22/11/2021

SEAMEO School Network Program provided by CRICED-University of Tsukuba

Mathematics Education to Develop Students Agency:
The Case of Fractions
Teaching Mathematics to Develop Mathematical Thinking as Higher Order Thinking:
How do you teach? Why? IILesson 7: Multiplication and Division of Fractions (2)Lectured by: Masami Isoda, Prof/PhD., University of Tsukuba, Japan
With support of: Wahid Yunianto, Dr., SEAMEO-QITEP, Indonesia
With participation of:
Ms. Mei Nakada, Japan
Mr. Diego Solis, Costa Rica

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22/11/2021

SEAMEO School Network Program provided by CRICED-University of Tsukuba

Mathematics Education to Develop Students Agency:
The Case of Fractions
Teaching Mathematics to Develop Mathematical Thinking as Higher Order Thinking:
How do you teach? Why? II

Summary
Lectured by: Masami Isoda, Prof/PhD., University of Tsukuba, Japan
Reference:
Masami Isoda (2013). Fraction for Teachers: Knowing What before Planning
How to Teach. (2nd edition) Tokyo: CRICED, University of Tsukuba***To appear as a book with contributors.

Page 53.b, 54.a

Various Meanings of Fraction:

Number and its extension
 Existence and Necessity

Various Meanings of Fraction
Problematic caused by Dividing Fraction
Missing of the whole, difficulty of equally dividing, a pair of numbers, and less than one
Diving fraction is discussed as actions of equally dividing. It is good for learning thehalf and quarter. However, dividing into three, five, seven,…. we have to use the tools
for equally dividing such as a set of parallel lines with the same distance.
Dividing fraction enhances the shading activity with counting numbers of numerator
and denominators. Then, Dividing fraction produces the source of various
misconceptions because there is no discussion of whole, and it is a pair of numbers
not the number itself. It is a kind of ordered pair of numbers. For seeing it as a number,
we have to discuss equivalence class as Equivalent fraction.
Dividing fractions can not compare if the whole is not the same. To compare, we haveto fix the whole and add the information of quantity or denomination: Then, it isQuantity Fraction.

Beyond Dividing Fraction(In Math, verification of well defined fordelimiting as for Algebraic construction of
the number system) Compare and Equivalence Establish Four Arithmetic operations
See Previous Course:
Teaching Mathematics to Develop MathematicalThinking as Higher Order Thinking:
How do you teach? Why? I
Halves of square and circle cannot compare without fixing
measurement because bothfigures are representatives ofsimilar square and circle.
Let’s fold or draw the equivalence of fraction for a half.

What are missing?

Task sequence by using what already learned and Progressive renovationof the Meanings of Fractions
Fraction as Quotient (Ans. of Division)

Rational
NumberDividing fraction can not extend more than one without fixing the whole because it isa part-whole model.
Area model and shading model are usually used as for explaining procedures ofoperations by teachers however many teachers lack these understanding and missing
to notice necessary ideas. It supports students understanding of procedure
momentary however we are not sure these models/representations to produce theoperations by and for themselves.

for Students Agency: learned how to learn

Multiplication Division(how manychildren)(how manycandies for eachchild)
Minutes
Problematic ofmultiplication(how many candies foreach child)
xDistance
dividedby time
(how manychildren)

Quantities

Partitive Division Situations Division ofTo find how many candies foreach child
Dividing Fraction(Part-Whole relationship,
Equally dividing a whole but itis action and not the number)

Quotative Division
Situations
To find how many children
Operational Fraction(Based on Unit Fraction: Scaling
number line by the unit fraction,
it can compare)

Ratio Fraction as Ratio
Ratio (Rate) for
different quantity

Different
Quantities Related Speed(km) : (hour)

Population Density(person): (km2)
Division ofSame
Quantities
No Related!

Ratio (times of base)for same quantity
Part to Whole(Boys) : (Human)
Likely dividing fraction

Proportionalityx

Operational Fraction, Fraction as Quotient (Value of Division),
Fraction as Ratio
Dividing Fraction is based on Partitive Division.

Operational Fraction is based on Quotative Division.
Operational Fraction enhance Unit Fraction.
Unit Fraction is necessary for Improper Fraction and Mixed Fraction.
Unit Fraction is necessary for considering Equivalent Fraction.
Fraction is the Value of Division which is called Fraction as Quotient.
Fraction as Quotient is necessary to recognize fraction as number.

Grade 2

Grade 3

If multiplicands
increase by 1,products
increase by thenumber ofmultiplierGrade 5
Proportion

Part to Part in Whole(Boys) : (Girls)

Grade 7 y=ax2

22/11/2021

Various Representations or Models for Fraction
Proportionality is a key for understanding
multiplication and division.
Let’s Develop Students who Draw Diagram by andfor Themselves
Proportional number line with Tape Diagram is the tool for students
who develop multiplication and division by and for themselves if they
are able to draw.

Number and its extension
 Existence and Necessity(In Math, verification of well defined fordelimiting as for Algebraic construction ofthe number system) Compare and Equivalence Establish Four Arithmetic operations
See Previous Course:
Teaching Mathematics to Develop MathematicalThinking as Higher Order Thinking:
How do you teach? Why? I

Proportional number line makes visible Rule of Three on Table.
Proportionality is also represented by relationship between theexpressions of multiplication and division.

To show proportionality on the proportional number line, the rule ofthree and the relationship between expressions, arrows are necessary!
Isoda&Olfos (2021)

References

Area diagram is also useful if it is drawn with quantity or denomination.
If not, it provide the source for misconceptions based on dividing
fraction such as which part is the whole?

The textbook used in this course must be
cited as shown below:

Masami Isoda et al. (2010). Mathematics Education Theories for Lesson Study: Problem Solving Approach and the Curriculum through Extension and Integration, Journal of Japan Society ofMathematical Education, 92(11). 1-158.
Shin Hitotsumatsu et al. (2005). Study with your friends: Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda et al. (2010). Study with your friends: Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda (2012). Introductory Chapter: Problem Solving Approach to Develop Mathematical Thinking. In Isoda, M. & Katagiri, S. edited (2012). Mathematical Thinking: How to Develop it in theClassroom. Singapore: World Scientific. 1-28.
Masami Isoda, Tenoch Esaú Cedillo Ávalos, et al. (2012). Matemáticas para la educación normal (11vols). Estado de México : Pearson Educación de México.
Tenoch Cedillo, Masami Isoda, et al. (2013) Matemáticas para la educación normal : guía para el aprendizaje y enseñanza de la aritmética, Estado de México : Pearson Educación de México.

Masami Isoda (2013). Fraction for Teachers: Knowing What before
Planning How to Teach. Tokyo: CRICED, University of Tsukuba.

Masami Isoda (2015). The Science of Lesson Study in the Problem Solving Approach. In Inprasitha, M. et.al. edited. Lesson Study: Challenges in Mathematics Education. Singapore: World Scientific.
pp.81-108.
Masami Isoda (2015). Dialectic on the Problem Solving Approach: Illustrating Hermeneutics as the Ground Theory for Lesson Study in Mathematics Education. In Sung, J. edit. Selected Regular Lecturesfrom the 12th International Congress on Mathematical Education. Cham: Springer. pp.355-381.
Masami Isoda & Aki Murata edited (2011). Study with your friends: Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Aki Murata, Aida Istino Yap edited (2015). Study with your friends: Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda & Aki Murata edited (2021). Study with your friends: Mathematics for Elementary School (12 vols). Tokyo: Gakkotosyo.
Masami Isoda, Raimundo Olfos. Edited. (2021). Teaching Multiplication with Lesson Study. Cham: Springer. DOI: 10.1007/978-3-030-28561-6

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