Presentation on theme: "Mathematics Foundations Number Sense"— Presentation transcript:
1 Mathematics Foundations Number Sense A California Department of Education Child Development Division Presentation
2 IcebreakerUse an icebreaker of your choice. There are two icebreaker slides at the end of the presentation for optional use.See Activity 1 for complete directions for the Take a Stand icebreaker.
3 Did you like math when you were in school? Take a StandDid you like math when you were in school?Activity #1This slide was developed by Region 3.Consider engaging in this activity prior to beginning math content.Ask participants if they liked math when they were in school.Have them think about their experiences with math content, teachers and past experiences.Place numbers 1 through 5 up high on the wall around the room with as much space as possible in between numbers so participants can have focused conversations and not be distracted by other groups.Ask participants to go stand by the number that best describes their experiences with math.For example:5 could be an A student; excelled, loved it, overall positive experiences etc. Add your own verbiage to each number.4 could be a B student; probably did well, did not love it, was not bad at it3 could be a C student mediocre experiences that left them lukewarm2 could be a D student that did poor1 would be an F student hated it, every aspect of it didn’t get it, tried again and again negative experiences overall.Once in their groups have participants share why they chose this number and discuss feelings centered around content, teachers and past experiences.After discussion reconvene and keep people in their like groups, then ask a participant from each group to share out a collective summary of their groups experiences. Some people may want to have more than one person share. Facilitator can decide how to structure this portion.Summary: Have fun with this exercise these are just guidelines add your own twist to setting it up. Once participants begin to talk with others in their group they will realize they have had similar experiences and will freely share among each other.What we found is that the discussion will bring up powerful emotional experiences around math. Administrators and teacher leaders need to acknowledge their feelings in order to become comfortable with understanding and teaching math to preschool teachers. This exercise can also be conducted at teacher professional development sessions.Enjoy! This activity was fun and very therapeutic for CPIN participants.
4 OutcomesBecome familiar with the research in early childhood and mathematicsDescribe what the implications of this research and instruction are for young children and teachersBecome familiar with the preschool learning foundations in mathematics with an emphasis on number senseUnderstand the links between the mathematics research in California’s Child Development Division’s initiatives (Preschool Learning Foundations-Mathematics, PreK Guidelines & Desired Results)Identify classroom practices that support all children’s mathematical growth and developmentRead slide
5 Norms Start on time and end on time. Turn off cell phones or to vibrate.Help the group stay on task.Listen to thoughts and ideas of others.Contribute your thoughts and ideas.Use this slide as desired, and/or chart the norms.
6 Parking LotPlease write questions on post-its and place them on chart paper titled “Parking Lot.”Slide optional: Place Parking Lot posters in the room.Participants may post their questions in the Parking Lot. Note that facilitators will try to answer as many questions as possible. Any unanswered questions will be sent to CDD for answers.
7 AgendaResearchWhat research tells us about the importance of early mathematicsNational Council of Teachers of Mathematics - 5 Big IdeasCalifornia’s Preschool InitiativesThe Prekindergarten Learning and Development GuidelinesThe Early Childhood Environment Rating ScaleThe Desired Results Developmental Profile-RevisedPreschool Learning FoundationsResearch to PracticeExploring Number Sense in the classroom
8 Lessons Learned from Research Gaps are strikingLess is moreConnect informal and school mathMeet the needs of all learnersInclude geometryUse learning trajectoriesDoug Clements, 2007 CPIN presentation
9 Gaps are Striking School Mathematics is improving but Not Working Well Enough for Enough Students 2007Internationally our studentsare not mathematicallycompetitiveHIGHER20 countriesSAME14 countriesLOWER7 countriesHIGHER9 countriesSAME2 countriesLOWER37 countriesUnited StatesThis slide shows how our 8th grade students are doing in comparison to other countries in the Nation. This study comes from a study done in about 10 years ago. However there are studies completed within the last few years indicating the same results. we have changed, because we have not changed the way we teach. There are 20 countries ahead of us, 14 scored the same and 7 countries are lower.PURPOSE: This slide provides evidence of the current status of mathematics achievement.The slide shows the United States’ relative position in the Grade 8 portion of Third International Mathematics and Science Study (1995); 7 countries are below the U.S. in performance, 14 are statistically no different from us, and 20 countries are above our performance.SPEAKING POINTSInternational achievement data comes most recently from the Third International Mathematics and Science Study, show that our eighth grade students’ performance must improve to be on a par with the students in the high performing countries of the world.The performance of students at grade 4 was higher, relative to other countries in this international study, than U.S. 8th grade students. At both grade levels, the Canadian performance was better than that of the US.REFERENCESSee “Additional Resources” file: TIMSS Fact SheetThis information is taken from Pursuing Excellence: A Study of U.S. Eighth-Grade Mathematics and Science Achievement in International Context and the companion documents for grades 4 and 8.Third International Mathematics and Science Study:Grade 8, 1994–1995 and 200799
10 Comparisons: SES & Ethnicity Low SES have gaps - Level 2:79% of children with mothers with bachelor’s degree passedvs.32% of those whose mothers with less than a high school degreeEthnic groups70% of Asian and 66% of non-Hispanic white children passed42% of African American, 40% of Hispanic, 48% of Hawaiian Native or Pacific Islander, and 34% of American Indian or Alaska NativeAmerica’s kindergarteners enter with informal math competencies and 94% passed a simple level 1 test which was counting to 10 and recognizing numeral and shapes. 58% passed their level 2 test which included reading numerals, counting beyond 10, sequencing patterns and using non-standard units of length to compare objects.Various ethnic groups had significant differences.
11 International Comparisons There are cultural differences: High income and low income Chinese children performed higher than high income U.S. children.There are cultural differences as well as economic differences. In China when asked what they liked to do, children responded they liked playing with Legos, blocks, paper folding and playing outside. What do you think children in the U.S. said? Watch TV, play video games, computers, etc. Chinese families clear off the table after dinner and sit around and solve math problems for fun.
12 Using ResearchEarly mathematics research helps teachers be more effective and have more fun teaching math.
13 Core Mathematic Content Areas PreK - 8th grade Number and OperationsGeometryMeasurementAlgebraData Analysis & ProbabilityNational Council Teachers of Mathematics (NCTM)This organization has made an impact on the field. NCTM is the public voice for mathematics education. Their mission is “to provide leadership, vision, and professional development to support teachers in ensuring equitable mathematics instruction for all students.” (Taken from NCTM Web site.)NCTM Principles and Standards for School Mathematics organizes into five content areas for PreK - 8th grade.Numbers & Operations: Numbers can be used to tell us how many, describe order, and measure; they involve numerous relations and can be represented in various ways.Geometry: Geometry can be used to understand and represent the objects, directions, location in our work and the relationship between them. Geometric shapes can be described, analyzed and transformed.Measurement: Comparing and measuring can be used to specify “how much” of an attribute and object possess such as length or weight. Measures can be determined by repeating a unit or using a tool.Algebra: Patterns can be used to recognize relationships and can be extended to make generalizations.Data Analysis and Probability: Data analysis can be used to classify, represent and use information to ask and answer questions.
14 NCTM Focal Points Number and Operations Geometry Measurement National Council Teachers of MathematicsThis same group of experts developed “focal points” in an effort to solve the problem of vast, but shallow, curriculum. At each grade level, NCTM recommends a cluster of concepts, skills, and procedures that form the foundation for higher level mathematics. They identify the most important mathematical topics for each grade level, Pre-K through 8th grade.NCTM’s focal points are the key mathematical topics of instruction for each grade level Prek - 8th grade.Preschool focal points are number and operations, geometry and measurement.
15 NCTM Focal Points for Prekindergarten Handout 1Optional: This was from Region 7.Handout #1- NCTM Focal Points PrekindergartenIn the packet, the focal points for prekindergarten are included. Take a moment to study this sheet and, with the table, discuss focal points and how they play out in the classroom. Note what skills are included in each area.The categories data analysis, number and operations, and algebra are on the left hand side of the page. NCTM suggests these are not areas of study; rather they should be integrated into the three main areas, as it is taught.
16 Research Bites In the table groups, open the research bite. Share information and discuss how this informs participant’s work.There will be an opportunity to share insights with the large group.TRAINER’S NOTE: Materials will be on the table in an envelope.Activity-2 Sharing the ResearchAfter 5 -7 minutes of table talk, bring participants back to attention.Go to the next slide and ask which table/s had research bite #1.Ask someone from that table to read it and then show the following slide.
17 Research Bite #1 Infants are born with the ability to understand numerical ideas “Children possess and build mathematical competencies from their first year and keep on learning mathematical ideas throughout their preschool years.”Clements, D.H. & Sarama, J. “Creative Pathways to Math,” Scholastic Early Childhood Today Journal, 2003.Have table read research bite information and share insights.Research shows that many mathematics concepts, at least in the intuitive beginnings, are developed before school. Infants spontaneously use the ability to recognize and discriminate small numbers of objects.
18 For example…For example, when placing three pictures showing 1, 2, or 3 dots placed in front of a six-month-old child and three drumbeats are sounded, the child will most likely focus on the picture with three dots!Preschoolers possess informal mathematical abilities (i.e. develop number and geometry abilities, counting objects, making shapes, and use math knowledge every day). Educators must nurture these informal math abilities, or risk missing opportunities to support children in the early stages of their mathematical development.
19 Research Bite # 2“High quality teaching in mathematics is about challenge and joy, not imposition and pressure. Math instruction includes providing a lot of unit blocks, along with loads of time to use them. It invites children to experience mathematics as they play in, describe, and think about their world.”Clements, D.H., “Mathematics in the Preschool,” Teaching Children, 2001.By observing children’s play, teachers can find opportunities for mathematical instruction.Provide books illustrating different block arrangements.Post pictures in the block center.Offer different objects for children to measure when they are comparing sizes.Teachers can support child by entering into a conversation to discuss and clarify; i.e., when two children talk about having the biggest building teacher says, “You have a very tall building and Chris’s building seems to be very wide. It is big in a different way.”Offer an “I wonder” question, such as “I wonder if you can make a building as tall as your block building using the Duplo blocks,” or “I wonder if the building is as tall as the yardstick.”
20 Research Bite #3“Teachers should provide time, materials, and support for children to engage in play, to nourish their interest in exploration and manipulation of mathematical ideas.”National Association for the Education of Young Children (NAEYC) & National Council of Teachers of Mathematics (NCTM) Early Childhood Mathematics: Promoting Good Beginnings, A joint position statement of NAEYC & NCTM, Washington D.C.TRAINERS NOTE: Provide this joint statement if desired.In 2002, NAEYC published a joint position statement with the National Council for Teachers of Mathematics, entitled Early Childhood Mathematics: Promoting Good Beginnings. This joint statement was based on the ongoing research. The statement is very clear that mathematics instruction takes place in play. Additionally, effective mathematics teaching requires an understanding of what children know and need to learn, and challenges and supports them to learn it.
21 Research Bite #4“Preschool children explore a variety of mathematical ideas during play including comparison, estimation, patterns, symmetry, and spatial relationships.”K.-H., Seo. “What Children’s Play Tells Us about Teaching Mathematics,” Young Children, January, 2003.The joint position statement of NAEYC and NCTM points out, “Play does not guarantee mathematical development but it offers rich possibilities and to realize these possibilities teachers should provide not only ample time and materials, but also support for children to explore mathematical ideas during play.”We will look at how teachers can support children by asking questions and other strategies.
22 Research Bite #5Find the mathematics in, and develop mathematics from, children’s activity.Help children extend and “mathematize” every day activities from building blocks, art, songs, and puzzles.Create invitations to mathematical activities based on children’s experiences and interests.Clements & Sarama, Building Blocks Real Math Pre K Math Program, 2002.These are the basic guidelines from Dr. Clements’ program:Present real-life mathematical problems - Children need to see how math is used in their daily lives.Build on intuitive, informal strategies - As children play and say things about their play, teachers need to find the teachable moment and add mathematics to the moment. Remember the block play example?Noticing and using math vocabularyIntegrate concepts and skills - Throughout the day, create opportunities for all math strands to be presented through arranging our environments, the activities we plan, and what children do on their own.Practice at the problem-solving level - Takes math to a deeper level than just counting or naming shapes. Ask children, “How did you solve that?”
23 How we teach How we assess About Children DRDP-R California Preschool Learning Foundations,Volume 1PrekGuidelinesPreschoolFramework,Volume 1Now we will look at:About Children (based on the foundations)How we teach (The Prek Guidelines provide five guidelines for math in quality programs.)The Preschool Curriculum Framework is currently being developed and will be a critical tool for designing and implementing an effective learning program. It will explain how to use the foundations to teach and set up the environment. CDD does not prescribe or develop a curriculum for preschools. That is a local decision.DRDP-R determines how to assess what children have learned when mathematical invitations are provided in the classroom through planned activities and unstructured play.Early Childhood Environmental Rating Scales (ECERS) helps assess environments to insure that teachers create invitations for children to have mathematical experience through materials provided and adult interactions.ECERS-R
24 California Preschool Learning Foundations Handout 2AHandout #2A - The handout can be used as a learning aid.Highlight what is included in the handout and give participants a few minutes to read.The purposeThe definition - foundations define the knowledge, skills, and understandings that children can acquire at each age level with the support of quality programs.The age level informationWhat foundations are and what they are not24
25 California’s Preschool Learning Foundations - Mathematics Number SenseAlgebra and Functions (Classification and Patterning)MeasurementGeometryMathematical ReasoningThese are the five mathematic strands in the foundations for preschoolers.
26 Includes notes for children with disabilities Handout 2BMap of the FoundationsMathematicsStrandDomainAgeFoundationsSubstrandHandout #2BWalk participants through the structure of the map.There is a version of the map that has blank call outs.Includes notes for children with disabilities
27 Prekindergarten Learning & Development Guidelines Chapter 6 - Curriculum: Mathematics Learning and DevelopmentThe guidelines were published in 2000 by CDE press. (Hold up trainer’s copy.)This resource was published as a response to the need for clear guidelines about what constitutes high-quality programming across a broad spectrum of curriculum and practice in preschool.This book is intended as a resource for teachers “to support them in setting up environments, selecting appropriate materials, supporting children’s self-initiated play, and learning and planning and implementing teacher-guided learning activities.”Share the five guidelines from the mathematics section of Chapter 6, Curriculum, pages
28 Prekindergarten Learning & Development Guidelines for Mathematics Program develops and builds on children’s existing informal mathematical knowledge, recognizing that children enter preschool with different experiences in mathematics.Teacher-guided and child-initiated activities are integrated in a mathematically rich learning environment, using multiple instructional approaches.This slide presents sign language for numbers 1 & 2.1st Mathematics Guideline - When we want to know where our children are, we use the DRDP-R math measures to get a snapshot of a child’s development. We then design activities to accommodate children’s individual needs.2nd Mathematics Guideline - Effective programs include a balance of teacher-guided and child -initiated activities. Teachers are continually aware of mathematical moments and windows of opportunity in all areas throughout the day, not just during “math” time. We must create mathematical invitations to children through the set up of the environment, the “mathematizing” of teacher vocabulary and the activities we plan and provide. In language and literacy, we talk about activities being “playful and purposeful.” The same holds true for mathematics.
29 Prekindergarten Learning & Development Guidelines for Mathematics The program implements a mathematics curriculum that lays the foundation for children’s success in mathematics in elementary school.The program identifies clear, age appropriate goals for mathematics learning and development.The program establishes a partnership with parents and other caregivers in preparing children for mathematics learning.Sign language for numbers 3-5.3rd Mathematics Guideline - All children come to preschool with some degree of informal mathematical ability. The purpose of a strong preschool mathematics program should be to enhance and strengthen children’s informal mathematical abilities.4th Mathematics Guideline - Programs should have stated program goals for mathematics and staff should monitor implementation of the mathematics goals and revise them if necessary.5th Mathematics Guideline - Programs provide parents with family education classes (hold a family math night where activities are set up for adults to participate as a way of introducing the importance of mathematics), newsletters, and other forms of communication on how they can support preschool mathematics.
30 Prekindergarten Learning & Development Guidelines Children singing with teacher..one potato, two potato, three potato four.(Replace this PC version of the movie clip with the Quicktime version if running on a Mac.)
31 Early Childhood Environmental Rating Scales Revised (ECERS-R) Handout 3Early Childhood Environmental Rating Scales Revised (ECERS-R)Item 26 - Math/Number:Different types of materials for math/number help children to experience counting, measuring, comparing quantities, recognizing shapes, and become familiar with the written number.Handout #3 This handout is intended as a reference for participants to see that the ECERS addresses mathematics in the classroom environment.The ECERS-R is part of the Desired Results System. It assesses the environment to ensure materials, activities, and interactions are appropriate and well-planned for children.There is an item that specifically addresses mathematics.TRAINERS NOTE: This comes from the Clarification Notes on page 53 in the ECERS-R booklet.
32 Desired Results Developmental Profile-Revised (DRDP-R) Measures 22 & 23DRDP-RAn observational assessment instrumentDevelopmental continuumFor all children, including English- language learners and children with disabilitiesMeasures 22 & 23 are in folders on the table.The DRDP-R is a valid and reliable teacher observation tool that is on a developmental continuum. Teachers use it to measure children’s progress. Teachers use the DRDP to assess and document what children have actually learned.
33 Desired Result 2: Children are effective learners Indicator: MATH-Preschoolers demonstrate competence in real-life mathematical conceptsMeasure 22: Number sense: Understands quantity and countingMeasure 23: Number sense: Math operationsMeasure 24: ShapesMeasure 25: TimeMeasure 26: ClassificationMeasure 27: MeasurementMeasure 28: PatterningThese are all the math measures- Read Slide:Remember for preschool the NCTM focal points are Number & Operations and Geometry which are assessed by measures 22, 23, & 24.
34 Number and OperationsNow we will look at Number and Operations which is one of the most important math strands.NCTM uses the term Number and Operations and in the Preschool Learning Foundations the term Number Sense is used. The terms are different but what is included in these areas is similar.
35 Number and Operations Handout 4 Handout #4 –NOTE: WE ARE SEEKING PERMISSION FOR THIS HANDOUT. IF NOT RECEIVED, SLIDE WILL BE OMITTED.This graphic comes from the book Engaging Young Children in Mathematics. Trainers may consider using this slide in another spot in the presentation and coming back to it.TRAINERS NOTE: This is a handout in the folder. Decide how much time is required to cover the handout. Participants can discuss at the table, or use it as a reference. Ask them, “What would counting look like in the classroom?“ or “What about comparing and ordering?” etc. These questions will help participants realize that they already know some things about this information.Number and Operations is made up of much more than counting.Clements, D.H., Engaging Young Children In Mathematics, Lawrence Erlbaum Associates, 2004.
36 Number and Operations in Play Counting:Recognizing quantities by sight without actually counting objectsReading numbersPutting objects together or taking objects apartAdding or taking away numbersSeo, K.-H. “What Children’s Play Tells Us about Teaching Mathematics,” Young Children, January, 2003.This slide describes what we teachers would see children doing in classrooms in their play.TRAINERS NOTE: Participants may have discussed some of these examples during the previous slide.Teachers often spend a lot of time on counting and numeral recognition. It is important to distinguish numeral recognition as recognizing the symbol used to represent a number. Additionally, teachers need to understand that there is more to know about Number and Operations.
37 Activities for Multiple Goals Make and imagine small collections of items nonverballyCount by ones to tenKnow the last counting words tells “how many”Count out (produce a collection)Subitize - quickly “see” and label with a numberIdentify whether collections are the “same” number or which is “more” visuallyClements, D. H., & Sarama, J., Building Blocks Real Math PreK Curriculum, 2007.When teachers know the developmental sequence for counting, then they can plan activities that address multiple goals which will meet the needs of diverse learners no matter where they are on the counting developmental continuum.1st bullet: For example, a child who rolls a number cube and says “1”, takes one without counting.
38 Recognition of Small Number Early number is not counting, but recognition of numberRecognition of number is not numerals but the understanding of discreet quantitiesEarly number is not counting, but recognition of number. Remember the infants with the three dots where there is another example from the research done by by Starkey, Spelke & Gelman, This comes from page 17, Counting section in Engaging Young Children in Mathematics by Dr. Doug Clements.Also, recognition of number is misinterpreted by many as meaning reading numerals “3” or “2.” Instead, it means seeing a small group of objects and understanding how many in that group. (NOTE: Apprehend means to understand – Doug Clements)
39 Clements, D.H., Engaging Young Children In Mathematics, 2004, pg. 17. When six-month-old infants are shown a series of pictures that always depict three (3) objects they eventually get bored, but… CLICK TO NEXT SLIDE.Clements, D.H., Engaging Young Children In Mathematics, 2004, pg. 17.
40 Clements, D.H., Engaging Young Children in Mathematics, 2004, pg. 17. If the configuration of the objects depicted in the pictures is then changed from three (3) to two (2) or four (4), infants notice the change and become interested again. They can “see” small configurations of objects nonverbally. This is called subitizing. By 18 months of age, infants can notice which of two collections contains more objects (Cooper 1984). This provides and early perceptual basis for number, but it is not yet “number knowledge.”Clements, D.H., Engaging Young Children in Mathematics, 2004, pg. 17.
41 Recognition of Small Number Simple, but continuous, intervention makes a big difference - Hannula’s 3- year-oldsLarge differences in spontaneous recognition of number1st bullet: This was a study where a teacher decided to talk differently with the children. Instead of adult saying, “You drank a lot of milk,” she said, “You drank three cups of milk.” Or instead of ”You picked up so many blocks,” she said, “You picked up five blocks.”2nd bullet: Doing this will make a large difference in spontaneous recognition of number.
42 Recognition of Small Number Teachers can make huge differences by providing thousands of spontaneous experiences!Another example of ways teachers can make a difference is by giving a purpose to counting. For example, ask the child, “How many blocks do I have in my hand? Help me count.” Move object from hand to table deliberately. “one…two…three…four” then ask, “How many do we have?”
43 How many triangles did you see? That’s Subitizing!
44 Learning Trajectory for Subitizing Small collection namerMaker of small collectionsPerceptual subitizer to 4Perceptual subitizer to 5Perceptual subitizer is a child that can quickly determine the group has four without using any math processes such as counting or adding 2 +2.The next step would be for a child to use the pattern of the group such as a domino pattern to determine the number. This would be conceptual subitizer.This is in the pink handout on Subitizing. It has some activities you can use right away. Take a look at that. So now I would like to show you a video of a teacher playing with subitizing. I had explained what it was and that it was important skill for children to know. So she decided to try it. Surprisingly most of the children could not subitize larger than 2 – 3 objects. Interesting.Remember she has never done this before.44
45 Simple, but Continuous, Intervention Makes a Difference! Now we are going to watch a teacher interacting with children in the sandbox and see what she does to support the children’s mathematical thinking and learning.(Replace this PC version of the movie clip with the Quicktime version if running off a Mac)
46 Developmental Sequence of Counting Activity # minutesEach table has an envelope containing the developmental sequence cards.2.Participants take out the cards and as a group discuss the content and lay them out in a developmental sequence.3.You can do a call and response if you like or ask a table to volunteer what they had for the first level, then ask another table what they had for the second level etc.4.The answers are provided on slides 44 & 45.
47 Please Close Folders and Notes Group ActivityPlease Close Folders and NotesTake cards from envelopeAs a group, place cards in a developmental sequence
48 Developmental Sequence of Counting Saying number words in sequence. May omit some numbers when reciting the number wordsCounts a small set of objects (five or six) but may not have one-to-one correspondenceMay count correctly a larger set of objects (about ten) by keeping track of counted and uncounted objectsUnderstands that the number name of the last objects counted represents the total number of objects in the groupResource: California DRAFT Preschool Mathematics FrameworkExamples:1st bullet: Child’s counting may go like this: one, two, three, five, six, seven, ten.2nd bullet: Child may point to more than one object when saying one number word, or say a number word without pointing to an object.3rd bullet: Child may point or move objects while counting.4th bullet: This concept is called cardinality. For example, child repeats the last number when asked “How many?”
49 Developmental Sequence of Counting Knows to say the number words one to ten in the correct order, but is still learning the sequence between ten and twentyCreates set with a certain number of objectsKnows to say the number words up to twenty correctlyResource: California DRAFT Preschool Mathematics FrameworkExamples:1st bullet: Child may omit some teen words like 13, 14, 17, 18, 19.2nd bullet: When asked, “Give me two spoons,” the child counts out two spoons from the drawer of silverware.
51 Levels of Math Operations Comparing & Ordering - Can tell which group has more or lessAdding to/Taking away - Solve problems with three to five objectsComposing and Decomposing - Can develop the ability to recognize that three and two are “hiding inside” fiveGrouping - Can count groups of objectsClements, D.H., Engaging Young Children in Mathematics, 2004, pgWhen looking at math operations, consider these levels (reference pg , Engaging Young Children in Mathematics):Give an example of each:Comparing and Ordering - By age four, children use matching to create a collection equal to one that has already been constructed (Piaget & Szeminska, 1952)Adding to/Taking away to solve problems - For example, a child might say, “We need two more plates ‘cuz there are only two on the table and four of us.”Composing and decomposing - Example: children can see that five is 3+2, 4+1, 5+0, etc.Grouping - Child can count small group of objects. It is the operation (process) of combining objects into sets each having the same number of objects. Comparing - Ask if a child can tell you which plate has more crackers.
52 Meet JameyTRAINER’S NOTE: Show the preschool observation clip of James using magnet numbers on the white board. The clip is about two minutes long.Find the clip in the Preschool observation vignettes - second clip.When the video clip ends ask participants to review the number sense foundations and identify what they observed as it relates to the foundations. (3-5 min.)
53 Handouts 5 & Measures 22 & 23 Activity 4 - 30 minutes Handout # 5 Let’s PlayIn folder on the table DRDP-R Measures 22 & 23Number Sense foundationsNotify participants that they will have 15 minutes to complete this activity.Distribute one math game to each table.Participants follow the directions to their game.After playing the game, talk about the foundations, developmental progression for counting, DRDP-R measures involved in the game. Use the worksheet to structure the discussion.Participants will write a goal for a child in their class. For example, “Maria will count to five using one-to-one correspondence.”Later participants will be asked to present the activity to the group. Remind them to be ready to share the math concepts in the activity.
54 Let’s Play Follow the directions to the game. After playing the game, talk about the foundations, developmental sequence, DRDP-R measures involved in the game. Use the handout to structure the discussion.Write a goal for a child in your class. For example, “Maria will count to five using one-to-one correspondence.”Later participants will be asked to present the activity to the group. Be ready to share the math concepts in the activity.Each table has a Ziploc bag with materials.Open the bag and share the materials with table mates.There is a direction sheet to follow. Play with the materials for 10 minutes.Discuss with your table mates the math concepts that are involved in the activity.Are there any modifications to make and why?The debrief can take place later in the presentation.54
55 Teachers are the key! Read slide. Because teachers are the key, they need to know the importance of mathematics, and how to provide developmentally appropriate materials, activities and interaction to support ALL learners.
56 The Importance of Questions Mathematics is about thinking, not just doing something with manipulatives.How did you know?Why did you do it that way?How did you figure that out?What else would work like this?What would happen if?Handout 6Handout #6 OptionalTeachers can encourage children to reflect on their play and construct mathematical meaning from their experiences by asking questions such as…Refer to list.Teachers can challenge children to go beyond what they already know, and help them construct more advanced mathematical understanding.These are great questions for literacy language as well.
57 The Power of Asking “Why?” and “How did you know?” “You present problems, and they figure out what to do. Then you ask what process they used. I’m amazed they learned to! They’ll use this knowledge to answer science questions. They really do critical thinking. Asking ‘How do you know?’ Starting at Prek is very powerful!”- Anne, preschool teacher in the TRIAD research projectClements & Sarama - U.S. Dept. of Education's IES-funded TRIAD research projectTRAINER’S NOTE: This is a story from a teacher named Anne shared during a session with Dr. Clements. The quote speaks to the importance of asking “why” and “how did you know” questions.
58 In Classrooms that Promote Number Sense, Teachers Will… Activity 5 - Charting: What Teachers do.Directions are also on the next slide on the screen as well as the notes section for trainers.
59 What Teachers Do to Support Number Sense in the Classroom… Move to the assigned chart.Take five minutes to record ideas.Move to next chart when signaled.Add ideas to next chart and continue on at the signal.At the last chart, circle the top three ideas.Share top three ideas with group.Place charts around the room and place one of the signs next to each chart or write the information directly on the chart. (20 minutes)Assign tables to a chart and give participants five minutes to record ideas. Move to next chart. Continue until all have had a chance to record ideas. Participants end at their chart and then circle the top three ideas.Explain the charts and give an example for each chart.Interaction: Teachers talk with individuals and groups - For example, “We don’t have enough spoons for everyone. How many more spoons do we need, so everyone has one?” Ask why and how did child know that. THERE IS AN ADDITONAL SIGN ABOUT RESEARCH FOR THIS CHART.Daily Routines - Arrival and departure/transitions/mealtime/rest time/ clean upEnvironment - What kinds of materials are in the various interest areas, including outside?Activity - Teacher directed/child initiated including small group and large group activitiesSmall group/Individual instruction – Dr. Clements emphasizes that the research says this is the most important. TRAINERS NOTE: Highlight this when presenting. It is also on the sign by the charts.
60 Activities for Multiple Goals Ability to recognize and duplicate a small collection nonverballyCount by ones to tenKnow the last counting words tells “how many”Count out (produce a collection)Subitize - quickly “see” and label with a numberIdentify whether collections are the “same” number or which is “more” visuallyClements, D. H., & Sarama, J., Building Blocks Real Math PreK Curriculum, 2003.Once teachers know the learning trajectory, plan activities that address multiple goals which will meet the needs of diverse learners.1st bullet: For example a child rolls a number cube and says, “one”, takes one object without counting.
62 Tips for Supporting ALL Learners Handout 7Tips for Supporting ALL LearnersAssess what the child knows – scaffold.Slow down! Emphasize accuracy with counting.Guide the child’s hand while counting, if they are working on one-to-one correspondence.Repeated practice.Provide “wait” time.Handout #72nd bullet - Children who make a mistake counting are 20% more accurate the next time they try if they slow down.
63 Tips for Supporting ALL Learners Make it concrete! Count “real” objects.Ask the child to make a verbal plan. “Let’s count them by starting at the top.”Move items as they are counted.Involve the child’s whole body as much as possible.Simplify vocabulary and directions
64 Tips for Supporting English Learners Use concise language - speak clearlyUse oral descriptions when talking about conceptsModel and act out - Total Physical Response (TPR)Consider stages of language developmentThe previous strategies will work for English learners, and here are additional strategies to support children who are English Learners.1st bullet: When children are learning English, especially when working with teachers who don’t speak the home language of the child, teachers need to provide a time for the child to process and respond.4th: Teachers also need to consider the child’s stage of development. Level of responses should be according to level of language.
65 Let’s Play - Part 2 Now revisit your game and consider the following: How might you document children’s growth and development?How might you adapt this game for children with disabilities?What are some strategies for English Learners?Refer to the back side of your Let’s Play worksheet tofacilitate your discussion and record your ideas.Provide participants with an additional 15 minutes to complete part 2 of their Let’s Play worksheet.Option: Provide participants with a specific child scenario. You can use the scenarios from the other modules if you want.65
66 Handout 5 Activity takes 10 min. Participants complete the rest of the worksheet in their table groups.
67 In Classrooms That Promote Number Sense, You Will See Children… Activity 6- What Children Do…Participants are now going to consider what children do in the classroom that promotes number sense and use the two DRDP-R measures that assess number sense.Process:In table groups, participants review the measures and the number sense foundations.Brainstorm what children might be doing at each developmental level.Write one idea for each developmental level on the measure pages provided.Ask participants to consider: parts of the daily routine, interactions with adults, independent play or with a peer, and the various interest areas including outside time.
68 DRDP-R Measures 22 and 23 Handout 8 Review the measures at the table. Brainstorm what children might do at each developmental level.Record ideas on the measure pages provided.How might you document children’s learning?Handout #8Measures 22 and 23 are the two Number Sense measures on the DRDP-R.Ask participants to consider:Parts of the daily routineInteractions with adultsWhat children will do on their own or with a peer in the various interest areas, including outside
69 Taking It Back to the Classroom Consider:The researchThe activitiesThe games played and displaysConversationsHandoutsIdeas that were new or importantHandout 9Handout #9Give participants a few minutes to write down observations they want to remember.Encourage them to post this handout in the classroom or somewhere regularly visible.
70 Five Little Seashells Handout 10 Five little seashells sleeping on the shore.(Hold up 5 fingers, bending down one for each verse)Swish! went a big wave, and then there were four.(Move arms, palms up, to make the waves)Four little seashells quiet as can be.Swish! went a big wave, and then there were three.Three little seashells pearly and new.Swish! went a big wave, and then there were two.Two little seashells having great fun.Swish! went a big wave, and then there was one.One little seashell lying in the sun.Swish! went a big wave, and then there were none.Five little seashells gone out to sea.(Point out to sea)Handout #10This is the optional closing activity before evaluations.
71 CDE Web SiteAt the Web address, the underlined Preschool Learning Foundations link leads to the 192 page publication providing easy access to the chapters and sections.The Appendix, on pages , provides a summary list of the foundations.Frequently Asked Questions (FAQ) are posted on the Web site. Questions can be sent toTRAINER’S NOTE:Share the information in the first bullet.Provide the Web handout.Pages in the Appendix, provide a summary list of the foundations, excluding the examples and other material. (Hold one up to show it is only a few pages.)Remind participants of the preschool foundation’s address on the previous page and on the CDE Web site.Many questions were asked during the extensive public review process. CDE has developed a set of Frequently Asked Questions (FAQ) to provide the answers to those questions. The FAQ document is a living document. Over time, new questions will be added and existing responses may be clarified as needed.
72 The entire document is online at the California Department of Education Web site. Visitors to the site can look at a specific section or download the entire document. This slide shows the Web page’s design. The Appendix contains a summary list of the foundations, excluding the examples and other material. The foundations are also available for purchase through CDE press.
73 To Purchase the Preschool Learning Foundations Book The Preschool Learning Foundations publication is available for purchase from the CDE Press for $19.95 plus shipping and handling.Ordering information can be found at the CDE Web site or by callingRemind participants of the handout or brochure in the folder with ordering information.
74 To Purchase PEL GuideNOW in Spanish!Preschool English Learners: Principles and Practices to Promote Language, Literacy and Learning publication is available for purchase from the CDE Press for $15.95.Ordering information can be found at the CDE Web site or by callingAppendix A is translated into Chinese, Hmong, Korean, Spanish, Tagalog, and Vietnamese, and is available on the CDE Web site.
75 Did you like math when you were in school? Take a StandDid you like math when you were in school?Have participants revisit this actitivty.Ask participants how they feel about math now.Have them think about their experiences with math content, teachers and past experiences.Place numbers 1 through 5 up high on the wall around the room with as much space as possible in between numbers so participants can have focused conversations and not be distracted by other groups.Ask participants to go stand by the number that best describes their experiences with math.For example:5 could be an A student; excelled, loved it, overall positive experiences etc. Add your own verbiage to each number.4 could be a B student; probably did well, did not love it, was not bad at it3 could be a C student mediocre experiences that left them lukewarm2 could be a D student that did poor1 would be an F student hated it, every aspect of it didn’t get it, tried again and again negative experiences overall.Once in their groups have participants share why they chose this number and discuss feelings centered around content, teachers and past experiences.After discussion reconvene and keep people in their like groups, then ask a participant from each group to share out a collective summary of their groups experiences. Some people may want to have more than one person share. Facilitator can decide how to structure this portion.Summary: Have fun with this exercise these are just guidelines add your own twist to setting it up. Once participants begin to talk with others in their group they will realize they have had similar experiences and will freely share among each other.What we found is that the discussion will bring up powerful emotional experiences around math. Administrators and teacher leaders need to acknowledge their feelings in order to become comfortable with understanding and teaching math to preschool teachers. This exercise can also be conducted at teacher professional development sessions.Enjoy! This activity was fun and very therapeutic for CPIN participants.
76 CPIN - RegionContact info added here.You can announce your next event or ???
77 Please take a minute and complete the evaluation before leaving. Thank You for Coming!Please take a minute and complete the evaluation before leaving.Regional leads can customize this page.
79 Grade Yourself in MathOn a scale of “1” (A+) to “5” (F), how did you perform in math in school?In your math group, talk about experiences with Math in school.Be ready to share with the whole group.Activity #1OPTIONAL ICEBREAKER: This icebreaker comes from region 7. Attitudes around math affect how math is taught. If teachers feel incompetent in math, they tend to shy aware from those activities. Remember that attitudes show through to students even as young as preschool. For teachers who will do professional development around math, make sure to have a discussion about participants’ attitudes.79
80 The following slides are for the debrief of activities and the Make and Take directions. Place them where you think they fit best.
81 How Did the Activity Support Mathematical Thinking and the Foundations? Share your experience with the activity & materials used.What foundations did the activity address?Are there any changes or additions to make?Debrief math activities at the tables. Do a “show and tell” of materials and have participants take the foundations and apply them to the activities.Debrief conducted near the end of the day may require a few minutes to consider what math concepts are addressed. Participants might know more now. Encourage them to use the graphic handout to help think about the concepts.Complete math activities at the same time or intersperse them throughout the session.Remind participants that the activity directions are in the handouts. Consider stapling them as a separate handout, and distribute at the end.
82 Math Make and TakeThis is a quick make and take activity to place into the presentation where desired.Display the cards so participants can see the dots go in a row. Explain that developmentally children can “see” (subitze) easier if the dots are in a row. Later, participants can make sets where the dots are in a domino configuration to change the level.There are suggestions on the handout for different levels.Materials needed: 5 blank cards per person, 15 sticky dots, and a sandwich bag in which to place individual materials.TRAINER’S NOTE: The activity takes less than 5 minutes to make. Trainers may want to give participants time to play the game with a partner. Provide direction sheet to the game in the baggie as well.
83 How to Play “Peace” Each player has 5 shuffled cards. Place cards face down in a pile in front of you.Turn one card over and partner turns one over.Whoever has the card with the most dots takes the pair.Continue playing until someone has all the cards.This game is played with a partner. Tell participants they may have played the game “War,” but we call it “Peace.”OPTONAL ACTIVITY: This activity could also be used with the Let’s Play worksheet.Participants would play Peace and complete the worksheet. This would be a quick practice before playing the math game and completing the worksheet.