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How to plan a lesson for developing Mathematical Thinking Kyozai Kenkyu ???? Research Subject Mater A priori Analysis vs Planning on aims Masami Isoda CRICED, University of Tsukuba

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Title: How to plan a lesson for developing Mathematical Thinking Kyozai Kenkyu ???? Research Subject Mater A priori Analysis vs Planning on aims Masami Isoda CRICED, University of Tsukuba


1
How to plan a lesson for developing Mathematical
Thinking Kyozai Kenkyu ???? Research Subject
Mater A priori Analysis vs Planning on aims
Masami Isoda CRICED, University of Tsukuba
It is a general question that can not be answered
without restriction.
APEC-KKU Conference 16.8.2007
Knowing and embedding the aims of education in
the lesson plan by the classroom problem is a key
for improvement.
Where do Mathematics problems come from?
Some classroom problems come from the extension
in curriculum.

2
The Perspectives of Describing Mathematical
thinking in relation to the future of
mathematics learning on the document by MEXT
(1999)
  • Mathematization Reorganization of experience
    through the reflection
  • The ways of conceptual development
  • Refutation is acceptable.
  • The World of Invariant Mathematics
  • Mathematics is the pattern of science.
  • Mathematical Ways of Thinking
  • G. Polya
  • Learning how to develop mathematics

3
Problem Solving Approach a model of the lesson to
develop Mathematical Thinking
  • Teachers begin by presenting students with a
    mathematics problem employing principles they
    have not yet learned.
  • They then work alone or in small groups to devise
    a solution.
  • After a few minutes, students are called on to
    present their answers the whole class works
    through the problems and solutions, uncovering
    the related mathematical concepts and reasoning.
  • (from Teaching Gap. J.Stingler J. Hiebert)
  • Problem Posing
  • Predict the methods for Solution
  • Solving
  • Discussion
  • Reflection

In the process, children learn mathematics but
this process is not aimed to represent it. It
focuses on how to guide childrens activity.
4
Previously learned (Known Task)
5
Problem Solving Approach a model of the lesson to
develop Mathemetical Thinking by Child Centered
Approach
Phase Teacher Children
Problem Posing Posing the problem with the aim Given the problem without knowing the aim
Prediction Guiding childrens approach Waking both known and unknown in each Child
Solving Supporting individual works Clarifying and bridging known and unknown by each child.
Discussion Guiding discussion to the aim Working on the aim of bridging between known and unknown by all
Reflection Guiding the reflection Valuing the aim
6
Necessary Activity for developing the lesson
  • Plan the lesson with following problems
  • Find the following pairs of numbers
  • ?? ?-?
  • X Y X-Y

What kinds of thinking?
What if? What if not?
Generalization
(0,0)
(1,?)
(1/2,?)
(1,1/2)
Speicalzation
Yx/(x1)
(3,3/4)
7
Necessary Activity for developing the lesson
Mathematical Thinking What If
Correct Answer
Mathematics Problems
Other Approaches
Wrong Answers
Conditions of selection
Developing New Prob.
What can children learn from the process?
The Aim of Lesson, Mathematical Value
Anticipating Childrens Activity
Embedding the aim
For controlling childrens Activity on the aim
Classroom Problems in order / sequence (ways of
posing in the process of teaching)
  • Problems are used for children enabling them to
    engage in rich mathematical activity and lean
    from the reflection.

8
(1/2, 1/3) (1/n, 1/(n1))
Finding examples (0,0), (1,1/2), . Solving
generally y x/(x1)
Finding the aims
Find following pairs of numbers ?? ?-? X Y X-Y
  • Number Pattern
  • Invariant vs Variant
  • Specialization
  • Inductive Reasoning
  • Generalization
  • Power of Symbol
  • Sequence, Integra

Introduction of Symbol
(1/?)(1/?) (1/?)-(1/?)?
Why this problem is interesting for teachers? For
children?
?? ?-? Guessing examples (0,0),
(1,1/2),.Finding general pattern
(1/?)(1/?) (1/?)-(1/?) How to represent
general pattern?
(1/?)(1/(?1)) (1/?)-(1/(?1))
How can children recognize it as problematical?
9
In case teacher does not have information of
children. For example. Seiyamas lesson
????
How can children recognize this very strange
phenomenon?
10
Necessary Activity for developing the Lesson Plan
Classroom Problems in order (ways of posing in
process)
How to be clear of the aim and the value of math.
in process.
Set Childrens Activity through the questions
Plan of Activity, Questioning for Interaction and
the Black Board Writing
Planning the Black Board writing with
consideration of classroom interaction is the way
to consider real classroom setting.
11
Where do problems come from?
  • It is true that some mathematics problems
    originated from daily situation but the problem
    for each lesson does not always come from daily
    situation.
  • Most of mathematics problems are sited in the
    textbooks, source books and exercise books.
  • Thus, mathematics problems come from textbooks
    (or curriculum).
  • Enabling students to recognize the aims and
    contexts, teachers should embed the aim of the
    lesson into their classroom problems on their
    teaching plan.
  • Even if problems are described in daily
    situations, we are not sure that students can use
    the related mathematical ideas on the daily
    situation if students cannot recognize aims and
    contexts to use it.

12
How to use students misconceptions or wrong
ideas as the key problem in the mathematics
classroom?
But, a child answer is ..
Wow,Teacher used Childrens idea!
How many?
How to calculate?
? ??? ????? ??????? ????????? ???????????
What is the teachers or your expectation?
Oh, NO! If you are the teacher, what can you do?
? ??? ????? ???????
Why did teacher ask this question?
Important Math.Thinking Generalization,
Application, Value of them Faster, Easier,
Reasonable ,
13
How can we use students misconceptions in the
mathematics classroom? Teachers Theory for
Problem Solving Approach
Why do teachers feel that a misconception/misunder
standing is a problem? If it is not expected, it
must be a problem but it is expected The
teaching approach itself included the pedagogical
value/educational aim.. For teaching Ways of
Math. Thinking, Math. Communication,
and Developing Math. Using others idea is
basic reasoning
But, a child answer is ..
Wow,Teacher used Childrens idea!
How many?
How to calculate?
? ??? ????? ??????? ????????? ???????????
What is the teachers expectation?
Oh, NO! If you are the teacher, what can you do?
? ??? ????? ???????
Why did teacher ask this question?
Important Math.thinking Generalization,
Application, Value of them Faster, Easier,
Reasonable ,
What is Teachers Theory? the theory for
developing children, not for observing children
like Jean-Henri Fabre (a famous entomologist,
famous observer). It is the theory of supporting
the development of teachers eyes for educating
the children. Its usually developed with
teachers who are working on the Lesson Study
(Plan Do See).
Here, we use the terms Meaning and Procedure in
planning the lesson to develop childrens
mathematical knowledge based on the curriculum
14
Based on the nature of mathematics learning
Because it is the evidence of Thinking
Mathematically
  • Why should we use misconceptions?
  • Students use what they already learned.
  • Students try to use easier procedure.
  • Explain misconception with meaning and procedure
    from what students learned before.
  • Dialectic ways of discussion for teaching
    mathematical thinking and developing mathematical
    ideas.
  • Categorizing students ideas from the meaning and
    procedure.

Because it is the nature of Mathematics
Curriculum including the extension sequence
15
Explain Misconception with Meaning and Procedure
from What Students Learned Before.
16
To introduce parallel lines, Mr. Masaki started
by drawing a sample lattice pattern. The
following process shows how students develop the
idea of parallel with no instruction on the
definition of parallel.
  • Task 1. Lets draw the sample 1 lattice pattern
  • Task 2. Lets draw the sample 2 lattice pattern

Dialectic Discussion What? Why?
Synthesis Define the parallel line based on the
difference
17
To introduce parallel, Mr. Masaki started by
drawing a sample lattice pattern. The following
process shows how students develop the idea of
parallel with no instruction on the definition of
parallel.
Way of drawing 1 Procedure a ?Way of Drawing A
Task 2 If you want to draw the model, draw lines
spread evenly apart from 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 apart.
  • Even if teacher explained many times, there are
    diversity of childrens understanding.
  • Because children can not distinguish special
    ideas and general ideas.

18
Where does it come from?? explains ?
. .Schoenfeld, A (1986) Isoda, M.(1991, 1996)
Situation Meaning Procedure Explanation Appropriateness
I Introduction of calculation in vertical notation using whole numbers (integer) The meaning of a decimal notation system is based on the procedure of keeping decimal points in alignment. (The meaning and procedure match) Appropriate
II Becoming proficient in whole numbers When children become proficient, they no longer need to think about the reason they follow that procedure. As a result, the procedure is simplified from the alignment of the decimal points to one of right-side alignment. Valid by Proficiency (Procedure itself has meaning in some cases)
III Application to the decimal numbers (No meaning) Align to the right and write The procedure for whole numbers is generalized for decimal numbers. Inappropriate (Contradiction)
Write 23 5
Decimal notation system meaning
? (Forgotten)
Align to the right and write
Originated from Extension from Whole Number to
Decimal Number
Programmed Emergence of Misconception in
Curriculum/Teaching (Anyone cannot avoid
Epistemological Obstacle)
19
Meaning and Procedure (Isoda, 1991)
  • Meaning (here, Conceptual or declarative
    knowledge) refers to contents (definitions,
    properties, places, situations, contexts, reason
    or foundation) that can be described as is
    For example, 23 is the manipulation of ??????.
    The meaning can also be described as 23 is
    ??????, and as such explains conceptual or
    declarative knowledge.
  • Procedure (here, Procedural knowledge) on the
    other hand refers to the contents described as
    if., then do The procedure is used for
    calculations such as mental arithmetic in which
    calculations are done sub-consciously. For
    example, if it is 2x3, then write 6 or if it
    is 23, then write the answer by calculating the
    problem as ??????.
  • If we understood well, meaning and procedure are
    easier translatable in our mind automatically.
  • So many cases, they are not translatable easily.

20
Procedurization of meaning (Isoda,
1991) (Procedurization of Concept)
  • Procedures can be created based on meaning.
  • For example, when tackling the problem how many
    dl are in 1.5l? for the first time, a long
    process of interpreting the meaning is applied
    and the solution 1.5l is 1l 5dl is found.
    Additionally, this can be applied to other
    problems such as how many dl are in 3.2l? with
    the answer being 3.2l is 3l 2dl. Not before too
    long, children discover easier procedures by
    themselves. Simultaneously, children realize and
    appreciate the value of acquiring procedures that
    alternate long sequential reasoning to one
    routine which does not need to reason.
  • Many teachers believe that the procedure should
    explain based on the meanings but it should be a
    kind of preferred alternation because of the
    simplicity and earliness. Based on the value of
    mathematics, simplicity, we finally develop
    proceedings

21
Proceduralization of meaning (Isoda, 1991)
  • Preferred alternation from meaning to procedure
    based on faster and easier

22
Meaning entailed by procedure Conceptualization
of procedure (Isoda, 1991)
  • Meaning can be created based on procedure
  • In the first grade, like in the operation
    activity where ?????? means 32, children learn
    the meaning of addition from concise operations
    and then become proficient at mental arithmetic
    procedures (the procedurization of meaning). At
    that point, calculations such as 423 and 222
    are done more quickly than counting, which is
    seen as a procedure.
  • Further, in the second grade, comparing with
    several additional situation, only repeated
    addition problems lead to the meaning of
    multiplication. It is here where the specific
    addition procedure known as repeated addition
    (cumulation) is added as part of the meaning
    (meaning entailed by procedure). The reason such
    situations become possible is that children
    become both proficient at calculations and
    familiar enough with the procedure to do it
    instantly as well as the meaning of situation.
  • Children who are not familiar with the procedure
    resort to learning addition and multiplication at
    the same time, which in turn makes it more
    difficult for children to recognize that
    multiplication can be regarded as a special
    circumstance of addition.

23
Procedurization of Concept Conceptualization of
Procedure on the Extending Curriculum Sequence
(Isoda,1996)
More General Case
Procedure of C
Meaning of C
General Case
Procedure of B
Meaning of B
Special Case
Meaning of A
Procedure of A
24
Dialectic Ways of Discussion for Teaching
Mathematical Thinking and Developing Mathematical
Ideas. -Beyond the Contradiction-
25
Dialectic Discussion 1/2 1/3?
If you are correct, then what will happen? If
1/21/32/5, then 1/21/2 ?
1/2 1/32/5
1/2 1/35/6
contradiction
Prioritize procedure without meaning type 112,
235 then 2/5
Secure procedure and meaning type


Parallel Deadlock
Prioritize procedure with confused or ambiguous
meaning type
26
Why Dialectic Discussion? Because it is the
nature of Math and it is important for Human
communication.
  • Ancient Greek If your saying or result is given,
    then
  • Ways of communication If your saying it true,
  • Socrates Method (in German Pedagogy) on Platos
    School
  • Indirect proof of Pythagorean school
  • Analysis and Synthesis of Euclid and Pappus
  • Renaissance
  • Rene Descartes
  • Analysis on Geometry If we have conclusion
    (construction), .
  • Analysis on Algebra If we have the answer x, .
  • Fermat Bernoulli,
  • Analysis on Calculus If we have the limit, .
  • Mathematical logic of discovery focused on the
    function of counter examples
  • Hegelians Karl Popper, Imre Lakatos

27
Contradiction
  • Two strategies against the parallel discussion.
  • What if As idea is correct?
  • If 1/21/32/5 is correct, then 1/21/22/41/2.
  • Facilitating awareness through application of
    tasks in different situations and examples

28
Categorizing Students Ideas from the Meaning and
Procedure.
29
Previously Learned Task The problem to confirm
the previously learned procedure and the meaning
that forms the base of todays target task
  • When children who have knowledge of basic
    division work out the equation, 1600400 is done,
    the following is reviewed
  • Take away 00 and calculate procedure
  • Explain A as a unit of 100 (bundle) meaning
  • Substitute A for a 100 yen coin and explain
    meaning

30
Target Task (Unknown Task) Unknown problem to
press for application of the previously learned
meaning and procedure
  • The target problem presented is 1900400, which
    presents a problem for some children and not for
    others as to how to deal with the remainder.
  • Answer to the equation using a procedure in which
    the meaning is lost.
  • Apply A (Take away 00) and make the remainder 3.
    Because the meaning is detracted, the children do
    not question the remainder of 3 Half of the
    class
  • Answer to the question when procedures have
    ambiguous meanings.
  • Using A (Take away 00) and B (Unit 100), the
    remainder was revised to 300. However, because
    the meaning was ambiguous, it was changed to 400
    Several students.
  • Answer to the question when the procedure is
    ambiguous.
  • A (Take away 00) was used, but here a different
    procedure was selected by mistake. No students
    question the quotient 400 Very few students
  • Answer to a question that confirms procedural
    meanings.
  • Using A (Take away 00) , an explanation of the
    quotient and remainders from the meaning of B
    (Unit 100) and C (100 yen).

31
Target Task (Unknown Task) Categorizing ideas by
Meaning and Procedure.
  • What does each child know and how does he/she
    apply it?
  • Answer to the equation using a procedure in which
    the meaning is lost.
  • Apply A (Take away 00) and make the remainder 3.
    Because the meaning is detracted, the children do
    not question the remainder of 3 Half of the
    class
  • Answer to the question when procedures have
    ambiguous meanings.
  • Using A (Take away 00) and B (Unit 100), the
    remainder was revised to 300. However, because
    the meaning was ambiguous, it was changed to 400
    Several students.
  • Answer to the question when the procedure is
    ambiguous.
  • A (Take away 00) was used, but here a different
    procedure was selected by mistake. No students
    question the quotient 400 Very few students
  • Answer to a question that confirms procedural
    meanings.
  • Using A (Take away 00) , an explanation of the
    quotient and remainders from the meaning of B
    (Unit 100) and C (100 yen).

Type 1. Solutions reached through the use of
procedures without meaning Prioritize procedure
without meaning type
Type 2. Solution reached through the use of
procedures with meaning Prioritize procedure
with confused or ambiguous meaning type
Type 2. Solution reached through the use of
procedures with meaning Prioritize procedure
with confused or ambiguous meaning type
Type 3. Solution reached through the use of
procedures backed by meaning Secure procedure
and meaning type
32
Previously learned (Known Task)
33
To introduce parallel, Mr. Masaki started by
drawing a sample lattice pattern. The following
process shows how students develop the idea of
parallel in case of the no instruction of the
definition of parallel.
Way of drawing 1 Procedure a ?Way of Drawing A
Task 2 If you want to draw the model, draw lines
spread evenly apart from 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 apart.
  • Even if teacher explained many times, there are
    diversity of childrens understanding.
  • Because children can not distinguish special
    ideas and general ideas.
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