APEC-Tsukuba Intrenational Conference
Developing mathematical thinking through reflective experience
December 10, 2007

Mathematics Education
andReflective Experience: The Significance of “Unlearning” in Mathematics Education

College of Education , Ibaraki University

Hiroshi Nemoto

1 Acquisition of Viable Knowledge
1-1 Flexible Understanding and Viable Knowledge: Thinking Like a Human
1-2 Presentiments of the Future and Mathematical Activities: Transcending the Sensory Domain

2 Reflective Experience and Mathematical Activity
2-1 Mathematical Activity and Insight: Observing That Which Keep out of Sight
2-2 The Importance of Reflective Experience: Connecting with the Future

3Suggestions for Improving the Teachingof Arithmetic/Mathematic
3-1 The Importance of Meaningful Experience: Reflect on oneself (Introspection)
3-2 Improving Arithmetic/Mathematics Teaching: Encouragement of Unlearning

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The Course of Study (Gakusyusidouyouryo in Japanese)for Lower Secondary School
Overall Objectives

For the students to understanding deeply the fundamental
concepts, principles, and rules relating to numbers, quantities,figures, and so forth. For students to acquire methods of mathematical expressions and strategies, and to improve their ability
to relate phenomena mathematically. For students to enjoy
mathematical activities, to appreciate the importance of mathematical approaches and ways of thinking, and to inculcate inthem the right attitudes necessary to make use of mathematics.

Problem : Which is larger, 5/13 or 11/19?

Chap.1.

Both fractions are approximately 0.5 (1/2).
5/13 → if the numerator were 6, 6/13 < 6/12 (0.5).
(∵ the denominator is larger.)
And since 5 is one less than 6, 5/13 < 6/13.
Therefore, 5/13 < 0.5.
11/19 → if the numerator were 9, 9/19 ≒ 0.5.
11/19 > 9/19 (≒ 0.5). (∵ 11 is two greater than 9.)
Therefore, 11/19 > 0.5.
∴ 5/13 < 0.5 <11/19, and therefore, 5/13 < 11/19.
Both fractions are approximately 0.5 (1/2).
5/13 → If the fraction were 5/10, 5/10＝1/2. 5/13 < 1/2.
(∵ the denominator is 13.)
11/19 → 10/20 is also equal to 1/2, 11/19 > 1/2.
(∵ the numerator is larger than 10 and its denominator is smaller.)
Therefore, 5/13 < 1/2 < 11/19. Consequently, 5/13 < 11/19.

This shows just how flexible human thought is.

2fundamentals/basics in life
Common to both reasoning
appropriate standards to judge… the action of setting some sort of a standard is simplistic,even in our daily lives, but it is happening naturally.

viable knowledge… the setting of a certain standard (viewpoint) for making a judgment,
which is used when pursuing ordinary thoughts, is fundamental/basic
viable knowledge, and even in learning the size relation of fractions, it isimportant to be aware of the fact that you are acquiring such knowledge.

evolvability
Knowledge is what prompts the subject (person) into acting according
to the situation. Knowledge can be thought of as something having
built-in evolvability, in the sense that it spurs new actions.
Generally speaking, the methods and concepts that are taught
are alive in the minds of students, and therein lies evolvability.
… a close relationship is forged between concepts and between
methods, and those concepts and methods take root deep in the mind,
resulting in the potential for self-evolvability.
Ryoichiro Sato

Mathematical facts can be conveyed through language, and students can probably learn
that. However, it is wrong to readily think that mathematics learned this way is transformedin their knowledge. Furthermore, it must be noted that this is nothing more than ready-madeknowledge prepared by adults. If it does not have any impact on the future action ofstudents, then it cannot necessarily be called viable knowledge.

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Cognitive Conflict
2 is less than 3

1/2 is greater than 1/3

The real problem is not one of “form” (i.e.
the fraction x/y), but rather it is a matter ofentering into a world of numbers that havea different relational structure that what
was known thus far.
One difficulty, no doubt, is that mathematics requires such refined andcomplicated symbolic techniques for its expression. The teacher who
becomes so utterly engrossed in mastering techniques for operating withsymbols that he forgets their conceptual background is fated to lose theinterest of his pupils and will also leave them with some of the misconceptionsof mathematics that are so current today. (R. L. Wilder)the initial teaching of fractions
magnificence of mathematical activity that ventures into unknown worlds

Problem
1. Compare the size of fractions that havethe same denominators.
(2/9 5/9) , (4/5 3/5)
2. Compare the size of fractions that havethe same numerators.
(1/2 1/3) , (2/5 2/9)
Fig.1.

linguistic knowledge

The generality of a) and b) is
not thought to berecognizable simply with apassing look at Figure 1. !!

●Among fractions with same denominator, the fraction with the larger numerator
is greater. …… a)●Among fractions with same numerator, the fraction with the smaller denominatoris greater. …… b)●There are many that are the same size even though they have different numeratorsand denominators. …… c)4

Chap.2.

Observation andthe Representation Process

Looking back upon what one did isnothing more than thinking about
what significance the action
you just did holds foryourself.

Among fractions with samenumerator, the fraction withthe smaller denominator isgreater. Etc.,

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Mathematical activity &. Knowledge
Knowledge is organized within the individual (student)
the careful observation of a factactions and mental activity
Human beings have the ability to engage in action
within the mind and action transcending facts.

(1) The success or failure of problem solving does not necessarily depend only on
the memorizing of algorithm-like knowledge.
(2) The recognition of a general mathematical fact (prior knowledge)
accompanies the non-verbal aspect of knowledge in that area.
(3) The organization of knowledge signifies an overall systematic understanding,and the representation process is crucial therein. It is mathematical behavior
that underpins this. In particular, mental behavior is key.

reflective experience &. physical experience… carefully observing an object andperceiving the mathematic relation,
universal nature, and generality that lie
behind it… the partial and sensory
relation to an eventthe verbal memory of a) and b)● You should focus on either the numerator ordenominator to judge fraction size. (Re1)● You can judge size by transforming fractions into
the same numerator or denominator. (Re2)

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Reflective experience(1) What constitutes experience?
(2) What decides the value of experience?
(3) What is the difference between experience and mere activity?
(4) What is meaningful experience?

tryinginner
enemy

Subject

object(leaner)< <<< <<
undergoing << << <Return wave

Transcending“I’ve Done ItBefore”

Which is larger, 5/13 or 11/19 ?
●Fractions given the same
numerator and denominatorare greater than then original
fractions ． (d)ab ≦a+mb+ma0＜ b ＜ 1( b＞a＞0, m≧0)

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Suggestions for Improving theTeaching of Arithmetic/Mathematics

Chap.3.

Meaningful Experience
Mathematics instruction is focused too much on mathematical content andnot enough on mathematical behavior. If we want our students to become
active learners and doers of mathematics rather than mere knower ofmathematical facts and procedures, we must design our instruction to help
develop their metacognition.
Leroy G. Callahan and J. Garofalo・make sense that links to the future・action where a conscious effort is made to discover a rule

Symbolic Initiative
Man possesses what we might call symbolic initiative; that is, he canassign symbols to stand for objects or ideas, set up relationships between
them and operate with them on a conceptual level. …… However, much ofour mathematical behavior that was originally of the symbolic initiative typedrops to the symbolic reflex level. …
R.L.Wilder
・ … a place must be provided for experience that promises highlevel standards, and behavior that can foresee leaps to the future.

8A
Problem ： Show that the area of square ABCD in thefollowing diagram is equal to the shaded area.

roBr ; radius of the circle

Darea of the inscribed square2 r 2 (= (√2･r ) 2 )Carea of the shadeda result will be obtained
after carrying out theproper calculation. …

Hippocrates’ Lunes
ABC
right triangle ABC“The area of the figure that is enclosed by the half circle that forms the diameterof the hypotenuse and the half circle that forms the diameter of the two sides
that are between the right angle of the right triangle is equal to the area of thatright triangle.”

This fact should also hold true even forright isosceles triangles.

9AB

M

C

The figure in the problem will beachieved anytime, anywhere, and byanyone by rotating around point A as
Athe center.

B (M′)D (M)CThis sort of understanding is formed
clearly within the mind …

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Improving Arithmetic/Mathematics Teaching
Encouragement of Unlearning

Actively Monitoring One’s Own Progress
Trying to understand What You Are・ the labor of making sense in one’s own way for self-action.

Questioning Yourself About What You Are Doing・It is also about questioning yourself about what you are doing
while enabling yourself to think.

Trying to Put What You Know to Work・In short, the meaning of the phrase “It is better to know than not toknow” is slighting different. What is important is how that knowledge
was come to be known.

The Importance of the Word “Oneself”
Reflect on Oneself （Introspection）
The point of evaluation is self-evaluation. … Taking responsibility forwhat you have done is a requirement for living in society, but appropriate
self-evaluation is required to ensure this. The learning of mathematics can bethought of as something that gives students the ability to do that and further
refine it themselves.

Let him know that he is expected to be making discoveries all the time;
not merely that the best established law is not complete, but that is thevery simplest things it is not so much what he is told by a teacher, butwhat he discovers for himself, that is of real value to him, that becomespermanently part of his mental machinery. Educate through theexperience already possessed by a boy; look at things from his point ofview --- that is, lead him to educate himself.
J. Perry11

I learned many things at the university,
but I had to unlearn them later.
Among fractions with same denominator,(Helen Keller)

the fraction with the larger numerator
is greater.
……

Tsurumi said he came to understand her words as follows.
I imagined I had knit a sweater in a conventional manner and thenunraveled it into the yarn that comprised it and knitted it again while
holding it up to my own body. The knowledge learned at universities is, ofcourse, necessary.
However, it is of no use if it is simply memorized. Thatr
which results from unlearning such knowledge becomes flesh and blood.
o…… the spirit of this unlearning and the reflective
experience for making what you have learned a partof you are one in the same.

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