14/06/2022

Universityof Tsukuba
Japan

Mathematics Education to Develop Students Agency:
Part III: 1. Problem Solving Approach,Lesson 4. Dialectic Approach and Task Sequence

Lesson Study : DesignApproaches based oncontent of teaching

Review of task sequence for PSA

Masami Isoda
PhD/Prof, h.c.PhD (KKU), h.c.Prof(USIL)
Director of CRICED, University of Tsukuba, JapanIf we make it in order,
it becomes beautiful.

Beautiful!

Not beautiful

Task 1Model Curriculum SEA‐BES: CCRLS for Math'If your saying is true, what will happen?'

Mathematical Thinking : What it isMotto of thecourse: By using
what welearned, wecontinue
furtherlearning

Universityof Tsukuba
Japan

Task sequence on the textbook
Part I. Number and Operations: Lesson 2~10

Repetition by using what students learned.

Task 2
Appreciate learned, again and again….

1

task sequenceAtiude,tMathematical
WayofThinking,Thinking

andIdeas

Acquisition

 Imagine your students canapply it or not. If not, set
Task 1 for reviewing thecomplementary of what
they should already learned.
That is a most necessaryJapanese Textbook (Gakkko Tosho) is anpreparation for today’s
exemplar which provide theclass.
 Then, plan to do discussion
appropriate task sequence.
by using students’ solution
Further
Step: General mannerof the task 2 (problem)
Design unit plan or curriculum via using terminology
which is existed on youragiri&K(1987):
taagiri(2012)

Design PSA’s task sequence

what already taught.

Through repetitions ofappropriate (well designed)

Reflection

preparing tomorrow’s class Enjoy problem solving on
your textbook by yourself
for knowing how we cansolve it by only using

Couse of Study by MEXT, JAPAN (2017)

First Step: Local manner for

Couse of Study by MOE, JAPAN (Shimada, 1956)
KatIsoda

How students are able toreflect and appreciate by andfor themselves?

incontent

Contradiction which is a part of Problem Solving Approach
Authentic
Math. Act.
Oriented

Universityof Tsukuba
Japan

Done by teachers who

Mathematics is the subject
Students do Math. likewell Know Mathematical
Mathematician without Teachers for overcoming contradiction
Activity

Math.
Critique

Reference

Appreciationteachers need to enjoy theactivity to design task
sequence for which
students can thinkmathematically bythemselves.

Dialectic Approach (since Isaoda 1991): For Overcoming

Math. Value &Attitude

Universityof Tsukuba
JapanMath. Thinking
and Ideas

Education

textbook which has appropriate
task sequence:

Students Agency inGeneral

Objectivesof Mathematics
In case, we do nothave theIf your saying istrue, what will
happen?

DialecticApproach
Designing up to overcoming contradictio

Problem Solving Approach

Shift to Designing Task‐sequence.

PFroor b
stluedments, any task is open as long as
students use what they learned
Designing Task only
Change the task to Open‐EndedOpen Approach

Shift it to Students Centered

Math treats only true.
Exam.
Injection Approach
Teacher Centered
Oriented
Done by teachers who do not design task and lesson by themselves
Approaches should be selected by teachers depending on what students we would like todevelop. Good teachers can manage various approaches and control them as theirpreference because they know these as parameters depending on the objective of the class.

1

14/06/2022

University
Mathematics should not have the contradiction;of Tsukuba
JapanHowever, School Mathematics Curricula have huge numberof contradictions from the perspective of learners.

Process of Mathematization and Extension & Integration under
the perspective of Conceptual and Procedural Knowledge

Universityof TsukubaJapan(Isoda & Olfos, 2021; In Japanese Isoda, 1991): Learners met the difficulties because they could only
think by using what they already learned as long as we ask students to use what they learned.
Do Division by
Multiplication?

That is a difficulty of learning mathematics for students!
Ex. the extension of numbers
 Multiplication increase the number! → NO! Not always!
 Subtract smaller number from lager number! → NO! Any number can subtract!

𝟐𝟓+ 𝟑X1= 𝟒?
𝟐3𝟓Could we teach these conclusions (no parts) from the beginning? We could not, right?
What we can do is to develop students who challenge these contradictions!
It is the nature for mathematics which was known as a Hegelian, Imre Lakatos (1976) on his book ‘Proofs
and Refutations.’

Where shall we pu t thedecimal point in?

Product is lager than…

Dialectic which has been known by Ancients Greece: it begins from “If your saying (conclusion) is true,
what will happen?” In mathematics, it is a kind of action through analysis and embedded into
mathematics system as the proof by contradiction (reductive absurdum/reductio ad absurdum) and the proof
by contraposition. In German didactics, it is Socratic method of eliciting ideas latent in the mind of anotherby a series of questions and answers, called the maieutic method.

23x4= 23+23+23 +23?
or= 20x4+3x4?

Thus, the principle of curricula is necessary such as: Mathematization (as reinvention by Freudental, H. 1973):

23x4= 20x4+3x4?
or on Column
Row: 4x3 & 4x2?

Reorganize the experience in real world and in mathematics: the means of the thinking on the lower level become the subject matterof the thinking on the upper level.

 Extension and Integration in Japanese curriculum since 1968 to develop mathematical thinking.
The principles positively treat the contradictions as a part of mathematical activity in classroom for theopportunity to develop mathematical thinking.

Multiplication is
NOT the addition
but a binary
operation.

2x3= 2+2+2?
or= 3+3?

If your saying (learned/thinking) is true, what will happen?

5

Where does it come from?②
explains ① .
.

Schoenfeld, A (1986).

Isoda, M.(1991,1996)
Situation
I
Introduction of
calculation invertical notation
using wholenumbers (integer)
II
Becoming
proficient inwhole numbers

Meaning

Decimalnotationsystemmeaning(A)? (Forgotte n)(C)

If your saying is true, what will happen? For example….

AppropriatenessProcedure

Explanation

Write
23+5

The meaning of a decimal
notation system is basedon the procedure of
keeping decimal points inalignment. (The meaningand procedure match)

Appropriate

When children become
proficient, they no longer
need to think about thereason they follow thatprocedure. As a result, theprocedure is simplified
from the alignment of thedecimal points to one ofright-side alignment.

Valid by
Proficiency(Procedure
itself has
meaning insome cases)(B)

Align tothe rightand write

Where does it come from?②
explains ① .

Universityof Tsukuba
Japan

Situation
I
Introduction of
calculation invertical notation
using whole
numbers (integer)
II
Becoming
proficient inwhole numbers

Meaning

Decimalnotation
systemmeaning? (Forgotte n)(C)III
Application to
the decimalnumbers(No meaning)

Universityof Tsukuba
Japan

Schoenfeld, A (1986)

Isoda, M.(1991,1996)
AppropriatenessProcedureExplanation

Write
23+5

The meaning of a decimal
notation system is basedon the procedure of
keeping decimal points inalignment. (The meaningand procedure match)

Appropriate

When children become
proficient, they no longer
need to think about thereason they follow thatprocedure. As a result, theprocedure is simplified
from the alignment of thedecimal points to one ofright-side alignment.

Valid by
Proficiency(Procedure
itself has
meaning insome cases)(B)

Originated from
Extension from Whole Number to Decimal NumberIII(No meaning)
Align to the The procedure for whole
InappropriateApplication
tois generalized for (Contradiction)
Programmed
Emergence ofright
Misconception
in Curriculum/Teaching
and write numbers
the decimal
decimal numbers.
Anyone cannot avoid: It is an Epistemological Obstacle.
7numbers(A)

Review ofPart I. Lesson
14~17 6

Align tothe rightand write

Align to the The procedure for whole
right and write numbers is generalized for
decimal numbers.

Inappropriate(Contradiction)

Background 82

14/06/2022

To introduce parallel, Mr. Masaki started by drawing asample lattice pattern. The following process shows how
students emagent the idea of parallel in case of the no
instruction of the definition of parallel. (Isoda, 1991, 1996)
Sample 1

Task 1. Let’s draw
the sample 1
lattice pattern

Way of drawing
pattern 1

Sample 2

Way of Drawing A:
Even intervals along theedges

Way of Drawing B:

Universityof TsukubaJapan

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.

Sample 1

Way of drawing
pattern 1

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.

Sample 2

Using the right diagonalline as the base

Way of Drawing A: Evenintervals along the edges

Moment of Dialecti

A lot of examples

Way of drawing
pattern 1

Sample 2

Using the right diagonalline as the base

Way of Drawing B:
Even intervals from theline

Synthesis: Define the parallel
line based on the difference

To introduce parallel, Mr. Masaki started by drawing asample lattice pattern. The following process shows how
students develop the idea of parallel in case of the noinstruction of the definition of parallel. (Isoda, 1996)

Universityof Tsukuba
JapanSample 1

Way of Drawing A:
Even intervals along theedges

Dialectic Discussion:“What? ”“Why?”

Even intervals from theline

Synthesis: Define the parallel
line based on the difference

Even if teacher explained many times,
there are diversity of children’s
understanding and ways of utilizing
learned ideas.
Why? Because not yet learned children
can not distinguish special ideas
and general ideas.

To introduce parallel, Mr. Masaki started by drawing asample lattice pattern. The following process shows how
students emagent the idea of parallel in case of the no
instruction of the definition of parallel. (Isoda, 1991, 1996)

Task 1. Let’s draw
the sample 1
lattice pattern
Task 2. Let’s draw
the sample 2
lattice pattern

Using the right diagonalline as the baseTask 2. Let’s draw
the sample 2
lattice pattern
Dialectic Discussion:“What? ”“Why?”

Universityof Tsukuba
Japan

Background

Review Part II. Lesson 1. What is fraction?

Universityof Tsukuba
JapanIn mathematics, if we find the contradiction, we have to think𝟐m𝟑

Way of Drawing B:
Even intervals from thelineIsoda, M. (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as the Ground Theoryfor Lesson Study in Mathematics Education. DOI 10.1007/978‐3‐319‐17187‐6_21

Read Appendix

If your saying is true, what will happen? For example….

123

14/06/2022

Review Part II. Lesson 1. What is fraction?

Universityof Tsukuba
Japan

Review Part II. Lesson 3. : Addition with different denominator

Universityof Tsukuba
Japan

In mathematics, if we find the contradiction, we have to think𝟐m𝟑

If your saying istrue, what will
happen? Forexample….

Isoda, M. (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as the Ground Theory
for Lesson Study in Mathematics Education. DOI 10.1007/978‐3‐319‐17187‐6_21

Read Appendix
Background

Review Part II. Lesson 3. : Addition with different denominator

Isoda, M. (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as the Ground Theory
for Lesson Study in Mathematics Education. DOI 10.1007/978‐3‐319‐17187‐6_211314

Universityof Tsukuba
JapanUniversityof TsukubaJapan

Bordwriting (Bannsho): Sample• Need the copy from my book

Isoda, M. (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as the Ground Theoryfor Lesson Study in Mathematics Education. DOI 10.1007/978‐3‐319‐17187‐6_21

Background15164

14/06/2022

Dialectic Approach (since Isaoda 1991): For Overcoming
Contradiction which is a part of Problem Solving ApproachDone by teachers whoMath.
CritiqueMath. Value &Attitude

Math. Thinking
and Ideas

Mathematics is the subject
Students do Math. likewell Know Mathematical
Mathematician without Teachers for overcoming contradiction
Activity

Students Agency inGeneral

AuthenticMath. Act.
Oriented

Universityof TsukubaJapan

If your saying istrue, what will
happen?

DialecticApproach
Designing up to overcoming contradictio

Problem Solving Approach

Shift to Designing Task‐sequence.

PFroor b
stluedments, any task is open as long as
students use what they learned
Designing Task only
Change the task to Open‐Ended

Open Approach

Shift it to Students Centered

Math treats only true.
Exam.
Injection Approach
Teacher Centered
Oriented
Done by teachers who do not design task and lesson by themselves
Approaches should be selected by teachers depending on what students we would like todevelop. Good teachers can manage various approaches and control them as theirpreference because they know these as parameters depending on the objective of the class.

5