on the use of computational tools to promote students’ mathematical thinking

8/3/2019 On the use of computational tools to promote students mathematical thinking

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COMPUTER MATH SNAPSHOTS

ON THE USE OF COMPUTATIONAL TOOLS TO PROMOTE

STUDENTS MATHEMATICAL THINKING

MANUEL SANTOS-TRIGO

Mathematics Education

Center for Research and Advanced Studies

Av. IPN 2508, Sn Pedro Zacatenco, 07360 Mexico City, Mexico

E-mail: [email protected]

This column will publish short (from just a few paragraphs to ten or so

pages), lively and intriguing computer-related mathematics vignettes.

These vignettes or snapshots should illustrate ways in which computer

environments have transformed the practice of mathematics or math-

ematics pedagogy. They could also include puzzles or brain-teasers

involving the use of computers or computational theory. Snapshots are

subject to peer review. In this snapshot students employ dynamic

geometry software to find great mathematical richness around a see-

mingly simple question about rectangles.

Computer Math Snapshots

Editor: Uri Wilensky

Center for Connected Learning and Computer-Based Modeling

Northwestern University, USA

E-mail: [email protected]

1. INTRODUCTION

Recent curriculum proposals recognized the importance for students

of using computational tools to comprehend mathematical ideas

and solve problems [National Council of Teachers of Mathematics

This report is part of a research project that involves high school students use of

computational tools in problem solving activities. The author acknowledges the

support received by Conacyt, reference #47850.

International Journal of Computers for Mathematical Learning (2006) 11:361376

DOI: 10.1007/s10758-006-9105-8 Springer 2006


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(NCTM), 2000]. Since different tools may offer students distinct

opportunities to represent, explore and solve mathematical prob-

lems, it becomes important to identify and discuss types of math-

ematical reasoning that students might develop as a result of using

particular tools. In order to examine the potential of using a par-

ticular tool it is important to discuss questions such as: What types

of task or problem representations are relevant for students in

identifying, constructing, and explaining mathematical relation-

ships? To what extent does the use of particular tools favor the

identification and exploration of mathematical relations? Which

attributes of mathematical thinking can be enhanced through theuse of technology in mathematical problem solving? In this snap-

shot we discuss aspects of mathematical practice that emerge while

solving a problem that involves the construction of a rectangle with

the help of computational tools. We show that thinking of different

approaches to solving the problem represents an opportunity for

students to identify and explore diverse mathematical relationships.

In this context, thinking about the construction of a rectangle in

terms of its properties provides a platform for students to formulate

and pursue related questions.

1.1. The Context

The task discussed in this report came from a problem-solving

seminar in which senior high school students worked on a series of

problems in 2 h weekly sessions over the course of one semester. We

used Dynamic Geometry software (Cabri) and hand calculators to

understand, represent, and work on textbook problems. Studentswere encouraged to construct dynamic representations of the

problems. This enabled them to identify and explore relevant

questions which led them to recognize mathematical relationships.

This report focuses on describing problem-solving approaches ra-

ther than analyzing, in detail, student problem-solving behaviors.

Thus, the focus is on characterizing the students approaches that

appeared and were discussed as a group or within a learning

community that promoted the participation of all its members. We

show the relevant questions and mathematical activities that appearin problem-solving episodes: problem formulation; understanding of

the task; solution to the problem; relation to similar problems; and

reflections.

MANUEL SANTOS-TRIGO362


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1.2. First Episode: Problem Formulation

What conditions or properties do we need to represent a rectangle?How can we draw a dynamic representation of a rectangle? What

type of relationships can we identify between elements or attributes

(perimeter, area) within a dynamic representation of a rectangle?

These types of questions are important in helping the students to

think of a rectangle in terms of its properties to represent those

properties dynamically. Indeed, Laborde and Capponi (1994) stated

that the use of dynamic software (in our case Cabri) creates an

environment in which students develop the notion of figure by

focusing on underlying relationships rather than on the particulars ofa specific drawing. For example, some properties that are relevant to

the construction of a rectangle are that such a figure has pairs of

congruent parallel sides, four right angles, perpendicular sides, and

equal diagonals, and attributes like area or perimeter. In this context,

a question related to the conditions to draw a rectangle was posed by

one of the students:

Can we construct a rectangle if we know only its perimeter and

one of its diagonals?

1.3. Second Episode: Understanding and Making Sense of the Task

This involved discussing questions such as:

What does it mean geometrically to know the perimeter of a

given rectangle? How many rectangles can we construct with fixed perimeters?

From a family of rectangles with fixed perimeters, how can weidentify the one that has a given diagonal?

How can we represent algebraically all rectangles of a given

perimeter? How can we represent algebraically the diagonal of a rectangle

in terms of its side?

What does it mean algebraically to determine a rectangle, given

its perimeter and its diagonal?

This discussion led the students to work on two dynamic represen-tations of the problem and one algebraic approach. One dynamic

representation relied on representing the perimeter as a segment that

was used to generate a family of rectangles with fixed perimeters.

ON THE USE OF COMPUTATIONAL TOOLS 363


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Another dynamic representation involved drawing a family of tri-

angles with perimeters equal to the sum of two sides of the rectangle

plus the length of the diagonal.

1.4. Third Episode: The Role of the Tools in Helping Students

Represent and Solve the Task

1.4.1. Rectangles with Fixed Perimeters

A key aspect in this approach was to construct, using dynamic

geometry software, a family of rectangles with fixed perimeters.

Questions that guided this construction included: How can one have a geometric representation of the perimeter

or a diagonal?

What information do the perimeter and the diagonal provide

about the sides of the rectangle?

How is the perimeter information related to the diagonal?

The procedure used to build a family of rectangles with fixed

perimeters is described next:

(1) Represent the semi-perimeter of the rectangle as segment AB

and choose point Q on it. Thus, segments AQ and QB generate

sides EH and EF of a rectangle. With this information

students could draw the corresponding rectangle EHGF

(Figure 1). This is a dynamic construction in the sense that the

sides of drawn rectangles correspond to a particular position of

Q on segment AB (any position of Q determines a particular

rectangle).

(2) Students should notice that by moving point Q along segmentAB, a family of rectangles with fixed perimeters is generated. As a

result, they begin to explore questions like: What is the locus of

point G when point Q is moved along AB? The software becomes

a powerful tool to determine the path left by point G when Q is

moved along AB (Figure 2).The locus of point G when point Q is

moved along AB is the segment ST, and when point Q becomes

point B, then ET will become segment AB. Similarly, when point

Q coincides with point A, segment ES becomes AB. There is an

infinite number of rectangles having fixed perimeters, since, foreach position of Q, there is a corresponding rectangle.

(3) Within this dynamic representation, students observe that any

rectangle inscribed in triangle ETS will have a perimeter equal to



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2(AB). With the use of the software it is easy to determine the

dimensions of the rectangles sides inscribed in the right triangle

EST. They can also observe that they could inscribe two

congruent rectangles in triangle EST, except when the rectangle

becomes a square (Figure 3).

Students complete the construction of the rectangle by drawing a

circle centered on point E and a radius the length of the

given diagonal. The intersections of the circle with segment ST

represent the vertices of the required rectangles. Figure 3 shows that

rectangles EONM and EHGF are sought rectangles. What happens

when the circle does not intersect segment ST? Is there any rela-

tionship between the perimeter and the diagonal to insure the

construction of the rectangle? Students observe that when the circle

with center E and radius of the given diagonal intersects segment STat only one point G, the rectangle is a square (EHGF) (Figure 4)

and any other circle with radius less than diagonal EG will not

intercept segment ST. When the circle only touches one point of

1x

1

y

E

A BQ

H

F G

S

T

Figure 2. Locus of vertex G when point Q is moved along AB.

1 x

1

y

E

A BQ

H

F G

1x

1y

E

A BQ

H

F G

Figure 1. Moving point Q along segment AB generates a family of rectangles

with fixed perimeters.



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segment ST, then the radius of this circle represent the minimum

value of the diagonal to guarantee the existence of the desired

rectangle.

1.4.2. Construction of the Rectangle via the Construction of TrianglesA group of students explores the construction of the rectangle in

terms of drawing a family of triangles with a perimeter equal to the

sum of two sides of the rectangle plus the length of the given diag-

1x

1

y

E

A BQ

H

F G

S

T

N

UM

OW

Z

Diagonal segment

3.03 cm

1.47 cm

1.47 cm

3.03 cm

2.15 cm

2.35 cm

Figure 3. Construction of rectangles inscribed in triangle EST.

Diagonal length

2.20 cm

2.20 cm

1x

1

y

E

A BQ

H

F G

S

T

Figure 4. The length of the diagonal determines the existence of one, two or no

rectangles.



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onal. In this case, the given diagonal AB represented a fixed side ofthe triangle and the other two sides of the triangle as the semi-

perimeter of the rectangle (segment PR). The main steps to achieve

the construction are summarized next:

(1) Segment AB represents the given diagonal, and segment PR is the

rectangles semi-perimeter and Q any point on PR. Draw two

circles, one with its center at A and radius PQ and the other with

its center at B and radius QR. These two circles are intersected at

C and C and triangle ABC is drawn. Indeed, for each position of

point Q a triangle is generated (Figure 5, left).(2) By moving point Q along PR, students observe that vertex C left a

particular path. What is the locus of points C and CwhenpointQis

moved along PR? Students use the available software to determine

that the locus of points C and C when Q is moved along PR is an

ellipse (foci A and B and constant PR) (Figure 5, right).

Again, there are infinite numbers of triangles such that AC + CB

is constant; however, students were interested in finding the triangle

with angle ACB equal to a right angle. To find it, students drew acircle centered on the middle point of the diagonal AB, with a radius

of half the diagonal. That is, they used a result previously studied: if a

triangle is inscribed in a circle and one of its sides is a diameter, then the

opposite angle to this side is a right angle. Thus, the intersections of

the ellipse and the circle determine the vertices of the right triangles

(Figure 6, left). Figure 6, right, represents two positions of the

desired rectangle.

While examining the dynamic representation, students should

notice that there are cases in which angle ACB never becomes a rightangle (Figure 7). In this case it can be observed that the circle, with

center at mid-point of AB and radius of half AB, does not intersect

the ellipse.

A B

P R

Q

C'

C

A B

P R

Q

C'

C

Figure 5. Left Generating a family of triangles; right What is the locus of point

C when point Q is moved along PR?



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1.4.3. Algebraic Approach

Yet another approach used by some students to draw the rectan-

gle involved the use of algebraic procedures. They used x and y

to represent the sides of the possible rectangle and expressed

the corresponding equations associated with the given conditions(Figure 8).

How can the expression be represented graphically? How can the

system of equations be solved? How can we interpret the solutions?

Students can now recognize that one expression represents a line and

the other a circle with center the origin and radius the value of the

diagonal (D). By taking particular values for P and D (Figure 9),

students can represent the system geometrically.

y x P

2

x2 y2 D2

A

P RQ

C'

C

D

D'

A B

P R

Q

C'

C

D

D'

B

Figure 6. Left Identifying right triangles; right drawing two rectangles.

AB

P RQ

C'

C

Figure 7. Case without right angle.



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To solve the system of equations, students could use their calcu-

lators and discuss conditions for the existence of solutions. They

could then recognize that the circle might intersect the line at one

point, in the case of a square, two points, as in Figure 8, or have no

intersection points. Figure 10 shows the general solution of the sys-

tem and a particular solution when P takes the value of 14 and D the

value of 5.Students may then conclude that, to guarantee the solution of the

system, it must hold that 8d2)p2 0; that is, d! pffiffi8

p : Figure 11 showsthat, when d pffiffi

8p , the system has one solution and it means that the

constructed figure will be a square.

Figure 12 represents the graph of y x 7 and x2 y2 14= ffiffiffi8p 2: The coordinates of the intersection points represent thedimensions of the square with perimeter 14 and diagonal 14ffiffi

8p .

1.5. Fourth Episode: Relations to Similar Problems

Based on the solution to the task that involves inscribing a family of

rectangles in a right triangle, a question was posed to the students:

2 x y P

x

2

y

2

D

2

Symbolic representation

y

x

D

Pictorial representation

Figure 8. Translating properties of the rectangle into algebraic expressions.

x

y

y = -x + P/2

x2 + y2 = D2

1

1

Figure 9. Graphic representation of the equations.



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Does the area of those inscribed rectangles change or is it constant as

is the value of their perimeter? If the area changes then where does it

reach its maximum value? Using the software, students calculate the

area of some inscribed rectangles to investigate the behavior of the

areas of these rectangles.

Students would notice that, if they moved point C along segment

ST, the areas associated with the corresponding rectangles changed(Figure 13). Here, students represented the area variation through

the use of two related representations: a discrete table and a graph

representation (Figure 14).

Figure 10. Solution of the system of equations.

Figure 11. When d pffiffi8

p ; there is only one solution to the system.

Figure 12. The system has one solution when P = 14 and D = 14ffiffi8

p .



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It is important to mention that students could generate the graph

of the area function without relying on its algebraic expression.Based on this graphical representation, students could observe that

any line parallel to the X-axis intersects the graph either at two

points, or one point or there is no intersection at all. Here, they

x

y

y = -x + 7

14.00 cm

10.00 cm2

( 5.00, 2.00 )

A T

S

CD

Bx

y

y=-x+7

14.00 cm

12.00 cm2

( 3.00, 4.00 )

A T

S

CD

B1

1

1

1

Figure 13. Identification of area changes.

1

1

x

y

y = -x +7

14.00 cm

12.25 cm2

1

23456789

10

(

1.00 6.00 6.00

2.00 5.00 10.003.00 4.00 12.003.50 3.50 12.254.00 3.00 12.004.50 2.50 11.255.00 2.00 10.005.50 1.50 8.256.00 1.00 6.006.50 0.50 3.25

Side AB (Side AD

Area ofABCE

y = -x2 + 7x

( 3.50, 12.25 )

( 3.50, 3.50 )

3.50 cm

3.50 cm

A T

S

CD

B

Figure 14. Graphical representation of the rectangles areas.



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would notice that, when there is only one point of intersection, the

parallel line is tangential to the function area and, in this case, there

is a maximum value. The table in Figure 14 shows that, for

x = 3.50 (the length of side AB), the graph reaches its maximum

value of 12.25.

In an algebraic context, when the given perimeter is 14 units, the

equation of the line becomes y = )x + 7, whose domain is the

interval (0, 7). Thus, for any x value on that interval, then the cor-

responding area of the inscribed rectangle with side x will be(x)()x + 7) = )x2 + 7x. Here, students could recognize that this

expression represented a parabola, and they utilized the calculator to

represent it graphically. On the graph, they located the maximum

value of the area directly (Figure 15).

Another question that students were asked to discuss was: What if

we now have the area (instead of the perimeter) and the diagonal as

data; can we also draw the corresponding rectangle?

Students could commence by assigning particular numbers to the

value of the area and the diagonal to represent graphically bothequations with the help of the software. Thus, they represent

graphically the circle x2 + y2 = 52 and y 12x

and identify the

intersection points as the vertices of the desired rectangle (Fig-

ure 16).

Students may observe that the intersections of line

y = )x + 7 and the graph of the area function (y 12x

) are ex-

actly the points that gave the dimensions of the rectangles that

were asked-for (Figure 17). That is, the rectangle with area 12 cm

and diagonal 5 cm has dimensions or sides of lengths 3 cm and4 cm.

The graphical representation leads them to explore the variation in

the perimeters of those rectangles whose one vertex lies on the area

Figure 15. Graphical representation of the function area.



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graph. Again, the software becomes a powerful tool to determine the

graph of the perimeter and find where this graph approaches its

minimum value (Figure 18).

x

y

y = -x + 7

( 4.00, 3.00 )

( 3.00, 4.00 )

14.00 cm

14.00 cm 12.00 cm2

12.00 cm2

y = 12/x

A

CD

B

P

Q1

1

Figure 17. Two rectangles with the given condition.

x

y

3.00 cm

4.00 cm

x2 + y2 = 52

( 4.00, 3.00 )

( 3.00, 4.00 )

5.00 cm

Area = 12.00 cm2

1

1

Figure 16. Construction of rectangles with given area and diagonal.



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1.6. Reflections

It is important to identify themes and features of mathematics

practice that appear as relevant during the development of the

problem-solving sessions:

(1) Formulation of questions: The use of the software demands that

students think of mathematical objects in terms of their properties.

This may encourage students to formulate and explore distincttype of questions. For example, the question Can we construct a

rectangle if we know only its perimeter and one of its diagonals?

might emerge when students discuss ways to represent this figure

using the dynamic software commands.

(2) Dynamic representations: The use of the software may help

students to identify and explore fundamental mathematical

relationships. For example, moving a point on a segment that

represents the semi-perimeter should lead students to recognize

that the locus of one vertex of the rectangle is a segment. Thissegment is the hypotenuse of a right triangle, and all rectangles

inscribed in that right triangle have the same perimeter

(Figure 2). Here, students would not only associate geometric

1x

1

y

14.15 cm

2x2 - xy + 24 = 0

Side AB Side BC PerimeterArea ofABCE

2.82 cm

4.26 cm

12.00 cm2

123456789

10

2.59 4.64 12.00 14.452.74 4.38 12.00 14.243.02 3.98 12.00 13.993.26 3.69 12.00 13.883.40 3.53 12.00 13.863.60 3.33 12.00 13.873.74 3.21 12.00 13.903.89 3.09 12.00 13.954.02 2.98 12.00 14.014.25 2.82 12.00 14.15

xy - 12 = 0

A

CD

B

Figure 18. Identifying the rectangle with minimum perimeter.



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meaning to a particular case (the family of rectangles with fixed

perimeters) but also pose and explore other questions: How does

the area of those rectangles change? Where does the area reach its

maximum value?

(3) Connections: Can we represent the problem in different ways?

Students are encouraged to look for different ways to represent

and solve the problems. For example, when students focus on

constructing a family of triangles by taking the given diagonal as

one side and the sum of the other two sides as one segment, then

they would recognize that they are dealing with the definition of

an ellipse. Here, they also rely on the use of a mathematical resultpreviously studied to identify from that family of generated tri-

angles those that were right triangles to construct the desired

rectangle. Thus, the use of the software offers students the

opportunity to apply and relate concepts that they may have

studied in subjects like algebra, analytic geometry, and calculus

(Santos-Trigo, 2006).

(4) Use of different tools: The use of calculators represents an

opportunity for students to relate particular representations of

the problem to a general model of the situation. The use of thistool is also important for dealing with algebraic operations and

visualizing graphically some mathematical relationships. Thus,

students are able to explain and express, in general terms, for

example, conditions for the construction of the rectangle and

then to validate results that emerge from representing the prob-

lem dynamically. That is, rather than privileging the use of one

tool, students should be encouraged to utilize more than one

computational tool to identify and support mathematical results

(NCTM, 2000).(5) Line of thinking: The systematic use of both dynamic software

and calculators helps students to recognize that initial questions,

posed by the students themselves, can be approached and

examined from different perspectives. As a result, they have the

opportunity to examine and contrast different types of repre-

sentations, which allows them to go beyond reaching a single

solution and to pose and explore other related questions. For

example, while working with dynamic representations, in addi-

tion to visualizing the behavior of some mathematical relations,

students could operate with quantities assigned to some parts of

the construction (segments lengths, angles measurements,



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perimeters, and areas) without using algebraic representations.

This process becomes important for students to identify mathe-

matical relations and their meaning.

(6) Mathematical richness: A simple question that involves the con-

struction of a rectangle provides conditions for students to explore

and reflect on the use of distinct concepts and representations to

examine mathematical relationships and their meaning. Here, the

students use of the tools seems to facilitate not only the identifi-

cation of interesting mathematical relations but also the meaning

associated with various types of representation of the problem.

In short, thinking of different ways to solve a problem, andexploring connections and relationships of the original statement,

seems to be a crucial activity that all students should engage in during

their mathematical learning (Schoenfeld, 1985).

REFERENCES

Laborde, C. and Capponi, B. (1994). Cabri-Ge ometre constituant dun milieu pour

l

apprentissage de la notion de figure ge

ome

trique.DidaTech Seminar

150:175218.

National Council of Teachers of Mathematics (2000). Principles and standards for

school mathematics.

Santos-Trigo, M. (2006). Dynamic representation, connections and meaning in

mathematical problem solving. For the Learning of Mathematics 26(1): 2125.

Schoenfeld, A.H. (1985) Mathematical Problem Solving. Orlando, FL: Academic

Press.


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