Interactive geometry labs â combining the US and ...

598 Int. J. Cont. Engineering Education and Life-Long Learning, Vol. 18, Nos. 5/6, 2008

Interactive geometry labs – combining the US and Russian approaches to teaching geometry Irina Lyublinskaya* Education Department, College of Staten Island, City University of New York, 2800 Victory Blvd., 3S-208, Staten Island, New York 10314, USA Fax: +718 982 -3743 E-mail: [email protected] *Corresponding author

Valeriy Ryzhik Physical Technical High School – Lycee 194021, ulitza Khlopina 8, Building 3, St Petersburg, Russia, Fax: +7 812 534 5817 E-mail: [email protected] Abstract: With the introduction of dynamic geometry software such as Geometer’ Sketchpad® and Cabri®, teaching geometry has changed dramatically. Mathematics educators are constantly faced with the challenge to re-think and reevaluate what they teach and how they teach. That particular goal was in the core of the collaborative US–Russian curriculum project supported by The Clay Mathematics Institute and Best Practices in Education, Inc. This project united two different styles of teaching geometry in one set of pedagogical materials: 22 interactive geometry labs that have been now successfully implemented in many US and Russian schools. These materials provide an integrated, dynamic environment for students to stretch themselves and discover mathematics in a unique way. These labs utilise technology and hands-on manipulative tools to discover geometric concepts and theorems. Dynamical geometry software significantly enhances the student’s experience. Keywords: constructivist approach; deductive reasoning; dynamic geometry software; interactive geometry; problem solving; Russia; the US; visualisation skills. Reference to this paper should be made as follows: Lyublinskaya, I. and Ryzhik, V. (2008) ‘Interactive geometry labs – combining the US and Russian approaches to teaching geometry’, Int. J. Continuing Engineering Education and Life-Long Learning, Vol. 18, Nos. 5/6, pp.598–618. Biographical notes: Irina Lyublinskaya received her master’s degree in Physics in 1986, a PhD in Theoretical and Mathematical Physics in 1991 from the Leningrad State University and has published substantially in that field. She has taught at the university as well as the high school level for over 20 years. In recent years, she has directed her professional endeavours to the curriculum development in the area of integrating technology into mathematics and science Copyright © 2008 Inderscience Enterprises Ltd.

Interactive geometry labs – combining the US and Russian approaches

599

education and to the professional development of mathematics and science teachers, conducting grant-funded workshops to help teachers learn to use educational technology. Lyublinskaya has received several national awards for excellence in high school mathematics and science teaching, and has published multiple articles and seven books about the teaching of mathematics and science. Valeriy Ryzhik has been teaching mathematics for almost 50 years at K-12 and college levels. In 1974, Ryzhik received his PhD in Mathematics Education with specialisation in teaching geometry from the Leningrad Pedagogical Institute. For his contribution to the education, he has been awarded Presidential Award for Excellence in Teaching Mathematics. Many of his students became prominent mathematicians all around the world. For example, recent recipient of the Fields Medal Grisha Perelman, who proved Poincaré Conjecture, is one of his former students. Since 1980, Ryzhik has been working on writing geometry textbooks for Russian schools. He is a Co-author of geometry textbooks adopted by Russian Academy of Science and Ministry of Education for seven – 12th grades geometry courses. Ryzhik published multiple books and articles about teaching mathematics in elementary and secondary school. He regularly presents at the Russian and international conferences on issues in teaching mathematics.

1

Introduction

The current changes in teaching secondary mathematics can be compared to a ‘crawling revolution’. This is not due to the administrative reforms but more so due to the growth of information technology, especially educational software used in secondary education. While we are fortunate to witness the birth of this exciting process, we are the educators who have to look for new ideas and new methodologies of teaching and learning mathematics. The use of a dynamic mathematics environment could and should completely change the way mathematics is taught. That particular goal was in the core of the collaborative US–Russian curriculum project resulting in a set of unique interactive geometry labs. The geometry labs combined different styles of teaching geometry in one set of pedagogical materials (Lyublinskaya and Ryzhik, 2003). In the US students are increasingly encouraged to experiment, make conjectures and discover mathematical concepts. In this constructivist model, students are taught to investigate and explore geometric theorems visually, and to formulate predictions about the properties of geometric shapes. The Russian approach to teaching geometry encourages students to use multi-level thinking and deductive reasoning skills. This approach allows the teacher to use the student conjectures and geometric observations to motivate them to think spatially and to write convincing arguments that support these new hypotheses. The students test their understanding at a higher level by considering more complex problems that require multiple level thinking. We have written 22 laboratories that utilise technology and hands-on manipulative tools to discover geometric concepts and theorems (Lyublinskaya et al., 2003). The student projects we developed are focused upon multi-level thinking skills. While technology is not required to do these labs, dynamical geometry software significantly enhances the student’s experience.

600

I. Lyublinskaya and V. Ryzhik

During 2002–2004, interactive geometry labs have been piloted in Russia and the US. Three public high schools have been chosen in Russia. Two schools were selected in St Petersburg – Lyceum #261, a secondary school specialised in mathematics and physics, and Gymnasium #597, secondary school specialised in technology and informatics, both are college preparatory schools. The third school – Lyceum #8, was selected in Sosnovyi Bor, a small town in the suburbs of St Petersburg. In the USA, pilots have been conducted at the Peddie School in Hightstown, New Jersey, a private college – preparatory boarding school, and at the Paul Robeson High School in Brooklyn, New York, a public school with over 80% of the students at poverty level or below. The two urban schools in St Petersburg were similar to the private school in New Jersey in terms of students’ population – all three being college preparatory selective schools. The school in Sosnovyi Bor has large percentage of students whose parents are blue collar workers, so it is similar to the school in Brooklyn. The pilot studies have been conducted with students in grades 8th–10th. The classroom experience of using these projects in Russian and the US schools showed that students came away with a ‘signature’ experience, which was memorable, applicable and challenging. We were also pleased to see that this project addressed many of our curricular goals to provide an integrated, dynamic method for students to stretch themselves and discover mathematics in a unique way. In addition, students who participated in the pilot study in the US showed significantly better performance on standardised exams, such as New York City Regents and on Scholastic Aptitudes Tests used nationally in the US for students applying to college (SATs), compared to a control group of students in the same schools who did not use these materials.

2

Philosophy of teaching geometry: a joint US–Russian position

Geometry is the only area of mathematics that develops students’ visual and spatial thinking. There are three major components in teaching geometry: development of spatial thinking, development of logical reasoning and development of ability to link studied material to real-life applications. In order to address all of these components in this set of materials, we included problems that generally require four steps of solution. 1

The problem is posed as a real-life or a ‘make-believe’ word problem. One of the goals of the suggested approach is the development of the students’ spatial thinking. Thus, the illustration to the problem is not provided. Students are expected to visualise the problem and to answer the problem in their mind without any additional materials. At this step, the strong emphasis is on the development of spatial thinking. Not all students will be able to complete this step; however, it is important for them to try, and if possible, to come up with a visual image of the problem, and conclude a conjecture based on this image, even if it is wrong.

2

The next step is to search for a conjecture with the help of non-technology manipulative tools. Students can use paper and pencil, transparencies with drawings, rulers, protractors, graph paper, etc. This step allows students to test their visualisation and refine their initial conjecture.


601

3

The third step requires students to use dynamic geometry software to visualise the final answer and to search for additional evidence concerning the final answer. Experienced users can make constructions from the blank document. Sketchpad documents that include all necessary constructions are provided for the novice users. The conjecture is finalised at this step.

4

The last step of the solution is a proof of a formulated conjecture. Students are expected to reason and justify their conjecture at the level appropriate for their geometry course.

Students are encouraged to explore each problem further. All explorations suggested in the problems are open-ended. These materials can be used as supplemental to any course of geometry, ranging from middle-school explorations of geometry to a high school systematic course of geometry. In order to illustrate this approach, several problems are discussed below.

3

Examples of developed materials

3.1 The assembly line problem A uniformly expandable robotic arm is secured between a stationary control box mounted on the factory floor and an airplane engine moving on a straight assembly line to the side of the control box. A support piece with a wheel under it is placed below the middle of this robotic arm. What can you say about the path that the wheel will follow as the engine moves along the assembly line?

3.1.1 Pre-requisite knowledge Vocabulary: midpoint, mid-segment and similar triangles. Key property: The mid-segment of a triangle is parallel to a side of the triangle.

3.1.2 Visualisation Students are expected to visualise the robotic arm that connects the stationary point (the control box) and another point (the airplane engine) moving along the line (the assembly line) then describe it. Thus, students will formulate initial conjecture about the locus of the midpoint (the support piece) at this stage.

3.1.3 Physical modelling At the next stage, students will explore the problem without technology in order to refine their initial conjecture. Students can use a variety of approaches with different materials such as ruler, compass, graph paper, rubber band, etc. to model the problem. Here are two possible approaches that can be used for the non-technology explorations. 1

Students can use two pencils, a pushpin, a ruler and a rubber band. Make three holes through a rubber band – at each end and at the midpoint. One end of the rubber band is affixed with the rubber band to a board, so it stays at the same place at all times. One pencil will be placed through a whole at the other end of the rubber band, and

602

I. Lyublinskaya and V. Ryzhik another pencil will be placed through a hole in the middle. Use the ruler to trace a straight line with the 1st pencil and observe the line that the 2nd pencil is tracing. Students will easily see the locus of the midpoint of the stretching rubber band in this case.

2

Using pencil and paper students can draw a line a and a point A that does not belong to the line. Then they can draw a segment that connects point A to a line a at some point and find a midpoint, M1, of that segment. They can draw another segment that connects point A to a line a and find midpoint, M2, of the 2nd segment. Repeating this procedure two or more times, students will be able to see the locus of the midpoints of these segments.

3.1.4 Computer exploration Students can construct a geometric model of this problem or use pre-made Geometer’s Sketchpad® document that allows them to move the point X representing the airplane engine along the line a representing the assembly line and to trace the point M representing the support piece using command Trace from the Display menu (Figure 1). Students could also select points X and M and choose command Locus from the Construct menu to see the locus of the midpoint M. This will allow them to finalise the conjecture for the problem. Figure 1

The sketchpad snapshot of the assembly line problem (see online version for colours)

3.1.5 Answer The path of the support piece is the mid-segment of the triangle; the mid-segment of the triangle is parallel to the side of the triangle.


603

3.1.6 Proof Consider a line a, a point A that does not belong to the line a, and a point X that moves along the line a. Let X1, X2 and X3 be three different positions of point X on the line a. Let M1, M2 and M3 be corresponding positions of the midpoint of segment AX (Figure 2). We will prove that points M1, M2 and M3 belong to the same line. In order to do that it is sufficient to prove that M3 belongs to the line passing through points M1 and M2. Let b be the line passing through the points M1 and M2. We will prove that M3 belongs to the line b. By construction M1M2 is a mid-segment of ǻAX1X2, so M1M2||a, e.g. b||a. By construction M1M3 is a mid-segment of ǻAX1X3, so M1M3||a. Then, M1M3||b. Since line M1M3 has a common point with the line b and is parallel to the line b, and then it is the same line as the line b. Thus, any midpoint of segment AX belongs to the line b, that is parallel to the line a, and segment connecting any two midpoints is a mid-segment of a corresponding triangle. Figure 2

Illustration to the proof for the assembly line problem

3.1.7 Further explorations 1

If the support piece is placed at another point under the robotic arm, what can you say about the path of the wheel attached to it? How do the properties of the path depend upon the position of the support piece?

2

For any given position of the support piece compare the area of the region formed by two different positions of the robotic arm and the assembly line with the area of the region formed by the same positions of the robotic arm and the wheel’s path.

3

Consider a line a and a stationary point A not on line a. Let point X be on the line a and let point Y be the point on the ray XA such that |AY|/|AX| = k, where k is constant and point A lies between points X and Y. What is the path of the point Y when the point X is moving along the line a?

4

Consider triangle 'ABC with point X on the median CM. Let A1 be the point of intersection between lines CA and BX, and B1 be the point of intersection between lines CB and AX. How does the line A1B1 move when point X is moving along the median CM?

604


3.2 The art project problem As part of an art project, a fourth-grade student decorates a rectangular sheet of paper with bent pipe cleaners. She gathers several pipe cleaners, all the same size, and bends each one at a different location to form right angles. Then, she takes a bent pipe cleaner and staples its tips so they touch two adjacent edges of the paper. Only the tips of the pipe cleaner touch the edges of the paper. The two edges of the pipe cleaner are parallel to the edges of the paper. One by one, the child takes her bent pipe cleaners and overlaps them according to the same rule: the tips of each pipe cleaner touch the same two adjacent edges. A point sits at the bend of each pipe cleaner. When viewed collectively on the paper, what path will these points form?

3.2.1 Pre-requisite knowledge Vocabulary: right isosceles triangle, parallel lines and corresponding angles, slope. Key property: The sum of the distances from a point on the hypotenuse of a right isosceles triangle to the legs is constant.

3.2.2 Visualisation Students are expected to imagine in their minds placing bent pipe cleaners in the corner of the rectangular piece of paper in such a way that the tips of the pipe cleaners touch adjacent edges of the paper, and describe what path the points at the bend will form. Thus, they will formulate their initial conjecture.

3.2.3 Physical modelling There are several approaches that could be used to help develop a visual sense of the problem. In each case, students should be gathering evidence that leads to a conjecture. Two approaches are described. 1

It is possible to model this problem using pipe cleaners, and follow the directions described in the problem. Make an observation about the path that emerges.

2

A coordinate version of this problem can be investigated by drawing x- and y-axes to represent the edges of the paper. Let a be the length of the pipe cleaner. Locate ordered pairs (x, y) such that x + y = a. For instance, if the length of the original segments is 6, you could use (0, 6), (1, 5) (2, 4) (3, 3) and so on. The points plotted will be collinear. Students could show that the points are collinear using an algebraic approach with slope, or by using a number of different geometric proofs.

3.2.4 Computer exploration The construction for this problem requires advanced skills and it is time consuming. The pre-made Geometer’s Sketchpad® document allows students to use a slider to change the point where pipe cleaners are bent and then to use action buttons to place the pipe cleaners to fill the right angle. Students can also use animation button and trace the point of bending to see the path that is formed by this point in order to finalise their conjecture (Figure 3).

Interactive geometry labs – combining the US and Russian approaches Figure 3

605

The sketchpad snapshot of the art project problem (see online version for colours)

3.2.5 Answer The points at the bend will form a segment, which is a hypotenuse of the right isosceles triangle with the legs along the edges of the paper. Symbolically, the locus of the point T is the segment AB.

3.2.6 Proof Given AOB = 90q. By construction, OA = OB = C. Let T is such point that the sum of distances from T to the lines OA and OB remains constant = C > 0. Let prove that point T moves along segment AB. Construct TK A OA and TL A OB. From the rectangle OKTL, we get TK = OL and TL = OK, so TK + TL = TK + OK. Since OA = OB, then OAB = OBA = 45q. Then, 'AKT is isosceles right triangle and TK = AK. Then, TK + TL = AK + OK = OA, which proves that segment AB is a locus of point T (Figure 4). Figure 4

Illustration to the geometric proof for the art project problem

606


3.2.7 Alternate proof Use the coordinate method. Let the point O be the origin with the coordinates (0, 0), ray OA be the y-axis and ray OB be the x-axis. Consider any point T(x, y) such that TK + TL = C. Since, TK + TL = OK + OL = x + y, then the equation of the locus is x + y = C or y = C x, which is an equation of a line. Additional restrictions follow from the condition that point T is inside the right angle AOB x t 0 and y t 0. Then, the locus is a segment of the line y = C – x in the first quadrant, which is the segment AB (Figure 5). Figure 5

Illustration to the coordinate method proof for the art project problem


Is your solution unique? Can you prove it or find another solution?

2

Students now decided to use a parallelogram shaped poster board. What is the shape of the bounded region if the students fill the acute angle following the same method? Obtuse angle?

3

Students decided to fill the acute angle of the parallelogram shaped poster board but to change the method. They now bend each pipe cleaner so that at the edges of the poster board pipe cleaners form right angle. What is the shape of the bounded region now?

3.3 The battlefield problem At a Revolutionary War site along the Delaware River in New Jersey, there is a marker describing the battle with the British forces. General George Washington is believed to have commanded his troops from that spot. Assume that prior to the battle, his two chief commanders were stationed on opposite sides of a large square with General Washington at its centre. If you knew only the location of the two commanders and where the general was standing, how could you restore the square?


607

3.3.1 Pre-requisite knowledge Vocabulary: square, perpendicular bisector, diagonal. Key property: Rotational symmetry of a square.

3.3.2 Visualisation Students are expected to visualise General Washington at a central point, with two commanders located on either side of him, imagine a possible way of constructing the square from this information, and describe or draw a diagram of what has been visualised. Students will formulate initial conjecture (in this case construction method) at this stage of the problem investigation.

3.3.3 Physical modelling There are several approaches that could be used to help develop a visual sense of the problem. In each case, students should be gathering evidence that leads to a conjecture. Two approaches are described below. 1

Be sure that students realise that the two commanders do not need to be located at vertex points. Therefore, begin with labelling three points to represent General Washington (W) and his two commanders (C1, C2). Draw a line through C1W and locate C1c such that C1W C1c W . Repeat a similar construction so that C2 W Cc2 W . Draw lines C1C1c and C2 C1c . Using construction tools construct the perpendicular from W to line C1Cc2 and label the intersection point R. Segment WR represents 0.5 the length of the side of the original square. There are several ways at this point to determine the vertices. Overlaying a transparency with two perpendicular lines and the 45q angle lines drawn, will help students locate the vertices. They could also locate the vertices by copying segment WR onto ray RC1 and ray RCc2 and repeating a similar construction on line C2 C1c (Figure 6).

2

A second approach relies upon the rotational symmetry of a square. Locate General Washington’s position (W) and his first commander (C1). Use a piece of paper and mark-off two points from a corner of the paper so that they are the same distance as from W to C1. Rotate the paper 90q, 180q, and 270q in order to locate C1c , C1s , and C1cs , respectively. Repeat this process using the distance from W to his second

commander C2 to locate Cc2 , Cs2 and Ccs2 . Draw lines C1Cs2 , C1c Cs2 , C1sC2 and C1csCc2 . The intersection of the pairs of lines represents the vertices of the original square (Figure 7).

608


Figure 6

Physical modelling using square central symmetry (see online version for colours)

Figure 7

Physical modelling using square rotational symmetry (see online version for colours)

3.3.4 Computer exploration The solution to the problem is unique only if the points W, C1, and C2 are not collinear. If all three points are on the same line, there are an infinite number of squares that can be constructed according to the conditions of the problem. Thus, it is suggested that students use pre-made Geometer’s Sketchpad® document that lock the points C1 and C2 to assure that there is no co-linearity. There are two documents that can be used with this problem. The first document just provides positions of George Washington and his two commanders. The square is hidden. To enhance visualisation, students can use the command, Show All Hidden, from the Display menu. The second document gives the students a hint about the rotational symmetries of the square.


609

3.3.5 Answer Different construction methods can be used to restore the original square. Two possible methods that are based on the ideas described in the physical modelling section are presented below:

Method 1 (Figure 8) 1

construct line C1W and mark point C1c so that C1W C1c W

2

construct line C2W and mark point Cc2 so that C 2 W Cc2 W

3

construct lines C1Cc2 and C2 C1c

4

construct line WR A C1Cc2 and WS A C2 C1c . Here, R is a point of intersection between C1Cc2 and WR and S is a point of intersection between the line C2 C1c and WS

5

construct points A and B on the line C1Cc2 such that RA = RB = WR

6

construct points C and D on the line C2 C1c such that SC = SD = WS

7

ABCD is a searched square.

Figure 8

The battlefield problem – construction method based on square central symmetry (see online version for colours)

Method 2 (Figure 9) 1

Construct images of the point C1 by rotation clockwise about the point W at 90q, 180q and 270q. These are points C1c , C1s and C1cs .

2

Construct images of the point C2 by rotation clockwise about the point W at 90q, 180q and 270q. These are points Cc2 , Cs2 and Ccs2 .

3

Construct the following lines: C1Cs2 , C1c Ccs2 , C1sC2 and C1csCc2 . The points of intersection between each pair of these lines are the vertices of the square ABCD.

610 Figure 9

I. Lyublinskaya and V. Ryzhik The battlefield problem – construction method based on square rotational symmetry (see online version for colours)

3.3.6 Proof Let us prove that ABCD is a square. Based on the first construction method (Figure 8) 'WC1c C2 WC1c C 2

'WC1Cc2 (SAS)

WC1Cc2 . Then, C1c C2 || C1Cc2 . Since, RS A C1c C2 RS A C1Cc2 and WS = WR

as corresponding altitudes in congruent triangles 'WC1c C2 and 'WC1Cc2 . By construction, |CD| = 2|WS| and |AB| = 2|WR| = 2|WS| |CD| = |AB|. Since |AB| = |CD| and AB__CD, then ABCD is a parallelogram. Since, the median line of the parallelogram RSAAB, then ABCD is a rectangle. By construction, |CD| = |RS|. Since the median line of the rectangle is equal to its side, |AD| = |RS| and |RS| = |CD| |AD| = |CD|, then ABCD is a square with the points C1 and C2 located on the opposite sides of this square and the point W located at the midpoint of the median line, so it is in the centre of the square ABCD. Based on the second method of construction (Figure 9) C1c is an image of C1 and Ccs2 is an image of Cs2 of rotation about the point W at 90q, so the line C1c Ccs2 is an image of rotation of the line C1Ccs2 about W at 90q. Thus, C1c Ccs2 A C1Ccs2 A=90° , since the angle between the line and its image is equal to the angle of rotation. Same way, B = C = D = 90q, so ABCD is a rectangle. Since, the line C1c Ccs2 is a rotational image of the line C1Cs2 about W, then the distance from the point W to the line C1c Ccs2 is the same as the distance from the point W to the line C1Cs2 . Then, |WK| = |WN|. In the same way, we can prove that the distance from the point W to lines C1sC2 and C1csC2c are the same, so |WL| = |WM| = |WK|, where WK A C1Cs2 , WN A C1c Cs2 , WL A C1cs Cc2 and WM A C1s C2 . Then, in a quadrilateral WNAK three angles are right, so KWN = 90q,

and since |WK| = |WN|, WNAK is a square. Same way we can prove that WMDN = WLCM = WKBL = WNAK |AB| = |BC| = |CD| = |DA| and ABCD is a square.


611


What if the commanders were on adjacent sides of the square around George Washington rather than on opposite sides? Explain how you can or cannot restore the square.

2

What if you were given one point on each of three different sides of the square, but no centre point? Explain how you can or cannot restore the square.

3

What if you were given one point on each side of the square, but no centre point? Explain how you can or cannot restore the square.

3.4

The pirate problem

In 1785, pirates in the Caribbean Ocean buried a treasure chest full of gold doubloons and silver on an island. In order to locate the treasure later on, they used two large rocks and an old palm tree as markers. Your sister discovered the notes that the pirates had written describing the location of the treasure, and the two of you went off in search of the gold and silver. “From the palm tree, walk directly to the Falcon-shaped rock and count your paces to the rock. Turn a quarter circle to the right and go the same number of paces, and put a stick in the ground. Return to the palm tree and walk directly to the Eagle-shaped rock again counting your paces to the rock. Turn a quarter circle to the left and go the same number of paces, and put another stick in the ground. Connect the two sticks with a rope and dig in the middle.”

You discovered the rocks, but the palm tree was no longer there. Where should you dig to find the treasure?

3.4.1 Pre-requisite knowledge Vocabulary: right angle, midpoint of a segment, clockwise and counter clockwise rotation. Key properties: the mid-segment of the trapezoid is an average of the bases. An alternative solution approach could require analytic geometry.

3.4.2 Visualisation Students are expected to imagine the position of the treasure in relation to the rocks in their minds first and then describe or draw the image of what they visualised. Students will formulate their initial conjecture based on this visualisation.

3.4.3 Physical modelling There are several approaches, which could be used to help develop a visual sense of the problem. In each case, students should be gathering evidence that leads to a conjecture. Instructor may use the provided Geometer’s Sketchpad® template or reproduce the diagram of the rocks on a larger sheet of paper to make the constructions for the two approaches described below.

612


Make the constructions using appropriate construction tools. Each group will choose a random location for the tree and find a location for the treasure. Have students compare their solutions to the location of the treasure by overlapping the templates. Regardless of the original tree location, the treasure is always at the same location. Locate a fixed point to represent the palm tree. Label it point P. Using tracing paper (or a transparency), draw two rays at right angles to one another to represent the paces. Scale the axes with 0.5 cm increments. Overlay the tracing paper on the original template. Mark the location of stick #1. Repeat this process on the 2nd rock to locate stick #2. Draw a segment between the two sticks and fold the template to locate the treasure.

3.4.4 Computer exploration In this particular problem, experienced users could make constructions from the blank document. However, it is suggested that instructors provide students with pre-made Geometer’s Sketchpad® document that defines the locations of two rocks. This will facilitate easier comparison of results between the students. This sketch asks the student to follow the pirate’s instructions to locate the treasure. The student is expected to choose an arbitrary location for the tree, and then construct points and segments as needed. The points F and E are fixed at coordinates 0,0 and 4,0, respectively, so that the student cannot accidentally move them. Each student or student group should obtain the same coordinates for the treasure, regardless of the location of the tree chosen. This template requires the student to be able to use the software to construct segments and midpoints, and to use the Transform menu to rotate objects around a centre point. Novice Sketchpad users can explore the problem right away with the pre-made sketch that has complete constructions and requires only basic knowledge of the use of the software. Students can move the tree point and observe immediately that the position of the treasure is independent from the position of the tree (Figure 10). Students are also expected to describe the position of the treasure point in relation to the two given rock points. Mathematical formulation of the answer to the problem provides a conjecture that students have to justify as the last step of problem solving. Figure 10

Sketchpad snapshot of the pirate problem (see online version for colours)


613

3.4.5 Answer The location of the treasure does not depend upon the location of the tree. The treasure is on the perpendicular bisector of Falcon- and Eagle-shaped rocks and is 0.5 the distance between the rocks.

3.4.6 Proof The following proof is for the case when S1, S2 and P are on the same side of the line EF. Draw the line EF. Construct perpendiculars from the points S1, S2 and T to the line EF (Figure 11). Then, AS1 A EF, TD A EF, CS2 A EF AS1»» CS2 AS1S2C is a trapezoid (if AS1 = CS2, then AS1S2C is a rectangle). TD A EF and TS1 = TS2 TD is a median of the trapezoid TD=AS1+CS2/2. Figure 11

Illustration to the proof for the pirate problem

By construction 'EAS1 and 'EPB are right triangles with equal hypotenuse, since ES1 = EP by construction. Since, AS1 A EB and ES1 A EP AS1E = PEB 'EAS1 = 'EPB AS1 = EB. Using similar method it can be shown that 'FCS2 = 'FPB CS2 = FB. Then, TD= AS1 CS2 2 EB FB 2 1 2 EB . Since, the distance EF is constant and the distance TD is constant. Furthermore, since TD is a median of the trapezoid, CD = DA. From congruence of triangles above, PB = CF = EA. Then, FD = DE, so D is a midpoint of EF. Then, location of the point T is unique. The treasure is located on the perpendicular bisector to the segment EF at the distance equal to the 0.5 of EF.


If the locations of the rocks change, how does the location of the treasure change?

2

The squares PES1C and PFS2D are drawn on the sides of triangle PEF formed by the palm tree and the rocks. Draw medians and altitudes of the triangles CDP and PEF from the vertex P. Use the dynamic geometry software tools to explore the relationships between the medians and the altitudes of the triangles CDP and PEF.

614

4


Discussion

Our goal was not only to develop the set of materials that can be used in geometry courses, but also to promote a unique ‘hi-tech’ approach to teaching and learning geometry. We felt that all four steps of the approach are important for students’ development of geometric thinking. However, teachers at different pilot schools used these problems in different ways, not necessarily following the suggested approach to problem solving. Despite this, the results of using these problems in both countries were encouraging. While visualisation is a common practice in courses of geometry in Russian schools, for the US students and their teachers this was a new experience. Students commented that the few minutes at the beginning of the problem when they had to think about it with closed eyes helped them to focus on the problem and start thinking of the conjecture for the problem. Figure 12 shows a student at The Young Women's Leadership School in Harlem, New York, NY, imagining what a buried-treasure map might look like before trying to decipher directions to the location of the loot in the Pirate Problem. Figure 12

A student at the young women’s leadership school at Harlem, New York City, describes visual image of buried treasure in the pirate problem (see online version for colours)

The difference between approaches in Russian and the US schools was also observed in the physical modelling stage of the problem exploration. In Russian schools practically all students used compass and ruler constructions on the paper. For example, for the pirate problem an 8th grade student from St Petersburg school #261 suggested the following construction: “I constructed two points representing two rocks, then I constructed one circle of arbitrary radius with the center in one point, and another one with the center in another point, so that these two circles intersect. I chose one of the intersections of circles as the position of the palm tree. Then I followed the instructions and found the position of the treasure. I repeated the same process with circles of different radius and found out that the position of the treasure was the same as before.” (Ivanov et al., 2003)


615

The US students used a much larger variety of materials at this non-technology stage of the problem exploration. A 9th grade student at the Peddie School in Hightstown, NJ, where this problem was assigned as an independent project, baked a cake with a hidden ring in it. Other students had to use geometry construction on the top of the cake to determine the location of the ‘treasure.’ Students marked different possible positions of the tree and determined the point where to ‘dig for treasure.’ In Paul Robeson High School, Bedford-Stuyvesant, New York City, NY, 10th grade students explored the battlefield problem by drawing the position of George Washington and his commanders on the floor and used ropes to model the whole problem (Figure 13). Figure 13

Physical modelling of the battlefield problem by students from Paul Robeson high school at Bedford-Stuyvesant, New York City (see online version for colours)

Classroom observations in all schools showed that students are more motivated and engaged when solving real-context problems in the dynamic geometry environment. Students also check their constructions, determine and correct any construction mistakes more often than during non-technology approach to the problem. The results of implementation of these problems in Russian schools showed that 80% of students can develop correct conjecture for the problem in 5–10 min if using Geometer’s Sketchpad®. However without Sketchpad, only 25% of students were able to solve these problems independently and the first correct answer from students was received only after 30 min of work. “I like using Geometer’s Sketchpad® for solving geometry problems because I can make very precise constructions that are easy to change, and I can move, rotate, and stretch my sketches to verify my solution,”

wrote 9th grade Russian student from St Petersburg school #261 about his experience with the Sketchpad. After completing technology-based exploration, students are expected to reason and justify their conjecture at the level appropriate for their geometry course. Several students at the Peddie School commented that making constructions in Sketchpad during technology investigations helped them later to develop the proof of the conjecture.

616


Many teachers found the use of these problems beneficial for their students’ motivation and engagement in learning geometry. Teachers in both countries noted that students liked working with Sketchpad. All Russian teachers commented that the use of dynamic mathematics environment is mostly efficient when working with optimisation problems, locus problems, and problems that require consideration of many cases before conjecture could be formulated. These are exactly the types of problems that we chose when developing these materials. Finding an answer to these problems is usually difficult and time consuming, but with the help of dynamic geometry software it is much easier and takes much less time. The US teachers emphasised the benefits of using The Geometer’s Sketchpad® in problems that require difficult geometric constructions. They noted that students are more motivated to complete constructions within the dynamic geometry environment rather than using paper and pencil, they spend much less time for Sketchpad constructions, and that provides more time for discussions, analysis, and proofs. While working with a set of developed materials, some Russian teachers deemphasised the non-technology modelling step. In two St Petersburg schools, students were encouraged to model problems in the Sketchpad right after visualisation step, and emphasis was placed on the proofs. However, in school #597 while all students were able to find the answers to the problems at the technology exploration stage, many students were not able to develop proofs for their conjectures. On the other hand, in school #261, all students actively used Sketchpad to explore problems and to find solutions and at the same time majority of the students demonstrated ability to develop formal proofs of the conjectures found in technology explorations. The approach to solving these problems was completely different in school #8 of Sosnovyi Bor, where students would go right into proofs before ever exploring problem in the Sketchpad. That happened due to the fact that this school had less access to computers and strong traditional paper and pencil approach to teaching geometry. In the US schools, the process of modelling without and with technology was both strongly emphasised. Students spend most of the time on these two steps. At the Peddie School, the justification was also an important part of the problem solving, although only honours level students were required to provide formal proofs. In regular track geometry classes, usually students were asked to justify their solutions informally or proofs were discussed with teacher’s help. In Paul Robeson High School, the justification step was de-emphasised completely during the first year of the pilot. Classroom observations and meetings with the teachers showed that the majority of teachers at Paul Robeson High School did not have sufficient background in geometry, and could not provide proofs to the problems themselves. In addition, the NYC Regents exam does not require geometry proofs. Thus, teachers did not feel the need to push students to justify their answers, and most of the time they skipped the last step. Working with these teachers in monthly professional development workshops allowed us to help them with content area, and during the second year of the pilot testing we saw more emphasis on justification. There were still no formal proofs, but teachers did emphasise the importance of justifying results, which was a significant change in teachers’ attitudes towards teaching geometry (Figure 14). Even with this limited exposure, students at Paul Robeson High School performed significantly better on NYC Regents exams compared to other students in this school who did not experience interactive geometry labs in their mathematics classes.

Interactive geometry labs – combining the US and Russian approaches Figure 14

617

Teacher from Paul Robeson high school at Bedford-Stuyvesant, New York City, provides proof for the battlefield problem (see online version for colours)

Testimonials from the US teachers and administrators (BPE, 2001–2004) have been encouraging: “I’ve have heard from students who have an aptitude for other subjects like history, but never felt they were good at mathematics. The Interactive Geometry approach was able to tap into my students.” natural talent and curiosity and make them feel that, “Wow, I can do mathematics.” (David Lewine, High School Mathematics Teacher, The Young Women’s Leadership School, Harlem, NYC) “When tenth-grade students were asked to reconstruct the complex piratetreasure lesson lab for an exam, almost everyone – 28 or 30 out of my 34 students – could remember the original problem, use logical reasoning to reconstruct the diagram, apply a new set of directions, and explain the new situation properly in geometric terms. It was amazing.” (Erick Delcham, Mathematics Teacher, Paul Robeson High School, Bedford-Stuyvesant, NYC) “Interactive Geometry is not just about geometry. The whole process can be used in other courses throughout our curriculum. It is really a programme to help students become better problem solvers. If Paul Robeson teachers want to move up the ladder and teach higher-level mathematics, they can not do it here without learning the Interactive Geometry approach. Interactive Geometry makes teachers think about what they are teaching.” (Sheila McClintockFraser, Assistant Principal for Mathematics, Paul Robeson High School, Bedford-Stuyvesant, NYC)

Today these materials have been successfully used in hundreds of schools in both countries, at middle and high school levels. The materials have been published electronically by the Best Practices in Education, Inc. Teachers interested in using these materials in the classroom can contact the authors of this article in order to obtain these materials.

618


Acknowledgements This project was sponsored by The Best Practices in Education, Inc., non-profit operating foundation that identifies and introduces into the US classrooms excellent mathematics education practices from abroad to improve teaching and learning in grades pre-K to 12, especially in urban schools and by Clay Mathematics Institute, a privately funded operating foundation dedicated to increasing and disseminating mathematics knowledge.

References BPE – Best Practices in Education, Inc. (2001–2004) Project Interactive Geometry. Available at: http://homepages.nyu.edu/~gr659/BPE/projects/InteractiveGeometry/index.html. Ivanov, S.G., Lyublinskaya, I.E., Ryzhik, V.I., Armontrout, R., Corica, T. and Boswell, L. (2003) ‘Investigations using Geometer’s Sketchpad’, Computer Technology in Education, Vol. 3, pp.14–20. Lyublinskaya, I.E. and Ryzhik, V.I. (2003) Investigations for the Geometer’s Sketchpad (p.44). St Petersburg: Informatizatzija Obrazovanija (Computers in Education). Lyublinskaya, I., Armontrout, R., Ryzhik, V., Boswell, L. and Corica, T. (2003) Interactive Geometry Labs, CD-ROM. New York, NY: Best Practices in Education.

Interactive geometry labs â combining the US and ...

Interactive geometry labs â combining the US and ...

Suggest Documents