Math and the Gifted Learner

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Math and the Gifted Learner

CLIU 21 – Gifted SymposiumUnwrapping the Potential

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0011 0010 1010 1101 0001 0100 1011Agenda

• Goals• Why Alternatives to Acceleration?• What Works

– Open Questions– Parallel Tasks

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0011 0010 1010 1101 0001 0100 1011Challenge vs Acceleration

• Common Core Standards• Research

– Gifted Students– Brain and Learning

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0011 0010 1010 1101 0001 0100 1011Common Core Standards

• Much more rigorous– Shift in when concepts are introduced– Most noticeable in K – 8

• More depth, fewer concepts in most grades• Progressions across grade levels more

coherent• Standards for Mathematical Practice

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Common Core Standards for Mathematical Practice

#1 Make sense of problems and persevere in solving them.

Can the student– Consider analogous problems?– Monitor and evaluate their progress, changing

course if necessary?– Explain correspondences between the different

mathematical representations?– Identify correspondences between different

approaches?

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Common Core Standards for Mathematical Practice

#2 Reason abstractly and quantitatively.

Can the student– Decontextualize AND Contextualize?– Create a coherent representation of the problem?– Attend to the meaning of quantities, not just

compute them?

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Common Core Standards for Mathematical Practice

#3 Construct viable arguments and critique the reasoning of others.

Can the student explain– What his/her solution is?– Why his/her solution works?– How someone else’s solution works and why?

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Common Core Standards for Mathematical Practice

#4 Model with mathematics.Can the student

– Apply the mathematics to solve problems in real-world situations?

– Can they use tools such as diagrams, two-way tables, graphs, flowcharts and formulas?

– Routinely interpret their results in context, reflect on whether the results make sense and revise model if necessary?

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Common Core Standards for Mathematical Practice

#5 Use appropriate tools strategically.Can the student

– Make sound decisions about which mathematical to use in the situation?

– Use technology to • help visualize the results to analyze, explore, and

compare• to explore and deepen understanding of mathematical

concepts

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Common Core Standards for Mathematical Practice

#6 Attend to precision.Can the student

– Communicate precisely to others?– Use clear definitions in discussions and their

own reasoning?– Use symbols, units of measure, labels

consistently and appropriately?

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Common Core Standards for Mathematical Practice

#7 Look for and make use of structure.

Can the student– Discern patterns or structure?– Can they see complicated things as being one

and as being composed of simpler things?

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Common Core Standards for Mathematical Practice

#8 Look for and express regularity in repeated reasoning.

Can the student– Notice repetition in calculations and look for

general methods and shortcuts?– Maintain oversight of the process while

attending to the details?– Evaluate reasonableness of intermediate

results?

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0011 0010 1010 1101 0001 0100 1011Pause and Reflect

• How do our current practices in mathematics instruction for gifted students align with these expectations?

• What questions do these Standards raise?

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0011 0010 1010 1101 0001 0100 1011What Does the Research Say?

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0011 0010 1010 1101 0001 0100 1011Research – Differences

• Pace at which they learn• Depth of their understanding• Their interests

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0011 0010 1010 1101 0001 0100 1011Research – Needs

• Unable to explain their solution• De-emphasis on right answers• Uneven pattern of development: concepts

vs computation• Individual attention AND opportunities to

work in groups

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0011 0010 1010 1101 0001 0100 1011Research – What Works

• Explain their reasoning orally & in writing• Flexible grouping• Inquiry-based, discovery learning

– Open-ended problems– Problems with multiple solutions or multiple

paths to a solution• Higher level questioning

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0011 0010 1010 1101 0001 0100 1011Research – What Works Cont’d.

• Differentiated assignments• Activities completed individually & in

groups• Use of manipulatives and “hands-on”

activities• Analyzing errors• Technology

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Research – Curriculum • Consider

– Depth– Breadth– Pacing

• ALL Students– Reasoning– Real-world Problem Solving– Communication– Connections

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Two Specific ExamplesOpen Questions and Parallel Tasks

• Provide tasks within each student’s zone of proximal development

• Each student has opportunity to make a meaningful contribution

• On topic, addressing same standards; level of depth or complexity changes

• Common Core: Standards for Mathematical Practice (especially #1 & 3)

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Math Experience #1A problem

Replace the boxes with values from 1 to 6 to make each problem true. You can use each number as often as you want. You cannot use 7, 8, 9, or 0.

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0011 0010 1010 1101 0001 0100 1011Reflection

• How did you chose your numbers?• Think about your students.

– What would their answers tell you about their weaknesses or strengths?

– How might you challenge a strong student who picks ‘easy’ numbers?

– What supports could you give students who are struggling with this task??

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0011 0010 1010 1101 0001 0100 1011Math Experience #2

• A task - Choose one of the following tasks and use the grid of dots given.• Option 1 – Make as many shapes as you can on

the grid with an area of 12. The corners of the shapes must be dots on the grid.

• Option 2 – Make as many rectangles as you can on the grid with an area of 12. The corners of the rectangles must be dots on the grid.

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Reflection

• Which option did you choose? Why?• Think about your students.

– What would their choice tell you about their weaknesses or strengths?

– How might you challenge your stronger students with this task?

– What adaptations or supports could you give students who are struggling with this task??

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Key Elements• Big Ideas

– The focus of instruction must be on the big ideas being taught so that they are all addressed, no matter at what level.

• Choice– There must be some aspect of choice for the

student, whether in content, process, or product.• Pre-assessment

– Prior assessment is essential to determine what needs different students have.

Small, Marian. Great Ways to Differentiate Mathematics Instruction. Teachers College Press. 2009

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0011 0010 1010 1101 0001 0100 1011Open Questions

• Mathematically meaningful• Variety of responses and approaches

possible• Richer mathematical conversations• All students can participate• Build mathematical reasoning,

communication, and confidence

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0011 0010 1010 1101 0001 0100 1011Creating Open Questions

• Convert conventional questions to open questions by:– Turning around a question– Asking for similarities and differences– Replacing a number with a blank– Asking for a number sentence– Changing the question

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0011 0010 1010 1101 0001 0100 1011Parallel Tasks

• Sets of two or three tasks• Same ‘big idea’, standards• Close enough in context that they may be

discussed simultaneously – questions asked fit both tasks

• Lead to discussion of important underlying mathematical ideas

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0011 0010 1010 1101 0001 0100 1011Creating Parallel Tasks

• Identify the big idea and standards• Identify developmental differences• Develop similar contexts and common

follow up questions– Can use a task readily available and alter it for a

different development level (up or down)

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0011 0010 1010 1101 0001 0100 1011Things to Remember

• Deeper learning is important• Open questions must allow for correct

responses at a variety of levels• Parallel tasks allow struggling students to

succeed and challenge proficient students• Both should be constructed so all students

can participate in follow up discussions

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0011 0010 1010 1101 0001 0100 1011Other Simple Possibilities

• Give students problems with errors in the solution. Students need to find error, correct it and explain why the error occurred.

• Require students to find more than one solution to a problem.

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0011 0010 1010 1101 0001 0100 1011Resources for Math

• NCTM Illuminations– http://illuminations.nctm.org/

• Inside Mathematics– http://insidemathematics.org/

• NRICH – http://nrich.maths.org/public/

• HoodaMath– http://www.hoodamath.com/

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Resources for Math• CLIU Content Networking Groups Wiki

– http://cliu21cng.wikispaces.com/• Print Resources

– Van De Walle, John A., Karen S. Karp, LouAnn H. Lovin, and Jennifer M. Bay-Williams. Teaching Student-centered Mathematics. Second ed. Vol. I, II & III. New York: Pearson.

– Small, Marian. Good Questions: Great Ways to Differentiate Mathematics Instruction. New York: Teachers College, 2009. Print.

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0011 0010 1010 1101 0001 0100 1011Thank you!

Cathy EndersCarbon Lehigh Intermediate Unit #21Curriculum & Instruction/Educational Technologies [email protected]