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Developing Deep Understanding
of Mathematics Teaching
LI Shiqi
East China Normal University
[email protected]
1
Outline
 Background: General situation in China
 Challenges: 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 grade
Teaching research group at school
School 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 suggestions
Lesson 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 / quantitative
 Who?
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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Some data from students group：
Group
a
b
c
∠A
∠B
∠C
c/sinC
b/cosA
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
a/sinA a/conB b/sinB
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Following teaching steps:
 Teacher let students make conjecture, and
he wrote the correct conjecture on
blackboard
 Next step: Teaching on to apply the law
Doubt: Is there any vital problem in
the process of teaching design?
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Case 2: Situated teaching
“The minimum distance for fire fighting”
A
B
C
Doubt: How to set situation for teaching?
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Case 3: Introduce the concept of midpoint
connector of trapezoid from the one of
triangles
14
15
16
17
Making connection between concepts !
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Case 4: Some improvements of indicators
in evaluation form for lesson observation
 Form 1
 Form 2
 Form 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
Admin
Knowledge
Type
A．Administrative
B．Mechanist
C．Remembering
D．Explanative
E．Reasoning
F．Criticizing
Freq
16
14
12
38
13
0
Percent
17.2%
15.0%
12.9%
40.9%
14.0%
0
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Mr. B’s Questioning Analysis
Admin
Knowledge
Questioning Number：46
Type
Freq
A. Administrative
18
B. Mechanist
4
C. Remember
2
D. Explatative
13
E. Reasoning
9
F. Criticizing
0
Percent
39.1%
8.7%
4.3%
28.3%
19.6%
0
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Mr. A, B’s Questioning Comparison
Number
A. Adminstrative
B. Mechanist
C. Remember
D. Explanative
E. Reasoning
F. Criticizing
Freq
93
16
14
12
38
13
0
Mr. A
Percent
100%
17.2%
15.0%
12.9%
40.9%
14.0%
0
Freq
46
18
4
2
13
9
0
Mr. B
Percent
100%
39.1%
8.7%
4.3%
28.3%
19.6%
0
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be
r
in g
izi
Cr
itic
g
ng
on
in
Re
as
lan
at
ion
em
at
ive
an
ist
str
ec
h
Ex
p
Re
m
M
Ad
nin
i
50%
Mr.A
40%
Mr.B
30%
20%
10%
0%
Questioning comparison
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60%
50%
Reasoning
Explanative
40%
30%
20%
10%
0%
Mr. A
Mr. B
Complicated questioning
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30%
Remembering
20%
Mechanist
10%
0%
Mr. A
Mr. B
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 directly
From Conjecture to
proving
Situated problem
As application of theorem
As the introduction to
theorem
Knowing theorem
Reciting
Read text
Rephrasing theorem
Word 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 principle
Thoroughly, deeply and
clearly explain
Less explain and more
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practice
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 !
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