Finding Focus for Finding Focus for Mathematics Instruction Mathematics Instruction

Grades K-2Grades K-2

March 25, 2010March 25, 2010 Huron Area Technical Huron Area Technical CenterCenterMarch 31, 2010March 31, 2010 Tuscola Technical Tuscola Technical Center Center

April 14 and 26, 2010April 14 and 26, 2010 Huron ISD Huron ISD Central OfficeCentral Office

Thumb-Area Student Achievement Model and Huron Intermediate School District

Getting SB-CEUs?Getting SB-CEUs?

Online evaluation Online evaluation REQUIRED for REQUIRED for credit:credit:

e-mail address e-mail address last 4 digits of SS# last 4 digits of SS#

on sign-in sheeton sign-in sheet enter PIC # in enter PIC # in

profileprofile Shirley will e-mailShirley will e-mail

Today’s GoalsToday’s Goals

Become familiar with Mathematics CurriculumBecome familiar with Mathematics Curriculum Deepen understanding of Big Math IdeasDeepen understanding of Big Math Ideas Use strategies for explicit vocabulary Use strategies for explicit vocabulary

instruction instruction Understand instructional implications of Understand instructional implications of

researchresearch Use assessment in a 3-tier process Use assessment in a 3-tier process Support instruction for intervention and Support instruction for intervention and

enrichmentenrichment

What Are the Focal Points?What Are the Focal Points?

What Do the Focal Points Look What Do the Focal Points Look Like?Like?

1.1. Work in grade-level teams.Work in grade-level teams.

2.2. Find the focal points for your grade.Find the focal points for your grade.

3.3. Sort the GLCE topics according to Sort the GLCE topics according to focal point. Make a separate pile for focal point. Make a separate pile for “leftover topics.”“leftover topics.”

Compare your sort to MDE list. Compare your sort to MDE list.

Re-arrange if necessary.Re-arrange if necessary.

How Should the Focal Points How Should the Focal Points Impact Instruction?Impact Instruction?

Based on nationally-recognized topicsBased on nationally-recognized topics Related to GLCEs and MEAPRelated to GLCEs and MEAP (MEAP assessed in 3(MEAP assessed in 3rdrd for 2 for 2ndnd grade): grade):

Core expectationsCore expectations Must be related to a focal pointMust be related to a focal point No more than 20 per gradeNo more than 20 per grade Assessed with two items, all studentsAssessed with two items, all students

Extended core expectationsExtended core expectations Not related to a focal pointNot related to a focal point Assessed with no more than one item (sampled)Assessed with no more than one item (sampled)

DiscussionDiscussion

70-80% of instruction 70-80% of instruction should focus on GLCEs should focus on GLCEs related to focal points. related to focal points.

Use your textbook to think Use your textbook to think about your instruction. about your instruction. What topics should be What topics should be emphasized more? Less?emphasized more? Less?

Useful DocumentsUseful Documents

Math GLCEs Math GLCEs Assessed with NC Assessed with NC DesignationsDesignations www.mi.gov/mathematicswww.mi.gov/mathematics

Mathematics Focal Mathematics Focal Points K-8 Alignment Points K-8 Alignment (11-11-09 from SAM)(11-11-09 from SAM) http://www.hisd.k12.http://www.hisd.k12.

mi.us/SAM/main.htmlmi.us/SAM/main.html

Common Core Standards Common Core Standards InitiativeInitiative

Coalition of states who have all Coalition of states who have all agreed to adopt the same state agreed to adopt the same state standardsstandards

48 states, 2 territories (Puerto Rico 48 states, 2 territories (Puerto Rico and the U.S. Virgin Islands), and the and the U.S. Virgin Islands), and the District of ColumbiaDistrict of Columbia

Who’s missing?Who’s missing? Alaska and TexasAlaska and Texas

http://www.corestandards.orhttp://www.corestandards.org/g/

Common Core DraftCommon Core Draft

Common Core Standards Common Core Standards Initiative Revised DRAFT 3-10-Initiative Revised DRAFT 3-10-

20102010 Match CCSI Match CCSI

standards to Focal standards to Focal Points and GLCEsPoints and GLCEs

Make new piles if Make new piles if neededneeded

No GLCE for a CCSI No GLCE for a CCSI standard? Check standard? Check another grade.another grade.

GRADE KGRADE K Current MI Focal PointsCurrent MI Focal Points 3-10-2010 Revised DRAFT CCSI3-10-2010 Revised DRAFT CCSI

RepresentinRepresenting, g, comparing, comparing, ordering ordering whole whole numbersnumbers

Count, write, order Count, write, order numbersnumbersCompose & decompose Compose & decompose numbersnumbersAdd & subtract numbersAdd & subtract numbersExplore number patternsExplore number patterns

Number namesNumber namesCounting to tell number of objectsCounting to tell number of objectsComparing & ordering numbersComparing & ordering numbersComposing & decomposing Composing & decomposing numbers; addition and subtractionnumbers; addition and subtractionComposing & decomposing tenComposing & decomposing ten

Describe Describe shapes and shapes and spacespace

Explore & describe Explore & describe shapesshapesExplore &describe Explore &describe geometric patternsgeometric patterns

Shapes and their attributesShapes and their attributesSpatial reasoningSpatial reasoning

Ordering Ordering objects by objects by measurable measurable attributesattributes

Explore other Explore other measurement attributesmeasurement attributesExplore concepts of timeExplore concepts of time

Direct measurementDirect measurementRepresenting & interpreting dataRepresenting & interpreting data

Grade 1 – Grade 1 –

GRADE 1GRADE 1 Current MI Focal PointsCurrent MI Focal Points 3-10-2010 Revised DRAFT 3-10-2010 Revised DRAFT CCSICCSI

Whole Whole number number relaationshiprelaationships-grouping s-grouping tens and tens and onesones

Count, write, order numbers Count, write, order numbers Place valuePlace value

Numbers up to 100Numbers up to 100

Addition & Addition & subtractionsubtraction

Add & subtract whole numbersAdd & subtract whole numbersWork with moneyWork with moneySolve problemsSolve problems

Add & subtract in base tenAdd & subtract in base tenSolve problems with Solve problems with addition and subtractionaddition and subtraction

Linear Linear measurememeasurementnt

Estimate and measure lengthEstimate and measure lengthTell timeTell time

Measure lengthMeasure lengthTell timeTell time

ConnectionsConnections Create and describe shapesCreate and describe shapes Shapes, attributes, spatial Shapes, attributes, spatial reasoningreasoning

ConnectionsConnections Use pictographsUse pictographs Representing and Representing and interpreting datainterpreting data

GRADE 2GRADE 2 Current MI Focal PointsCurrent MI Focal Points 3-10-2010 Revised DRAFT 3-10-2010 Revised DRAFT CCSICCSI

Understand Understand base-ten base-ten system and system and place-value place-value conceptsconcepts

Count, write, order whole Count, write, order whole numbersnumbersUnderstand place valueUnderstand place value

Numbers up to 1000Numbers up to 1000

Addition Addition and and subtraction subtraction facts recall facts recall including including multi-digit multi-digit factsfacts

Addition and subtraction of whole Addition and subtraction of whole numbersnumbersTell time & solve time problemsTell time & solve time problemsMeasure, add, subtract lengthMeasure, add, subtract lengthSolve measurement problemsSolve measurement problemsRecord, add, subtract moneyRecord, add, subtract money

Adding and subtraction in base Adding and subtraction in base tentenSolving problems with addition Solving problems with addition and subtractionand subtractionLength measurementLength measurementTime and moneyTime and money

Composing Composing and and decomposidecomposingng

geometric geometric shapesshapes

Identify and describe shapesIdentify and describe shapesWork with unit fractionsWork with unit fractions

Shapes, their attributes, spatial Shapes, their attributes, spatial reasoningreasoning

ConnectionConnectionss

Understand multiplication & Understand multiplication & divisiondivision

ConnectionConnectionss

Create, interpret, solve problems Create, interpret, solve problems involving pictographsinvolving pictographsUse coordinate systemUse coordinate system

Represent and interpret dataRepresent and interpret data

Common Core (CC) – Common Core (CC) – When???When???

OLD Race to the Top:OLD Race to the Top: Adopt CC in June and roll out in AugustAdopt CC in June and roll out in August Current MEAP/MME through Spring, 2014Current MEAP/MME through Spring, 2014 New test beginning 2014-2015New test beginning 2014-2015 Application not approved; MI will re-submitApplication not approved; MI will re-submit

Best Guess:Best Guess: MI will adopt Common CoreMI will adopt Common Core At least 4 more years of Current MEAP likelyAt least 4 more years of Current MEAP likely

MI Participates in Two Multi-MI Participates in Two Multi-state Assessment state Assessment

CollaborativesCollaboratives SMARTER (p. 52)SMARTER (p. 52)

Summative Multi-state Assessment Resources for Teachers Summative Multi-state Assessment Resources for Teachers and Educational Researchersand Educational Researchers

MI is a lead stateMI is a lead state Develop summative assessments based on the CCK-12 Develop summative assessments based on the CCK-12

Standards in ELA and mathematicsStandards in ELA and mathematics MOSAIC (p. 52)MOSAIC (p. 52)

Multiple Options for Student Assessment and Instruction Multiple Options for Student Assessment and Instruction ConsortiumConsortium

““Interim benchmark and formative assessments . . . Interim benchmark and formative assessments . . . designed to complement a summative assessment system” designed to complement a summative assessment system” (i.e., SMARTER)(i.e., SMARTER)

Also collaboratively develop a curriculum framework and Also collaboratively develop a curriculum framework and instructional support and integration materialsinstructional support and integration materials

RecommendationsRecommendations

Make changes now Make changes now to reflect Focal to reflect Focal Points and Points and core/extended core/extended GLCEsGLCEs

When Common When Common Core is adopted by Core is adopted by MI, develop a MI, develop a transition plan for transition plan for your districtyour district

What does it take to pass the What does it take to pass the Math MEAP?Math MEAP?

FALL FALL 20092009

Lowest Raw Lowest Raw Score for Level 2Score for Level 2

Percent to Percent to PassPass

Grade 3Grade 3 15 out of 4515 out of 45 33%33%

Grade 4Grade 4 19 out of 5319 out of 53 36%36%

Grade 5Grade 5 21 out of 5221 out of 52 40%40%

Grade 6Grade 6 19 out of 5219 out of 52 37%37%

Grade 7Grade 7 19 out of 5319 out of 53 36%36%

Grade 8Grade 8 19 out of 5119 out of 51 37%37%

What does this mean?What does this mean?

Percent Proficient is meaningless Percent Proficient is meaningless Cut scores are low because of the Cut scores are low because of the

composition of the testcomposition of the test

Look at Item Analysis:Look at Item Analysis: Not reliable to item or GLCE levelNot reliable to item or GLCE level Look at groups of items – by Focal Point Look at groups of items – by Focal Point

or Topicor Topic

Looking at your item Looking at your item analysisanalysis

80% or better80% or better 60-79%60-79% 59% or lower59% or lower Prioritize by topic Prioritize by topic

or focal pointor focal point Track multi-year Track multi-year

trendstrends

2009 Grade 3 MEAP (Gr. 2 2009 Grade 3 MEAP (Gr. 2 content)content)

Base ten number systemBase ten number system Count, write, order Count, write, order

wholeswholes

Working with geometric Working with geometric shapesshapes

Work with unit Work with unit fractionsfractions

Identify & describe Identify & describe shapesshapes

Addition, subtraction Addition, subtraction fluencyfluency

Add & subtract whole Add & subtract whole numbersnumbers

Measure, add, & Measure, add, & subtract lengthsubtract length

Record, add, & subtract Record, add, & subtract moneymoney

Solve Solve measurementmeasurement problemsproblems

Tell time, solve problemsTell time, solve problems

Focal Points Grades K-2Focal Points Grades K-2

Grade K Grade 1 Grade 2

Compose and decompose numbers

Understand and use place value

Understand and use place value

Add and subtract Add and subtract

Order objects by attribute

Measure lengths

Describe shapes and space

Compose and decompose shapes

MentallyMentally

Add 28 + 35Add 28 + 35

Mental Computation Strategies Mental Computation Strategies for 28 + 35for 28 + 35

Compensation – find a “friendly” numberCompensation – find a “friendly” number Think: 30 + 35 = 65; take away 2, so 63Think: 30 + 35 = 65; take away 2, so 63 OR: 30 + 33 = 63OR: 30 + 33 = 63

Decomposition – use place value positionsDecomposition – use place value positions Think: 8 + 5 = 13; 20 + 30 = 50; 50 + 13 = 63Think: 8 + 5 = 13; 20 + 30 = 50; 50 + 13 = 63 OR: 20 + 30 = 50; 8 + 5 = 13; 50 + 13 = 63OR: 20 + 30 = 50; 8 + 5 = 13; 50 + 13 = 63

Jump – begin with one numberJump – begin with one number Think: 28 + 5 = 33; 33 + 30 = 63Think: 28 + 5 = 33; 33 + 30 = 63 OR: 28 + 30 = 58; 58 + 5 = 63OR: 28 + 30 = 58; 58 + 5 = 63

11 Game11 Game

Use place value to Use place value to add and subtractadd and subtract

Two ways to play:Two ways to play: Bingo - each person Bingo - each person

has his own cardhas his own card Connect 4 - share a Connect 4 - share a

cardcard

Think - WriteThink - Write

Choose the Choose the addition/subtraction addition/subtraction focal point for your focal point for your grade. What are grade. What are the critical ideas – the critical ideas – the mathematics the mathematics that students must that students must understand – for understand – for that topic?that topic?

Read the Read the introductiointroduction.n.

Share with Share with a a neighbor.neighbor.

How many focal How many focal points are at your points are at your grade level?grade level?

The GLCE topics The GLCE topics are the same as are the same as the the Core and Core and Extended Extended DesignationsDesignations document from document from MDEMDE

National Math National Math Panel Panel Benchmarks are Benchmarks are checkpointscheckpoints

Benchmarks are Benchmarks are often a grade or often a grade or two past where two past where the topic is the topic is typically taughttypically taught

Find this chart Find this chart for each focal for each focal point at your point at your grade.grade.

The columns are The columns are the same as the the same as the 11” x 17” K-8 11” x 17” K-8 Alignment chartAlignment chart

Three Sections for Each Focal Three Sections for Each Focal PointPoint

From the 3-10-2010 Revised DRAFT of the Common Core Standards Initiative

Explore the Explore the Finding Focus Finding Focus DocumentDocument

There are two places in this document to There are two places in this document to find the list of GLCE topics for focal a find the list of GLCE topics for focal a point. Where are those two places?point. Where are those two places?

Find the “leftover” GLCEs for your grade. Find the “leftover” GLCEs for your grade. Compare the chart to the Compare the chart to the 11” x 17”11” x 17” K-8 K-8 Alignment.Alignment.

Find the addition and subtraction focal Find the addition and subtraction focal point for your grade. What number is it?point for your grade. What number is it?

Choose any focal point at your grade. Choose any focal point at your grade. Compare the GLCEs for the focal point to Compare the GLCEs for the focal point to the DRAFT CCSI standards.the DRAFT CCSI standards.

The Big Ideas are NOTThe Big Ideas are NOT Topics for instructional Topics for instructional

planningplanning GLCEs for assessing GLCEs for assessing

studentsstudents

Big Mathematical Big Mathematical Ideas and UnderstandingsIdeas and Understandings

The Big Ideas AREThe Big Ideas ARE The mathematics YOU The mathematics YOU

should keep in mind should keep in mind as you plan instructionas you plan instruction

Critical ideas that are Critical ideas that are true at all grade levelstrue at all grade levels

SILENT ReadingSILENT Reading

Find the “Big Mathematical Ideas and Find the “Big Mathematical Ideas and Understandings” for the focal point Understandings” for the focal point you brainstormed about earlieryou brainstormed about earlier

Read, re-read, highlight, and take Read, re-read, highlight, and take notesnotes

Add to the list you brainstormed of Add to the list you brainstormed of critical ideas for your gradecritical ideas for your grade

BreakBreak

Big IdeasBig Ideas

With your grade level, discussWith your grade level, discuss What stood out to you from the Big What stood out to you from the Big

Ideas?Ideas? What questions do you still have?What questions do you still have? Revise your notes.Revise your notes.

Arithmetic CombinationsArithmetic Combinations

Should children memorize the basic facts? Yes . . Should children memorize the basic facts? Yes . . but that is misleading as stated. Knowledge of but that is misleading as stated. Knowledge of

arithmetic concepts form an organizing arithmetic concepts form an organizing framework for storing arithmetic combinations . . framework for storing arithmetic combinations . . . Students with greater conceptual knowledge are . Students with greater conceptual knowledge are

more likely to use sophisticated strategies and more likely to use sophisticated strategies and retrieve combinations accurately. That is one retrieve combinations accurately. That is one

reason we do not prefer the term “fact” – reason we do not prefer the term “fact” – knowing an arithmetic combination well means knowing an arithmetic combination well means far more than knowing a simple, isolated “fact.”far more than knowing a simple, isolated “fact.”

- Sarama and Clements (2009), pp. 131 – 132- Sarama and Clements (2009), pp. 131 – 132

Partial Learning Progression for Partial Learning Progression for Arithmetic CombinationsArithmetic Combinations

Understand numerals and countingUnderstand numerals and counting Understand the structure of the teens numbersUnderstand the structure of the teens numbers Solve addition and subtraction with totals less Solve addition and subtraction with totals less

than 10than 10 Find “break-apart” partners (numbers <= 10)Find “break-apart” partners (numbers <= 10) Use the 10s structure to add/subtract with teen Use the 10s structure to add/subtract with teen

numbers: 10 + 2 = 12; 18 – 8 = 10numbers: 10 + 2 = 12; 18 – 8 = 10 Add and subtract with three addends using 10sAdd and subtract with three addends using 10s

4 + 6 + 3 = 10 + 3 = 13; 15 – 5 – 9 = 10 – 9 = 14 + 6 + 3 = 10 + 3 = 13; 15 – 5 – 9 = 10 – 9 = 1 Develop the “Break-Apart-to-Make-Tens” Develop the “Break-Apart-to-Make-Tens”

StrategyStrategy

Understand the Structure of Understand the Structure of Teen NumbersTeen Numbers

Nimble NumeracyNimble Numeracy (Fischer, 2002), p. 7 (Fischer, 2002), p. 7

Verbal Counting – 13 through 19Verbal Counting – 13 through 19

00 11 22 33 44 55 66 77 88 99

1010 1111 1212 1313 1414 1515 1616 1717 1818 1919

Place Value for the Place Value for the –teen–teen NumbersNumbers

How many ones?How many ones?

77

99

1414

1616

1010

tenstens ones ones

77

99

11 44

11 66

11 00

““Break-Apart” Partners for Break-Apart” Partners for Numbers through 10Numbers through 10

Partner HousesPartner Houses Math MountainsMath Mountains

- Math Expressions- Math Expressions Grade 1, Grade 1, Houghton Mifflin HarcourtHoughton Mifflin Harcourt

Break-Apart-to-Make-Tens Break-Apart-to-Make-Tens (BAMT)(BAMT)

Concrete PhaseConcrete Phase

Break-Apart-to-Make-Tens Break-Apart-to-Make-Tens (BAMT)(BAMT)

Representational PhaseRepresentational Phase

Clements and Sarama (2009), p. 86

Becoming Fluent with Becoming Fluent with Arithmetic CombinationsArithmetic Combinations

Guidelines to Achieving Fluency:Guidelines to Achieving Fluency:

1.1. Essential core skills onlyEssential core skills only2.2. Concepts and strategies firstConcepts and strategies first3.3. Distributed practiceDistributed practice4.4. Research-based strategiesResearch-based strategies

• softwaresoftware• increasing ratio reviewincreasing ratio review

5.5. Develop relationships and strategic thinkingDevelop relationships and strategic thinking• 8 = 2 x 48 = 2 x 4

- Sarama and Clements, pp139-140- Sarama and Clements, pp139-140

Neither memorization without understanding nor unsophisticated counting

procedures helps.

Developing Knowledge, Developing Knowledge, Fluency, and Adaptive Fluency, and Adaptive

ExpertiseExpertiseWhat does NOT workWhat does NOT work Timed tests were Timed tests were

negativelynegatively correlated correlated with knowledge of basic with knowledge of basic combinationscombinations

Flash card use was not Flash card use was not correlated with better correlated with better combination knowledgecombination knowledge

Basic-combination Basic-combination worksheets were worksheets were positively correlated, positively correlated, but only weaklybut only weakly

- - Sarama and Clements, pp. 135-Sarama and Clements, pp. 135-138138

What worksWhat works Break-apart-to-make-Break-apart-to-make-

tenstens 7 + 8 = 8 + 2 + 57 + 8 = 8 + 2 + 5 15 – 8 = 10 – 8 + 515 – 8 = 10 – 8 + 5

Tens complementsTens complements More predictive than More predictive than

work on doubles or work on doubles or smaller sumssmaller sums

nn + 1 + 1 before doublesbefore doubles

Structured discoveryStructured discovery

Math VocabularyMath Vocabulary

Targeted Vocabulary with Focal Targeted Vocabulary with Focal Points on Page 5Points on Page 5

Suggested Vocabulary at end of Suggested Vocabulary at end of documentdocument

How would you describe each list? How would you describe each list?

How would you use each list?How would you use each list?

Importance of Importance of Explicit Vocabulary InstructionExplicit Vocabulary Instruction

Find “Targeted Vocabulary” on Page 5

Number PatternNumber Pattern

num-ber pat-ternnum-ber pat-tern nəm-nəm-bər bər pa-pa-tərn tərn

1.1. Write the termWrite the term

2.2. Outline the termOutline the term-- category or category or

synonymsynonym details details

3.3. Add an exampleAdd an examplemath sentences math sentences

and personal and personal connections also connections also workwork

number pattern: a list of numbers that is created by repeating a rule over and over

number pattern- List

of numbers

created by repeating a rule

Rule:

•start with 1

•add 3

Pattern:

1, 4, 7, 10, 13, . . .

Equal ToEqual To

e-quale-qual ē-ē-kwəl kwəl

Other ways to say “equal to”Other ways to say “equal to” equalsequals = =

http://www.etymonline.com/http://www.etymonline.com/

equal to: having the same amount

equal to

Any strong

connection

•Examples and non-examples

•Multiple examples

Alternate Ideas:

- Story problem with matching equation

- Dictionary definition sentence

My sister gets more allowance than I do. That‘s not fair.

The number of spoons should be equal to the number of bowls in ice cream.

5 = 3 + 2

7 = 6

Place ValuePlace Value

The The valuevalue of the digit depends on its of the digit depends on its placeplace in the number in the number

Explicit Vocabulary Instruction Explicit Vocabulary Instruction – Word Banks– Word Banks

Students see relationships between Students see relationships between wordswords

Students practice using math Students practice using math languagelanguage

Create a word Create a word bank for place bank for place valuevalue

Share at your Share at your tabletable

Definitions by MurphyDefinitions by Murphy

shin bone - a device for finding shin bone - a device for finding furniture in a dark roomfurniture in a dark room

fine - a tax for doing wrongfine - a tax for doing wrong

tax - a fine for doing welltax - a fine for doing well

Textbook AnalysisTextbook Analysis

How does your textbook present new How does your textbook present new terms?terms?

Does the teacher’s guide include any Does the teacher’s guide include any strategies for explicit vocabulary strategies for explicit vocabulary instruction?instruction?

LunchLunch

Finding Focus Finding Focus DocumentDocument– – Instructional Implications Instructional Implications

SectionSectionIn grade-level groupsIn grade-level groups Each person reads the “Instructional Each person reads the “Instructional

Implications” for a different focal pointImplications” for a different focal point Share the highlights with your groupShare the highlights with your group

Something newSomething new Something not newSomething not new Something you question or wonder aboutSomething you question or wonder about

AssessmentAssessment

Think-Pair-Share: Think-Pair-Share: In what ways do you assess your In what ways do you assess your

students’ learning?students’ learning? What is the purpose of each method?What is the purpose of each method?

3.3

Different assessments serve different purposesDifferent assessments serve different purposes

AssessmentAssessment PurposePurpose ExampleExample

CBM Measure CBM Measure (universal screening)(universal screening)

Identify students at riskIdentify students at riskEvaluate core instructionEvaluate core instructionMonitor overall progressMonitor overall progress

Quantity DiscriminationQuantity DiscriminationMixed ComputationMixed Computation

Formative classroom Formative classroom assessmentassessment

Inform instructionInform instructionProvide student feedbackProvide student feedback

Teacher observationTeacher observationExit questionsExit questions

Summative classroom Summative classroom assessmentassessment

Evaluate studentsEvaluate studentsEvaluate core instructionEvaluate core instruction

Quizzes and testsQuizzes and testsProjectsProjects

District assessmentsDistrict assessments Evaluate studentsEvaluate studentsEvaluate programsEvaluate programs

Common “short-cycle” Common “short-cycle” assessmentsassessmentsCommon course examsCommon course exams

State assessmentsState assessments Evaluate studentsEvaluate studentsEvaluate schoolsEvaluate schools

MEAPMEAPMMEMME

Single-skill or multi-Single-skill or multi-skill assessments skill assessments (mastery (mastery measurement)measurement)

Identify strengths and Identify strengths and weaknessesweaknessesGroup students by needGroup students by needMonitor specific progressMonitor specific progress

MMPI diagnostic MMPI diagnostic assessmentsassessmentsDistrict assessmentsDistrict assessmentsClassroom assessmentsClassroom assessments

CBM v. Mastery CBM v. Mastery MeasurementMeasurement

Curriculum-based Curriculum-based measurement (CBM):measurement (CBM):

Measures general progressMeasures general progress Brief, timed “probes”Brief, timed “probes” Selected skills Selected skills

representative of the representative of the whole year’s curriculumwhole year’s curriculum

Less useful for diagnosing Less useful for diagnosing specific needsspecific needs

Mastery measurement:Mastery measurement: Determines specific Determines specific

needs (diagnostic pre-needs (diagnostic pre-assessment)assessment)

Tracks mastery of each Tracks mastery of each skill or standardskill or standard District short-cycle District short-cycle

assessmentsassessments NWEA also an exampleNWEA also an example

Usually not timedUsually not timed Does not assess Does not assess

overall “health”overall “health”

Diagnostic Inventories from Diagnostic Inventories from MMPIMMPI

Choose addition or subtractionChoose addition or subtraction Do #1, 5b, 6, 9, 11a, 12bDo #1, 5b, 6, 9, 11a, 12b Print from Print from www.michiganmathematics.orgwww.michiganmathematics.org

Math Measures Used for Math Measures Used for Universal Screening in SAMUniversal Screening in SAM

MeasureMeasure SourceSource TimeTime

Tests of Early Tests of Early Numeracy (TEN)Numeracy (TEN)Grades K-1; 4 measuresGrades K-1; 4 measures

AIMSwebAIMSweb 1 min. 1 min. each;each;

individually individually administeradministereded

Mixed Mixed ComputationComputation Grades Grades 1-61-6

AIMSwebAIMSweb 2 – 4 min.;2 – 4 min.;

paper/paper/pencilpencil

Algebra MeasuresAlgebra Measures Grade 6 – 8 and Algebra Grade 6 – 8 and Algebra 11

AAIMS – AAIMS – Iowa Iowa State State UniversityUniversity

5 – 7 min.;5 – 7 min.;

paper/paper/pencilpencil

Progress MonitoringProgress Monitoring

Tests of Early NumeracyTests of Early Numeracycreated by and stored at created by and stored at

AIMSwebAIMSweb

They are All They are All FluencyFluency MeasuresMeasures

Each takes one minuteEach takes one minute Individually administeredIndividually administered Correct/errors reportedCorrect/errors reported All four measures are done three All four measures are done three

times in K times in K andand first grade first grade

Oral CountingOral Counting

Number IdentificationNumber Identification

Quantity DiscriminationQuantity Discrimination

Missing NumberMissing Number

Math Calculation Fluency first Math Calculation Fluency first gradegrade

Math Calculation Fluency 2Math Calculation Fluency 2ndnd gradegrade

OPTIONAL Mathematics OPTIONAL Mathematics Concepts and Applications Concepts and Applications

Probes Probes (M-CAP Grade 2+)(M-CAP Grade 2+)

M-CAP Grade 2 SampleM-CAP Grade 2 Sample

Track skill mastery for each studentTrack skill mastery for each student Track class mastery for each skillTrack class mastery for each skill State assessment data State assessment data Demographic information and grades from Demographic information and grades from

an SIS an SIS District-administered tests District-administered tests Daily classroom assessments given by Daily classroom assessments given by

teachersteachers

Supporting InstructionSupporting Instruction

Conceptual Conceptual understanding and understanding and procedural fluencyprocedural fluency

Materials and Materials and resourcesresources

Scheduling optionsScheduling options Educator’s round Educator’s round

tabletable

Conceptual Understanding and Conceptual Understanding and Procedural FluencyProcedural Fluency

Developing computational skill and developing Developing computational skill and developing understanding are often seen as competing understanding are often seen as competing

for attention in school mathematics. But for attention in school mathematics. But pitting skill against understanding creates a pitting skill against understanding creates a false dichotomy. Understanding makes it false dichotomy. Understanding makes it

easier to learn skills, while learning easier to learn skills, while learning procedures can strengthen and develop procedures can strengthen and develop

mathematical understanding.mathematical understanding.

- Kilpatrick and Swafford, 2002,- Kilpatrick and Swafford, 2002, p. 13p. 13

Skills vs. ProcessesSkills vs. Processes

Both produce some form of productBoth produce some form of product Both require multiple opportunities for Both require multiple opportunities for

review and practice over timereview and practice over time A “process” is more complex A “process” is more complex Step-by-step procedures and Step-by-step procedures and

“processes” may need even more “processes” may need even more distributed practicedistributed practice

PLEASE try to build distributed practice PLEASE try to build distributed practice into you annual planinto you annual plan

Materials and ResourcesMaterials and Resources

Getting the most from Getting the most from your textbook seriesyour textbook series

For remediation, consider resources For remediation, consider resources from a previous gradefrom a previous grade

Ancillary materials contain extension, Ancillary materials contain extension, expansion, or intervention lessons expansion, or intervention lessons useful for remediation or enrichmentuseful for remediation or enrichment

EthnomathematicsEthnomathematics(fancy word for learning math outside of (fancy word for learning math outside of

school)school) How to Create a How to Create a

Mathematical Mathematical Environment at HomeEnvironment at Home

InvestigationsInvestigations ActivitiesActivities ExplorationsExplorations GamesGames How to avoid-How to avoid-

Flashcards, textbooks, Flashcards, textbooks, memorization, and testsmemorization, and tests$17.00

Technology OptionsTechnology OptionsSimulations.Simulations. Distance Distance

LearningLearningWeb Based Web Based InstructionInstruction-Kuto SoftwareKuto Software-Number WorldsNumber Worlds

-Building -Building BlocksBlocks

Word Word processing, data processing, data base, base, PowerpointPowerpoint

scanners, MP3scanners, MP3

Interactive A.P.Interactive A.P.

Interactive Interactive within HISD, within HISD, Virtual H.S.Virtual H.S.

PLATO,PLATO,

Study IslandStudy Island

Education 2020Education 2020

Microcassettes- Microcassettes- Mp3 digital Mp3 digital recordersrecorders

Textbook Textbook WebsitesWebsites

Types of Instructional Types of Instructional SoftwareSoftware

TutorialsTutorials Drill and practiceDrill and practice SimulationsSimulations Problem-solving Problem-solving DiscoveryDiscovery GamesGames

8.6

Computer-Aided Instruction Computer-Aided Instruction Research:Research:

Individualized instructionIndividualized instruction Timely feedbackTimely feedback

Programs:Programs: No clear evidenceNo clear evidence Odyssey from Compass LearningOdyssey from Compass Learning

http://www.compasslearningodyssey.com/http://www.compasslearningodyssey.com/ “Solutions” “Solutions” Cognitive Tutor from Carnegie LearningCognitive Tutor from Carnegie Learning

http://www.carnegielearning.com/software_features.cfmhttp://www.carnegielearning.com/software_features.cfm

IlluminationsIlluminationshttp://http://www.illuminations.nctm.orgwww.illuminations.nctm.org

Illuminations: Illuminations: ConcentrationConcentration

NCTM E-ExamplesNCTM E-Examples

www.nctm.orgwww.nctm.org Standards and Standards and Focal Points Focal Points under Principles under Principles and Standards, and Standards, choose “e-choose “e-Examples”Examples”

National Library of Virtual National Library of Virtual ManipulativesManipulatives

www.nlvm.usu.eduwww.nlvm.usu.edu

Project InteractiveProject Interactive

http://www.shodor.org/interactivate/http://www.shodor.org/interactivate/

Project Interactive: Shape Project Interactive: Shape SorterSorter

KeyMath3:KeyMath3:An Assessment-Intervention An Assessment-Intervention

SystemSystem Concepts, skills, and problem solving Concepts, skills, and problem solving K-5 content (remediation through Grade 8)K-5 content (remediation through Grade 8)

““Essential Resources” is the intervention Essential Resources” is the intervention componentcomponent Stand-alone or with the assessment componentStand-alone or with the assessment component Scripted lessons for small-group interventionScripted lessons for small-group intervention Lessons 30-40 minutes eachLessons 30-40 minutes each

SchedulingSchedulingOptionsOptions

Block Scheduling (1 ½ hour math sections)Block Scheduling (1 ½ hour math sections) Common grade schedules to allow sharing Common grade schedules to allow sharing

between classroomsbetween classrooms Extra helpers in the classroom (co-Extra helpers in the classroom (co-

teachers, aides, special education, teachers, aides, special education, Chapter)Chapter)

Before and after school, Saturday schoolBefore and after school, Saturday school During “specials” timeDuring “specials” time Lunch buddiesLunch buddies

Educators’ Round Table Educators’ Round Table OptionsOptions

Meet with colleagues Meet with colleagues from your districtfrom your district

Exchange ideas with Exchange ideas with teachers from other teachers from other districtsdistricts

Ask specific Ask specific questions of the questions of the presenterspresenters

Explore materials or Explore materials or websiteswebsites

BreakBreak

ReflectionReflection

What ideas did you learn from the What ideas did you learn from the educators’ round table?educators’ round table?

What questions do you still have?What questions do you still have? What will you do next?What will you do next?

Student Errors withStudent Errors withMulti-Digit Addition and Multi-Digit Addition and

SubtractionSubtraction247247

++ 175 175

312312

6262

-- 4646

2424

701701

-- 32 32

579579

Many of these errors stem from children’s Many of these errors stem from children’s treatment of multidigit numbers as a series treatment of multidigit numbers as a series

of single-digit numbers, without of single-digit numbers, without consideration of their place value and role consideration of their place value and role in the mathematical situation. Thus U.S. in the mathematical situation. Thus U.S.

children learn to carry out the steps of the children learn to carry out the steps of the algorithm, but do not develop conceptual algorithm, but do not develop conceptual understanding of place value. This is a understanding of place value. This is a

national problem. national problem.

- (Clements and Sarama (2009), p. 92)- (Clements and Sarama (2009), p. 92)

Standard written algorithms intentionally relieve Standard written algorithms intentionally relieve the user of thinking about where to start, what the user of thinking about where to start, what place value to assign to digits, and so forth. place value to assign to digits, and so forth. This This is efficient for those who already understand, but is efficient for those who already understand, but often has negative effects on initial learning.often has negative effects on initial learning.

In comparison, mental strategies are derived from In comparison, mental strategies are derived from and support underlying concepts. Conventionally and support underlying concepts. Conventionally taught students usually take a long time to taught students usually take a long time to master algorithms and often never master them. master algorithms and often never master them. Students learn better if mental computation is Students learn better if mental computation is taught and performed before written algorithms taught and performed before written algorithms (and practiced throughout education), along with (and practiced throughout education), along with appropriate work with concrete materials and appropriate work with concrete materials and drawings.drawings.

- (Clements and Sarama (2009), p. 94)- (Clements and Sarama (2009), p. 94)

What do you think?What do you think?

Modeling Addition and Modeling Addition and SubtractionSubtraction

““Add To” – Result UnknownAdd To” – Result Unknown

Three bunnies sat on the grass. Two Three bunnies sat on the grass. Two more bunnies hopped there. How more bunnies hopped there. How many bunnies are on the grass now?many bunnies are on the grass now?

““Take From” – Result Take From” – Result UnknownUnknown

Five apples were on the table. I ate Five apples were on the table. I ate two apples. How many apples are on two apples. How many apples are on the table now?the table now?

Put Together / Take Apart – Put Together / Take Apart – Total UnknownTotal Unknown

Three yellow cars and two blue cars Three yellow cars and two blue cars are on the table. How many cars are are on the table. How many cars are on the table?on the table?

Modeling Multi-Digit Addition Modeling Multi-Digit Addition and Subtractionand Subtraction

Addition

With Regrouping

Base 10 Blocks

Allan and Scott

Problem: 37 + 26

Build 37 on the top portion of the Addition Mat.

How many tens? Record it.

How many ones? Record it.

How much is that?

Build 26 on the center portion of the Addition Mat.

How many tens? Record it.

How many ones? Record it.

How much is that?

Combine the ones, pulling them to the bottom of the sheet.

How many ones? (thirteen) Train them and compare the train with a ten rod.

Are there enough ones to exchange for a ten? (yes)

Place the ten rod where you want to record the exchange in the algorithm.

Make the exchange and record the thirteen.

Combine the tens, pulling them to the bottom of the sheet.

Are there enough tens to exchange for a hundred? (no)

How many tens? (six) Record the sixty.

How many ones? (three) Record the three.

How much altogether? (sixty-three)

PracticePractice

2828 4242 3737

++ 4343 ++ 5151 ++ 4545

Take Away Subtraction

With Regrouping

Base 10 Blocks

Allan and Scott

Problem: 32 - 17

Build 32 on the top portion of the Subtraction Mat.

How many tens? Record it.

How many ones? Record it.

How much is that?

3 2

Write minus seventeen on the middle portion of the mat.

3 2

- 1 7

Is it possible to take away seven ones? (no)

Exchange a ten for 10 ones and place them on the ones portion of the mat.

Record the exchange.

3 2

- 1 7

1 22

Remove seven ones and place them below the line on the small sheet.

23

- 1 7

1 22

How many ones left on the Subtraction Mat? (5)

Pull them to the bottom portion of the Mat and record.

31 222

- 1 7

5

Remove one of the tens and place it below the line on the small sheet.

31 222

- 1 7

5

How many tens left on the Subtraction Mat? (1)

Pull it to the bottom portion of the Mat and record.

31 222

- 1 7

51

Practice and DiscussPractice and Discuss

4242 5353 6262

-- 2424 -- 4646 -- 2727

ReflectionReflection

What is one idea What is one idea you will use in the you will use in the next week?next week?

What are 2-3 items What are 2-3 items you will share or you will share or discuss with your discuss with your colleagues?colleagues?

How will this How will this happen?happen?

Contact InformationContact Information

Jenni TrusockJenni Trusock989/269-6406989/269-6406 jtrusock@hisd.k12.mi.usjtrusock@hisd.k12.mi.us

Kristen LegaultKristen Legault 989/269-6404989/269-6404 legault@hisd.k12.mi.uslegault@hisd.k12.mi.us

Craig WalterCraig Walter 989/269-6406989/269-6406 cwalter@hisd.k12.mi.uscwalter@hisd.k12.mi.us

Joanne ZangJoanne Zang 989/269-6406989/269-6406 jz@hisd.k12.mi.usjz@hisd.k12.mi.us

ReferencesReferences

Auman, Marueen and Valette, Debbie (2009) Auman, Marueen and Valette, Debbie (2009) Step Up to Writing in MathStep Up to Writing in Math. Longmont, . Longmont, CO: Sopris West Educational Services.CO: Sopris West Educational Services.

Clements, Douglas H. and Sarama, Julie Clements, Douglas H. and Sarama, Julie (2009) (2009) Learning and Teaching Early Math: Learning and Teaching Early Math: The Learning Trajectories Approach The Learning Trajectories Approach. New . New York, NY: Routledge.York, NY: Routledge.

Fischer, Phyllis E. (2002) Fischer, Phyllis E. (2002) Nimble Numeracy: Nimble Numeracy: Fluency in Counting and Basic ArithmeticFluency in Counting and Basic Arithmetic. . Farmington, ME: Oxton House Publishers. Farmington, ME: Oxton House Publishers.

ReferencesReferences

Fuson, Karen C. (2006) “Teacher’s Guide.” Fuson, Karen C. (2006) “Teacher’s Guide.” Math Math ExpressionsExpressions, Grade 1, Volume 1. Boston, MA: , Grade 1, Volume 1. Boston, MA: Houghton Mifflin Company.Houghton Mifflin Company.

Kilpatrick, J. and Swafford, J. (2002)Kilpatrick, J. and Swafford, J. (2002) Helping Children Helping Children Learn Mathematics. Learn Mathematics. Washington, DC: National Washington, DC: National Academy Press. Academy Press.

Marzano, Robert J.; Pickering, Debra J.; Pollock, Jane E. Marzano, Robert J.; Pickering, Debra J.; Pollock, Jane E. (2001) (2001) Classroom Instruction that WorksClassroom Instruction that Works. . Alexandria, VA: Associagtion for Supervision and Alexandria, VA: Associagtion for Supervision and Curriculum Development.Curriculum Development.

Sarama, Julie and Clements, Douglas H. (2009) Sarama, Julie and Clements, Douglas H. (2009) Early Early Childhood Mathematics Education Research: Childhood Mathematics Education Research: Learning Trajectories for Young ChildrenLearning Trajectories for Young Children. New York, . New York, NY: Routledge.NY: Routledge.

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