Loading...

PPT – 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 PowerPoint presentation | free to download - id: 6b7d5c-MzIzZ

The Adobe Flash plugin is needed to view this content

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.

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

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.

Previously learned (Known Task)

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

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)

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.

(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?

In case teacher does not have information of

children. For example. Seiyamas lesson

????

How can children recognize this very strange

phenomenon?

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.

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.

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 ,

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

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

Explain Misconception with Meaning and Procedure

from What Students Learned Before.

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

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.

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)

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.

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

Proceduralization of meaning (Isoda, 1991)

- Preferred alternation from meaning to procedure

based on faster and easier

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.

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

Dialectic Ways of Discussion for Teaching

Mathematical Thinking and Developing Mathematical

Ideas. -Beyond the Contradiction-

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

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

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

Categorizing Students Ideas from the Meaning and

Procedure.

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

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).

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

Previously learned (Known Task)

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.