How to plan a lesson for developing Mathematical Thinking Kyozai Kenkyu 教材研究 Research Subject Mater A priori Analysis vs Planning on aims Masami Isoda

How to plan a lesson for developing Mathematical Thinking

Kyozai Kenkyu教材研究

Research Subject Mater

A priori Analysis vs Planning on aims

Masami Isoda

CRICED, University of Tsukuba

Where do Mathematics problems come from?

Knowing and embedding the

aims of education in the lesson plan by the classroom

problem is a key for

improvement.

Some classroom problems

come from the

extension in curriculum.

　

It is a general question that can not be answered without restriction.

APEC-KKU Conference

16.8.2007

Math.Edu.Univ.of Tsukuba

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 reflectionThe ways of conceptual developmentThe ways of conceptual development

Refutation is acceptable. The World of Invariant Mathematics:

Mathematics is the pattern of science.Mathematics is the pattern of science.Mathematical Ways of Thinking:

G. PolyaLearning how to develop mathematicsLearning how to develop mathematics

Math.Edu.Univ.of TsukubaProblem Solving Approach

a model of the lesson to develop Mathematical Thinking

Problem PosingPredict the

methods for Solution

Solving DiscussionReflection

•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)

In the process, children learn mathematics but this process is not aimed to represent it. It focuses on how to guide children’s activity.


Target (extending) Task

Appreciation and sense of achievement

Confirming and understanding the extended meaning and procedure.

Reflection/ Summary

Previously learned (Known Task)

Previously learned Procedures and Meanings

Recall, Confirmation and Understanding.

It goes well!!Sense of efficacy.

ConflictExposure of gaps in procedure and meaning

No meaning and procedure type

Prioritize meaning without procedure (or confused) typeSecure procedure and meaning type

Prioritize procedure with confused or ambiguous meaning type

Prioritize procedure without meaning type

Comparison to previously learned knowledge “hmmmm,” “what?”

Students become aware of the gaps and differences with knowledge previously learned.: Concern, uneasiness and conflict

Aiming to eliminate gaps and conflict

Reproducing and reconsidering proceduresHow did you do that?

Reproducing and reconsidering meaningsWhy did you think that way?

Emortional AspectsWhat is confusing or troubling you?

“What?” and “Why?”Asking themselves and others again

Facilitate developmental discussion based on meanings and procedures previously learned, and eliminated gaps.


Problem Solving Approacha model of the lesson to develop Mathemetical Thinking by Child Centered Approach

Phase Teacher Children

Problem Posing

Posing the problem with the aimthe aim

Given the problem without knowing the aim

Prediction Guiding children’s approach

WakingWaking both known and unknown in each Child

Solving Supporting individual works

Clarifying and bridgingClarifying and bridging known and unknown by each child.

Discussion Guiding discussion to the aim

Working on the aimWorking on the aim of bridging between known and unknown by all

Reflection Guiding the reflection Valuing the aimValuing the aim


Necessary Activity for developing the lesson

Plan the lesson with following problems Find the following pairs of numbers

□×△ ＝ □－△

X ×Y 　＝ X － Y

(0,0) (1,△ )

(1,1 ／ 2)

(1 ／ 2 ，△ )

Ｙ＝ｘ／ ( ｘ＋１）

(3 ， 3 ／4)

What if?What if

not?Generalization

Speicalzation

What kinds of thinking?

Math.Edu.Univ.of TsukubaNecessary Activity for developing the lesson

Problems are used for children enabling them to engage in rich mathematical activity and lean from the reflection.

Mathematics Problems

Classroom Problems in order / sequence(ways of posing in the process of teaching)

The Aim of Lesson,Mathematical Value

Anticipating Children’s Activity

Correct AnswerOther Approaches

Wrong Answers

For controlling children’sActivity on the aim

Embedding the aim

What can What can children learn children learn from the from the process?process?

Conditions of selection Developing New Prob.

Mathematical Thinking

What If



Classroom Problems in order / sequence(ways of posing in the process of teaching)



Find following pairs of numbers

□×△ ＝ □－△

X ×Y 　＝ X － Y

Finding examples

(0,0), (1,1/2), ……….

Solving generally

y= x/(x+1)

-4 -3 -2 -1 1 2 3 4 x

-4

-3

-2

-1

1

2

3

4

y

O

(1/2, 1/3)…

(1/n, 1/(n+1))Finding the aims

•Number Pattern •Invariant vs Variant

•Specialization

•Inductive Reasoning

•Generalization

•Power of Symbol

•Sequence, Integra

□×△ ＝ □－△Guessing examples;(0,0), (1,1/2),….Finding general pattern

(1/□)×(1/ )△ ＝ (1/□) － (1/ )?△Introduction of Symbol

(1/□)×(1/(□ ＋ 1)) ＝ (1/□) － (1/ （□＋1))

(1/□)×(1/ )△ ＝ (1/□) － (1/ ) △ How to represent general pattern?How to represent general pattern?

Why this problem is interesting for teachers?

For children?

How can children recognize it as problematical?How can children recognize it as problematical?


In case teacher does not have information of children. For example. Seiyama’s lesson

3 8 　

6 2

7 5

→ Children

→ Teacher

→ Children

□ ＋□＋□＝？


Classroom Problems in order(ways of posing in process)



How can How can children children recognize this recognize this very strange very strange phenomenon?phenomenon?

Math.Edu.Univ.of TsukubaNecessary Activity for developing the Lesson Plan

Classroom Problems in order(ways of posing in process)

Plan of Activity, Questioning for Interactionand the Black Board Writing

How to be clear of the aim and the value of math. in process.

Set Children’s Activity through the questions

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).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 many? How to calculate? ●●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

What is the teacher’s or your expectation?Why did teacher ask

this question?

But, a child answer is

…..

Oh, NO!If you are the teacher, what can you do?

Wow,Teacher used Children’s idea!

Important Math.Thinking; Generalization, Application, ……Value of them; Faster, Easier, Reasonable , ……

How to use students’ misconceptions or wrong ideas as the key problem in the mathematics classroom?

How can we use students’ misconceptions in the mathematics

classroom?Teacher’s Theory for Problem

Solving Approach

●●●●

●●●●●●●●●●●●

How many? How to calculate? ●●●●

●●●●●●●●●●●●

●●●●●●●●●●●●●●●●●●●●

What is the teacher’s expectation?Why did teacher ask this question?

But, a child answer is

…..

Oh, NO!If you are the

teacher, what can you

do?

Wow,Teacher used Children’s idea!

Important Math.thinking; Generalization, Application, ……Value of them; Faster, Easier, Reasonable , ……

What is Teacher’s 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.It’s usually developed with teachers who are working on the Lesson Study (Plan Do See).

Why do teachers feel that a misconception/misunderstanding is a problem?If it is not expected, it must be a problem but it is expected…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, andDeveloping Math. Using other’s idea is basic reasoning

Here, we use the terms: Meaning and Procedure in planning the lesson to develop children’s mathematical knowledge based on the curriculum

Math.Edu.Univ.of TsukubaBased on

the nature of mathematics learning

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 evidence of

Thinking Mathematically

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. Let’s draw the sample 1 lattice pattern

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

Sample 1 Way of drawing pattern 1

Sample 2 Using the right diagonal line as the base

Way of Drawing A: Even intervals along the edges

Way of Drawing B: Even intervals from the line

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.

Even if teacher explained many times, there are diversity of children’s understanding.

Because children can not distinguish special ideas and general ideas.



Way of Drawing A: Even intervals along the edges Way of Drawing B:

Even intervals from the line

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.


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

IIBecoming 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

(C)

(A)

(B)

Where does it come from?② explains ① .

.Schoenfeld, A (1986)Isoda, M.(1991, 1996)

Programmed Emergence of Misconception in Curriculum/TeachingProgrammed Emergence of Misconception in Curriculum/Teaching(Anyone cannot avoid: Epistemological Obstacle)(Anyone cannot avoid: Epistemological Obstacle)

Originated fromOriginated fromExtension from Whole Number to Decimal NumberExtension from Whole Number to Decimal Number


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, 2+3 is the manipulation of ○○←○○○. The meaning can also be described as: “2+3 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 2+3, 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.


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

Procedurization of meaning (Isoda, 1991)

(Procedurization of Concept)


Proceduralization of meaning (Isoda, 1991)

Preferred alternation from meaning to procedure based on faster and easier

Proc

edur

e ba

sed

on

mea

ning

Procedurization

of Meaning

Getting expression basen on the meaning of problem

A 3/4m long iron bar weighs 2/5kg. How much does 1m of this iron bar weigh?

Representation


Meaning can be created based on procedure In the first grade, like in the operation activity where

“○○○←○○” means 3+2, 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 4+2+3 and 2+2+2 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.

Meaning entailed by procedureConceptualization of procedure (Isoda, 1991)


Procedurization of ConceptProcedurization of Concept Conceptualization of ProcedureConceptualization of Procedure

on the Extending Curriculum Sequence on the Extending Curriculum Sequence (Isoda,1996)(Isoda,1996)

Meaning of A Procedure of A

Meaning of B Procedure of B

Meaning of C Procedure of C

Special Case

General Case

More General

Case


Dialectic Ways of Discussion for Teaching Mathematical Thinking and Developing

Mathematical Ideas.-Beyond the Contradiction-

Math.Edu.Univ.of TsukubaDialectic Discussion

1/2 + 1/3=?

1/2 + 1/3=2/5 1/2 + 1/3=5/6

Prioritize procedure without meaning type1+1=2, 2+3=5 then 2/5

Secure procedure and meaning type

Prioritize procedure with confused or ambiguous

meaning type

+ =

If you are correct, then what will happen?

If 1/2+1/3=2/5, then 1/2+1/2 =?

contradiction

Parallel

Deadlock


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, thenWays of communication; If your saying it true, …

Socrates Method (in German Pedagogy) on Plato’s School

Indirect proof of Pythagorean schoolAnalysis and Synthesis of Euclid and Pappus

RenaissanceRene 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 examplesHegelians; Karl Popper, Imre Lakatos


Two strategies against the parallel discussion.What if A’s idea is correct?

If 1/2+1/3=2/5 is correct, then 1/2+1/2=2/4=1/2.

Facilitating awareness through application of tasks in different situations and examples

Children who are aware of the meaning of the procedure and children who are not aware of it contradict each other. Here, the discussion develops based on the ideas and concerns of children who have an ambiguous understanding of meaning or procedure.

The reactions of children who are no longer aware of the meaning of the procedure

The reactions of children who remain aware of the meaning of the procedure

Dialectic discussion that eliminates gaps in diverse ideas

In order to eliminate contradictions and gaps, it is necessary for children to persuade to revise other’s ideas.

Contradiction


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 today’s target task When children who have

knowledge of basic division work out the equation, 1600÷400 is done, the following is reviewed:

A)Take away 00 and calculate: procedure

B)Explain A as a unit of 100 (bundle): meaning

C)Substitute A for a 100 yen coin and explain: meaning

1600

16004004

0


Target Task (Unknown Task):Unknown problem to press for application of the previously learned meaning and procedure The target problem presented is 1900÷400,

which presents a problem for some children and not for others as to how to deal with the remainder.

a. 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

b. 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.

c. 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

d. 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?a. 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

b. 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.

c. 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

d. 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 3. Solution reached through the use of procedures backed by meaning: Secure procedure and meaning type

Type 2. Solution reached through the use of procedures with meaning: Prioritize procedure with confused or ambiguous meaning type

1600

16004004

0


Target (extending) Task

Appreciation and sense of achievement

Confirming and understanding the extended meaning and procedure.

Reflection/ Summary

Previously learned (Known Task)

Previously learned Procedures and Meanings

Recall, Confirmation and Understanding.

It goes well!!Sense of efficacy.

ConflictExposure of gaps in procedure and meaning

No meaning and procedure type

Prioritize meaning without procedure (or confused) typeSecure procedure and meaning type

Prioritize procedure with confused or ambiguous meaning type

Prioritize procedure without meaning type

Comparison to previously learned knowledge “hmmmm,” “what?”

Students become aware of the gaps and differences with knowledge previously learned.: Concern, uneasiness and conflict

Aiming to eliminate gaps and conflict

Reproducing and reconsidering proceduresHow did you do that?

Reproducing and reconsidering meaningsWhy did you think that way?

Emortional AspectsWhat is confusing or troubling you?

“What?” and “Why?”Asking themselves and others again

Facilitate developmental discussion based on meanings and procedures previously learned, and eliminated gaps.


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.

Even if teacher explained many times, there are diversity of children’s understanding.

Because children can not distinguish special ideas and general ideas.



Way of Drawing A: Even intervals along the edges Way of Drawing B:

Even intervals from the line

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.

Documents

How to plan a lesson for developing Mathematical Thinking Kyozai Kenkyu 教材研究 Research Subject Mater A priori Analysis vs Planning on aims Masami Isoda