This document discusses assessing high school students' ability in proving math statements and reclaiming its importance. It notes that proof is often neglected in school math standards, textbooks, and classrooms. Standardized tests do not assess proof, so teachers and students do not value it. The document analyzes examples of proofs in Indonesian math standards and textbooks, finding that they lack precision and reasoning. It assesses students' ability to prove theorems like the Pythagorean theorem and evaluate proposed methods. It concludes that providing good proofs in math learning experiences and assessing what is valued can help reclaim the importance of proof in school math.

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ASSESSING HIGH SCHOOL
STUDENTS’ ABILITY IN PROVING
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 MATH
State Math Standards or Curricula, Books, Classrooms
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NCTM’s
Process
Standards
[NCTM, 2013],
Recognize
reasoning and proof
as fundamental
aspects of
mathematics
Make and
investigate
mathematical
conjectures
Develop and
evaluate
mathematical
arguments and
proofs
Select and use
various types of
reasoning and
i
d
methods of proof
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Proof
neglection
Knuth (2002b)
... proof has become
“peripheral at best,
usually limited to the
domain of Euclidean
Geometry.”
G
t ”
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Why do teachers and schools neglect the
proving process?
Not assessed in the
[standardized] tests
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[Steen, 1999]
students and the
educators value
what the
assessment
t
assesses
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Indonesian School Math Standards
The verbs used in the
basic competence list
are mostly on the
application of th
li ti
f the
formula.
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DIAGNOSTIC FINDINGS
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“Proofs” in Most School Math Books
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.
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In Asia, most students can read proper math
proofs from Internet and English-spoken books
Some US school
math textbooks like
[Larson, 2001] are
written satisfy math
itt
ti f
th
standards.
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The Proof Fact in Math Classroom
“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
statement.
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Wu says about definition in [Kiang, 2012]
“Everything is
informal; it almost
never gives a
precise definition.”
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Math books should …
“(1) give precise enunciations
(1)
of definitions and
mathematical conclusions
(i.e., theorems),
(2) provide reasoning t
id
i to
support each mathematical
assertion,”
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Theorems Evaluated in this Study
• Pythagorean theorem,
• Triangle sum theorem,
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
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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 NEXT
Spread the good practices through many channels
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Provide Good
Proofs in Math
Learning
Experiences
Spread through Internet,
Internet
Social Media, public
forums, Personal
Learning Network,
Teacher Preparation
Programs
“It is important,
It important
therefore, to carefully
design examples and
procedures to guide
effective learning.”
g
~[Quinn, 2012]
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The Rules
If we fail to assess what
we value, the students
will value what we
assess only
What
Wh t we assess
defines what we value.
~G. Wiggins
Teaching 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 YOU
Iwan Pranoto
Mathematics Department
Institut Teknologi Bandung
Bandung, Indonesia
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