Describe the geometric reasoning that
you observe in the following “imaginary”
K-5 math classrooms video- clips.
You will have 30 seconds to view each clip and
type in your observations into the chat pod.
Students are eagerly identifying where they see
of shapes in their classroom.
Carlos-“ The door looks like a rectangle”
Alicia- “The clock on the wall is a circle”
Ibrahim-“The rug is kind of like an oval”
Students are on the floor using pattern blocks
to create tessellating patterns.
Students are putting together Tangram pieces
to create shapes.
Students are predicting which pentominoes will
fold up into a box. (Sunny and Mick, Thanks for the picture,)
Students are working in groups to place
shapes correctly in a Venn Diagram.
2 sides congruent
opposite sides parallel
Students are making figures on geoboards to
match given attributes.
For example, make a figure that has
 4 sides
 exactly two right angles
 1 pair of parallel sides
Presented by
Cynthia Carter and Landrea Miriti
November 15, 2007
( Imagine a nice graphic: People with flashlights searching in
the dark )


Historically, proof first visited in high school
geometry with little or no previous experience
NCTM’s 1989 standards
- rarely used the word, ‘proof’
- resorted to such euphemisms as
“validate”, “justify”, etc.
(Steen, 1999, p. 4)

Foundation for mathematical understanding as well
as for discovery and communication (Christou,
Mousoulides, Pittalis, & Pitta-Pantazi, 2004; NCTM, 2000; Steen, 1999;
Stylianides, 2007a, 2007b; Stylianides, A., Stylianides, G. & Philippou, 2005)

How/when do we move to "demonstrative
mathematics" in the K-12 curriculum? (C. Lee, personal
communication, 2007)

NCTM Reasoning and Proof process standard
PK-12 (NCTM, 2000)



“Any subject can be taught effectively in some
intellectually honest form to any child at any stage of
development.” (Bruner, 1960, p.33 )
Elementary children can reason deductively
& Voelz, 1997; Maher & Martino, 1996)
(Galotti, Komatsu,
“If students are consistently expected to explore, question,
conjecture, and justify their ideas, they learn that
mathematics should make sense, rather than believing
that mathematics is a set of arbitrary rules and formulas.”
(Reys, Lindquist, Lambdin, Smith, & Suydam, 2001, p. 81)

Children naturally want to know the “whys” (Reys el at., 2001)

Discontinuity exists regarding proof as
children move into secondary school (Stylianides,
2007b, p. 290)

“Thinking should be at the core of all school
learning …” (Goldenberg & Shteingold, 2002, p. 12)
Is this proof ?
Students are using geometer sketchpad to
explore the measures of the angles of
triangles. After finding the angle sum of
various triangles, many students come to the
conclusion that the sum of the angles of a
triangle is always = 180 degrees.
Type your answer in either the ‘yes’ pod or
the ‘no’ pod.
Type your ideas into one of the chat pods.
An mathematical argument that has the following components:




Foundation: the basis of the argument- definitions, axioms,
agreed upon truths
Formulation: the development of the argument- deductive
reasoning
Representation: how the argument is expressed- everyday
language, algebraic notation, pictures
Social Dimension: how it plays out in the social context of the
classroom community
(Stylianides, 2007a)

Why does an odd + odd = even?
Betsy’s argument: “All odd numbers if you
circle them by twos there’s one left over,
so if you..plus one , um or if you plus another
odd number, then the two ones left over will
group together,
and it will make an even number”
(Stylianides, 2007a, p. 11)
(Stylianides, 2007a, p. 11)

Stephanie, 4th and 5th grades in New Jersey school, found
all 3 cube tall stacks of 2 colors
“Dear Laura,
Today we made towers 3 high and with 2 colors. We
have to be sure to make every posible pateren. There are
8 patterns total. … I will prove it. If I put the towers in
color order …
If this doesn’t convince you I tell you more -- over -- …”
(Maher & Martino, 1996, p. 195)

“Fifth grader Naomi proudly proclaims that she has discovered a new
math rule: that whenever the perimeter of a rectangle increases, its
area also increases. She uses a table to demonstrate her rule.
Then, in asking whether the rule works in all cases, she and her
classmates search for a counterexample. e. g., length 6 and width 1
has perimeter of 14 units and an area of 6 square units.
(Goldenberg & Shteingold, 2002, p. 7)
#1
#2
#3
#4
Length
4 units
6 units
8 units
10 units
Width
2 units
3 units
4 units
5 units
Perimeter
12 units
18 units
24 units
30 units
Area
8 sq units
18 sq units
32 sq units
50 sq units
Thomas: ..all of these shapes are triangles!
Susanna: No! I disagree!. The triangles I know look like B , not
like L. Its too stretched out. I still think they’re just in the triangle
family.
Evan: If I’m all stretched out and turned upside down, I’m still
Evan.
(Schifter, Bastable, & Russell, 2002, p.82-84)




Help students discern the difference between deductive and
inductive reasoning
Assign logic problems and puzzles
Go beyond explorations, conjectures – ask why? how do you
know? …
Dynamic geometry software – add teacher-directed
questioning to cue students to seek justifications (Christou,
Mousoulides, Pittalis, & Pitta-Pantazi, 2004, p. 222)

“Students should see how learning proof relies and builds on
their own good sense.” (Goldenberg & Shteingold, 2002, p. 8)

IMAGES – Improving Measurement and Geometry in
Elementary Schools - http://images.rbs.org

MegaMath-http://www.ccs3.lanl.gov/megamath/gloss/math/mathtruth.html


www.mathforum.org
for Ask Dr. Math http://mathforum.org/dr.math/faq/faq.proof.html
Books and software – www.goenc.com and
www.pbs.org/teachers/bookslinks/bookspages/ma
th-archive.html












Bruner, J. (1960). The process of education. Cambridge, MA: Harvard University Press.
Christou, C., Mousoulides, N., Pittalis, M., & Pitta-Pantazi, D. (2004). Proofs through exploration in dynamic
geometry environments. Proceedings of the 28th Conference of the International Group for the Psychology of
Mathematics Education 2, 215-222. Bergen, Norway.
Galotti, K. M., Komatsu, L. K., & Voelz, S. (1997). Children’s differential performance on deductive and inductive
syllogisms. Developmental Psychology, 33(1), 70-78.
Goldenberg, P. E. & Shteingold, N. (2002). Mathematical habits of mind for young children. Retrieved October 9,
2007, from www2.edc.org/thinkmath/math%20Habits%20of%20Mind.pdg
Maher, C. A. & Martino, A. M. (1996). The development of the idea of mathematical proof: A 5-year case study.
Journal for Research in Mathematics Education, 27(2), 194-214.
National Council of Teachers of Mathematics. (2000). Principles and standards for school mathematics. Reston,
VA: Author.
Reys, R. E., Lindquist, M. M., Lambdin, D. V., Smith, N. L., & Suydam, M. N. (2001). Helping children learn
mathematics. New York: John Wiley & Sons, Inc.
Schifter, D., Bastable, V., & Russell, J. (2002). Examining the features of shape: casebook. Parsipany, NJ: Dale
Seymour.
Steen, L. A. (1999). Twenty questions about mathematical reasoning. In L. Stiff (Ed.) Developing mathematical
reasoning in grades K-12 (pp. 270-285). Reston, VA: National Council of Teachers of Mathematics.
Stylianides, A. J. (2007a). The notion of proof in the context of elementary school mathematics. Journal of
Educational Studies in Mathematics 65(1), 1-20.
Stylianides, A. J. (2007b). Proof and proving in school mathematics. Journal for Research in Mathematics
Education 38(3), 289-321.
Stylianides, A. J., Stylianides, G. J., & Philippou, G. N. (2005). Prospective teachers’ understanding of proof: what
if the truth set of an open sentence is broader than that covered by the proof? Proceedings of the 29th
Conference of the International Group for Psychology of Mathematics Education 4, 241-248. Melbourne,
Australia.