Fostering Mathematical Excellence in Early Childhood Education

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Fostering Mathematical Excellence in Early
Childhood Education

• Dr. Susan Looney
• looneyconsulting_at_comcast.net
• www.looneymathconsulting.com

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PSSM

• Most students enter school confident in their own
  abilities, and they are curious and eager to
  learn more about numbers and mathematical
  objects. They make sense of the world by
  reasoning and problem solving, and teachers must
  recognize that young students can think in
  sophisticated ways. Young students are active,
  resourceful individuals who can construct,
  modify, and integrate ideas by interacting with
  the physical world and with peers and adults.
  They make connections that clarify and extend
  their knowledge, thus adding new meaning to past
  experiences. They learn by talking about what
  they are thinking and doing and by collaborating
  and sharing ideas.

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- Andrews and Trafton, 2002

• When children make sense of mathematics, they
  develop deep understanding of important ideas by
  making connections with their informal
  mathematical knowledge and making connections
  among mathematical ideas.
• Children surprise us by learning mathematics
  beyond our perceptions and they learn it better.

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The Handshake problem in Kindergarten?????

• Start with 4 or 5 students in a group. How many
  handshakes will there be if each person in your
  group shakes the hand of every person once?
• Tell who was in your group.
• How did you get your answer?
• Try to show your solution on paper.
• How many handshakes will there be if 1 more
  person joins your group?
• What if 2 more join? 3 more join?
• Do you see a pattern in the numbers? Describe the
  pattern.

????? Joey Kim Connor Trina





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Partnership for the 21st Century Skills

• An organization leading the way ..

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Rainbow

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Learning and Innovation Skills

• Creativity and innovation
• Critical thinking and problem solving
• Communication and collaboration

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Counting on Frank by Rod Clement
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Problem Solving Choose a problem to solve. Show
and tell how you arrived at your answer.

• 1. The boy calculates that 24 Franks (his pet
  dog) could fit into his bedroom. What if, in
  addition to these 24 Franks, 30 Franks could fit
  into the boys parents bedroom, 25 in the living
  room, 10 in the bathroom, and twenty in the
  kitchen? How many Franks in all would fit into
  the boys house?
•  
• If there were 24 Franks in the bedroom,
  how many ears would there be? How many Frank paws
  would there be?
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• 2. If the boy accidentally knocks 15 peas off
  his plate each night, how many peas would he
  knock off in one week? Two weeks? What about for
  the last 8 years as stated in the story?
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• 3. The tree in the boys yard grows 6 feet
  each year. If he had grown at the same speed, he
  would be almost 50 feet tall. How old is the boy
  in the story? How tall would you be if you grew
  at this rate?

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Characteristics of Sense-making Classrooms

• Children take ownership of task and problems,
  letting them use their own approaches and
  strategies / over time with sharing, immature
  strategies become more mature
• Sufficient time repeated opportunities
• Reflect and communicate
• Variety of tools available at all times
• Teachers role of observer when to let children
  work and when to provide info and closure /
  questions and genuine interest
• Respect for childrens ideas and belief that all
  children can learn mathematics

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How do I Teach Students to make connections in
their solutions?

• After students complete a task, brainstorm what
  types of connections they can go back and add to
  their work.
• After students complete a task, request that they
  make an eye noticed statement about their
  solution.
• Ask good questions and give rich tasks that will
  provide opportunity for students to make
  connections.
• Spotlight students who make connections, either
  by giving them a mathematics award, or just a pat
  on the back.
• Give students opportunities to self assess, then
  revise their work.

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Motivating Students to keep thinking .

• Focus on the outcome vs. the performance.
• You answered the question vs. good job
• You did it! You wrote that!
• What you did vs. how you did
• Check-in points You did half of the problems.
• Rationale Releases a brain chemical called
  dopamine which increases motivation to continue
  because the students feels good.

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Characteristics of Rich Tasks

• Open ended
• High demands
• Novel
• What else???

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Consider the two classrooms.

• What skills are necessary for the students to be
  successful in classroom A? Classroom B?
•  
• How would you describe the level of challenge in
  Classroom A? Classroom B?

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Classroom A

• Classroom A
•  
• Check of last nights homework
• Demonstration of how to add with regrouping
• Students work on 20 similar problems
• Teacher reminds students how to compute
• Assigns remainder of problems for homework plus
  word problems at the end of the page
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• Word problem Sarah and John went apple picking.
  Sarah picked 23 apples. John picked 38 apples.
  How many apples did they pick all together?
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Classroom B

•  
• Students enter room and are asked to immediately
  begin working in small groups on the task as
  described by the teacher. They will have the
  entire period to work on this task, and they may
  quietly get whatever paper, tools, or
  manipulatives they will need to complete the
  task.
• Task Collecting Shells
• Paul and Amy love to collect seashells. The first
  time they went to the beach they each found 5
  shells. The second time they went to the beach
  Paul found 6 shells and Amy found 4 shells. The
  third time they went to the beach Paul found 15
  shells and Amy found 17 shells. Amy said that she
  found the most shells while Paul argued that they
  both found the same amount of shells. Who was
  right? Show how you know.
•  
• During class the teacher questions various
  groups, provides hints about how to proceed, but
  never shows students exactly how to go about
  solving the problem.
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• At the end of the period, none of the groups are
  finished with the task, but they are all actively
  engaged in problem solving.

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Depth of Knowledge (DOK)

• Level 1 (recall) conduct basic math
  calculations perform routine procedures
• Level 2 (skill / concept) solve routine
  multiple-step problems organize, represent, and
  interpret data
• Level 3 (strategic thinking) apply a concept in
  other contexts, compare, non-routine problems,
  strategic thinking, logical arguments
• Level 4 (extended thinking) apply math model to
  illuminate a problem, to inform, and to solve
  design, critique, prove

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Task Sort

•  
• Identify which Depth of Knowledge level for each
  of the tasks described below.
•  

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Task A Grade 1

• Sam bought 4 cookies for 10 cents a piece. How
  much money did he spend?
• Recall Skill Strategic thinking Extending
  thinking

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Task B Grade 1

• 3 7
• 4 6
•  
• Recall Skill Strategic thinking Extending
  thinking

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Task C Grade 3

• John is making rectangles using square tiles. He
  has 24 tiles. How many different rectangles can
  he build?
• Show all your work.
• Recall Skill Strategic thinking Extending
  thinking
•  

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Task D Grade 2

• Write a word problem that would be solved by this
  equation
• 23 45 68
• Recall Skill Strategic thinking Extending
  thinking
•  

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Task E Grade 2

• Using base-ten blocks, solve the following
  addition problem
•  
• 23 45 68
• Recall Skill Strategic thinking Extending
  thinking

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Task F Kindergarten

• Start with 4 or 5 students in a group. How many
  handshakes will there be if each person in your
  group shakes the hand of every person once?
• Tell who was in your group.
• How did you get your answer?
• Try to show your solution on paper.
• How many handshakes will there be if 1 more
  person joins your group?
• What if 2 more join? 3 more join?
• Do you see a pattern in the numbers? Describe the
  pattern
• Recall Skill Strategic thinking Extending
  thinking

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Questions for analyzing student work

• What can we say about each students response?
• What does this students response tell us he knew
  about ___?
• What questions might we ask the student to better
  understand his reasoning?
• What mathematical concepts or procedures might
  the student use to answer this question
  correctly?
• What might we do to further explore this
  students ability to communicate what he knows
  about ____?
• What other mathematical ideas could we assess
  with this task?
• In what ways could we use information learned
  from this task to plan for the next lesson with
  this students?

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Task Student work

• Look through the student samples and highlight
  where the student went beyond the demands of the
  task.
• Identify what makes each task a rich task

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Summary

• Choose rich tasks
• Find time to enrich and extend
• Explicitly teach children to persevere
• Allow for revisions
• Use student work to inform instructional
  decisions
• Celebrate and share excellence

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Conclusion

• The attitudes children develop in the early
  years will strongly influence their future
  mathematical performance. If children experience
  connections and sense-making mathematics, they
  will come to think of mathematics as a
  sense-making experience. If they experience
  interesting, powerful mathematics, they will come
  to view mathematics as the interesting and
  powerful endeavor it can be.

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Resources

• Problems from Exemplars www.exemplars.com
• Clement, Rod (1991). Counting on Frank.
  Milwaukee, WI Gareth Stevens Publishing.
• Principles and Standards for School Mathematics,
  NCTM 2000.
• Andrews and Trafton (2002). Little Kids
  Powerful Problem Solvers. NH Heinemann.
• Fosnot (2007). Investigating Number Sense,
  Addition, and Subtraction, Unit Beads and Shoes,
  Making Twos. NH Heinemann.
• Seeley (2009). Faster Isnt Smarter. CA Math
  Solutions.
• NCTM (2001). Assessment Sampler, Grades PreK 2.
  VA NCTM.
• http//www.21stcenturyskills.org/