# Intriguing Lessons about Teaching and Assessing Math around the World Steve Leinwand American Institutes for Research NCTM – April 28,

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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 its instruction that really matters A recognition that the U.S. doesnt 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

Todays agenda Glimpses of intriguing differences in: Curriculum Frameworks and Standards Instruction Assessment Teacher Support 4

Part 1 Curriculum Frameworks and Standards 5

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

How Singapore Organizes the Curriculum 7

8 8 Singapore logically builds-up math topics across grades: numbers I I

A composite of Singapore, HK, Korea Gr. 2Gr. 3Gr. 4Gr. 5 Fractions / concepts Fraction of a whole: Interpret a fraction as part of a whole Read and write fractions Compare and order unit fractions and like fractions. (denominators less than or equal to 12) Equivalent fractions: Recognize and name equivalent fractions Write the equivalent fraction of a fraction, given the denominator or the numerator Express a fraction in its simplest form Compare and order unlike fractions, including comparing fractions with respect to one half (denominators less than or equal to 12) Mixed numbers and improper fractions: Understand the concepts of mixed numbers and improper fractions Express an improper fraction as a mixed number, and vice versa, and expressing both in simplest form Fractions / arithmetic operations Addition and subtraction of two related fractions (one denominator a factor of the other) within one whole (denominators of given fractions should not exceed 12) Addition and subtraction of - like fractions - related fractions (denominators of given fractions should not exceed 12) Addition and subtraction of fractions with unlike denominators: Add and subtract fractions with unlike denominators 9

A composite of Singapore, HK, Korea Gr. 3Gr. 4Gr. 5Gr.6 PerimeterDevelop the concept of perimeter Measure and find the perimeter of 2 dimensional shapes Find one dimension of a rectangle given the other dimension and area and perimeter Understand the area and the circumference of circles Understand the concept of pi Find the area and the circumference of circles, semicircles, and quarter circles AreaDevelop the concept of area Compare areas, using improvised units Measure area in square centimeters (cm 2 ) and square meters (m 2 ) Apply the formula for area of squares and rectangles and composite figures made up of rectangles and squares Apply the formula for the area of triangles, parallelograms, and rhombuses 10

K-6 Density or A Glimpse at a Mile Wide Total No. of Topics Avg. No. of Topics per Grade Avg. No. of Grades/ Topic Avg. No. of Outcomes/ Grade (1)(2)(3)(4) Singapore40152.339 California4220 2.9 51 Florida54394.2107 Maryland46293.869 New Jersey50283.456 N. Carolina41182.636 Ohio48263.362 Texas40192.844 11

Singapore, Japanese and Hong Kong Math Standards at: http://hrd.apec.orghttp://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 childrens 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 Chinas 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)

A 2 nd peek at Chinas 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 18 Textbook Grade 1 # Topics # Les- sons Average Pages/ Lesson Pages of Develop- ment Pages of Exer- cises Other Pages Sing- apore 133415 174 (35%) 261 (53%) 62 (12%) U.S. (Scott Fores- man) 251574 145 (26%) 169 (30%) 250 (44%)

19 How about content balance at grade 1? TextbookNumberMeasurementGeometryAlgebraData Singapore 27 (79%) 5 (15%) 1 (3%) 01 (3%) Scott 106 (68%) 27 (17%) 11 (7%) 5 (3%)8 (5%) Everyday Math* 62 (56%) 17 (15%) 8 (7%) 2 (2%)7 (6%) *14 (13%) review and routines lessons

20 A Glimpse at Three Grade 6 Textbooks Textbook# Chapters # Lessons Total Pages Average Pages/ Lesson Pages of Development Pages of Exercises Other Pages Singapore 112440217 107 (27%) 132 (33%) 163 (41%) Scott 121587405 202 (27%) 216 (29%) 322 (44%) Everyday Math 101134114 299 (73%) 112 (27%) 0

21 How about content balance at grade 6? TextbookNumberMeasurementGeometryAlgebraData Singapore 15 (63%) 4 (17%) 3 (13%) 1 (4%) 1 (4%) Scott 64 (41%) 14 (9%) 15 (9%) 18 (11%)17 (11%) Everyday Math* 37 (33%) 5 (4%) 18 (16%) 19 (17%)23 (20%) *11 (10%) review lessons

But what about depth, rigor and connections? 22

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

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

And then there is Singapores visual approach to mathematics: The bar model 25

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27 http://www.thesingaporemaths.com/

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And if we crank it up a bit…. 33

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And 25! slides later: 36

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But what about the actual art of teaching? Heres a glimpse at the Japanese 6 th 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 pencils 012345678910 # of pens109876543210 Total cost 700670640610580550520490460430400 40

Lets take a peek 41

Part 3 Assessments 42

A Question of Balance % Multiple Choice % Multi-step % Finding an Intermediate Unknown Singapore – 6312519 Texas – 810062 Mass – 877135 Ohio – 87417 NAEP – 464154 NAEP - 860218

Massachusetts – Grade 3 – 2007 Alan has the number tiles shown below. 4 7 8 1 Use all of Alans 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 Characteristic MassachusettsHong Kong MCAS Spring 2007 Released Items 2007 Territory Wide Spring System Assessment Items by strand** Number: 17 (49%) Measurement: 4 (11%) Geometry: 4 (11%) Data: 6 (17%) Algebra: 4 (11%) Number: 15 (42%) Measurement: 12 (33%) Geometry: 7 (19%) Data: 2 (6%) Algebra: 0 (0%) Multiple choice items 25 (71%) 5 (14%) Constructed respon se items 10 constructed response (all but two are short closed constructed response) 31 constructed response (all but two are short closed constructed response) 49

Mass vs. Hong Kong – Grade 3 Characteristic MassachusettsHong Kong MCAS Spring 2007 Released Items 2007 Territory Wide Spring System Assessment Items with graphics 20 (57%)25 (69%) Items within real world contexts 19 (54%)21 (58%) Items by computational difficulty Low: 34 (97%); Medium: 1 (3%); High: 0 (0%) Average (1, 2, 3) = 1.03 Low: 22 (61%); Medium: 14 (39%); High: 0 (0%) Average (1, 2, 3) = 1.39 Items by cognitive complexity*** Level 1: 23 (66%) Level 1+: 0 Level 2: 12 (34%) Level 2+: 0 Level 3: 0 Average: 1.34 Level 1: 12 (33%) Level 1+: 4 (11%) Level 2: 16 (44%) Level 2+: 3 (8%) Level 3: 1 (3%) Average: 1.68 50

A Singapore Grade 6 Hard Problem You Wont See on U.S. Grade 8 State Assessments Source: Singapore MOE

Part 4 Teacher Support 52

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

Singapore primary teacher preparation: a focus on pedagogical content Teaching and Learning of Primary Mathematics I (Numbers) Overview of the Singapore Primary Mathematics Curriculum; Preparation of Scheme of Work and Lesson Plans; Pedagogical Strategies and Psychological Theories; Teaching of Whole Numbers, Fractions, Decimals, Percentage, Ratio and Direct Proportion, Rate and Speed. [Common pupils errors will be dealt with in the teaching of various topics.] Teaching and Learning of Primary Mathematics I I (geometry, measurement, statistics, algebra) Teaching Problem Solving and Investigations; Mathematical Communication; Teaching of Geometry, Money and Measures, Mensuration, Graphical Representation and Statistics, Algebra. [Common pupils errors will be dealt with in the teaching of various topics.] 54

Singapore primary teacher preparation: a focus on pedagogical content (cont) Teaching and Learning of Primary Mathematics III (e.g. assessment) This course covers two broad areas: (a) Various traditional assessment modes in Mathematics and the use of these modes in schools to assess pupil performance, in particular, the planning and construction of test items and (b) Practice of teaching skills, including catering for pupils of mixed abilities. Teaching and Learning of Primary Mathematics IV (e.g., IT) This course provides further pedagogical skills for those who will go deeper into teaching mathematics, especially at upper primary levels. Topics include: Games in Mathematics; Challenging Problems in Upper Primary Mathematics. Student teachers will 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 lessons impact (with others educators) 5.Revise the lesson 56

Chinas 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 dont 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 http://hrd.apec.org Singapore Bar Model: http://www.thesingaporemaths.comhttp://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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