Introduction: 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.
Kindergarten 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”
1st grade Students are on the floor using pattern blocks to create tessellating patterns.
2nd grade Students are putting together Tangram pieces to create shapes.
3rd grade Students are predicting which pentominoes will fold up into a box. (Sunny and Mick, Thanks for the picture,)
4th grade Students are working in groups to place shapes correctly in a Venn Diagram. 2 sides congruent opposite sides parallel
5th grade 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
Where’s the proof? In search of proof in the K-5 classroom Presented by Cynthia Carter and Landrea Miriti November 15, 2007 ( Imagine a nice graphic: People with flashlights searching in the dark )
History of proof K-12 • 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)
Why proof? • 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)
Why proof in elementary school? • “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 (Galotti, Komatsu, & Voelz, 1997; Maher & Martino, 1996) • “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)
Why proof in elementary school?… • 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)
but what is proof? 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.
What is an appropriate description of proof for K-5 level ? Type your ideas into one of the chat pods.
A definition of proof 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- deductivereasoning • 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)
Example1- Betsy, 3rd grade • 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)
Example 2-Case by case • 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 posiblepateren. 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)
Example 3 – Proof by counterexample • “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)
How do we foster development of proof at the elementary level?- classroom conversations 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)
Fostering development of proof… • 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)
Websites • IMAGES – Improving Measurement and Geometry in Elementary Schools - http://images.rbs.org • MegaMath-http://www.ccs3.lanl.gov/mega-math/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/math-archive.html
References • 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.