finding focus for mathematics instruction grades k-2 march 25, 2010huron area technical center march...
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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 [email protected]@hisd.k12.mi.us
Kristen LegaultKristen Legault 989/269-6404989/269-6404 [email protected]@hisd.k12.mi.us
Craig WalterCraig Walter 989/269-6406989/269-6406 [email protected]@hisd.k12.mi.us
Joanne ZangJoanne Zang 989/269-6406989/269-6406 [email protected]@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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