Florida K-8 Florida K-8 Mathematics Mathematics StandardsStandards

April 29, 2008April 29, 2008

Grade 4Grade 4

Adapted from a presentation given by Julie Kay Dixon, Ph.D, UCF – a member of the K-8 Writers Group

A student said this…A student said this…

When asked to compare 4/5 and 2/3, When asked to compare 4/5 and 2/3, a student said, “I know that 4/5 is a student said, “I know that 4/5 is greater than 2/3.”greater than 2/3.”

How would you respond?How would you respond?

Hopefully you would ask the Hopefully you would ask the student how he or she knew.student how he or she knew.

Perspective…Perspective…

The student said…The student said…

I made both fractions using manipulatives. I I made both fractions using manipulatives. I knew that 4/5 was bigger because 4/5 has 4 knew that 4/5 was bigger because 4/5 has 4 pieces and 2/3 only has 2 pieces and since 4 pieces and 2/3 only has 2 pieces and since 4 is greater than 2 then 4/5 is greater than 2/3.is greater than 2 then 4/5 is greater than 2/3.

What would this response tell you?What would this response tell you?

Perspective…Perspective…

Would you ask this student to Would you ask this student to compare 2/5 and 1/2?compare 2/5 and 1/2?

According to the intent of the new According to the intent of the new standards, the answer should be yes. standards, the answer should be yes. This problem is appropriate for a This problem is appropriate for a student in grade 3. student in grade 3.

Perspective…Perspective…

Developing the Developing the StandardsStandards The new Florida K-8 Mathematics Standards

are framed by the recently released NCTM Curriculum Focal Points for Prekindergarten through Grade 8 Mathematics and informed by the Singapore Standards, the SSS Grade Level Expectations, and standards from other states that received high grades for rigor, focus, specificity and clear progression of content.

There are clear differences between the new

standards and the 1996 K-8 mathematics SSS.

Developing the Developing the StandardsStandards The “framers,” a group that

represented K-12 teachers, K-12 mathematics supervisors, mathematicians, and mathematics educators, were convened to address issues related to the current standards and to establish a framework for the design of the new standards. The framers recommended that the Curriculum Focal Points be used as the foundation for the new K-8 standards.

Developing the Developing the StandardsStandards The “writers,” a group that

represented the same set of stakeholders, were convened to generate the revised standards. The writers of the K-8 standards had the task of actualizing the intent of the Curriculum Focal Points within a set of grade-level specific standards.

Developing the Developing the StandardsStandards

September 2006: Framers met with “experts” to learn about task and conceptualize new standards.

October 2006 - January 2007: Writers wrote draft of standards.

February - March 2007: New standards posted for public review period.

April - May 2007: Standards revised by writers and representation from framers based on comments received during review

September 2007: Standards approved by State Board of Education.

Who were the Who were the “experts”?“experts”?

Dr. Barbara Reys: Center for the Study of Mathematics Curriculum (CSMC); shared a review of 42 state’s mathematics standards.

Dr. Jane Schielack: Chaired NCTM committee that wrote the Curriculum Focal Points.

Dr. Kaye Forgione: Senior Associate of Mathematics Benchmarking Initiative with Achieve, Inc.

Dr. Alan Ginsburg: US Dept. of Education, What the United States can Learn from Singapore’s World-class Mathematics System.

Dr. R. James Milgram: Wrote the California Mathematics Standards.

Describing the Describing the StandardsStandards Big Ideas---Standards which are aligned with the

Curriculum Focal Points. – They should be the primary focus of mathematics instruction

for each grade level, K - 8. – There are three Big Ideas for each grade. – The Big Ideas are not the same for each grade. – Instructional time may not be evenly divided among the three

Big Ideas. The order of the Big Ideas does not determine the

order of instruction nor does it indicate that one idea requires greater instructional emphasis.

Describing the Describing the StandardsStandards Supporting Ideas---standards that serve one or more of the

following purposes: – Establish connections to and between the strands of

mathematics as defined by NCTM;

– Prepare students for future mathematics teaching and learning; and

– Address gaps in instruction that are important to the understanding, fluency, and application of mathematics ideas to problem solving.

The Supporting Ideas are not less important than the Big Ideas, but are key components to a structurally sound mathematics education.

Describing the Describing the StandardsStandards Access Points

– Written for students with significant cognitive Written for students with significant cognitive disabilities to access the general education disabilities to access the general education curriculumcurriculum

– Reflect the core intent of the standards with reduced Reflect the core intent of the standards with reduced levels of complexitylevels of complexity

– Include three levels of complexity: participatory, Include three levels of complexity: participatory, supported, and independent with the participatory supported, and independent with the participatory level being the least complexlevel being the least complex

Describing the Describing the StandardsStandards Access Points

– The Access points were not written by the Mathematics Standards Writing Committee and are not intended for mainstream students.

Describing the Describing the StandardsStandards Coding Scheme for Kindergarten through

Grade 8

MA.MA. 5.5. A.A. 1.1. 11

SubjectSubject Grade-Grade-LevelLevel

Body of Body of KnowledgKnowledg

ee

Big Idea/ Big Idea/ SupportinSupportin

g Ideag Idea

BenchmarBenchmarkk

Describing the Describing the StandardsStandards

Body of Knowledge Key:A - AlgebraC - CalculusD - Discrete MathematicsF - Financial LiteracyG - GeometryP - ProbabilityS - StatisticsT - Trigonometry

Describing the Describing the StandardsStandardsGrade LevelGrade Level Number of Number of OldOld

GLE’sGLE’sNumber of Number of NewNew

BenchmarksBenchmarks

KK 6767

11stst 7878

22ndnd 8484

33rdrd 8888

44thth 8989

55thth 7777

66thth 7878

77thth 8989

88thth 9393

Describing the Describing the StandardsStandardsGrade LevelGrade Level Number of Number of OldOld

GLE’sGLE’sNumber of Number of NewNew

BenchmarksBenchmarks

KK 6767 1111

11stst 7878 1414

22ndnd 8484 2121

33rdrd 8888 1717

44thth 8989 2121

55thth 7777 2323

66thth 7878 1919

77thth 8989 2222

88thth 9393 1919

Describing the Describing the StandardsStandards Old Standards had an average of 83.3

Grade Level Expectations (GLEs) per grade.

The new Standards have an average of 19 benchmarks per grade.

Intent of the Intent of the StandardsStandards What is the importance of having

fewer expectations per grade????

Intent of the Intent of the StandardsStandards A member of the Florida Department

of Education shared a reaction by a teacher during an open forum regarding the new Florida standards. The teacher looked at the short list of curricular topics in a grade and said,

“I can teach this in 20 days, what do

I do the rest of the year?”

Intent of the Intent of the StandardsStandards How do we help teachers with similar

views come to understand what is meant by facilitating “deep understanding, mathematical fluency, and an ability to generalize” (NCTM, 2006, p. 5)?

Describing the Describing the StandardsStandards To enable the development and

mastery of a few key concepts in each grade level it was necessary to make decisions about the placement of topics. As a result, some topics are not introduced until later grades. This does not necessarily mean that students are incapable of learning at an earlier grade. Instead, it is an attempt to streamline the focus of content at each grade level.

For Example…For Example…

Old Standards New Standards

Division of Whole Numbers (1 digit divisor and up to 3 digit dividends) - Explains, demonstrates, and solves real world problems using manipulatives, drawings, and algorithms

For Example…For Example…

Old Standards New Standards

Division of Whole Numbers (1 digit divisor and up to 3 digit dividends) - Explains, demonstrates, and solves real world problems using manipulatives, drawings, and algorithms

Divide multi-digit whole numbers fluently, including solving real-world problems, demonstrating understanding of the standard algorithm and checking the reasonableness of results.

For Example…For Example…

Old Standards New Standards

Addition and Addition and subtraction of subtraction of decimals and decimals and common fractions common fractions in grade 4in grade 4

For Example…For Example…

Old Standards New Standards

Addition and Addition and subtraction of subtraction of decimals and decimals and common fractions common fractions in grade 4in grade 4

Develop an Develop an understanding of understanding of and fluency with and fluency with addition and addition and subtraction of subtraction of fractions and fractions and decimals in grade decimals in grade 55

Big Ideas for Fourth Big Ideas for Fourth Grade:Grade:

1:1: Develop quick recall of multiplication Develop quick recall of multiplication

facts and related division facts and facts and related division facts and fluency with whole number multiplicationfluency with whole number multiplication

2: 2: Develop an understanding of Develop an understanding of decimals, decimals,

including the connection between including the connection between fractions and decimalsfractions and decimals

3: 3: Develop an understanding of area and Develop an understanding of area and

determine the area of two-determine the area of two-dimensional dimensional objectsobjects

Fourth Grade Fourth Grade Supporting IdeasSupporting Ideas

Algebra:Algebra: – Generate algebraic rules and use all Generate algebraic rules and use all

four operations to describe patterns, four operations to describe patterns, including nonnumeric growing or including nonnumeric growing or repeating patternsrepeating patterns

– Describe mathematics relationships Describe mathematics relationships using expressions, equations, and using expressions, equations, and visual representationsvisual representations

– Recognizes and writes algebraic Recognizes and writes algebraic expressions for functions with two expressions for functions with two operationsoperations

Fourth Grade Fourth Grade Supporting IdeasSupporting Ideas

Geometry & Measurement:Geometry & Measurement: – Classify angles of two-dimensional shapes Classify angles of two-dimensional shapes

using benchmark angles (i.e., 45using benchmark angles (i.e., 45°, 90°, 180°, °, 90°, 180°, and 360°)and 360°)

– Identify and describe the results of Identify and describe the results of translations, reflections and rotations of 45, translations, reflections and rotations of 45, 90, 180, 270 and 360 degrees, including 90, 180, 270 and 360 degrees, including those with line and rotational symmetry.those with line and rotational symmetry.

– Identify and build a three-dimensional object Identify and build a three-dimensional object from two-dimensional representations of that from two-dimensional representations of that object and vice versa.object and vice versa.

Fourth Grade Fourth Grade Supporting IdeasSupporting Ideas

Numbers and Operations:Numbers and Operations: – Use and represent numbers through Use and represent numbers through

millions in various contexts, including millions in various contexts, including estimation of relative sizes of amounts or estimation of relative sizes of amounts or distancesdistances

– Use models to represent division as the Use models to represent division as the inverse of multiplication, as partitioning, or inverse of multiplication, as partitioning, or as successive subtractionas successive subtraction

– Generate equivalent fractions and simplify Generate equivalent fractions and simplify fractionsfractions

– Determine factors and multiples of Determine factors and multiples of specified whole numbersspecified whole numbers

Fourth Grade Fourth Grade Supporting IdeasSupporting Ideas

Numbers and Operations:Numbers and Operations: – Relate halves, fourths, tenths and Relate halves, fourths, tenths and

hundredths to decimals and hundredths to decimals and percentspercents

– Estimate and describe Estimate and describe reasonableness of estimates; reasonableness of estimates; determine the appropriateness of determine the appropriateness of estimate versus an exact answerestimate versus an exact answer

Describing the Describing the StandardsStandards Mathematics instruction at each

subsequent grade will continue to use concepts and understandings learned in earlier grades as needed.

When asked at a recent Florida Council When asked at a recent Florida Council of Teachers of Mathematics meeting, a of Teachers of Mathematics meeting, a representative from FCAT said, representative from FCAT said, “students would still need to know “students would still need to know concepts from previous grades. They concepts from previous grades. They just won’t be tested in isolation.” just won’t be tested in isolation.”

Describing the Describing the StandardsStandards Some prerequisite knowledge and

skills, not specifically identified in the standards, may need to be added to the curriculum to meet the standards.

Students who move to Florida from other states may need exposure to topics not addressed at their grade of entry.

Real-World ProblemsReal-World Problems

To the extent possible, it is expected that the relevance of mathematics would be made clear to students by illustrating how mathematics is used in the real world. To this end, the curriculum should include real-world contexts in addition to mathematical contexts. The overall goal is to help students relate mathematics to the real world and their experiences.

Remarks are provided to:Remarks are provided to:

Clarify what is described in the standards.

Provide context to be addressed as part of

the standards.

Provide examples of the types of problems

that the standards address.

Provide content limits when deemed

appropriate.

RemarksRemarks

Remarks were not included with the

standards presented to the State

Board of Education.

Remarks are currently included in

course descriptions.

Important LinksImportant Links

Florida Mathematics Standards & Course Florida Mathematics Standards & Course Descriptions: Descriptions: – http://http://www.floridastandards.orgwww.floridastandards.org

Florida Department of Education, Office of Florida Department of Education, Office of Mathematics and ScienceMathematics and Science– http://www.fldoestem.orghttp://www.fldoestem.org

Florida Council of Teachers of MathematicsFlorida Council of Teachers of Mathematics– http://www.fctm.nethttp://www.fctm.net

National Council of Teachers of MathematicsNational Council of Teachers of Mathematics– http://www.nctm.orghttp://www.nctm.org

Santa Rosa County Mathematics Santa Rosa County Mathematics DepartmentDepartment– http://www.santarosa.k12.fl.us/currinst/http://www.santarosa.k12.fl.us/currinst/

Next steps should Next steps should include:include:

Statewide communication regarding new standards

(ongoing). A comprehensive crosswalk between the new and

existing standards (currently available in draft form).

District-by-district plans for transitioning to the new standards (work together!).

District curriculum plan for each grade level, K – 8 Professional development for teachers in order to

provide tools and knowledge necessary to implement new standards with success (ongoing)

Assessment…

How will it change?

FCAT Crosswalk~

Impact on Assessment

Grade 4Selection from a

PowerPointPresented by

Linda OaksfordTest Development Center

Big Ideas ~ Grade 4

1. Multiplying multi-digit whole numbers~ check for reasonableness of products

2. Decimals (through thousandths) and fractions3. Area of two-dimensional shapes

based on unit square, including rectangle formula

Grade 4 ~ MA.4.A.4.3

Supporting Idea: Algebra

Recognizes and writes algebraic expressions for functions with two operations

two

MA.4.A.4.3 Sample

Alex is 4 years more than twice as old asSam. Which expression shows Alex’s age,using s for Sam’s age?

A) (4 + 2) x s C) 4s x 2B) (s + 4) x 2 D) 2s + 4

Previous Benchmark: MA.D.1.2.1

The student describes a wide variety of patterns and relationships through models, such as manipulatives, tables, graphs, and rules using algebraic symbols.

(Student extended patterns and relationships; limited to one operation)

Sample: MA.D.1.2.1

The students in Mrs. Jay’s class need a notebook for every subject. According to the table what would be the price of 6 notebooks?

Notebooks

Price

4 $12

5 $15

6

7 $21

A)16 C)19

B) 18 D)20

As of 2011. . .As of 2011. . .NOT assessed at 4th gradeNOT assessed at 4th grade

Measurement of weight/mass, Measurement of weight/mass, capacity, temperature, capacity, temperature, (perimeter), etc.(perimeter), etc.

CoordinatesCoordinates Measures of central tendency – Measures of central tendency –

mean, median, mode, and rangemean, median, mode, and range Probability – likeliness of outcomeProbability – likeliness of outcome Data AnalysisData Analysis CombinationsCombinations