Academic presentation for college course (paper and pencil design)

Math Content Part 1Ernestine Saville-BrockJanuary 20131Urgency in the teaching of mathematicsThe United States suffers from innumeracy in its general population, math avoidance among high school students, and 50% failure among college calculus students (Reuben Hersh ) Too many children choose their college major and their career paths based upon how many math courses they need to take. (Boaler, 2008)Urgency in the classroomTeachers need to see themselves as mathematicians. If we foster environments in which teachers can begin to see mathematics as mathematizing-as constructing mathematical meaning in their lived world-they will be better able to facilitate the journey for the young mathematicians with whom they work. (Fosnot)Good afternoon, CoachesGood afternoon, Instructional Coaches. We are going to explore developing early number concepts, number sense, and an introduction to early addition and subtraction.

Let take a minute and greet at least three others in the room.

Activity- Quick Peek Activity

Quick Peek- What visual clues did you rely on (triangular, square, or identical sets)?Is this activity similar to any you or your teachers use?4Agenda for the sessionAfternoon MeetingPattern visualizationQuick Peek GameNumber Sense and CountingQuickwriteNumber RelationshipsCounting StoriesModels MatterTen FramesCrazy Mixed Up NumbersMath RacksEarly addition and subtraction

Objectives for instruction and expected results and/or skills developed from learning. 5How many dots are there?6How many dots are there?*07/16/96*##SubitizingSubitizingThe ability to just see itNaming amounts based on patterns not countingChildren must explore quantity before they can countA fundamental skill in the development of students understanding of number- Baroody, 1987A complex skill that needs to be developed and practiced through experiences with patterned setsAids students in developing more sophisticated and efficient strategies for counting and learning number combinations.

Relative vocabulary list. 8Brain research effecting teaching and learning (Sousa)Creating and using conceptual subitizing patterns help young students develop the abstract number and arithmetic strategies they will need to master counting.

Information is most likely to be stored if it makes sense and has meaningBrain Research contdToo often, mathematics instruction focuses on skills, knowledge and performance but spends little time on reasoning and deep understandingJust as phonemic awareness is a prerequisite to learning phonics and becoming a successful reader, developing number sense is a prerequisite for succeeding in mathematicsThe Power of Groups- Chunk the DotsSpeedy Pictureswww.fi.uu.nl/rekenweb/en

A list of procedures and steps, or a lecture slide with media.11Number SenseTo achieve in mathematics, students must acquire a good sense of numbers early in their academic career. Bradley S. Witzel

Quickwrite- What is number sense? Share ideasOf the 22 kindergarten CCSS, 14 can be directly linked to elements of number sense. Conclusion to course, lecture, et al. 12Discussion of Number SenseHowden, 1989Good intuition about numbers and their relationshipsDevelops gradually as a result of exploring numbers, visualizing them in a variety of contexts, and relating them in ways that re not limited by traditional algorithmsGersten and Chard, 1999A childs fluidity and flexibility with numbers, the sense of what numbers mean and an ability to perform mental mathematics and to look at the world and make comparsionsAn opportunity for questions and discussions.13Number RelationshipsSpatial visualizationOne and two more; one or two lessBenchmarks of 5 and 10Part-part-whole relationship

Spatial visualization- visualizing a patternOne more or less more than the ability to count on 2 or back 2 Knowing that 7 is 1 more than 6 and 2 less than nineBenchmarks- 5 and 10Part-part-whole- to conceptualize a number as being made up of 2 or more parts is an important relationship that can be developed by numbers. 7 can be thought as a set of 3 and a set of 4 or a set of 5 and a set of 2CCSS 1.OA.3- apply properties of operations as strategies to add and subtract.Break apart numbers- 7 + 8 why it is useful?How can I break up 7 + 8 to make it easier?14Spatial Relationships recognizing how many without counting by seeing the visual pattern. One & Two More, One & Two Less this is not the ability to count on two or count back two, but instead knowing which numbers are one more or two less than any given number. Benchmarks of 5 and 10 since 10 plays such an important role in our number system (and two 5s make up 10), students must know how numbers relate to 5 and 10. Part-Part-Whole to conceptualize a number as being made up of two or more parts is the most important relationship to develop.Van De Walle, 2006How do children develop skill with counting and number concepts?Key IdeasCounting is the basis of childrens ability to add and subtractDeveloping strategies for keeping track of the objects being counted is an important landmark in student learningThe structure of a ten frame can help students develop an understanding of number conceptsCounting and Number SenseCounting is a skill that is easy to take for granted, but it is the foundation for understanding number and learning addition and subtraction.

Two areas of developmentFluency with verbal countingAbility to accurately and reliably count collections of objectsSkill with the first doesnt guarantee skill with the second Activity- Counting StoriesRead the student vignettes Discuss the vignettes with a partnerSort the vignettes into two piles:Students with counting fluencyStudents who have difficulty countingConsider the following questions:Does the student have fluency with verbal counting? What is the evidence?Does the student have the ability to accurately count each collection of objects? What is the evidence?

CountingBrainstorm a list of tasks or activities that might help students who have not yet mastered counting tasks?

Students develop their understanding of counting and number primarily through practice. Teachers and researchers have found that providing a variety of meaningful counting experiences is the key to helping children develop their counting abilities. Developing Number ConceptsCounting builds to a mental visualization of a number line

K

K-2 use number paths.

Activity- Circle 77 is a quantity. Circle 1-7. 20Models MatterTo build strong number sense is to introduce models with structures that can develop an understanding of number relationships. And to emphasize important numerical understandings such as organizing with ten. One model is a ten frame.

Ten frame may help students to see numbers in relation to 5 and 10, which is a helpful strategy for gaining fluency with addition and subtraction.

Ten FramesVan de Walle (2009) suggests that teachers should provide students with practice using ten frames rather than teaching procedures.

Crazy, Mixed-up NumbersAll students make their ten frames show the same number.Teacher calls out random numbers between 0-10After each number, children change their ten frames to show the new number.

ANIMALS ON BOARD by Stuart Murphy

More than just a problem solving-How did you decide how to change the ten frame?Before changing the ten frame, how many counters need to be added or removed?25Add and subtract within 20, demonstrating fluency for addition and subtraction within 10. Use strategies such as counting on; making ten (e.g., 8 + 6 = 8 + 2 + 4 = 10 + 4 = 14); decomposing a number leading to a ten (e.g., 13 4 = 13 3 1 = 10 1 = 9); using the relationship between addition and subtraction (e.g., knowing that 8 + 4 = 12, one knows 12 8 = 4); and creating equivalent but easier or known sums (e.g., adding 6 + 7 by creating the known equivalent 6 + 6 + 1 = 12 + 1 = 13).- CCSSFluencyAccuracy- speaking a foreign languageFlexibility- when cooking not having an ingredient but can substituteEfficiently- when driving, the best route26Strategies NOT Memorization

- Counting on- Making ten- Decomposing a number leading to a ten- Using the relationship between addition and subtraction- Creating equivalent but easier or known sumsCounting vs. Collections"Tasks that encourage students to think in collections provide rich opportunities for them to construct abstract mathematical relationships and become powerful problem solvers." --Wheatley & Reynolds, 2010

The math rackis a tool developed at the Freudenthal Institute in the Netherlands by Adrian Treffers to support the natural mathematical development of childrenin Dutch means calculating frame or arithmetic rack.looks like a counting frame but is designed to move children away from counting each bead.looks like an abacus but it is not based on place value.*07/16/96*##Features of the math rackThe beads are red and white.There are two rows of beads.There are five red beads and five white beads on the top row, and the same on the bottom.There are ten beads total on the top row, and ten beads on the bottom row.There are ten red beads and ten white beads on the rack.There are twenty beads altogether.

*07/16/96*##How the MathRack Can HelpWhat do you notice? explore the tool and learn the built-in structure before you have them use the tool.Show me On one row, "Show me __." Have them show a certain number. Some may count one-by-one but the structure of the tool allows for more advanced strategies.Flash Forward show a certain number on the MathRack for a few seconds and have them determine which number was flashed.How the MathRack Can HelpShow me and Flash Forwardbuilds Spatial and Benchmarks of 5 & 10

Just One More show a certain number on the MathRack and have the kids build theirs to show "one more" or "two less."builds One/Two More or Less

Show Me using two rows, "Show me ___." builds Part-Part-Whole

Peek-a-Boo cover the right side of the MathRack, push some beads over so kids can see them and ask how many beads are hiding.builds Part-Part-Whole (is a Missing Part activity) and Benchmarks of 5 & 10Quick ImagesHow many beads are there?

How do you know?

*07/16/96*##How many beads?Read this side*07/16/96*##How many beads?*07/16/96*##How many beads? How do you know?*07/16/96*##How many beads? How do you know?*07/16/96*##How many beads? How do you know?*07/16/96*##How many beads? How do you know?*07/16/96*##Turn and talk What are all the possible ways children will figure out how many?*07/16/96*##Developing the landmark strategiesSubitizingUsing the 5-structureUsing the 10-structureCounting onDoubles and near-doublesCompensationSkip countingPart/whole*07/16/96*##Contexts for the MathRackmathematical meaning in their lived worldBunk bedsDouble-decker busBookshelves*07/16/96*##The Double-Decker Bus

*07/16/96*##Games with the MathRackHow many empty seats on top?

3 on top7 on top2 on top8 on top6 on top

Day 5

*07/16/96*##Games with the MathRackPassenger Pairs matching game:Moving from the bus story to a model of the context*07/16/96*##Games with the MathRackRack Pairs matching game:

Moving away from the context

*07/16/96*##Games with the MathRackBus Stops game

How many on the bus as it pulls away from the bus stop?How do you know?

+58*07/16/96*##Games using the MathRackBus Stops game

How many on the bus as it pulls away from the bus stop?How do you know?

- 4 11*07/16/96*##Fluency and FlexibilityFluency- efficient and correctFlexibility- multiple solution strategies determined by the problemFluency is the by-product of flexibility. Assessing fluency by occasionally using timed tests is acceptable. Using timed tests as an instructional tool to build fluency is ineffective, inefficient, and damaging to student learning.

Focus on RelationshipsWhen we focus on relationships, it helps give children flexibility when dealing with their basic facts and extending their knowledge to new task. When we build a childs number sense it promotes thinking instead of just computing.Show 8 + 7 on your MathRack

Write down an equation that represents how you determine the total number of beads shown

What relationships did you use?

Model what you did on your Number PathWrite an equation- jumped to the abstractCRA- must include the pictoricalShow the rack, then the number path, then the equation51

Show 15 - 9 on your MathRack

Write down an equation that represents how you determine the total number of beads left

What relationships did you use?

Model what you did on your Number Path

ContextThere are 8 girls on Kylies bunk bed, some are on the top bunk and some are on the bottom bunk. Use your MathRack to show me how many might be on the top and how many might be on the bottom.

Take AwayThere are 15 people on the double-decker bus. At the first stop, 7 people got off the bus. How many people are still on the bus?

Finding the DifferenceAt Sierra's slumber party 15 girls were playing on her bunk bed.Some of the girls went to get snacks. There are 7 girls still on the bunk beds. How many girls went to go get snacks?

Adding UpThere are 7 people on the double-decker bus. At the next stop, some people got on the bus. Now there are 15 people on the bus.Questions