International Bench Marking Indiana

8/12/2019 International Bench Marking Indiana

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Table of Contents

Executive Summary ...................................................................................................... 1

Overview ....................................................................................................................... 3

Overview of Selected International Education Systems ....................................... 5

Singapore .................................................................................................................... 5

Finland ........................................................................................................................ 9

Reerences ................................................................................................................. 12

Te Features of the Standards .............................................................................. 13

Concepts across Grade Levels ......................................................................... 13

Use o Examples ................................................................................................ 14

LimitsWhat Is Not Covered ........................................................................ 14

Descriptions o Summary Expectations ........................................................ 15Use o Verbs: Do Verbs versus What .............................................................. 16

Use o Calculators ............................................................................................. 18

Te Process Standards .......................................................................................... 19

Indiana ...................................................................................................................... 19

Singapore .................................................................................................................. 19

Finland ...................................................................................................................... 21

Differences and Similarities in Process Standards .............................................. 21

Problem Solving ................................................................................................ 21Reasoning and Proo ........................................................................................ 22

Communication ................................................................................................ 22

Connections ...................................................................................................... 22

Representations ................................................................................................. 22

Estimation and Mental Computation ............................................................ 22

echnology ........................................................................................................ 22

Attitudes ............................................................................................................. 23

Metacognition ................................................................................................... 23Applications and Modeling ............................................................................. 23


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Table of Contents(continued)

K8 ............................................................................................................................... 25

Standard 1: Number Sense and Computation .................................................... 27

Assets ........................................................................................................................ 27DifferenceDevelopment o the our operations ............................................... 28

DifferenceFocus on multi-step problems ......................................................... 30

DifferenceDevelopment o the concept o quantity and relationshipsbetween quantities ................................................................................................... 32

Differencereatment o rates, ratios and percentsdevelopment oproportional thinking ............................................................................................. 35

Differencereatment o actors and multiples ................................................. 37

Differencereatment o rounding and estimating .......................................... 39

Differenceiming o odd and even numbers .................................................. 41

Additional considerations ...................................................................................... 42

Standard 2: Algebra and Functions ..................................................................... 43

Assets ........................................................................................................................ 43

Differencereatment o rules o arithmetic ..................................................... 43

DifferenceEarly work with equations ............................................................... 44

Differencereatment o number patterns ........................................................ 44

Additional considerations ...................................................................................... 46

Standard 3: Geometry and Measurement ........................................................... 48

Assets ........................................................................................................................ 48

DifferenceEmphasis on geometry and measurement ..................................... 48

Differencereatment o transormation and constructions ........................... 49

Differencereatment o money concepts ......................................................... 51

Differencereatment o grade 2 geometry ....................................................... 52

Differenceiming o symmetry ......................................................................... 53

Differenceiming o angles ................................................................................ 54

Differenceiming o units work ........................................................................ 55Additional considerations ...................................................................................... 55

Standard 4: Data Analysis and Probability ......................................................... 56

Assets ........................................................................................................................ 56

Differencereatment o probability .................................................................. 56

Differenceiming o data displays .................................................................... 58

Additional considerations .......................................................................................59


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Table of Contents(continued)

High School ................................................................................................................. 61

Comparison of Secondary Education Systems ................................................... 63

Multi-year programs versus courses ..................................................................... 63Age comparisons ..................................................................................................... 64

Different programs within the systems ................................................................ 64

Mobility within the systems ................................................................................... 65

Interdisciplinary courses ........................................................................................ 65

Alignment o curriculum and assessment ........................................................... 66

Critical Differences ............................................................................................... 67

Differenceiming o orks in the road .......................................................... 67

DifferenceChoices o academic programs ....................................................... 68DifferenceGradations o mathematics course content ................................... 68

DifferenceAlignment o coursework, syllabus and exam .............................. 70

Comparison of the Mathematics ......................................................................... 71

DifferenceContent over time ............................................................................. 71

Algebra 1 ............................................................................................................... 71

Comparison able ................................................................................................... 71

DifferenceTe development o Algebra I concepts ......................................... 72

Algebra II .............................................................................................................. 72

Comparison able ................................................................................................... 72

DifferencePlacement o complex numbers ...................................................... 73

Differenceopics covered under matrices ........................................................ 73

Differencereatment o polynomials beyond quadratics ............................... 75

Differencereatment o normal distribution ................................................... 75

Differencereatment o data and probability .................................................. 76

Geometry .............................................................................................................. 79

Comparison able ................................................................................................... 79

DifferenceContent in geometry courses ........................................................... 79

Pre-Calculus .......................................................................................................... 80

Comparison able ................................................................................................... 80

DifferenceContent in Pre-Calculus course ...................................................... 81

Appendix

Design and echnology Syllabus (Singapore)


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2008 International Benchmarking Indianas Academic Standards in Mathematics

National Center on Education and the Economy Page 1

EXECUIVE SUMMARY

With the emergence and growth o the global economy, many education policy makers have

turned to international comparisons to help guide the design and development o nationaland state perormance standards. Since the 1960s, the United States has participated activelyin international projects designed to provide key inormation about the perormance o itseducation system relative to systems in other countries. Tese comparisons shed light on ahost o issues regarding the rigor, depth, coherence, and content o the mathematics studentsare learning. Tey let U.S. policy makers examine different aspects o countries standards andcurricula, assess these systems perormances, and identiy potential strategies to improve studentachievement within our own system.

Tis study compares Indianas mathematics standards with those o Singapore and Finland, two

countries whose students score consistently high on the international mathematics assessments.

Overall, Indiana ared well. Te comparisons revealed high quality, rigorous mathematics taughtin a good progression. Te report also identifies areas where Singapore and Finland take differentapproaches, use a different timeline, or cover topics in more or less depth than the U.S. in generaland Indiana in particular.

MAJOR FINDINGS

Arithmetic operations are spread more evenly over grades K6 in Singapore and Finland,where they are introduced earlier and more incrementally. Indiana students (and U.S.students in general) are introduced to multiplication and division later but then areexpected to learn more per grade level in upper elementary to surpass the standards o theirinternational counterparts by the end o 6th grade.

Te international emphasis on solving multi-step problems is consistent with promotingconceptual understanding through thoughtul problem solving.

Te international standards provide a coherent progression and solid oundation in theconcept o quantity and units and in the relationships among quantitiesaddressingmeasurement in primary grades; with areas, rates, and variables in the upper elementarygrades; and unctions and graphs in middle school. Tis progression and emphasis builds astrong conceptual oundation that prepares students or work with the concepts o variableand unction in middle school and algebra courses.


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Page 2 National Center on Education and the Economy

o maximize efficiency rom upper elementary mathematics through algebra, there shouldbe a more explicit treatment o the application o the number properties and properties oequality in the standards to the progression o topics introduced in the upper elementarygrades and middle school. Tese are major organizing ideas that orm the oundation o thelanguage o algebra used in arithmetic.

Standards or high school in the U.S. must address three challenging policy issues. Teyshould speciy the mathematics content necessary or:

Students preparing to major in science, technology, engineering, ormath (SEM).

Students preparing to major in humanities, business, or social sciencein college (non-SEM). Tis content should make students eligible orcollege without remediation.

Students preparing or all levels or work in a high-tech economy.

Standards or the traditional high school mathematics courses could be refined andimproved. Te most urgent need is to redesign the college preparatory sequence ornon-SEM majors to equip them with more useul mathematics skills. Algebra IIdoes not address the needs o non-SEM majors. Indianas revised standards are a bigimprovement, but the pathways and options in the international systems offer more useulmathematics to more students, while still allowing university access to all.

Te presentation o the standards in the international benchmarking countries ocuses

concisely on the mathematical content, while emphasizing the metacognitive processesstudents should engage in. Te documents ocus on the what o the mathematics and onstudents development o habits o mind in studying and communicating mathematics. Incontrast, the U.S. tradition, as reflected in Indianas draf, ocuses on the kinds o problemsstudents should be able to get answers or, regardless o the mathematics they use to getthe answers.

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SectionA

O V E R V I E W


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Overview Page 5

O S I E S

A meaningul discussion o similarities and differences between Indianas mathematics standardsand those o Singapore or Finland requires a basic overview o the educational systems in which themathematics is taught.

SINGAPORE

Singapore has devised a very successul system o mass education that is both ree and universal. Tepublic education system is highly centralized, with curriculum and standards that are uniorm acrossall schools. National high-stakes assessment examinations measure achievement in about the 6, 10,and 12 year o education. Te centralized authoritythe Ministry o Educationis responsible orormulating and implementing educational policies, developing national curriculum rameworks andguidelines, and administrating national examinations in collaboration with the Cambridge GeneralCertificate o Education. Te educational system in Singapore is governed by the principle o meritocracy,

and merit is measured largely through the national examination system.

Primary Education

Education begins at a young age in Singapore. ypically, children attend two to three years okindergarten instruction, beginning as early as age 3. Te kindergarten years are servicedprivately, while compulsory public education begins at about age 5 or 6 when students enterprimary school.

At the completion o six years o primary education, students take the Primary School LeavingExamination (PSLE). Tis assesses students achievement levels and determines their suitability

or secondary education.

Singapore United States

Kindergarten,

Levels 13

Approximate

Age: 25

Kindergarten,

Level 1

Approximate

Age: 5

Primary,

Levels 16

Approximate

Age: 612

Elementary,

Levels 16

Approximate

Age: 612

Primary School Leaving Exam

(PSLE) No Exam


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Te Singapore system recognizes that some students may need special assistance to attaincompetence. Tese students do not attain mastery by ollowing Singapores regular program omathematics instruction, but ollow an alternative course. Beginning in grades 5 and 6, Singaporeidentifies its weaker students on the basis o a an examination. Tese students are taughtaccording to a special oundational 5th- and 6th-grade mathematics ramework. Students receive

special assistance by:more timeapproximately 30 percent more mathematics instruction than studentsin the regular track, and

exposure to the same mathematical content as students in the regular track, although ata slower pace.

Note: Tis report compares SingaporesMathematics Syllabus PrimaryLevels 16 to IndianasGrades 16 standards.

Secondary Education

Singapores secondary level entails our to five years o education roughly equivalent to that oU.S. grades 710. Based on their PSLE exam perormance, students enter one o our secondarystreams: Special, Express, Normal Academic, or Normal echnical. Special Stream students takeadvanced language or Higher Mother ongue. Afer our years, both the Special and ExpressStreams take the Singapore-Cambridge General Certificate Exam, Level O (GCE O). Aferour years, the Normal Academic Stream (NA) and the Normal echnical Stream (N) take theSingapore-Cambridge General Certificate Exam, Level N (GCE N). Normal Stream studentswho perorm well in the GCE N may continue with the program or a 5 year, moving on to takethe GCE O. Each secondary school offers all streams, and students are able to move rom onestream to another, based on merit.

Te our streams see a yearly distribution o students similar to that recorded in 2006:

Special Stream 9.1%

Express Stream 52.0%

Normal Academic Stream 24.5%

Normal echnical Stream 13.5%

Ministry of Education, 2006

Singapore also offers an Integrated Programme (IP) which is designed or students who aresel motivated, clearly university-bound, and ready to thrive in a less structured environment.Te program spans secondary and pre-university education and does not include intermediatenational examinations at the end o secondary school. In this program, time normally usedto prepare students or the GCE O is used to broaden students learning experiences. Te IPProgramme leads to a Baccalaureate Diploma or A-level Exam.


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Pre-University (or Post-Secondary) Education

Afer 4 to 5 years o secondary study and successul completion o the GCE O at age 16 or older,students are presented with a variety o options. For instance, they may pursue an academiccourse o study or a proessional centric course o study that ocuses on proessional-leveltechnical education. As determined by their GCE O scores, individual students continue theireducation either in pre-university or post-secondary institutions.

Pre-University

Junior Colleges (JCs) Centralized Institute (CI)

or the Millenia Institute

Post-Secondary

Polytechnic Institutes Institute o echnical Education

Academically inclined students with the necessary GCE O qualifications, will usually enter thepre-university system. Within the pre-university system there are different curricula. H1 and H2level courses are complementary; H1 is typically hal the breadth o an H2 course, although equalin depth. H3 courses are o the highest level and encourage critical thinking and may involve

research.

Polytechnic Institutes offer 3-year diploma courses or students who wish to pursue appliedand practice-oriented proessional training. Students must also have the necessary GCE OLevel qualifications to enter a polytechnic school. Polytechnic courses are ofen specializedand may include specific fields, such as marine engineering, business management, digitalcommunications, and the like. Polytechnic graduates with good grades also have the opportunityto pursue tertiary education at the universities.

University (34 years)

Pre-University Junior

Colleges (2 years)

Post-Secondary

Technical Education

(2 years)

Post-Secondary

Polytechnics (3 years)

Secondary School (46 years)

Primary School (6 years)

GCE O est GCE N est

echnical education is or students with GCE O or N Level certificates. echnical Institutes offer1- to 2-year technical and vocational courses. Students who do well can go to polytechnics or adiploma program. Qualified candidates may also progress to the universities or subsequent study.


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Overview Page 9

Regardless o the path takenwhether beginning in a echnical Institute or a Junior Collegeallstudents have the opportunity to apply themselves and work toward university-level studies. Allstudents must take the GCE A Level exam or entrance into the university.

Note: Te H1 mathematics as defined by the H1 testthe first level o the pre-university

mathematics courseswere used to compare to Indianas high school standards,along with Secondary 3 and 4.

FINLAND

Finlands education system is based on providing all children and young people with equal basiceducation services. In Finland, education is compulsory, starting rom the year the child becomes sevenyears old and ending when he or she reaches 16, a total o nine years o basics. Both municipal andprivate day-care services are available or children below the school age o seven. All 6-year-olds areentitled to at least one year o pre-school education beore beginning their basic education. Pre-schooleducation is available in school settings and in day-care centers.

Compulsory Basic Education (Ages 716)

Te Finnish National Board o Education outlines the learning goals and defines the maincontent o basic education. Te board sets the National Core Curriculum, the guidelines thatgovern all education providers. Te National Core Curriculum defines not only the goalsand content o the various subjects but also the cross-curricular themes. Tese themes areexpected to be integrated in a childs upbringing and education. Cross-curricular themes areresponsive to the educational challenges o the time. All education providers must draw up localmunicipality-specific (or school-specific) curricula as guided by the National Core Curriculumand any pertinent legislation.

Note: Because the education o a child in Finland begins at age seven, making exact correlationsto Indiana grade by grade becomes problematic. However, it is possible to concentrate onprogression and sequence; these will be the determining actors when comparing the twosystems. In this report, Indiana grades K8 are compared with Finland grades 18.

Upper Secondary Schools

Education afer primary school is divided into vocational and academic systems, according tothe old German model. raditionally, these two upper secondary systems did not inter-operate,

although today some restrictions have been lifed.

Approximately 92% o those who completed basic education in 2003 continued directly to uppersecondary school. ypically, students enter either a trade school (vocational upper secondary) oran academic-oriented General Upper Secondary program. rade school graduates may enter theworkorce directly afer graduation, while those in General Upper Secondary school do not studyvocational skills, as they are expected to continue to tertiary education.


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General Upper Secondary Education

Te General Upper Secondary program isdivided into courses. Each course consistso about 38 lessons. Tereore, year-longclasses are not required. In addition, allgeneral upper secondary schools are nownon-graded.

Te National Core Curriculum for UpperSecondary School requires a minimum o75 courses. O these, some are compulsory,some are considered specializations, andstill others are deemed applied courses.Applied courses may be integrated coursescombining elements rom differentsubjects, various methodological courses,or vocational skills.

Matriculation

Te final examination at the end o the General Upper Secondary programthematriculation examinationis drawn up by a centralized body. Tis body validates theindividual tests against uniorm criteria. Te matriculation examination consists o aseries o at least our tests. Every student must take the test in his or her mother tongue.Students may select three other tests to take, chosen rom possibilities including the

second national language, a oreign language, mathematics, sciences, and humanities. Inaddition, students are ree to take other optional tests.

Note: In this report, the Indiana high school standards were compared with compulsory coursesin the National Core Curriculum for Upper Secondary School.

Decision Making

Decisions regarding broad national objectives as well as the more specific aspects o education are

all decided by the government. By devising the National Core Curriculum, the Finnish NationalBoard o Education determines the objectives and core contents in the various subjects.

Beginning with the boards guidelines, education providers then draw up their own localcurricula. Te system must provide students with individual choices concerning studies,including the ability to utilize instruction given by other education providers, i necessary.


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Overview Page 11

A Special Note Regarding Teacher Education

When comparing the standards documents, it soon becomes evident that the National CoreCurriculum for Upper Secondary Schooloutlines Finlands mathematics objectives with a ratherbroad stroke and much less specificity than do Indianas Mathematics Standards. Te NationalBoard o Education recognizes the high caliber o Finlands teachers, most o whom possess anadvanced understanding o core concepts in mathematics and mathematics instruction. Hence,the document reflects teachers level o mathematical competency.

Tis high competency is developed within Finlands teacher education system. Finlands eacherEducation system has been identified as one o the top training programs in the world. (Barberand Mourshed, 2007) Finland begins by selecting only top candidates or their university system.Descriptive snippets o the selection process ollow:

multiple-choice examinations designed to test numeracy, literacy andproblem-solving skills top scoring candidates are then passed through a secondround in the selection process that is run by individual universities applicants are

tested or their communications skills, willingness to learn, academic ability andmotivation or teaching. (Barber and Mourshed, 2007)

In Finland, the teaching proession is a competitive field. Even afer intensive preparation andsuccessul completion o the training program, graduates apply to individual schools where theymust score well on additional exams in order to win a position. Young men and women workhard to gain admission to Finlands eacher Education system. And they continue to work hard ina culture that places high value on continual growth and improved practice, where proessionaldevelopment is regarded as essential and is organized extensively. (NBE, 2008)

Finlands teachers possess a level o competency that is reflected in the spare and open guidelines

provided in the National Core Curriculum. Teir high degree o proessionalism affords themleadership opportunities and a great deal o local control, not to mention instructional reedom.

teachers are considered pedagogical experts, and are entrusted with considerableindependence in the classroom, and also have decision-making authority as concernsschool policy and management. Tey are deeply involved in drafing the local curriculaand in development work. Furthermore, they have almost exclusive responsibility or thechoice o textbooks and teaching methods. (NBE, 2008)

Singapore has a similar screening system which is used to test and select applicants beore theyenter teacher education at the university level. Tis results in highly-trained teachers. Even so, the

Singapore Ministry o Education is more directive in its syllabi.


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Overview Page 13

F S

Indiana, Finland, and Singapore describe the requirements o its mathematics programs in different ways;however, similarities can be ound. First, Singapore calls its document a syllabus, rather than a set ostandards. Tis report looks at the SingaporeMathematics Syllabus Primaryand the SingaporeSecondaryMathematics Syllabusesplus the H1 test items used in the pre-university system. Te term standardsdoes not appear in Finlands title eitherits document is reerred to as the National Core Curriculum. Inthis report, both the National Core Curriculum for Basic Educationand the National Core Curriculum forGeneral Upper Secondarywere used.

When readers view the documents side by side, one o the first impressions they might notice is thelook o the standardsthat is, the ormat in which they are presented. When the Indiana Standardsfor Mathematicsare compared to SingaporesMathematics Syllabus Primaryand Finlands National CoreCurriculum for Basic Education, several differences become evident. Tese differences reflect not onlythe clarity o the presentations but the teachers assumed level o understanding o the mathematics.Some o the distinguishing eatures are discussed below.

Concepts across Grade Levels

One way Indiana represents its standards is on a multi-color chart, showing how conceptsplacevalue, comparing and ordering numbers, operations, and the rules o arithmetic, or instanceappear across the grade levels. Te specific mathematical concepts are a major organizing actorin the presentation o the standards.

INDIA NA

Tis approach allows teachers to see how students understanding o a concept builds rom oneyear to the next by highlighting the similar concepts that appear a year later or earlier.


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Use of Examples

Te Indiana Standards for Mathematicsalso include examples. Examples remove ambiguity sothat teachers can determine exactly what is meant by the standard.

INDIA NA

Number Sense and Computation Grade 1

1.1.2 Recognize numbers to at least 100 as groups o tens and ones.

Example:

Use base ten blocks to model 34 using 3 longs (tens) and 4 units o one; then using 34 unitso one; and finally, using 2 longs (tens) and 14 ones.

LimitsWhat Is Not Covered

As expected, the SingaporeMathematics Syllabus Primary guides teachers through themathematics to be covered in the primary and secondary years. What is especially noteworthyabout the portion o the syllabus shown below is that it goes a step urther and describes whatis notcovered in a given year.

SINGA PORE

Multiplication and Division Primary 1

Include:

multiplication as repeated addition (within 40)

use o the multiplication symbol () to write a mathematical statement or a given situation

division o a quantity (not greater than 20) into equal sets:

given the number o objects in each set

given the number o sets

solving 1-step word problems with pictorial representation

Exclude:

use o multiplication tables

use o the division symbol ()

By stating, when necessary, the work that should be excludedrom instruction, the Singaporesyllabus enables teachers to ocus their efforts and hone in on specific aspects o the concepts.Te Singapore syllabus makes clear what students are responsible or knowing and whenthey are expected to know it. Tis exclusionaryclarityserves to organize teachers instructionaround concrete, conceptual work so that students deepen their understanding o concepts ata reasonable pace. Tis also helps prevent the negative effects that can result when students arerushed through material that is beyond their level o comprehension.


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Use of Verbs: Do Verbs versus What

In the U.S., mathematics standards are ofen written in a style that ocuses on how studentswill do tasks rather than on what mathematics content they will learn. For example, U.S. mathstandards use a limited set o verbs to begin each standard. Tese words describe what studentsare expected to do: identiy, recognize, measure, find, or solve.

Singapore and Finland stress the content o their programs with simple, concrete descriptions othe mathematics. Because the ocus is on content, the work to be done is naturally revealed. Notethe two examples rom the SingaporeMathematics Syllabus Primary shown below.

SINGA PORE

Ratio Primary 6

Include:

expressing one quantity as a raction o another, given their ratio, and vice versa

finding how many times one quantity is as large as another, given their ratio, and vice versa

expressing one quantity as a raction o another, given the two quantities,

finding the whole/one part when a whole is divided into parts in a given ratio

solving word problems involving 2 pairs o ratios

SINGA PORE

Ratio, Rate and Proportion Secondary 1

Include:

ratios involving rational numbers

writing a ratio in its simplest orm

average rate

problems involving ratio and rate

On the same topic o ratio, the Indiana Mathematics Standards(see next page) defines theratio standardwhat the student needs to dowhile the example helps to illustrate themathematics content.


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Overview Page 17

Indiana has the do statements but with the added example illustrating the mathematics; thewhat becomes more transparent.

INDIA NA


6.1.7 Interpret ratios, model ratios, and use ratios to show the relative sizes o two quantities.Use the notations: a /b, ato b, and a: b.

Example:

A car moving at a constant speed travels 130 miles in 2 hours. Write the ratio o distanceto time and use it to find how ar the car will travel in 5 hours.

6.1.8 Recognize proportional relationships and solve problems involving proportionalrelationships.

Example

Sam made 8 out o 24 ree throws. Use a proportion to show how many ree throws Samwould probably make out o 60 attempts.

6.1.9 Solve simple ratio and rate problems using multiplication and division, includingproblems involving discounts at sales, interest earned, and tips.

Example:

In a sale, everything is reduced by 20%. Find the sale price o a shirt whose pre-sale pricewas $30.

Notice each o the Indiana standards ocuses on finding a single number. It is important toremember that there is a significant drawback to ocusing too much on the do. Te real crux

o the standard is the core mathematics. Do verbs can overshadow the corethe what omathematics. When this occurs, the mathematics becomes ragmented and easily subordinatedto teaching students how to get answers, without necessarily deepening their understanding.

In addition to the example problems shown in the above chart, some example questionscould also be included. Te right questions can extend students learning and deepen theirunderstanding. For example, in the ree throw problem illustrating standard 6.1.8, the ollowingquestions would enrich the example:

What is proportional to what?

What rates can you calculate?

What do the rates tell you?I Sam made 2 o his next 3 ree throws, what would happen to the rate?

Questions like these require more o students than simply calculating answers; they requirestudents to think deeply about what the core mathematics entails.


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Use of Calculators

Te Singapore syllabus provides substantial guidance on the use o calculators. In grades 5 and6, the syllabus states that calculator use is allowed unless otherwise stated. When it is prohibited,specific standards or student activities will include the phrase without using calculators.

For instance, the example below assures teachers that in grade 5 students are expected to add andsubtract proper ractions on their own, while it is acceptable or them to use calculators to addand subtract mixed numbers.

SINGA PORE

Fractions: Four Operations Primary 5(Calculator is allowed unless otherwise stated.)

Include:

addition and subtraction o proper ractions without using calculators,

addition and subtraction o mixed numbers,

multiplication o a proper ractions and a proper/ improper raction withoutusing calculators,

multiplication o an improper raction and an improper raction,

multiplication o a mixed number and a whole number,

division o a proper raction by a whole number without using calculators,

solving word problems involving the 4 operations.

Summary of the Three Documents

Te three documents, Singapores syllabi (Mathematics Syllabus Primary and SecondaryMathematics Syllabuses), Finlands National Core Curriculum (Basic Educationand GeneralUpper Secondary), and Indianas Mathematics Standards (K8 and Secondary) all have uniqueeaturesthe eatures being useul in different ways. Each document takes a unique approach toclariying mathematical concepts, instructional sequences, and expected student perormances atthe different grade levels.


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Overview Page 19

P S

Singapore, Finland, and Indiana all make somewhat comparable reerences to overall processes in theirstandards documents. Process standards reer to the major components that are critical to both the

effectiveness o a mathematics program and to a students ability to comprehend and ully engage withthe mathematics. Each document treats the subject differently, but there are enough similarities that adiscussion o the standards is merited.

INDIANA

Te Indiana mathematics standards or K8 divide process standards into five categories.

Problem Solving

Reasoning and Proo

Communication

Connections

Representation

Tree additional categories should be addressed at all grade levels in mathematics:

Estimation

Mental Computation

echnology

SINGAPORE

Te Singapore syllabus consists o several sections which correlate to Indianas process standards. Teseappear in the Aims o Mathematics Education in Schools and Mathematics Framework sections otheMathematics Syllabus Primaryand the Secondary Mathematics Syllabuses.

Aims of Mathematics Education in School

Tis section offers broad statements regarding the aims o the mathematics program. It answersthe question, What is the intent o mathematics learning and what will it enable students to doand achieve?


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MATHEMATICS S YLLABUS PRIMARY

Mathematics education aims to enable students to:

Acquire the necessary mathematical concepts and skills or everyday lie, and or continuous

learning in mathematics and related disciplines.Develop the necessary process skills or the acquisition and application o mathematicalconcepts and skills.

Develop the mathematical thinking and problem solving skills and apply these skills toormulate and solve problems.

Recognize and use connections among mathematical ideas, and between mathematics andother disciplines.

Develop positive attitudes towards mathematics.

Make effective use o a variety o mathematical tools (including inormation andcommunication technology tools) in the learning and application o mathematics.

Produce imaginative and creative work arising rom mathematical ideas.

Develop the abilities to reason logically, communicate mathematically, and learncooperatively and independently.

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Mathematics Framework

Tis section describes the importance o learning attitudes, metacognition, critical thinking,mathematical concepts, and required skills that are important components to achieving successin mathematics. (Te Mathematics Syllabus Primary includes a section in Part B titled Use oCalculator and echnology.) Te section describes the underlying, principle components that

rame an effective mathematics program: attitudes, metacognition, processes, concepts, and skills.

MATHEMATICS S YLLABUS PRIMARY

Metacognition

Mathematical

Problem

Solving

Attit

udes

Concepts

Pro

cesses

Skills

BeliefsInterest

AppreciationConfidence

Perseverance

Numerical calculationAlgebraic manipulation

Spatial visualizationData analysisMeasurement

Use of mathematical toolsEstimation

NumericalAlgebraic

GeometricalStatistical

ProbabilisticAnalytical

Monitoring of ones own thinkingSelf-regulation of learning

Reasoning, communication

and connectionsinking skills and heuristicsApplication and modelling


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Overview Page 21

FINLAND

Each o the Finland grade spans, 12, 35, and 69, have summary statements which describe theobjectives or that grade span.

FINLAND Grades 12

SummaryStatement

Te core tasks o mathematics instruction in the first and second grades are the development omathematical thinking; practice concentrating, listening and communicating; and acquisition oexperience as a basis or the ormulation o mathematical concepts and structures.

Objectives

Te pupil will:

learn to concentrate, listen, communicate, and develop their thinking they will derivesatisaction and pleasure rom understanding and solving problems

gain diverse experience with different ways o presenting mathematical concepts; in theconcept ormation process, the central aspects will be spoken and written language, tools,and symbols

understand that concepts orm structures

understand the concept o the natural number and learn the basic computational skillsappropriate to it

learn to justiy their solutions and conclusions by means o pictures and concrete modelsand tools, in writing or orally; and to find similarities, differences, regularities, andcause-and-effect relationships between phenomena

become practiced in making observations about mathematical problems that come up andare challenging and important rom their personal standpoints.

DIFFERENCES AND SIMILARITIES IN PROCESS STANDARDS

Problem Solving

Te Indiana Standards for Mathematicslists problem solving as a process standard. Singaporehas placed mathematical problem solving in the center o its mathematics ramework.Singapores philosophy is that mathematical problem solving is central to mathematics learning.Te development o mathematical problem solving abilities is dependent on five inter-related

components titled Concepts, Skills, Processes,AttitudesandMetacognition.

Finland also deals with problem solving but stresses becoming practiced in makingobservations about mathematical problems that come up and the ability to find similarities,differences, regularities, and cause-and-effect relationships between phenomena.

(Also see the Metacognition section on on page 23.)


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Overview Page 23

Attitudes

Both Singapore and Finland emphasize the importance o student attitudes toward mathematics.Singapore includes attitudes as an element o its mathematics ramework and also devotesa section o the introductory materials to the discussion o attitudes. In every grade leveloverview, Finland outlines objectives that involve attitudesstudent will derive satisactionand pleasure rom understanding and solving problems (12); gain experience in succeedingwith mathematics (35); and learn to trust themselves, and to take responsibility or their ownlearning in mathematics (69).

Metacognition

Singapore includes mention o metacognitionor thinking about thinking. Te discussionstresses that students need to be able to monitor their own thinking. It provides sometechniques that teachers can use to develop a students metacognitive awareness and to developproblem solving abilities. Finland places emphasis on students developing their own thinking

not just solving problems.

Applications and Modeling

Singapore discusses how applications and modeling play a vital role in the development omathematical understanding and competencies. Finland deals with modeling in grade 3 andabove with statements such as introduce the learning o mathematical models o thinking.


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Indianas development o operations with whole number is well donefirst starting with models,using number lines, then ocusing on automaticity.

In general, the topics covered in Number Sense and Computation in the Indiana Standards forMathematicsare equivalent to the content covered in Singapore and Finland. Te topics buildrom grade level to grade level and ollow a spiral approach.

DifferenceDevelopment of the four operations

In Indiana, as is generally the case throughout the United States, multiplication and division arenot introduced until grade 3. In Singapore, however, multiplication and division are specificallytaught beginning in grade 1, and in Finland, these operations are explicitly taught during the 12grade span.

FINLA ND

Numbers and Calculations Grades 12

multiplication and multiplication tables

division, using concrete tools

SINGA PORE

Multiplication and Division Primary 1

Include: multiplication as repeated addition (within 40),

use o the multiplication symbol () to write a mathematical statement or a given situation,

division o a quantity (not greater than 20) into equal sets:

given the number o objects in each set,

given the number o sets,

solving 1-step word problems with pictorial representation.

Exclude:

use o multiplication tables,

use o the division symbol ().

While at first this appears to be a rigor issue, this is not the case. By the end o grade 6,U.S. students are expected to master all our operations with whole numbers, ractions, anddecimals. By contrast, at grade 6, Singapore excludes certain sub-categories o division oractions, and Finland excludes both multiplication and division o ractions and decimalsor the grades 35 span.


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INDIA NA


5.1.5 Add and subtract decimals and ractions, including ractions with different

denominators and mixed number using a standard algorithmic approach.Example:

34

5 2

2

3= ?

5.1.6 Multiply ractions using a standard algorithmic approach.

Example:

Find3

4o

2

5. Explain when the product is smaller than the actors.

On the whole, Singapore has a larger number o more detailed operations standards than doesIndiana, including which concepts should not be included. Indiana has ewer standards but theyare shown with specific examples.

Suggestion: Tere is always a question as to the amount o specificity that is necessary, butconsider adding operations standards where appropriate and making them more specific.

DifferenceFocus on multi-step problems

Te Singapore curriculum emphasizes solving word problems, defining explicitly or teachersi the problems should be one-step or multi-step word problems. Tis progression is logicallydeveloped within their standards. Singapores emphasis on multi-step word problems requiresstudents to build and demonstrate a deeper understanding o the concepts.

SINGA PORE

Addition and Subtraction Primary 1

solving 1-step word problems involving addition and subtraction within 20.

Multiplication and Division

solving 1-step word problems with pictorial representation.


solving up to 2-step word problems involving addition and subtraction.

Multiplication and Division

solving 1-step word problems involving multiplication and division within themultiplication tables.

(continues)


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K8 Page 31

(continued)


solving up to 2-step word problems involving addition and subtraction.

Multiplication and Divisionsolving up to 2-step word problems involving the 4 operations.

Multiplication and Division: Whole Number Primary 4

solving up to 3-step word problems involving the 4 operations.

Multiplication and Division: Fractions

solving up to 2-step word problems involving addition, subtraction and multiplication.

Multiplication and Division: Decimals

solving up to 2-step word problems involving the 4 operations.

Percentage Primary 5solving up to 2-step word problems involving percentage.

Ratio

solving up to 2-step word problems involving ratio.

Volume of Cube and Cuboid

solving up to 3-step word problems involving the volume o a cube/ cuboid.

Speed Primary 6

solving up to 3-step word problems involving speed and average speed.

Data

solving 1-step problems using inormation presented in pie charts.

Finland does not have the same sequence o development as Singapore but does includeobjectives that iner students are solving problems.

FINLA ND

Description of Good Performance at the End of Second Grade

understand addition, subtraction, multiplication, and division and know how to apply themto everyday situations

Description of Good Performance at the End of Fifth Grade

know how to depict real-world situations and phenomena mathematically by comparing,classiying, organizing, constructing, and modelling

know how to present mathematical problems in a new orm; they will be able to interpret asimple text, illustration, or event and to make a plan or solving the problem


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K8 Page 33

Students study quantities and the relationships among quantities as the content o measurement.Tus, they study quantities measured by distance measures, time and money. Attention to units isgiven a critical ocus, and this early work with units develops the idea o variables. For instance,in early elementary, Singapore students work with compound units (relationships between units)starting in grade 3.

SINGA PORE

Length, Mass and Volume Primary 2

use o the appropriate measures and their abbreviations cm, m, g, kg, l,

solving word problems involving length/ mass/ volume.

Money

converting an amount o money in decimal notation to cents only, and vice versa.

Length, Mass and Volume Primary 3

conversion o a measurement in compound units to the smaller unit, and vice versa, solving word problems involving length/ mass/ volume/ capacity.

Time

conversion o time in hours and minutes to minutes only, and vice versa,

finding the duration o a time interval,

finding the starting time/ finishing time,

solving word problems involving addition and subtraction o time given in hoursand minutes.

Fluency with quantities, i.e. numbers with units, also supports learning ractions. Te unitsdenominate what is to be counted by the number and the number numerates how many. Tisis an important concept in studying ractions.

Rate and proportional relationships are built, in large part, on a oundation o work withquantities. In Singapore there is also more emphasis on the relationship between quantities. Inthis section on speed, notice the ocus on the relationship between distance, time, and speedrather than merely solving problems to find one rom the others.


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SINGA PORE

Distance, Time and Speed Primary 6

Include:

concepts o speed and average speed,

relationship between distance, time and speed

Distance = Speed ime

Speed = Distance ime

ime = Distance Speed

calculation o speed, distance or time given the other two quantities,

writing speed in different units such as km/h, m/min, m/s and cm/s,

solving up to 3-step word problems involving speed and average speed.

Exclude:

conversion o units, e.g. km/h to m/min.

From this work on the development o quantity, the conceptual oundation o unctions isbuilt. Tis is very much an engineering/science perspective on unctions, rather than a ormalmathematical conception. o an engineer, a variable is something you measure; it typically hasunits o measurement that define the units o the domain and the meaning o the numbers on thecoordinates o a graph. Functions are ormalized in electives in secondary, years 34 (roughly, atages 1516) or those who are specializing in SEM subjects.

In the United States, in general, students in earlier grades are not introduced to the concept o

quantity and relationships between quantities. Discussing a quantity as a magnitude using unitsand the numbers o units is sporadic in the U.S. Te Indiana Standards for Mathematicsdo notmake explicit the incorporation o numbers with units into their progression o concepts inelementary grades.

Suggestion: Consider building this coherence rom the early grades through minorrevisions to the measurement standards and then extending it in the standards relatingto rates, ratios, proportionality and linear unctions. See the next item, reatment orates, ratios and percentsdevelopment o proportional thinking, or more details onthat section.


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K8 Page 35

DifferenceTreatment of rates, ratios and percentsdevelopment of

proportional thinking

Te Singapore syllabus introduces rates, ratios and percents across our instructional years.Percents and ratio begins in grade 5 and is built on in grade 6. Problems o rate (ocusing on

speed) are also introduced in grade 6 with additional work on rate, ratio, and proportion inthe next two years o secondary school.

SINGA PORE

Rates, Ratios, and Percentage Primary 5

6 bullet points on percent

7 bullet points on ratio

Rates, Ratios, and Percentage Primary 6

3 bullet points on percent

5 bullet points on ratio

5 bullet points on speed

Rates, Ratios, and Percentage Secondary 1

4 bullet points on ratio, rate, and proportion

6 bullet points on percentage


1 bullet point on unctions and graphs

Rates, Ratios, and Percentage Secondary 2

2 bullet points on ratio, rate, and proportion

6 bullet points on percentage


With this more gradual accretion o concepts (and with the greater specificity about each conceptwithin the standards), Singapore avoids the difficult task o working with three conusinglysimilar ideas all in one year. Also Singapore spends time on the relationship between quantities.

In grades 6 and 7, Indiana students work with proportional situations within the contexts opercents, ratios, and rates. Although percents, ratios, and rates overlap, there are some placeswhere the concepts can be conused.


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INDIA NA


6.1.7 Interpret ratios, model ratios, and use ratios to show the relative sizes o two quantities.

Use the notations: a /b, ato b, and a: b.Example:

A car moving at a constant speed travels 130 miles in 2 hours. Write the ratio o distance totime and use it to find how ar the car will travel in 5 hours.

6.1.8 Recognize proportional relationships and solve problems involving proportionalrelationships.

Example:

Sam made 8 out o 24 ree throws. Use a proportion to show how many ree throws Samwould probably make out o 60 attempts.

6.1.9 Solve simple ratio and rate problems using multiplication and division, including

problems involving discounts at sales, interest earned, and tips.

Example:

In a sale, everything is reduced by 20%. Find the sale price o a shirt whose pre-sale pricewas $30.

Note that standard 6.1.7 specifies ratio and 6.1.8 specifies proportion. Yet the two examples areconceptually and structurally alike. It may not be clear to the reader that these two standardsare different in any interesting or important way.

Te wording in standard 6.1.9 specifies ratio and rate, but the accompanying example is a

percent problem.In preparation or algebra, students need to understand that a proportional relationship is arelationship between two variables where one is a constant multiple o the other: letyand xbetwo variables and mbe a constant. Iy= mx, thenyis proportional to x. Neither the SingaporeMathematics Syllabus Primarynor the Indiana Standards for Mathematicsexplicitly state thatstudents should understand this.

Suggestion: In order or teachers and students to recognize the overlaps and distinctionsamong the proportionally-based concepts o percent, ratio, and rate, add a level o

specificity across several grade levels.


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K8 Page 37

DifferenceTreatment of factors and multiples

In the Singapore syllabus, there is explicit treatment o actors, common actors, multiples, andcommon multiples.

SINGA PORE

Whole Numbers: Multiplication and Division Primary 2

recognizing the relationship between multiplication and division

Whole Numbers: Numbers up to 10,000 Primary 3

odd and even numbers

Fractions: Equivalent fractions Primary 3

writing the equivalent raction o a raction given the denominatoror the numerator

expressing a raction in its simplest orm

Whole Numbers: Factors and Multiples Primary 4

determining i a 1-digit number is a actor o a given number

listing all actors o a given number up to 100

finding the common actors o two given numbers

recognizing the relationship between actor and multiple

determining i a number is a multiple o a given 1-digit number

listing the first 12 multiples o a given 1-digit number

finding the common multiples o two given 1-digit numbers

Students are expected to learn and use the concepts o actors and multiples in grade 4, one yearollowing the grade 3 conceptual emphasis on multiplication and division o whole numbers.Tis work with actors and multiples orms the underpinnings o students work in arithmeticoperations with the traction orm that figures largely in grades 4 and 5.


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Below are the Indiana Standards for Mathematicsthat address actors and multiples.

INDIA NA


2.1.5 Identiy odd and even numbers and determine whether a set o objects has and odd oreven number o elements.


4.1.2 Compare and order ractions by using the symbols or less than ().


5.1.3 Identiy prime and composite numbers.

5.1.5 Add and subtractractions with different denominators and mixed numbers using astandard algorithmic approach.

Te Indiana Standards for Mathematicsdo not explicitly address divisibility, common actors,and common multiples, which are essential to calculating with ractions. Students compareand order ractions in grade 4 (where, although the example given involves simple ractions,the denominators are different). Ten go on to add and subtract ractions with differentdenominators in grade 5, without any explicit mention o actors and multiples in the standards.

Suggestion: Consider explicitly stating in the standards, probably in grade 4, whatstudents are expected to understand and be able to do with actors, multiples, commonactors, and common multiples.


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K8 Page 39

DifferenceTreatment of rounding and estimating

Finlands National Core Curriculum lists evaluating, checking, and rounding the results ocalculations in the core contents or grades 35. At grades 69, the core contents includerounding and estimation; using a calculator.

Te Singapore syllabus is explicit in various places about estimation and approximation skills.Rounding appears in grade 4, where skills include rounding off [whole] numbers to the nearest10 or 100 and rounding off [decimal multiplication and division] answers to a specified degreeo accuracy.

While Indiana students are required to perorm tasks that support the learning and use o roundingskills (count by hundreds, plot numbers on a number line, recognize real-world measurements asapproximations, etc.), the standards never explicitly require the students to round.

Suggestion: Consider speciying some rounding skills in the Indiana Mathematics

Standards, beginning at grade 4. Use examples that show that rounding is a buildingblock or understanding approximation and interpreting decimal results within ourbase-ten system and also reinorces other tools or approximation. Also include moreemphasis on checking and estimating computations.

DifferenceTreatment of decimals in grade 4

Te Singapore grade 4 standards are explicit in three places about decimals as an embodimento place value and working with decimals, which provide teachers with clarity on what studentsshould be able to do.

SINGA PORE

Decimals up to 3 Decimal Places Primary 4

Include:

Notation and place values (tenths, hundredths, thousandths)

Identiying the values o the digits in a decimal

Use o the number line to display decimals

Comparing and ordering decimals

Conversion o a decimal to a raction

Conversion o a raction whose denominator is a actor o 10 or 100 to a decimal

Rounding off decimals to the nearest [whole, tenth, hundredth]


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Indiana standard 4.1.3 addresses working with decimals, however,the standard itsel puts no particular ocus on place value.

INDIA NA

Number Sense and Computation Grade 44.1.3 Interpret and model decimals as parts o a whole, parts o a group, and points and

distances on a number line. Write decimals as ractions.

I students are modeling and interpreting decimal numbers, this work should include theunderstanding and use o the 10 to 1 relationship between each pair o adjacent places.

Also, in grade 5, Indiana students count, read, write compare, and plot decimals to 3 decimalplaces (5.1.1), compare and order decimals, ractions and percents (5.1.2), and add and subtractdecimals (5.1.5). Tis work builds on the grade 4 introduction to decimals, which is contained ina single standard.

Suggestion: Consider writing additional standards about decimals and tie them to placevalue explicitly.


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K8 Page 41

DifferenceTiming of odd and even numbers

At the end o grade 2, Finnish students are expected to know about odd and even numbers.Tey are also expected to work with multiplication and the multiplication tables, as well asdivision, using concrete tools. In Singapore, odd and even numbers are treated in grade 3, withwork in multiplication and division beginning in grade 1.

Indiana Standards for Mathematicsplace the understanding o odd and even numbers with skipcounting patterns in grade 2:

INDIA NA


2. 1. 5 Identiy odd and even numbers and determine whether a set o objects has an odd oreven number o elements.

2. 1. 2 Count by ones, twos, fives, tens, and hundreds to at least 1,000.

Tus, students are introduced to odd and even numbers in grade 2, a year beore multiplicationand division are introduced. o understand the underpinnings o the concept o odd andeven, rather than simply naming a number as odd or even, students will need the concept omultiplicationeven numbers are multiples o twoand divisionodd numbers make groupso two with a remainder.

Suggestion: Consider putting work with odd and even numbers in the same instructionalyear that students develop the concept o multiplication and division.


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Additional considerations

Specificity of Standards

In some cases, the Indiana Standards for Mathematicsspeciy the number ranges or thespecific standards (e.g., multiplication acts to 10) and in others these are not included.

Consider reviewing the standards with an eye to making them more specific about theranges o numbers that apply.

More Examples for Complex Standards

Some Indiana standards summarize a list o activities and subcategories related to aspecific concept, such as 4.1.1 which deals with counting, reading, writing, comparing andplotting whole numbers using words, models, number lines and expanded orm:

Consider providing two or three examples (rather than just one) to better illustrate therange o student activities and mathematical representations addressed in the standard.

Revise Example in Standard 4.1.5

Indiana standard 4.1.5 states that students will multiply numbers up to at least 100 by asingle-digit number and by 10 using a standard algorithmic approach. Te example that ispaired with this standard shows a multiplication problem with two 2-digit actors: 86 54.(Note that neither actor is 10.)

Consider changing the example to include a actor o 10.

Include Grade 2 Standard for Comparing and Ordering Numbers

Students at grade 1 are required to name numbers one more or one less than a given

number to 100 (Standard 1.1.3). Students at grade 3 are asked to compare and orderractions (Standard 3.1.4). At grade 2, although students are asked to use place value toshow numbers 10 more or 10 less than a given number, 10 to 90 (Standard 2.1.3), thestandard is in the place value category. Tis creates a grade level gap in the comparing andordering numbers subcategory.

Consider relocating Standard 2.1.3 to the comparing and ordering numbers sub-category.It is very similar to Standard 1.1.3. Ten write a new standard or the place valuesub-category to avoid a gap at grade 2: Recognize numbers to at least 1000 as groups ohundreds, tens, and ones.


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DifferenceEarly work with equations

Te Indiana Standards for Mathematicsintroduce algebra work in grades 1 and 2. Tis is donethrough the use o elements called open number sentences and single- and multi-step opennumber sentences (which are simple equations). From grades 14, Indiana students progressthrough the our arithmetic operations with writing and solving simple equations. Solvingthese equations consists o undoing an operation by knowing and using its inverse. In IndianaStandards for Mathematics, much o the conceptualization o pairs o inverse operations livesin Algebra and Functions.

Although it is not called algebra in Singapore, the undamentals are there. Tere is an explicitocus on inverse operations in grade 1 or addition and subtraction and in grade 2 ormultiplication and division.

So Singapore primary students do and undo operations under the headings o inverse operationsand two-step problems, while Indiana children do it under the heading o algebra.

Suggestion: Consider renaming open number sentences and single and multi-step opennumber sentences to simple equations in the primary grades and using examples toshow how simple they are.

DifferenceTreatment of number patterns

Te Indiana Standards for Mathematicsthat pertain to number patterns show a solid progressionin grades K4. However, they do not explicitly extend into the upper grades to tie together

number patterns with algebraic concepts.

INDIA NA

Number Sense and Computation Kindergarten and Grade 1

K.2.2 Create, extend, and give the rule or simple patterns with numbers and shapes.

1.2.2 Create, extend and give a rule or number patterns using addition.

Algebra and Functions Grades 24

2.2.2 Create, extend and give a rule or number patterns using addition and subtraction.

3.2.2 Create, extend and give a rule or number patterns using multiplication.4.2.2 Create, extend and give a rule or number patterns using multiplication and division


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FINLAND

Algebra Grades 69

study and ormulation o number sequences

Functions

know how to continue a number sequence according to the rule given and be able todescribe the general rule or a given number sequence verbally

Suggestion: Consider extending work with patterns into upper grades.

Additional considerations

Revise Example in Standards 2.2.3 and 3.2.3

Indianas grade 2 and 3 examples or standards 2.2.3 and 3.2.3 (the commutative andassociative properties) are more appropriate as illustrations o mental calculation.Currently, they read as ollows:

Example: Mentally add the numbers 5, 17, and 13, in this order. Now add them inthe order 17, 13, and 5. ell which order was easier and why.

Example: Multiply the numbers 7, 2, and 5, in this order. Now multiply them inthe order 2, 5, and 7. ell which was easier and why.

Consider changing these examples so they are less ocused on mental calculation andmore about the arithmetic property o commutativity by rewording them as ollows:

Example: Add the numbers 5, 17, and 13 in this order. Now add them in the order17, 13, and 5. Show that the results are the same. Explain why.

Example: Multiply the numbers 7, 2, and 5, in this order. Now multiply them inthe order 2, 5, and 7. Show that the results are the same. Explain why.


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K8 Page 47

Revise Example in Standard K.2.1

Te Number Sense and Computationstandards and accompanying examples orKindergarten speciy that student work with addition and subtraction is consistentlyconcrete: Model addition by joining sets o objects, model subtraction by removingobjects, states K.1.5. In standard K.1.3 students play a one more/one less game by using

actual dominoes.

In standard K.2.1, the example asks students to describe an addition relationship in anumber sentence: 3 + 1 = 4. It is not clear whether describe means in strictly verbaltermsTree and one more make our, although this would be consistent with the otherstandards and examples.

Assuming that standard K.2.1 means verbally and with concrete objects, consider addingverbally to the standard and speciying the use o objects, rather than reading ahundreds chart, as in the example.

Revise Example in Standard 6.2.1Te present wording o the 6.2.1 example is:

Example: Te area o a rectangle is 143 cm2and the length is 11 cm. Write anequation to find the width o the rectangle and use it to solve the problem.Describe how you will check to be sure that your answer is correct.

Consider changing it to:

Example: Te area o a rectangle is 143 cm2and the length is 13 cm. Write andsolve an equation or the width o the rectangle. Describe how you will check to besure that your answer is correct.

Tis uses more standard write and solve wording and yields a rectangle with a lengthgreater than its width.

Revise Example in Standard 8.2.3

Te present wording o the 8.2.3 example is:

Example: Use a scientific calculator to find the value o 3(2x+ 5)2.

Consider changing it to:

Example: Use a scientific calculator to expand 3(2x+ 5)2.

Also, consider reraining rom mentioning the use o a calculator, since 8.2.3 specificallyasks students to be able to simpliy algebraic expressions involving powers.


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Standard 3: Geometry and Measurement

Assets

TeIndiana Standards for Mathematicscover important concepts o transormations, symmetry,angles, lines, area, perimeter, surace area and volume. Mathematical tools and technology aresuggested to enable student understanding o some o the concepts.

DifferenceEmphasis on geometry and measurement

Singapore deals with geometry and measurement as separate topics. Measurement leads to thedevelopment o quantity. Te content covered in the Singapore syllabus gives more detaileddevelopment o the concepts with more connections to number and algebra.

Suggestion: Indiana specified a goal o depth over breadth. Tus decisions on when to

add more detail to the standards are difficult ones, but consider reviewing the geometryand measurement sections o the standards or depth, especially concerning themeasurement area and its relationship to developing the concept o quantity.


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K8 Page 49

DifferenceTreatment of transformations and constructions

Te Singapore standards have an emphasis on symmetry and transormations throughout thegrades. Te Singapore standards read as though they are laying the groundwork or later workwith transormations in grades K6.

SINGA PORE

Geometry: Patterns Primary 2

Include:

making/completing patterns with 2-D cut-outs according to one or two o the ollowingattributes

shape

size

orientation

colour

Geometry: Patterns Primary 3

Include:

conversion o units, e.g. km/h to m/min.

Geometry: Symmetry Primary 4

Include:

identiying symmetric figures,

determining whether a straight line is a line o symmetry o a symmetric figure,

completing a symmetric figure with respect to a given horizontal/vertical line o symmetry, designing and making patterns.

Exclude:

finding the number o lines o symmetry o a symmetric figure,

rotational symmetry.

Geometry: Tessellation Primary 4

Include:

recognising shapes that can tessellate,

identiying the unit shape in a tessellation,

making different tessellations with a given shape, drawing a tessellation on dot paper,

designing and making patterns.


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Te Finnish standards indicate there is regular, incremental work through the grades on conceptso transormation.

FINLA ND

Geometry Grades 12

simple reflections and dilations

Geometry Grades 35

dilations and reductions; similarity and scale

reflections across a line and around a point, symmetry, congruence, using concrete means

Geometry Grades 69

similarity and congruence

geometric construction

depictions o congruence: reflections, rotation, and transormation

TeIndiana Standards for Mathematicsalso include work on transormations and constructions.

INDIA NA

Geometry and Measurement Grade 2

2.3.2 Identiy and draw congruent two-dimensional shapes in any position.


3.3.2 Identiy and draw lines o symmetry in geometric shapes and recognize symmetricalshapes in the environment.


7.3.1 Identiy and use the ollowing transormations: translations, rotations and reflections tosolve problems.


8.3.2 Perorm basic compass and straight edge constructions: angle and segment bisectors,copies o segments and angles, and perpendicular segments. Describe and justiythe constructions.

Te topics o transormations and symmetry might be introduced over a series o years.

Suggestion: Symmetry has an important place in the coordinate plane. Finland hasstudents working with dilations well in advance o high school. Consider building theidea o symmetry and transormation across grade levels.


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K8 Page 51

DifferenceTreatment of money concepts

Singapore uses money to develop the concept o equivalence and operations. Te standards detailthe incremental conceptual work across the primary grades, including the concepts (equivalenceand operations) and skills (notation) that orm the basis o calculation with money:

SINGA PORE

Measurement: Money Primary 1

Include:

Identiying coins and notes o different denomination

Matching a coin/note o one denomination to an equivalent set o coins/notes o anotherdenomination

elling the amount o money

in cents up to $1.00

in dollars up to $100

Use o the symbols $ and

Solving o word problems involving addition and subtraction o money in dollars only(or in cents only).


Include:

Counting the amount o money in a given set o notes and coins

Reading and writing money in decimal notation

Comparing two or three amounts o money

Converting an amount o money in decimal notation to cents only, and vice versa

Solving word problems involving money in dollars only (or in cents only)


Addition and subtraction o money in decimal notation

Solving word problems involving addition and subtraction o money in decimal notation

Te Indiana Standards for Mathematicson money include the ollowing:

INDIA NA


1.3.3 Give the value o a collection o pennies, nickels, and dimes up to $1.00.


2.3.5 Find the value o a collection o pennies, nickels, dimes, hal-dollars, and dollars.


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K8 Page 53

DifferenceTiming of symmetry

Indiana students work with symmetry beore they study intersecting lines and angles.

INDIA NA


3.3.2 Identiy and draw lines o symmetry in geometric shapes and recognize symmetricalshapes in the environment.


4.3.1 Identify, describe, and draw parallel and perpendicular lines.

4.3.2 Identify, describe, and draw right angles, acute angles, obtuse angles, straight

angles, and rays using appropriate tools and technology.

Te Singapore standards explicitly introduce angles and line relationships in grade 3, and beginwork with symmetry in grade 4.

SINGA PORE

Geometry: Perpendicular and Parallel Lines Grade 3

Include:

Identiying and naming perpendicular and parallel lines

Drawing perpendicular and parallel lines on square grids

Geometry: Angles Grade 3Angle as an amount o turning

Identiying angles in 2-D and 3-D objects

Identiying angles in 2-D figures

Identiying right angles, angles greater than/ smaller than a right angle

Geometry: Symmetry Grade 4

Identiying symmetric figures

Determining whether a straight line is a line o symmetry o a symmetric figure

Completing a symmetric figure with respect to a given horizontal/vertical line o symmetry

Designing and making patterns

Suggestion: Consider introducing angles, and parallel and perpendicular lines beoreaddressing symmetry.


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DifferenceTiming of angles

Te Indiana Standards for Mathematicsintroduce angles in grade 4 (prior to that, K2 studentsclassiy and describe plane and solid figures in terms o their number o vertices).

INDIA NA


4.3.2 Identiy, describe, and draw right angles, acute angles, obtuse angles, straight angles,and rays using appropriate tools and technology.

Singapore introduces angles more gradually, starting in grade 3.

SINGA PORE

Geometry: Angles Grade 3

Angle as an amount o turning

Identiying angles in 2-D and 3-D objects

Identiying angles in 2-D figures

Identiying right angles, angles greater than/ smaller than a right angle

Geometry: Angles Grade 4

Using notation such as ABC and x to name angles

Estimation and measurement o angles in degrees

Drawing an angle using a protractor Designing and making patterns

Associating

1

4turn/right angle with 90

1

2turn with 180

3

4turn with 270

a complete turn with 360.

Suggestion: Consider introducing angles earlier and building conceptual knowledgemore gradually.


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K8 Page 55

DifferenceTiming of units work

Indiana grade 2 students begin work with units.

INDIA NA


2.3.3 Estimate and measure length to the nearest inch, oot, yard, centimeter, and meter,selecting appropriate units or the given situation, and use the relationships within theun

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