Fostering Fostering Mathematical Mathematical

Excellence in Early Excellence in Early Childhood EducationChildhood Education

Fostering Fostering Mathematical Mathematical

Excellence in Early Excellence in Early Childhood EducationChildhood Education

Dr. Susan LooneyDr. Susan Looneylooneyconsulting@comcast.netlooneyconsulting@comcast.netwww.looneymathconsulting.comwww.looneymathconsulting.com

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.

- 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.

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

Partnership for the 21st Century Skills

• An organization leading the way …..

Rainbow

Learning and Innovation Skills

• Creativity and innovation• Critical thinking and problem

solving• Communication and collaboration

Counting on Frank by Rod Clement

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 boy’s 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 boy’s house?

  If there were 24 Franks in the bedroom, how many ears would there be? How many Frank

paws would there be?    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?    3. The tree in the boy’s 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?

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• Teacher’s role of observer: when to let children work and

when to provide info and closure / questions and genuine interest

• Respect for children’s ideas and belief that all children can learn mathematics

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.

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.”

Characteristics of Rich Tasks

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

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?

Classroom A• Classroom A•  • Check of last night’s 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•  •  •  • Word problem: Sarah and John went apple picking. Sarah picked 23

apples. John picked 38 apples. How many apples did they pick all together?

•  

Classroom B 

• Student’s 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.

 • At the end of the period, none of the groups are finished with the task, but they

are all actively engaged in problem solving.

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

Task Sort Identify which Depth of Knowledge

level for each of the tasks described below.

 

Task A: Grade 1Sam bought 4 cookies for 10 cents a

piece. How much money did he spend?

Recall Skill Strategic thinking Extending thinking

Task B: Grade 13 + 7 = 4 + 6 = 

Recall Skill Strategic thinking Extending thinking

Task C: Grade 3John 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

 

Task D: Grade 2Write a word problem that would be

solved by this equation:23 + 45 = 68

Recall Skill Strategic thinking Extending thinking

 

Task E: Grade 2Using base-ten blocks, solve the

following addition problem: 

23 + 45 = 68

Recall Skill Strategic thinking Extending thinking

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

Questions for analyzing student work:

• What can we say about each student’s response?• What does this student’s 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 student’s

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?

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

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

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.”

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 Isn’t Smarter. CA: Math Solutions.

• NCTM (2001). Assessment Sampler, Grades PreK – 2. VA: NCTM.

• http://www.21stcenturyskills.org/