Quality of K-12 Mathematics and Science_ International Lessons by pptfiles

VIEWS: 0 PAGES: 61

• pg 1
```									  Intriguing Lessons about
Teaching and Assessing Math
around the World

Steve Leinwand
American Institutes for Research
SLeinwand@air.org

NCTM – April 28, 2012
What a great time to be
convening as teachers of
•
mathematics!
Common Core State Standards Initiative
• Quality K-8 instructional materials
• More access to material via the web than ever
• \$5 billion with a STEM RttT tie-breaker
• A president who believes in science and data
• The beginning of the end to Algebra II
• A long overdue understanding that it’s instruction
that really matters
• A recognition that the U.S. doesn’t have all the
And what can we learn from others?

• There is more than one way to do things.
• Like how a foreign language strengthens
ones understanding of English, so too can
international comparisons hold up mirrors.
• There might be good reasons why other
nations outperform the U.S.
• In answering “How is it done elsewhere?” we
can learn how to change what we do here.

3
Today’s agenda
Glimpses of intriguing differences in:
•   Curriculum Frameworks and Standards
•   Instruction
•   Assessment
•   Teacher Support

4
Part 1

Curriculum Frameworks
and Standards

5
How NCTM (and most states) Organizes
the Curriculum
• Five Content Strands   • Five Process Strands
– Numbers and            – Problem-Solving
operations             – Reasoning and
– Measurement              Proof
– Geometry               – Communication
– Algebra                – Connections
– Data and               – Representations
Probability
6
How Singapore Organizes the Curriculum

7
Singapore logically builds-up math topics

I
I

8 8
A composite of Singapore, HK, Korea
Gr. 2                           Gr. 3                         Gr. 4                       Gr. 5
Fractions /   Fraction of a whole:           Equivalent fractions:           Mixed numbers and
concepts      Interpret a fraction as part   Recognize and name              improper fractions:
of a whole                     equivalent fractions            Understand the concepts
Read and write fractions       Write the equivalent fraction   of mixed numbers and
Compare and order unit         of a fraction, given the        improper fractions
fractions and like             denominator or the              Express an improper
fractions.                     numerator                       fraction as a mixed
(denominators less than        Express a fraction in its       number, and vice versa,
or equal to 12)                simplest form                   and expressing both in
Compare and order unlike        simplest form
fractions, including
comparing fractions with
respect to one half
(denominators less than or
equal to 12)
Fractions /                                      Addition and subtraction    Addition and subtraction   Addition and subtraction
arithmetic                                   of two related fractions (one   of                         of fractions with unlike
operations                                   denominator a factor of the     - like fractions           denominators:
other) within one whole         - related fractions        Add and subtract fractions
(denominators of given          (denominators of given     with unlike denominators
fractions should not exceed     fractions should not
12)                             exceed 12)

9
A composite of Singapore, HK, Korea
Gr. 3                Gr. 4               Gr. 5                 Gr.6

Perimeter                        Develop the         Find one dimension
concept of           of a rectangle
perimeter            given the other    Understand the
Measure and find      dimension and        area and the
the perimeter of     area and             circumference of
2‑dimensional        perimeter            circles
shapes                                  Understand the
concept of pi
Area        Develop the          Apply the formula   Apply the formula
Find the area and
concept of area      for area of         for the area of
the circumference
Compare areas,        squares and         triangles,
of circles,
using improvised     rectangles and      parallelograms,
semicircles, and
units                 composite          and rhombuses
quarter circles
Measure area in        figures made up
square                of rectangles
centimeters (cm2)     and squares
and square
meters (m2)

10
K-6 Density or A Glimpse at a Mile Wide

Avg. No. of    Avg. No. of   Avg. No. of
Total No. of     Topics per     Grades/      Outcomes/

(1)           (2)            (3)           (4)
Singapore            40             15            2.3           39

California           42             20                          51
2.9

Florida              54             39            4.2           107

Maryland             46             29            3.8           69

New Jersey           50             28            3.4           56
N. Carolina          41             18            2.6           36

Ohio                 48             26            3.3           62

Texas                40             19            2.8           44
11
Singapore, Japanese and Hong Kong Math
Standards at: http://hrd.apec.org

12
Basic rationales for the new Chinese
math standards:
• We cannot do without mathematics in our daily living,
work and study.
• Contents of mathematics learning for school children
ought to be realistic, meaningful and challenging.
• Mathematical instructional activities should be based
on children’s cognitive developmental level and build
on past experiences.
• Modern information technology has great impact on
the values, objectives, contents and pedagogy of
mathematics education.
A peek at China’s approach:
• What is the approximate thickness of 1200 sheets of
paper? What is the approximate number of classes
that may be formed comprising of 1200 students?
What is the approximate length of 1200 footsteps?
• Estimate the number of words (or characters)
contained in one whole page of a newspaper.
nd
A 2  peek at China’s approach:

15
Initial Lessons
• Think holistically, and systemically, not
linearly and piecemeal
• Coherence and focus
• Learning progressions
• “Teach less, learn more”
• Provide examples

16
Part 2

Instruction

17
A Glimpse at Two Grade 1 Textbooks

#               Average     Pages of   Pages of
Textbook            # Les-                                     Other
Pages/      Develop-    Exer-
Grade 1   Topics    sons                                      Pages
Lesson       ment       cises

Sing-                                     174        261         62
13       34        15
apore                                    (35%)      (53%)      (12%)

U.S.
(Scott                                    145        169        250
25       157       4
Fores-                                   (26%)      (30%)      (44%)
man)

18
How about content balance at grade 1?

Textbook            Number             Measurement   Geometry   Algebra    Data

5            1
Singapore           27 (79%)                                        0       1 (3%)
(15%)        (3%)

27          11
Scott              106 (68%)                                      5 (3%)    8 (5%)
(17%)        (7%)

Everyday                                     17           8
Math*
62 (56%)                                      2 (2%)    7 (6%)
(15%)        (7%)
*14 (13%) review and routines lessons

19
A Glimpse at Three Grade 6 Textbooks

Average
#      Total               Pages of    Pages of    Other
Textbook   # Chapters                     Pages/
Lessons   Pages             Development   Exercises   Pages
Lesson

107         132         163
Singapore      11          24      402       17
(27%)       (33%)       (41%)

202         216         322
Scott          12         158      740       5
(27%)       (29%)       (44%)

Everyday                                                299         112
Math
10         113      411       4                                   0
(73%)       (27%)

20
How about content balance at grade 6?

Textbook                Number   Measurement   Geometry    Algebra     Data

4           3           1           1
Singapore             15 (63%)
(17%)       (13%)       (4%)        (4%)

14          15
Scott                 64 (41%)                               18 (11%)   17 (11%)
(9%)        (9%)

Everyday
5           18
Math*                 37 (33%)                               19 (17%)   23 (20%)
(4%)        (16%)
*11 (10%) review lessons

21
But what about depth, rigor and
connections?

22
U.S. Textbook Problems Emphasize
Mechanical Formulas:
Gr. 6 Pie Chart Requires Summing to a Total
Cost of Raising a Child to Age 18 (for each \$100)

23   23
Singapore textbooks uses structured variation and
multi-step Problems :
Gr. 6 Pie Chart Problem Incorporating Angles

24       24
And then there is Singapore’s visual
approach to mathematics:

The bar model

25
26
http://www.thesingaporemaths.com/

27
28
29
30
31
32
And if we crank it up a bit….

33
34
35
And 25! slides later:

36
37
But what about the actual art of teaching?

Here’s a glimpse at the Japanese 6th grade
lesson.

38
The problem- Grade 6
We bought pencils and ballpoint pens and the
total number of items was 10 and the price
was 460 yen.  The price of each pencil was 40
yen and the price of each ballpoint pen was
70 yen.
How many pencils and how many ballpoint
pens did we buy?

39
# of        0    1   2     3     4     5     6     7     8     9    10
pencils
# of pens   10   9   8     7     6     5     4     3     2     1     0
Total       700 670 640   610   580   550   520   490   460   430   400
cost

40
Let’s take a peek

41
Part 3
Assessments

42
A Question of Balance
% Finding an
% Multiple   % Multi-step   Intermediate
Choice                       Unknown

Singapore – 6            31            25             19
Texas – 8                100            6              2
Mass – 8                 77            13              5
Ohio – 8                 74            17
NAEP – 4                 64            15              4
NAEP - 8                 60            21              8
Massachusetts – Grade 3 – 2007
Alan has the number tiles shown below.
4     7      8      1
• Use all of Alan’s number tiles to make the four‑digit number
with the smallest value. Use each number tile only one time.
Write the number in the boxes below.

• What is the value of the digit 7 in the number you made?

44
Hong Kong – Grade 3 - 2007

45
Hong Kong – Grade 3 - 2007

46
Hong Kong – Grade 3 - 2007

47
Hong Kong – Grade 3 - 2007

48
Mass vs. Hong Kong – Grade 3
Massachusetts                              Hong Kong

MCAS Spring 2007 Released Items         2007 Territory‑Wide Spring System
Assessment
Characteristic
Items by strand**   Number:            17 (49%)              Number:            15 (42%)
Measurement:   4 (11%)                   Measurement:   12 (33%)
Geometry:         4 (11%)                Geometry:         7 (19%)
Data:                6 (17%)             Data:                2 (6%)
Algebra:            4 (11%)              Algebra:            0 (0%)

Multiple‑choice                             25 (71%)                                 5 (14%)
items

Constructed‑respo             10 constructed response                 31 constructed response
nse items                   (all but two are short closed           (all but two are short closed
constructed‑response)                   constructed‑response)

49
Mass vs. Hong Kong – Grade 3
Massachusetts                         Hong Kong

MCAS Spring 2007 Released       2007 Territory‑Wide Spring System
Characteristic                     Items                             Assessment
Items with graphics                       20 (57%)                            25 (69%)
Items within real‑world
19 (54%)                               21 (58%)
contexts
Items by computational  Low:                 34 (97%);      Low:                 22 (61%);
difficulty               Medium:           1 (3%);          Medium:           14 (39%);
High:                0 (0%)      High:                0  (0%)
Average (1, 2, 3) = 1.03            Average (1, 2, 3) = 1.39

Items by cognitive         Level 1:             23 (66%)    Level 1:             12 (33%)
complexity***              Level 1+:           0            Level 1+:           4 (11%)
Level 2:             12 (34%)    Level 2:             16 (44%)
Level 2+:           0            Level 2+:           3 (8%)
Level 3:             0           Level 3:             1 (3%)
Average: 1.34                        Average: 1.68
50
A Singapore Grade 6 Hard Problem You Won’t
See on U.S. Grade 8 State Assessments

Source: Singapore MOE
Part 4

Teacher Support

52
Primary education teacher Salaries
Ratio of salary after 15 years of experience to GDP per capita (2004)

53
Singapore primary teacher preparation:
a focus on pedagogical content
•   Teaching and Learning of                •   Teaching and Learning of Primary
Primary Mathematics I                       Mathematics I I (geometry,
(Numbers)                                   measurement, statistics, algebra)

Overview of the Singapore Primary           Teaching Problem Solving and
Mathematics Curriculum;                     Investigations; Mathematical
Preparation of Scheme of Work and           Communication; Teaching of
Lesson Plans; Pedagogical Strategies        Geometry, Money and Measures,
and Psychological Theories; Teaching        Mensuration, Graphical
of Whole Numbers, Fractions,                Representation and Statistics,
Decimals, Percentage, Ratio and             Algebra. [Common pupils’ errors will
Direct Proportion, Rate and Speed.          be dealt with in the teaching of
[Common pupils’ errors will be dealt        various topics.]
with in the teaching of various
topics.]

54
Singapore primary teacher preparation:
a focus on pedagogical content (cont)
•   Teaching and Learning of Primary            •   Teaching and Learning of Primary
Mathematics III (e.g. assessment)               Mathematics IV (e.g., IT)

This course covers two broad areas:             This course provides further
(a) Various traditional assessment              pedagogical skills for those who will
modes in Mathematics and the use                go deeper into teaching
of these modes in schools to assess             mathematics, especially at upper
pupil performance, in particular, the           primary levels.  Topics include:
planning and construction of test               Games in Mathematics; Challenging
items and (b) Practice of teaching              Problems in Upper Primary
skills, including catering for pupils of        Mathematics.  Student teachers will
mixed abilities.                                also undertake Independent Study
Topics which will enhance their
teaching repertoire.

55
Professional development:
Japanese lesson study is an application of quality
principles to improve teaching

1. Identify a teaching topic based on student needs
2. Plan a lesson with student learning as a focus
3. Teach a lesson with Japanese form (identify math
concept, present problems, discuss solutions,
summarize what is learned)
4. Evaluate lesson’s impact (with others educators)
5. Revise the lesson

56
China’s elementary specialists
• Chinese elementary teachers specialize by
teaching either mathematics or language but
not both.
• Most common model is teachers teach
mathematics grades 1-3 or 4-6.
• Teachers move from classroom to classroom
and young students stay in same room.
• Teachers don’t have their own home room,
they share a common teacher room promoting
teacher interaction.

57
The teaching hierarchy in Chinese cities
• Trainee teacher (3 years)
• Second rank teacher (if you qualify) – can
teach independently
• First rank teacher  - need to write articles
Then in Beijing or Shanghai: teach/mentor by:
• Master or senior teacher (2%)
• Leading or special teacher (0.4%)
• Super rank (te ji jiao shi) (0.15%) via contests!
58
Lots to think about.
Lots to consider emulating.
Lots of ways we can be more
effective.

Thank you.
SLeinwand@air.org
59
A Virtual Handout
• Lesson Study Plan and Video:
http://hrd.apecwiki.org/index.php/Classroom_Videos_fro
m_Lesson_Study
• What the US can Learn from Singapore:
http://www.air.org/files/Singapore_Report_Bookmark_Ver
sion1.pdf
• MA vs. Hong Kong:
http://www.air.org/files/AIR_Measuring_Up_Report_0427
091.pdf
• APEC Standards: http://hrd.apec.org
• Singapore Bar Model: http://www.thesingaporemaths.com

• Steve Leinwand:  SLeinwand@air.org
60