Teaching Young Learners with Teaching Young Learners with the Ohio Early Mathematics the Ohio Early Mathematics

Standards in MindStandards in Mind

Sponsored by Sponsored by

the Ohio Department of Jobs the Ohio Department of Jobs and Family Services and Family Services

in collaboration with in collaboration with the Ohio Department of the Ohio Department of

EducationEducationModule One

Overview of SeminarOverview of Seminar Module 1Module 1

• What are Standards, benchmarks and indicators?What are Standards, benchmarks and indicators?

• Supporting Children’s Early Concepts of NumberSupporting Children’s Early Concepts of Number

Module 2Module 2

• Early Addition and SubtractionEarly Addition and Subtraction

• Patterns and Algebraic ReasoningPatterns and Algebraic Reasoning

Module 3Module 3

• Geometric Reasoning through Children’s PlayGeometric Reasoning through Children’s Play

• Teaching Mathematics through Activity – Teaching Mathematics through Activity – Measurement & Data AnalysisMeasurement & Data Analysis

Themes for the SessionsThemes for the Sessions PlayPlay Inquiry & projectsInquiry & projects Home-school connectionsHome-school connections Adaptations for diverse learnersAdaptations for diverse learners Integration across the curriculum Integration across the curriculum The connected nature of mathematical The connected nature of mathematical

knowledge knowledge Role of conversation and questioningRole of conversation and questioning 3 levels of representational thinking3 levels of representational thinking

Overview of Module 1Overview of Module 1

What are the Ohio Early Learning What are the Ohio Early Learning Content Standards? Content Standards?

What are the benefits of using What are the benefits of using standards? standards?

What are some of the challenges What are some of the challenges of using standards?of using standards?

What are the Ohio early learning content standards?

How do standards, benchmarks, and indicators relate to one another?

How should teachers think about using them?

The Ohio StandardsThe Ohio Standards

ImagineImagine……

The Teacher as The Teacher as gardenergardener

The Learner as the The Learner as the growing plantgrowing plant

Standards as a grower’s Standards as a grower’s guideguide

Early content standards Early content standards as rootsas roots

Environment and Environment and circumstances as sun, circumstances as sun, water, soil, geography…water, soil, geography…

We all begin young and full of We all begin young and full of potentialpotential

    

Seeds and seedlings need much care.

The younger the plant, the greater the need for careful observation and responsiveness.

The child as learner needs similar support and responsiveness.

We need strong rootsWe need strong roots……

The roots of the plant develop first and are critical to further growth,

but they also continue their importance throughout the life of the plant.

an appropriate environmentan appropriate environment……

The sunlight, temperature, water, proximity and types of other plants, soil, fertilizer, and other environmental factors influence the development of the plant.

A child’s environment influences the development of his or her math thinking

……and a gardener who knows and a gardener who knows our unique needs.our unique needs.

A grower’s guide is a generic guide for how MOST plants grow. It provides benchmarks in the life of a plant, and it gives assistance to the novice gardener.

It can’t predict with certainty all the specific needs of one particular plant.

A skilled gardener adjusts directions in order to create the best environment for her particular plant.

There are many plants and There are many plants and many learners.many learners.

Ideas to Keep In MindIdeas to Keep In Mind……

We can’t depend on a guide to We can’t depend on a guide to provide the schedule of care for provide the schedule of care for each day.each day.

Too much of a good thing can be Too much of a good thing can be damaging.damaging.

Ideas to Keep In Mind Ideas to Keep In Mind (Cont.) (Cont.) ……

What worked with one plant will What worked with one plant will not necessarily work with another; not necessarily work with another; individualization is important.individualization is important.

Patience is important for the Patience is important for the gardener.gardener.

The standards are like theThe standards are like the grower’s guide.grower’s guide.

Teachers make educated Teachers make educated judgments about necessary judgments about necessary conditions for math learning. Each conditions for math learning. Each child is different.child is different.

We reap benefits when we opt not We reap benefits when we opt not to push children too hard or too to push children too hard or too quickly. Teachers need patience quickly. Teachers need patience and direction.and direction.

Using the Standards WiselyUsing the Standards Wisely

What is an early math What is an early math standard?standard?

Standards are what we expect Standards are what we expect students to know and be able to students to know and be able to use as they progress through use as they progress through school. school.

Standards outline the foundational Standards outline the foundational content and processes in content and processes in mathematics.mathematics.

Standards are both content and Standards are both content and processprocess

Number, number sense and Number, number sense and operationsoperations

MeasurementMeasurement

Geometry and spatial senseGeometry and spatial sense

Patterns, functions and algebraPatterns, functions and algebra

Data analysis and probabilityData analysis and probability

Mathematical processesMathematical processes

What is an indicator?What is an indicator?

Indicators are specific skills and Indicators are specific skills and understandings that students understandings that students demonstrate across the grade levelsdemonstrate across the grade levels

These indicators let us know that the These indicators let us know that the student is making progress toward student is making progress toward the benchmarksthe benchmarks

What are What are benchmarks?benchmarks?

Benchmarks are particular indicators Benchmarks are particular indicators that are “grouped” in developmental that are “grouped” in developmental chunks to indicate where students chunks to indicate where students should be by a particular grade. should be by a particular grade.

For early childhood math, the For early childhood math, the benchmark is second grade. benchmark is second grade.

Number, Number Sense and Number, Number Sense and Operations Standards Operations Standards

(by grade 12)(by grade 12)

Students will demonstrate number Students will demonstrate number sense, including an understanding of sense, including an understanding of number systems and operations and number systems and operations and how they relate to one another. how they relate to one another.

Students compute fluently and make Students compute fluently and make reasonable estimates using paper reasonable estimates using paper and pencil, technology-supported and pencil, technology-supported and mental methods. and mental methods.

22ndnd grade benchmark for grade benchmark for number standardnumber standard

There are 13 indicators that a student has There are 13 indicators that a student has reached the 2nd grade standard for reached the 2nd grade standard for number, number sense, and operation.number, number sense, and operation.

For example: recognize, classify, compare For example: recognize, classify, compare and order whole numbersand order whole numbers

Model, represent and explain subtraction Model, represent and explain subtraction as comparison, take-away, and part-to-as comparison, take-away, and part-to-wholewhole

Mathematical Processes Mathematical Processes StandardStandard

Students use mathematical Students use mathematical processes and knowledge to solve processes and knowledge to solve problems. Students apply problem-problems. Students apply problem-solving and decision-making solving and decision-making techniques, and communicate techniques, and communicate mathematical ideas.mathematical ideas.

Problem solving, Communication, Problem solving, Communication, Connections, Representation, & Connections, Representation, & Reasoning and ProofReasoning and Proof

Benchmark for process standardBenchmark for process standard

Example indicator (there are 9 all Example indicator (there are 9 all together): together):

Use a variety of strategies to Use a variety of strategies to understand problem situations, e.g., understand problem situations, e.g., discussing with peers, stating discussing with peers, stating problem in own words, modeling problem in own words, modeling problems with diagrams or physical problems with diagrams or physical materials, identifying a pattern.materials, identifying a pattern.

Achieving BalanceAchieving Balance

Children construct their Children construct their own knowledge own knowledge through playthrough play

• Teachers provide lots Teachers provide lots of time for free play of time for free play with math materialswith math materials

• Teachers place math Teachers place math related materials in related materials in every part of the every part of the roomroom

With IWith Intentional ntentional TeachingTeaching, teachers can , teachers can pay attention to pay attention to curriculum standards and curriculum standards and benchmarks as they benchmarks as they prepare environmentsprepare environments

• Teachers plan math Teachers plan math experiences based on experiences based on the standards, with the standards, with developmental levels developmental levels and culture in mind.and culture in mind.

• Teachers assess Teachers assess where children are where children are and provide next step and provide next step experiencesexperiences

BackmappingBackmapping

How do we know we are providing How do we know we are providing experiences that support students’ experiences that support students’ progress toward the benchmark?progress toward the benchmark?

We backmap!!We backmap!!

Backmapping is a regular routine where Backmapping is a regular routine where you look back at curricular experience and you look back at curricular experience and ask: which indicators have we been ask: which indicators have we been supporting? supporting?

BREAK TIMEBREAK TIME

Early number ConceptsEarly number Concepts

What understandings are young What understandings are young learners developing as they are learners developing as they are developing their concepts about developing their concepts about

number?number?

How do you support these How do you support these early competencies?early competencies?

Early number -- sortingEarly number -- sorting

Free sorting activities – what are they?Free sorting activities – what are they?

Teacher’s roles: providing sortable Teacher’s roles: providing sortable materials, conversation/questioningmaterials, conversation/questioning

“ “ What did you find?” What did you find?” •• “What kinds of “What kinds of groups did you make?” groups did you make?” •• “How did you “How did you put them together in different groups?” put them together in different groups?” •• “I see you sorted them into colors!” “I see you sorted them into colors!”

Structured Sorting Structured Sorting ActivitiesActivities

““In the Loop”: The leader (adult or In the Loop”: The leader (adult or child) places one piece in the loop. child) places one piece in the loop. Everyone else finds pieces from Everyone else finds pieces from their pile to add to the looptheir pile to add to the loop

““Guess my rule”: The leader places Guess my rule”: The leader places several pieces in a loop – everyone several pieces in a loop – everyone else guesses what s/he is thinking else guesses what s/he is thinking

As early number concepts As early number concepts develop furtherdevelop further

Classification – sorting according to Classification – sorting according to attributesattributes

PatternsPatterns

Comparisons of sets (more than & Comparisons of sets (more than & less than)less than)

Ordering Sets (smallest to largest)Ordering Sets (smallest to largest)

As early number concepts As early number concepts develop further (Cont.)develop further (Cont.)

ConservationConservation - Conservation is the - Conservation is the recognition that the number, length, recognition that the number, length, quantity, mass, area, weight, and quantity, mass, area, weight, and volume of objects and substances are volume of objects and substances are not changed by transformations in their not changed by transformations in their appearance.appearance.

Beginning to recognize how many in a Beginning to recognize how many in a small set without countingsmall set without counting

Early Counting Concepts and SkillsEarly Counting Concepts and Skills

What do you see when What do you see when children count? children count?

How is learning to count similar How is learning to count similar to learning to read?to learning to read?

Seven CandiesSeven Candies

While watching the children, notice what While watching the children, notice what they do to count the candiesthey do to count the candies

What do they show us about their What do they show us about their understanding of number and countingunderstanding of number and counting

Take notes about individual children’s:Take notes about individual children’s:

• Strategies Strategies

• Physical behaviorsPhysical behaviors

• Knowledge and competenciesKnowledge and competencies

• Misunderstandings or limited experienceMisunderstandings or limited experience

Seven CandiesSeven Candies

Seven CandiesSeven Candies

Seven CandiesSeven Candies

Seven CandiesSeven Candies

Seven CandiesSeven Candies

Seven CandiesSeven Candies

Developing Early Developing Early Concepts of NumberConcepts of Number

What do you do to foster children’s What do you do to foster children’s early development of counting?early development of counting?

Components of Early Number Components of Early Number Knowledge - Counting PrinciplesKnowledge - Counting Principles

Production of numbers (standard list of Production of numbers (standard list of counting words)counting words)

One-to-one correspondence (one object for One-to-one correspondence (one object for each number)each number)

Ordering or seriation (small to largest)Ordering or seriation (small to largest) Cardinality principle (last number is the number Cardinality principle (last number is the number

in the set)in the set) Which object in the set you start with doesn’t Which object in the set you start with doesn’t

matter matter Conservation– number stays constant even if Conservation– number stays constant even if

objects are rearrangedobjects are rearranged

Groups of 5 counters are arranged in the following Groups of 5 counters are arranged in the following 3 patterns.3 patterns. Discuss the students’ knowledge of Discuss the students’ knowledge of

counting principles on each of the following slidescounting principles on each of the following slides

A conversation with StephenA conversation with Stephen T: Are there more red, blue, or yellow T: Are there more red, blue, or yellow counters?counters? S: More blue.S: More blue. T: How do you know?T: How do you know? S: I can tell by looking.S: I can tell by looking. T: How many of each?T: How many of each? S: One, two, three, four, five... five red. S: One, two, three, four, five... five red. One, two, One, two,

three, four, five...five blue. three, four, five...five blue. One, two, three, four, five...five yellow.One, two, three, four, five...five yellow.

T: Five of each?T: Five of each? S: Yes.S: Yes. T: Do you still think there are more blue?T: Do you still think there are more blue? S: Yes, I can just see there's more blue.S: Yes, I can just see there's more blue.

A conversation with RebeccaA conversation with Rebecca

T: Are there more red, blue, or yellow T: Are there more red, blue, or yellow counters?counters?

R: They're the same.R: They're the same. T: How do you know?T: How do you know? R: I counted them.R: I counted them. T: How many of each?T: How many of each? R: One, two, three, four, five...Five red. R: One, two, three, four, five...Five red.

Five blue. Five yellow. Five blue. Five yellow. T: Five of each?T: Five of each? R: Yes. R: Yes.

Counting Principles: StephenCounting Principles: Stephen T: Here are some blocks in a row. T: Here are some blocks in a row.

Start with this one on the end and Start with this one on the end and count count them.them.

S: One, two, three, four, five, SIX. S: One, two, three, four, five, SIX. There There are six blocks.are six blocks.

T: What if you start at the other end T: What if you start at the other end of of the row and count them?the row and count them?

S: One, two, three, four, five, SIX. S: One, two, three, four, five, SIX. There There are six.are six.

Counting Principles: RebeccaCounting Principles: Rebecca T: Here are some red blocks in a row. T: Here are some red blocks in a row.

Start with this one on the end and Start with this one on the end and count count them.them.

S: (Touches each of the 5 blocks) One, S: (Touches each of the 5 blocks) One, two, three, five, six. Six red blockstwo, three, five, six. Six red blocks

T: Now count these blue blocks.T: Now count these blue blocks. S: (Touches each of the 4 blocks) One, S: (Touches each of the 4 blocks) One,

two, three, five. Five blue blocks.two, three, five. Five blue blocks.

Counting Principles: SallyCounting Principles: Sally

T: Here are some blocks in a row. T: Here are some blocks in a row. Start Start with the one on this end and with the one on this end and count them.count them.

S: One, two, three, four, five, six. S: One, two, three, four, five, six. There There are six.are six.

T: What if you start at the other end ofT: What if you start at the other end ofthe row and count them?the row and count them?

S: I already counted them! There are six! S: I already counted them! There are six!

Counting Principles: BrendaCounting Principles: Brenda

T: Here are some red blocks (4) in a T: Here are some red blocks (4) in a row. Start with this one on the row. Start with this one on the end and count them.end and count them.

S: (Points to each but says two S: (Points to each but says two numbers with each point) One, numbers with each point) One, two, three, four, five, six, seven, two, three, four, five, six, seven, eight. Eight red blocks.eight. Eight red blocks.

Number StandardNumber Standard

Age 3Age 3 Counts Collection of 1 Counts Collection of 1

to 4 itemsto 4 items Begins to understand Begins to understand

cardinalitycardinality Group recognition for Group recognition for

collections of 1 to 3collections of 1 to 3 Adds and subtracts Adds and subtracts

non-verbally low non-verbally low numbers numbers

Age 6Age 6 Counts and counts out Counts and counts out

collections up to 100 collections up to 100 using groups of 10using groups of 10

Group recognition for Group recognition for patterned collections patterned collections of up to 6 itemsof up to 6 items

Adds and subtracts Adds and subtracts using counting-based using counting-based strategies such as strategies such as counting on for counting on for numbers and totals numbers and totals less than 10less than 10