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Developing Deep Understanding

of Mathematics Teaching

LI Shiqi

East China Normal University

sqli@math.ecnu.edu.cn

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Outline

Background: General situation in ChinaChallenges: How to develop deep

understanding of assessment in teaching What to be focused in assessment? How to assess? Who do the assessment?

Summary

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General situations of teaching research activity (TRA) in China

There is a long history of teaching research activity in China Originally, it was a school-based teaching exchanges , but now it is extended to national wide academic activity There are some new trends developed

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TRA structure: levels of TRA

Parallel teaching group at every gradeTeaching research group at schoolSchool district Province / city level Area level (Northeast, East China, etc.) National level, organized by national

academic societies or associations

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Focus of TRA:

Lesson planning Teaching suggestionsLesson observation Discussion and reflection … …

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Aim: Improve practical teaching,

Teaching objects, Global Structure Steps and procedures, Teaching behaviors, Students response & achievement

then encourage education research

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Forms of teaching research:

Open lesson (公开课 )Model lesson (示范课 )Research lesson Teaching competition Lesson explanation (说课 )

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Challenges: How to develop deep understanding of assessment in teaching

What?

pedagogy focused / mathematics focused How?

qualitative / quantitativeWho?

expert’s / teachers’

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What: pedagogy focused/math focused

Belief: Math ideas and principles are the heart of math lesson. Between math and pedagogy, correct math is always put in the first place. Teacher must pay good attention to the math understanding and suitable treatment of teaching material.

Some cases of teaching: Teaching Sine Law with exploration Situated teaching Midpoint connectors: a teaching aid Some evaluation forms

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Case 1: Teaching with exploration

Process of teaching of Sine Law Students were grouped and draw own

triangles, measured its angles and sides; then computed some data such as: c/sin C， a/sin A, b/cos A, etc.

Some students report their results and fill them in a form

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Group

a b c ∠A ∠B ∠C c/sinC b/cosA a/sinA a/conB b/sinB

A 4.1 3.3 3.75 700 500 600 4.330 9.649 4.363 6.378 4.308

B 5.3 3.1 3.6 107.50 330 39.50 5.660 - 10.301 5.557 6.320 5.692

C 3 3 3 600 600 600 2.598 6 2.598 6 2.598

Some data from students group：

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Doubt: Is there any vital problem in the process of teaching design?

Following teaching steps: Teacher let students make conjecture, and

he wrote the correct conjecture on blackboard

Next step: Teaching on to apply the law

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A B

C

Doubt: How to set situation for teaching?

Case 2: Situated teaching

“The minimum distance for fire fighting”

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Case 3: Introduce the concept of midpoint connector of trapezoid from the one of

triangles

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Making connection between concepts !

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Case 4: Some improvements of indicators in evaluation form for lesson observation

Form 1Form 2Form 3

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How: qualitative/quantitative

Let qualitative and quantitative messages send suitable implications to teachers

A paper: Insight into mathematics teaching

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Case: A quantitative ways of analysis: Questioning analysis

A． Administrative Questioning B．Mechanist C． Remembering D． Explanative E． Reasoning F． Criticizing

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Questioning Analysis

A. Administrative Questioning Who has any new ideas about it? B. Mechanist How many auxiliary line are there? C. Remembering How did we prove it last time?

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Questioning Analysis

D. Explanative What is “base side” and what is “the third

side”? E. Reasoning Why do you draw such a auxiliary line? F. Criticizing

Why this is a wrong way? If so, what is your new suggestion?

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Mr. A’s Questioning Analysis

Question Number：93

Type Freq Percent

Admin A．Administrative 16 17.2% B．Mechanist 14 15.0%

C．Remembering 12 12.9% D．Explanative 38 40.9%

E．Reasoning 13 14.0%

Knowledge

F．Criticizing 0 0

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Mr. B’s Questioning Analysis

Questioning Number：46 Type Freq Percent

Admin A. Administrative 18 39.1%

Knowledge

B. Mechanist 4 8.7% C. Remember 2 4.3% D. Explatative 13 28.3% E. Reasoning 9 19.6% F. Criticizing 0 0

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Mr. A, B’s Questioning Comparison

Mr. A Mr. B

Freq Percent Freq Percent Number 93 100% 46 100%

A. Adminstrative 16 17.2% 18 39.1% B. Mechanist 14 15.0% 4 8.7% C. Remember 12 12.9% 2 4.3% D. Explanative 38 40.9% 13 28.3% E. Reasoning 13 14.0% 9 19.6% F. Criticizing 0 0 0 0

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0%

10%

20%

30%

40%

50% Mr.A

Mr.B

Questioning comparison

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0%

10%

20%

30%

40%

50%

60%

Mr. A Mr. B

Reasoning

Explanative

Complicated questioning

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0%

10%

20%

30%

Mr. A Mr. B

Remembering

Mechanist

Simple questioning

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Important behavioral differences between two teachers

Mr. A Mr. B

Definition introduction At the beginning After proof

Proving Just proving directlyFrom Conjecture to proving

Situated problem As application of theoremAs the introduction to theorem

Knowing theorem Reciting Read text

Rephrasing theoremWord by word same as on text

Right but flexible

“How many … ” Tell to students Hint

The difference to median

Tell to students Hint

Writing on chalkboard Formally Outline

Didactics principleThoroughly, deeply and clearly explain

Less explain and more practice

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Who: expert centered/teachers centered

A characteristic: teaching researchers play an important role

Change the pattern of “Teacher teaching and experts comment”: Lesson explanation: self description and reflection (Huang)

Online learning and assessing by teachers Yang: interesting research result: 3 rounds

action learning — not so good as expected

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A new trend: lesson explanation 说课

Teachers explain and reflect his/her own design of a lesson, its underlining ideas and related theories

An example:

Dr. HUANG Xinfeng’s work:

The sum of the first n terms of an arithmetic series

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For a general view, please read:

Peng, Aihui (2007): Knowledge growth of mathematics teachers during professional activity based on the task of lesson explaining, Journal of Mathematics Teacher Education, 10: 289 – 299

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Another kind of teacher’s reflecting activity: Online learning and assessing in Shanghai

Videotaped lessons are put online every three months or so. Teachers are required to observe and write own comments and questions online as a course work.

Teaching researchers will read such course work and send response to them.

Every teachers who finish the work will earn their training credits.

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A Case: Experts’ special research will give teachers more insights into practical teaching

YANG, Yudong (2005): Classroom Teaching Driven by Primitive Mathematics Ideas — An Action Research for Improving Mathematics Teaching, Journal of Mathematics Education (In Chinese), 14(2), 59-63.

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Interesting finding: three rounds of teaching improvement — not so perfect as expected

First round: teachers planed lesson and teach it himself —— there are some weaknesses

Second round: teachers improve their teaching with more comments and suggestions from experts etc. —— even less successful than the former one

Third round: teachers reflected their experience independently, adjust their lesson plan and teach again —— it seemed better and more successful

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Summary: Complementary & interdependent ways make lesson study assessing effective

Pay attention to both mathematics & pedagogy: keep right track of math teaching carefully

Apply both qualitative and quantitative evaluate ways: reveal and insight into the keys of teaching

Both experts and teachers do teaching assessment: improve practical teaching effectively

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Thank you for your attention !