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ASSESSING HIGH SCHOOL STUDENTS’ ABILITY IN PROVING MATH STATEMENTS AND

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MATH STATEMENTS AND RECLAIMING ITS IMPORTANCE

Iwan Pranoto

CoSMEd 2013

SEAMEO RECSAM, Penang, Malaysia

11-15 November 2013

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PROVING AND REASONING IN SCHOOL MATHState Math Standards or Curricula, Books, Classrooms

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NCTM’s Process Standards [NCTM, 2013],

Recognize reasoning and proof

Recognize reasoning and proof Make and Make and

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reasoning and proof as fundamental

aspects of mathematics

reasoning and proof as fundamental

aspects of mathematics

investigate mathematical conjectures

investigate mathematical conjectures

Develop and evaluate

mathematical

Develop and evaluate

mathematical

Select and use various types of

i d

Select and use various types of

i dmathematical arguments and

proofs

mathematical arguments and

proofs

reasoning and methods of proof

reasoning and methods of proof

Proof neglectionKnuth (2002b)

proof has becomeproof has become

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... proof has become “peripheral at best,

usually limited to the domain of Euclidean

G t ”

... proof has become “peripheral at best,

usually limited to the domain of Euclidean

G t ”Geometry.”Geometry.”

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Why do teachers and schools neglect the proving process?

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Not assessed in the [standardized] testsNot assessed in the [standardized] tests[standardized] tests[standardized] tests

[Steen, 1999]

students and thestudents and the

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students and the educators value

what the t

students and the educators value

what the tassessment

assessesassessment

assesses

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Indonesian School Math Standards

The verbs used in theThe verbs used in the

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The verbs used in the basic competence list

are mostly on the li ti f th

The verbs used in the basic competence list

are mostly on the li ti f thapplication of the formula.

application of the formula.

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DIAGNOSTIC FINDINGS

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“Proofs” in Most School Math Books

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Another way to prove the Pythagorean theorem is by putting a square on each side of the triangle. Observe the Figure 5.2 in detail.

In Asia, most students can read proper math proofs from Internet and English-spoken books

Some US school

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Some US school math textbooks like[Larson, 2001] are

itt ti f thwritten satisfy math standards.

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The Proof Fact in Math Classroom

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“Proof” is often confused with “illustration”

“Proof” is often confused with “illustration”

The student subjects thought that a great

number of examples are enough for proving the

validity of a mathematical

The student subjects thought that a great

number of examples are enough for proving the

validity of a mathematicalvalidity of a mathematical statement.

validity of a mathematical statement.

Wu says about definition in [Kiang, 2012]

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“Everything is informal; it almost

never gives anever gives a precise definition.”

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Math books should …

“(1) give precise enunciations

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(1) give precise enunciations of definitions and

mathematical conclusions (i.e., theorems),

(2) id i t(2) provide reasoning to support each mathematical

assertion,”

Theorems Evaluated in this Study

• Pythagorean theorem,

• Triangle sum theorem

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• Triangle sum theorem,

• Remainder theorem,

• Division with fractions,

• Derivative formula for algebraic expressions, etc.

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Determine the remainder of (x^20 -1) divided by (x^2 + 3x + 2).

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ASSESSING REASONING AND PROOF

An idea to assess proving ability [Gatot, 2013:50]

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Assessing students’ understanding on a proof of the angle sum of triangle

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A student’s proof

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WHAT’S NEXTSpread the good practices through many channels

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Provide Good Proofs in Math Learning Experiences

“It is importantSpread through Internet

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It is important, therefore, to carefully design examples and procedures to guide effective learning.”

Spread through Internet, Social Media, public forums, Personal Learning Network, Teacher Preparation Programs

g~[Quinn, 2012]

The Rules

If we fail to assess what Wh t

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If we fail to assess what we value, the students

will value what we assess only

What we assess defines what we value.

~G. Wiggins

Teaching and learningTeaching and learning for the test is bad, but can we make it at least teaching and learning for the test assessing

21st century skills?

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THANK YOUIwan PranotoMathematics DepartmentInstitut Teknologi BandungBandung, Indonesia