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Quality of K-12 Mathematics and Science_ International Lessons

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Quality of K-12 Mathematics and Science_ International Lessons Powered By Docstoc
					  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 
  answers.                                              2
  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 
                across grades: numbers

    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/
                     Topics         Grade        Topic         Grade


                     (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?     
  (grades 1-3)
• Estimate the number of words (or characters) 
  contained in one whole page of a newspaper. 
  (grades 4-6)
   nd
A 2  peek at China’s approach:
           (grades 10-12)




                                 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? 
  Explain your answer.



                                   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
                 Or Google:
•   Apec wiki lesson study
•   What the U.S. can Learn from Singapore
•   Massachusetts vs. Hong Kong + measuring up
•   The Singapore Maths.com




                                             61

				
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