Working Backward Strategy for increasing students' deductive reasoning skills in solving geometric problems Hee-chan Lew Korea National University of Education Santiago, Chile 24 August, 2013

 Korea is a small country in East Asia. But, we have the long history of more than 4000 years.  It has been surrounded by China, Russia, Japan.  Korean Peninsula has been divided since 1950: North and South. Korea

Korean students have demonstrated a high level of achievement in TIMSS and PISA for a long time:   TIMSS 2003, 2007, 2011,   PISA 2003, 2006, 2009 Despite of brilliant scores, the reports of TIMSS and PISA show the some negative results:   Lack of students’ mathematical attitude   Achievement gap induced by school locations. Current Issues in Mathematics Education

Lack of Mathematics Attitude ConfidenceLike learning mathematics Korea11%49/5023%50/50 International34%48% ConfidenceLike learning mathematics Value Korea3 %38/419 %40/4113 %41/41 International14 %28 %48 % TIMSS 2011 Mathematical Attitude (4 th Grade ) TIMSS 2011 Mathematical Attitude (8th Grade )

Achievement Gap PISA surveys shows that there were a big gap among 5 areas of village, small town, town, city and large city classified by number of their populations.

Why? Students think that mathematics is meaningless and is not important in their life, therefore we do not need to study mathematics hard. Students think that mathematics is meaningless and is not important in their life, therefore we do not need to study mathematics hard. But, how the city students’ score is so high? But, how the city students’ score is so high? It is because parents push their students to study hard. It is because parents push their students to study hard.

Deductive Reasoning Deductive reasoning (or proof) is a process to deduce a new result from assumptions existed in the problem, axioms, what was previously proven, etc. Since the 6th century BC, it has been the flower of mathematics and a mark to distinguish mathematics from other science.

Euclid’s Elements Euclid’s Elements written in BC 3rd century has been used as a textbook to develop students’ deductive reasoning over 2000 years. Euclid’s Elements written in BC 3rd century has been used as a textbook to develop students’ deductive reasoning over 2000 years. But, the axiomatic method used in Euclid’s Elements has been criticized for a long time. But, the axiomatic method used in Euclid’s Elements has been criticized for a long time. –No mathematical activities such as imagination, experiment, guess, trial and error, and mistake etc. (Clairaut, 1741; Lakatos, 1976; Vincent, 2005).

Current textbook The proof in current textbooks shows only final results of proving activities. The proof in current textbooks shows only final results of proving activities. Teachers explain the results step by step very kindly but, do not guide students to find the solution and to explain why the solution works by themselves. Teachers explain the results step by step very kindly but, do not guide students to find the solution and to explain why the solution works by themselves. This is not honest This is not honest

The Result of not honest way? Students have few meaning in the proof and lose ability and confidence in constructing proof eventually. Students have few meaning in the proof and lose ability and confidence in constructing proof eventually. I will show three examples that mathematics is treated imprudently from the thee levels of school: I will show three examples that mathematics is treated imprudently from the thee levels of school:  One from the university level  One from Junior high school level  One from elementary level

Fermat Point Fermat point is a point of a triangle in which the sum of lengths of three segments is minimum. Fermat point is a point of a triangle in which the sum of lengths of three segments is minimum.

Procedure to make Fermat Point Textbook introduce the procedure to find the Fermat Point. Textbook introduce the procedure to find the Fermat Point.

Procedure to prove Fermat Point(1) Then, textbook introduces the procedure to prove that the three lines make one point. Then, textbook introduces the procedure to prove that the three lines make one point.

Procedure to prove Fermat Point(2) Then textbook introduces the procedure that the sum of lengths of three segments is minimum Then textbook introduces the procedure that the sum of lengths of three segments is minimum

Tangent Line Draw a tangent line of Circle A through point B

Procedure to draw the tangent line

Trisecting a Square Equally Divide a square into three equal parts by using the method to divide a square into two equal parts and justify their way. Divide a square into three equal parts by using the method to divide a square into two equal parts and justify their way.

Procedure to trisect a square

Purpose of this talk It is to provide an "honest" learning environment to teach deductive reasoning for secondary students (9 & 10 graders). It is to provide an "honest" learning environment to teach deductive reasoning for secondary students (9 & 10 graders). This talk shows that the "analysis" method systemized by Greek mathematician Pappus in AD 3rd century might be the answer. This talk shows that the "analysis" method systemized by Greek mathematician Pappus in AD 3rd century might be the answer.

New strategy In order to improve deductive reasoning ability and confidence, an "active justification" to find the heuristics for solution and to explain the reason by students themselves should be required rather than a "passive justification" to accept or to follow teacher’s explanation or persuasion In order to improve deductive reasoning ability and confidence, an "active justification" to find the heuristics for solution and to explain the reason by students themselves should be required rather than a "passive justification" to accept or to follow teacher’s explanation or persuasion This talk introduces one possible strategy for active justification: "Analysis" with dynamic geometry software. This talk introduces one possible strategy for active justification: "Analysis" with dynamic geometry software.

Analysis The analysis method which is the oldest mathematics heuristics in the history of mathematics assumes “what is sought as if it were already done and inquire what it is from which this results and again what is the antecedent cause of the latter and so on, until by so retracing the steps coming up something already known or belonging to the class of first principles.” (Hearth, 1981, p.400) The analysis method which is the oldest mathematics heuristics in the history of mathematics assumes “what is sought as if it were already done and inquire what it is from which this results and again what is the antecedent cause of the latter and so on, until by so retracing the steps coming up something already known or belonging to the class of first principles.” (Hearth, 1981, p.400)

Synthesis The synthesis as the reverse of the analysis takes as already done that which was last arrived at in the analysis and arrives finally at the construction of what was sought by arranging in their natural order as consequences what before were antecedents and successively connecting them one with another. The synthesis as the reverse of the analysis takes as already done that which was last arrived at in the analysis and arrives finally at the construction of what was sought by arranging in their natural order as consequences what before were antecedents and successively connecting them one with another.

Dialectic integration Greek mathematicians thought the dialectic integration of analysis and synthesis as a substance of mathematical thought. Greek mathematicians thought the dialectic integration of analysis and synthesis as a substance of mathematical thought. However, Euclid’s Elements considered only synthesis to reduce theorems from the foundation like axioms and the givens as a way to guarantee the truth of mathematics. However, Euclid’s Elements considered only synthesis to reduce theorems from the foundation like axioms and the givens as a way to guarantee the truth of mathematics.

Design of an instructional scheme Problem situation Problem situation There are two kinds of geometry problems: Proof problem or construction problem: There are two kinds of geometry problems: Proof problem or construction problem: –Prove something A under the condition(s) of B. –Construct something A under the condition(s) of B In order to deductive reasoning ability, we need to start at the proper problem situation In order to deductive reasoning ability, we need to start at the proper problem situation Depending on the kind of problem, two kinds of different analysis are used. Depending on the kind of problem, two kinds of different analysis are used.

Two kinds of analysis  The first is for proof problem. It is to find justification process by getting a series of previous sufficient conditions of the conclusion to be deduced under the assumption that what is required to be deduced is already done.  The second is for construction problem. This is to find the construction process by getting a series of necessary conditions from the assumption that what is required to be constructed is already constructed.

Design of an instructional scheme Four phase problem solving process Four phase problem solving process First is “understanding” phase First is “understanding” phase Second is “analysis” phase to assume what to be solved is done and to find the construction ideas by a series of necessary conditions Second is “analysis” phase to assume what to be solved is done and to find the construction ideas by a series of necessary conditions Third is “synthesis” phase to construct a deductive proof as a reverse of the analysis Third is “synthesis” phase to construct a deductive proof as a reverse of the analysis Finally, “reflection” phase to reflect on whole problem solving process. Finally, “reflection” phase to reflect on whole problem solving process.

Design of an instructional scheme Dynamic geometry Dynamic geometry Analysis method is very difficult to be applied in the paper and pencil environment because various dynamic operations are required. Analysis method is very difficult to be applied in the paper and pencil environment because various dynamic operations are required. It might be because of the lack of proper dynamic tools that the analysis known well by Greek mathematicians has not emphasized in schools since Greek era. It might be because of the lack of proper dynamic tools that the analysis known well by Greek mathematicians has not emphasized in schools since Greek era.

Teacher’s roles Technical assistant: provide students information about mathematical knowledge and computer environment Technical assistant: provide students information about mathematical knowledge and computer environment Counselor: advise some ideas in solving process Counselor: advise some ideas in solving process Collaborator: cooperate with students in solving process Collaborator: cooperate with students in solving process

Understanding Draw a tangent line of Circle A through point B

Analysis Assume that line BC is a tangent line of the circle A

Analysis Draw segments AC and AB Triangle ABC is a right triangle Let D is a midpoint of the segment AB DC = DA = DB

Synthesis Draw a segment AB and let D is a midpoint of AB Draw a circle D with radius DA and let C is an intersection point of the two circles BC is the tangent line to find

Reflection Let students to find another method. Let students to find another method. Let them to write the construction process on the paper Let them to write the construction process on the paper Let them to make an encapsulization of the whole process. Let them to make an encapsulization of the whole process. Let them to test the construction process by dragging the constructed object Let them to test the construction process by dragging the constructed object Let them to discuss something difficult during their problem solving process Let them to discuss something difficult during their problem solving process

Understanding Construct an inscribed circle to the fan shaped figure OAB Construct an inscribed circle to the fan shaped figure OAB

Analysis Assume that circle O is the inscribed circle. Assume that circle O is the inscribed circle.

Analysis Let point F be a tangent point of the circle E and the arc AB. Let point F be a tangent point of the circle E and the arc AB. Draw a perpendicular line to the radius OF at Point F Draw a perpendicular line to the radius OF at Point F

Analysis Extend line OB to make line OB 1 Extend line OB to make line OB 1 Extend line OA to make line OA 1 Extend line OA to make line OA 1 Then the circle E is an inscribed circle of the triangle OA 1 B 1 Then the circle E is an inscribed circle of the triangle OA 1 B 1 Now, students can find the way to draw the inscribed circle Now, students can find the way to draw the inscribed circle

Synthesis Draw line OC to bisect angle AOB. Draw line OC to bisect angle AOB. Let C be the intersection point of line OC and arc AB Let C be the intersection point of line OC and arc AB At C, draw the perpendicular line to line OC At C, draw the perpendicular line to line OC Extend line OB Extend line OB Let D be the intersection point of line OB and the perpendicular line Let D be the intersection point of line OB and the perpendicular line

Synthesis At D draw the line to bisect angle ODC. At D draw the line to bisect angle ODC. Let E be the intersection point of line OC and the angle bisector at D Let E be the intersection point of line OC and the angle bisector at D Draw a circle with radius CE Draw a circle with radius CE

Reflection

Understanding There is a triangle ABC. Make three equilateral triangles ABD, BCE, AFC by using each side of the given triangle ABC. Then prove that quadrilateral BEFD is a parallelogram

Analysis DF=BE and BD=FE To prove that quadrilateral BEFD is a parallelogram we have to show DF=BE and BD=FE To prove that DF=BE, we have to find two congruent triangles of which one side is DF and BE respectively. To prove that DF=BE, we have to find two congruent triangles of which one side is DF and BE respectively. But, only △ ADF and △ BCE are triangle having DF and BE as a side. But, these are not congruent

Analysis How to find the two triangles? How to find the two triangles? Here teachers have to remind students that BE=BC Here teachers have to remind students that BE=BC Students can find the two triangles, Students can find the two triangles, △ ADF and △ ABC, in which DF=BC=BE. We have to prove that DF=BC

Analysis To prove that DF=BC, we have to show that To prove that DF=BC, we have to show that △ ADF and △ ABC are congruent. Why? Students can find the reason based on the fact that DA=BA, AF=AC, ∠ DAF= ∠ BAC. The reason comes from that △ ADF and △ ABC are equilateral.

Synthesis 1. △ ABD, △ BCE, △ AFC are equilateral triangles 2. ∠ FCA and ∠ ECB are 60° and ∠ FCB is common ( ∠ DAB and ∠ FAC are 60° and ∠ FAB is common) 3. ∠ ACB = ∠ FCE and AC = FC, BC = EC ( ∠ BAC = ∠ DAF and AC = AF, AB = AD)

Synthesis 4. △ ABC ≡ △ FEC ( △ ABC ≡ △ ADF) 5. AB = FE (DF=BC) 6. BD = FE (DF=BE) 7. Quadrilateral BEFD is a parallelogram

Trisecting a Square Equally Divide a square into three equal parts by using the easy method to divide a square into two equal parts and justify their way. Divide a square into three equal parts by using the easy method to divide a square into two equal parts and justify their way.

Analysis Let’s assume that E is the trisecting point of AD. It is important that E is the assumed point rather than a final mathematical solution. Let’s assume that E is the trisecting point of AD. It is important that E is the assumed point rather than a final mathematical solution.

Analysis The ratio value of EG and GFG is 1: 2. Intuitively this is clear to students. They can use 180 degree rotation of △ MEG and translation of the rotated triangle. The ratio value of EG and GFG is 1: 2. Intuitively this is clear to students. They can use 180 degree rotation of △ MEG and translation of the rotated triangle.

Analysis On the other hand, let H be a point of the intersection of EF and DB. As a similar reason as the previous activity, students can understand that H is a point to divide EF is the ration of 1:2. On the other hand, let H be a point of the intersection of EF and DB. As a similar reason as the previous activity, students can understand that H is a point to divide EF is the ration of 1:2.

Synthesis Here, what is important for students is how to get E to trisect AD. If we can find G or H, we can find E. But, problem at this moment is that G and H actually do not exist because EF does not exist actually. How to find such a point(s)? Here, what is important for students is how to get E to trisect AD. If we can find G or H, we can find E. But, problem at this moment is that G and H actually do not exist because EF does not exist actually. How to find such a point(s)?

Synthesis It is to draw two lines BD and MC and to find the intersection point G (or H). It is to draw two lines BD and MC and to find the intersection point G (or H).

Synthesis If we draw a perpendicular line to AD passing through G (or H), the points I and J of intersection of BC and AD are points to trisect AD and BC. IJ is the answer to trisect a square. If we draw a perpendicular line to AD passing through G (or H), the points I and J of intersection of BC and AD are points to trisect AD and BC. IJ is the answer to trisect a square.

Reflection We ask students to extend the trisecting method to divide the square into four equal parts. We ask students to extend the trisecting method to divide the square into four equal parts. It is to draw IC and to find its intersection point K with BD and justify the method. It is to draw IC and to find its intersection point K with BD and justify the method.

Reflection We can continue this process to find the way to divide square into five and six equal parts by using analogical thinking and justification We can continue this process to find the way to divide square into five and six equal parts by using analogical thinking and justification

DG as a tool for analysis DG can help students draw a precise figure. DG can help students draw a precise figure. To find more easily a series of necessary conditions of final conclusion, students have to show relations among components in the problem situation.To find more easily a series of necessary conditions of final conclusion, students have to show relations among components in the problem situation. In a figure roughly drawn on the paper it is very difficult for students to find the relation.In a figure roughly drawn on the paper it is very difficult for students to find the relation.

DG as a tool for analysis DG has a measure function. DG has a measure function. –It makes students determine the good starting point for analysis by measuring length or angle etc continuously while dragging the point continuously. –In paper circumstance, it is very difficult for students to make a precise measuring enough to find the good starting point for analysis.

DG as a tool for analysis DG is dynamic DG is dynamic –It can make student perform various experiment to find necessary conditions by drawing, erasing and manipulating figures easily as well as dynamically. –In paper and pencil circumstance, it is almost impossible to perform analysis because the figure drawn on the paper cannot be manipulated

DG as a tool for analysis DG is a reflective tool DG is a reflective tool If the relation among components is preserved while dragging the picture constructed by synthesis, the construction process can be considered as a right procedure.If the relation among components is preserved while dragging the picture constructed by synthesis, the construction process can be considered as a right procedure. In paper and pencil circumstance, there is no way to check whether the construction process is right or not.In paper and pencil circumstance, there is no way to check whether the construction process is right or not.

Conclusion Today, I introduced the way to increase deductive reasoning. It is the analysis methods systemized by Greek mathematician Pappus. Today, I introduced the way to increase deductive reasoning. It is the analysis methods systemized by Greek mathematician Pappus. In the late of 1980s, Cabri and GSP were designed as a dynamic tool for students to investigate the properties and relation within and between figures through operating figures on the computer screen directly. In the late of 1980s, Cabri and GSP were designed as a dynamic tool for students to investigate the properties and relation within and between figures through operating figures on the computer screen directly.

Conclusion DG was proposed as the dynamic teaching method for Euclidean geometry by Clairaut, French mathematician in the 17 th century DG was proposed as the dynamic teaching method for Euclidean geometry by Clairaut, French mathematician in the 17 th century But, then there was not a proper dynamic tool But, then there was not a proper dynamic tool

Conclusion In DG, students can make a conjecture to geometric properties and confirm them informally In DG, students can make a conjecture to geometric properties and confirm them informally and they can improve their deductive reasoning skills by using the analysis method. and they can improve their deductive reasoning skills by using the analysis method.

Conclusion DG is a very excellent tool for the analysis method which is a good mathematical strategy proposed by Greek mathematicians but forgotten for a long time because of maybe a lack of proper tool. DG is a very excellent tool for the analysis method which is a good mathematical strategy proposed by Greek mathematicians but forgotten for a long time because of maybe a lack of proper tool.

Conclusion In mathematics textbooks, the conclusion is a conclusion. In mathematics textbooks, the conclusion is a conclusion. In mathematics education, the conclusion should be a starting point rather than a conclusion. In mathematics education, the conclusion should be a starting point rather than a conclusion. The analysis method combined with DG can provide a better and safe route from the starting point to the development of deductive reasoning of normal students. The analysis method combined with DG can provide a better and safe route from the starting point to the development of deductive reasoning of normal students.

Conclusion The analysis method is being introduced to Korean textbooks. The analysis method is being introduced to Korean textbooks. I am not sure that this methodology can make students to like mathematics and to think mathematics meaningful. I am not sure that this methodology can make students to like mathematics and to think mathematics meaningful. At tis moment it is an honest way to teach deductive reasoning. At tis moment it is an honest way to teach deductive reasoning.

Thank you very much for your attention!!