Working Backward Strategy for increasing students'

deductive reasoning skills in solving geometric problems

Hee-chan LewKorea National University of

EducationSantiago, 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

Confidence Like learning mathematics

Korea 11% 49/50 23% 50/50International 34% 48%

Confidence Like learning mathematics

Value

Korea 3 % 38/41 9 % 40/41 13 % 41/41International 14 % 28 % 48 %

TIMSS 2011 Mathematical Attitude (4th Grade)

TIMSS 2011 Mathematical Attitude (8th Grade)

Achievement GapPISA 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.

But, how the city students’ score is so high?

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.

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. 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

The Result of not honest way?

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: 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.

Procedure to make 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.

Procedure to prove Fermat Point(2)

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.

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).

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

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)

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.

Dialectic integration 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.

Design of an instructional scheme

Problem situation 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

Depending on the kind of problem, two kinds of different analysis are used.

Two kinds of analysisThe 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 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

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.

Design of an instructional scheme

Dynamic geometry 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.

Teacher’s roles Technical assistant: provide students

information about mathematical knowledge and computer environment

Counselor: advise some ideas in solving process

Collaborator: cooperate with students in solving process

UnderstandingDraw a tangent line of Circle A

through point B

AnalysisAssume

that line BC is a tangent line of the circle A

AnalysisDraw seg-

ments AC and AB

Triangle ABC is a right triangle

Let D is a midpoint of the seg-ment AB

DC = DA = DB

SynthesisDraw 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 them to write the construction

process on the paper 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 discuss something difficult during their problem solving process

Understanding Construct an inscribed circle to the

fan shaped figure OAB

Analysis Assume that circle O is the inscribed

circle.

Analysis 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

Analysis Extend line OB to

make line OB1 Extend line OA to

make line OA1 Then the circle E

is an inscribed circle of the triangle OA1B1

Now, students can find the way to draw the inscribed circle

Synthesis Draw line OC to

bisect angle AOB. Let C be the

intersection point of line OC and arc AB

At C, draw the perpendicular line to line OC

Extend line OB Let D be the

intersection point of line OB and the perpendicular line

Synthesis At D draw

the line to bisect angle ODC.

Let E be the intersection point of line OC and the angle bisector at D

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 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.

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? Here teachers have

to remind students that BE=BC

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 △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 equilat-eral 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 paral-lelogram

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.

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.

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.

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.

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)?

Synthesis 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.

Reflection 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.

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

DG as a tool for analysis 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.

• 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.

– 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

– 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

• 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.

Conclusion 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.

Conclusion DG was proposed as the dynamic

teaching method for Euclidean geometry by Clairaut, French mathematician in the 17th century

But, then there was not a proper dynamic tool

Conclusion 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.

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.

Conclusion In mathematics textbooks, the

conclusion is 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.

Conclusion 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.

At tis moment it is an honest way to teach deductive reasoning.

Thank you very much for your attention!!