30/06/2022

Mathematics Education to Develop Students Agency
Part III: Shape and Figure
Figura and AtributeStudent 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)
(1)

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

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

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

1

This lecture use SEA-BES: CCRLS to explain objectives

Appreciation(6)

2Review of Part III, Lesson 4

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.

Long term task sequence(7)

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

Acquisition

Another terminology is theterminology to explain
conceptual differences ofeach materials. It explainstask sequence.

(4)

341

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

Mathematization from L1 to L2

Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.
van Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

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)attribute

Properties Proposition(平面図形:plane figure)

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 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
aredifferentfigure
Properties
offigureare(shape). Properties of figure
attributeof 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 5and 2018

 Japanese had alreadydeveloped the task sequenceof it from 1934 textbook up to
1950s curriculum reform
because they had been
distinguished shape, figure andplane figure in Japanese at thebeginning of school system in
1870s.
 If we use appropriate
terminology to distinguishconceptual differences, we canproduce appropriate task
sequence for students’ activity.
 What terminology is the key
on those two pages’ activity?
Shape, Attribute via
collecting,(9)touching,stacking and making

6

Why this activity
is necessary?
Those pages are NOT
independently existed.
It has task sequence
beyond chapters andgrades as well as othercontent-strands.

Let’s explain byusing terminology.
Did you use?
Shape, Attribute via tracing, cuttingand drawing(10)

https://support.gakuto.co.jp/
r2s_sansu_translation/782

30/06/2022

On L2, the names of figure are given

Shape on L1 hashidden attribute:
Ex. The a and h aretriangular (三角) andthe c is possible.

sTask 3Teaching is theactivity for students
to be able to thinkon upper level.
Triangular Rice ball.
(三角おにぎり)

Task 2

Figures on L2 have each name withattribute:
Ex. The a and k are triangle.

Task 1
910

Plane figure on L4 can extend: Ex. The a, f and k are triangle.

How technical terms are introduced.
Which level is it?
It is preparation of L3.

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

However, it’s still L2.
Why?
For shifting L3, students are necessary tolearn the technical terms which explain theattributes (components) of figure on L2.

shttps://support.gakuto.
co.jp/r2s_sansu_translation/

Ex. Technical term to explain attribute:side, right angle, same length of sides,angle, vertices, ….
On L2, the attribute (components) isbelonging to figure.

s

On L3, The figure is given by the propertiesof its components: It is the properties offigure, not the attribute of figure.
On L3, the figure is given by components asconditions. Then, they are able to compare
the condition of figures.

It is necessity to learn technical terms in mathematics for students

11

The ‘corner’ is the attribute of shape for 12
concrete materials, if it is angle, how different?

330/06/2022

How do you explain the objective of each task?
Let’ read!

Categorization

For What?

13

14

Mathematization (Freudenthal, 1973) and van Hiele Levels (van Hiele,1986)
Freudenthal defined Mathematization by the re-organization of (mathematical) experience by mathematical means.
van Hiele Levels (Level of Thinking by Freudenthal): Teaching is the activity for students to be able to think on upper level.

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)attribute

Properties Proposition(平面図形:plane figure)

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 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
rectanglerectangle
arefiguredifferent
figureare different(shape).
(shape).
of figure
PropertiesPropertiesof figure areareattributeof figure
attributeof figure(shape).
(shape).
Square is rhombus.

Square is rhombus.
Square is rectangle.
Squareis isrectangle.
Rhombussquare if ... .
Rhombussquare ifif ...
Rectangle isis rhombus... ..
Rectangle is rhombus if ... .
By Isoda such as 201515
and 2018

4