reflection

L2(5)
All these are
the content oflearning!!

L3

Concrete
ObjectFigure(図形)Properties Proposition

Acquisition(平面図形:plane figure)(4)3

attribute

Attribute belonging
each figure is treatedas the properties offigure
Figures arerecognized byproperties.

Contradiction between levels
Round shape is not a circle
Round shape is not a circle(asfigure).
(as figure).

Fold
shapesandandfind
Fold roundround shapes
finda circle
wasa circle
whenwhenit wasitfolded
foldedjust overlap.
just overlap.
Circle isis drawn
drawn byCircle
bycompasswith centerandcenterradius.andcompass
withradius.
How can we provide our students forthe opportunity to think by and forthemselves, and learning how to?

Contradiction is theopportunity to think anddiscuss mathematically
Square,rhombusand
andSquare, rhombus
rectangle
are differentfigure(shape).
rectangle
aredifferentfigure
PropertiesPropertiesof figure are(shape).
of figureattributeof figure(shape).
areattributeof figure(shape).
Square is rhombus.

Squareisrectangle.
rhombus.
Square is(8) if ... .
Squareisisrectangle.
RhombussquareRectangle is... .
Rhombusis rhombus
square ifif ...
Rectangle is rhombus if ... .
By Isoda such as 2015 4and 20181

06/07/2022

On L2, the names of figure are given

Pose questions (task) todo activities: Why?

Make an order!
ab

How technical terms
are introduced.
Shape on L1 has hidden
attribute:
Ex. The a, h and J are
triangular (三角) and thec is possible.

scde

fghi

Triangular Rice ball.
(三角おにぎり)Figures on L2 have each name withattribute:
Ex. The a, J and k are triangle.

How technical terms areintroduced.
5Plane figure on L4 can extend: Ex. The a, f, J and k are triangle.

6

Why is angle introduced like this?

If you would liketo know task
sequence by thepage by page:s

https://support.gakuto.
co.jp/r2s_sansu_transla
tion/

7

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06/07/2022

How do use each of these for the discussionsof parallel line and perpendicular line?

A Dialectic by the problem solving
approach to introduce parallel line.

Way of drawing 1: Procedure a→Way of Drawing A; Task 2
If you want to draw the model,
draw lines spread evenly apartfrom the top edge of the paper.

Way of drawing 1: Procedure b→Way of Drawing B; Task 2
If you want to draw the model,
draw lines spread evenly apartfrom diagonal.
9How do use each of these for the discussionsof parallel line and perpendicular line?

10

How do use each of these for the discussionsof parallel line and perpendicular line?

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Mathematization (Freudenthal, 1973) and van Hiele Levels (van Hiele,1986)

Produce Plane by Figure

Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.

Tessselationvan Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

Discuss figure
bycomponents.
Discuss
figures by therelationships.

Le Object
velOperation/
Means

Nature of language:
ways of thinking

L1

Shape (形)Concrete object istreated by thehidden attribute ofeach shape

L2

L3

Concrete
ObjectFigure(図形)Properties Proposition(平面図形:plane figure)13

attribute

Attribute belonging
each figure is treatedas the properties offigureFigures arerecognized byproperties.

Contradiction between levels
Round shape is not a circle
Round shape is not a circle(asfigure).
(as figure).

Fold
shapesandandfind
Fold roundround shapesfinda circle
wasa circle
whenwhenit wasitfolded
foldedjust overlap.
just overlap.
Circle isis drawn
drawn byCirclebycompasswith center
andcenterradius.andcompass
withradius.
How can we provide our students forthe opportunity to think by and forthemselves, and learning how to?

Contradiction is theopportunity to think anddiscuss mathematically
Square,rhombusand
andSquare, rhombus
rectangle
are different
figure(shape).
rectangle
aredifferent
figureProperties
offigureare(shape). Properties of figure
attributeof figure(shape).
areattributeof figure(shape).
Square is rhombus.

Squareisrectangle.
rhombus.
Square is
Squareisisrectangle.
Rhombussquare if ... .
Rectangle is... .
Rhombusis rhombus
square ifif ...
Rectangle is rhombus if ... .
By Isoda such as 201514
and 2018

4