Dynamic Geometry Sofware Research Papers

An Alternative Approach to Proof in Dynamic Geometry

1998, From Lehrer, R. & Chazan, D. (Eds). (1998). Designing Learning Environments for developing Understanding of Geometry and Space. Mahwah, NJ: Lawrence Erlbaum Publishers, pp. 369-393.

In this chapter, the traditional approach of introducing proof in geometry as a means of verification is critiqued from a philosophical as well as a psychological point of view, and in its place an alternative approach to the... more

Type 3: False Conclusion. Another approach is to give students a dia- gram like that in Fig. 15.3a for any arbitrary triangle ABC, followed by a “proof” that it then follows that CA = CB (i.e., that any triangle is isosce- les). Children are then told that the obviously false conclusion arises be- cause of the inaccuracy of the diagram (one of the points D and F always falls inside and the other outside, as shown in Fig. 15.3b), and are cau- tioned to be careful about how a figure looks or is drawn—that, in fact, all sketches are essentially unreliable and we should only rely on our power of reasoning.

A CRITICISM OF THE TRADITIONAL APPROACH TO THE TEACHING OF PROOF IN GEOMETRY

FIG. 15.4 Underpinnings of personal conviction, When investigating the validity of an unknown conjecture, mathe- maticians normally not only look for proofs, but also try to construct counterexamples at the same time by means of quasi-empirical testing be- cause such testing may expose hidden contradictions, errors, or unstated assumptions. Frequently, the discovery /construction of counterexamples necessitates a reconsideration of old proofs and the construction of new ones. Personal certainty consequently also depends on the continued ab- sence of counterexamples in the face of quasi-empirical evaluation. More generally, the attainment of personal conviction depends on positive justi- fication and/or negative falsification (see Fig. 15.4).

Looking Back. Notice that the property of equal adjacent sides (or ar axis of symmetry through one pair of opposite angles) was not used at all In other words, we can immediately generalize the result to a “perpendic ular” quad as shown in Fig. 15.5b. Furthermore, note that the general re sult was not suggested by the purely empirical verification of the origina conjecture. Even a systematic empirical investigation of various types o quadrilaterals would probably not have helped to discover the genera case because most people would probably have restricted their investiga tion to the more familiar quadrilaterals such as parallelograms, rectangles rhombuses, squares, and rectangles. (Note that from the preceding expla nation we can also see that EFGH will always be a parallelogram in an} quadrilateral.)

Formulation. — If equilateral triangles DAB, EBC, and FCA are constructe on the sides of a right triangle ABC with ZB = 90°, then DC, EA, and F are concurrent (see Fig. 15.7). If we call the observed point of concurrenc O, then it looks as if the six angles formed at O are each equal to 60°. B measurement and transformation on Cabri or Sketchpad, this can easil be confirmed (see Fig. 15.7a), In other words, quadrilaterals ADBO, BECC and CFAO must be cyclic because the exterior angles are equal to the of posite interior angles (60°). We can now use this observation to produc the following explanation. (Note that this illustrates another function ¢ quasi-empirical testing and exploration, namely, assistance in the discov ery /invention of a deductive explanation.)

detinition to crossea quaaruateralis. (1 nis 1s a strategy oiten used in mat ematics when extending certain concepts beyond their original domai1 Suppose we walk clockwise from A to B, B to C, and so on, arour the perimeter of the convex quadrilateral shown in Fig. 15.11. The intern angle at each vertex will then be the angle through which the next sic must be rotated clockwise (with the vertex as rotation center) to coincic with the previous side. In the same way we can now determine the inte nal angles of the crossed quadrilateral by walking around the perimeter. shown. This leads us to the surprising conclusion that the two reflexir angles indicated at A (360° - ZBAD) and D (360° - ZADC) are the “inte nal” angles of the crossed quadrilateral ABCD. We can now also calculate the sum of the interior angles of a cross«

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Engaging in problem posing activities in a dynamic geometry setting and the development of prospective teachers’ mathematical knowledge

by Atara Shriki

2010, The Journal of Mathematical Behavior

In the present study we explore changes in perceptions of our class of prospective mathematics teachers (PTs) regarding their mathematical knowledge. The PTs engaged in problem posing activities in geometry, using the “What If Not?”... more

This article appeared in a journal published by Elsevier. The attached copy is furnished to the author for internal non-commercial research and education use, including for instruction at the authors institution and sharing with colleagues. Provided for non-commercial research and education use. Not for reproduction, distribution or commercial use.

The distribution of the negated attributes and alternatives chosen by the PTs for further investigation. Table 2

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Mediational activities in a dynamic geometry environment and teachers' specialized content knowledge

by Arthur B. Powell

A B S T R A C T Dynamic geometry environments can support learning of geometry through mediating learners' activity. To understand how dynamic geometry environments mediate the activity of mathematics teachers, we used Rabardel's... more

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Examining Technology-Mediated Communication Using a Commognitive Lens: the Case of Touchscreen-Dragging in Dynamic Geometry Environments

by Oi-Lam Ng

In this paper, I introduce Sfard's discursive framework to examine secondary-school calculus students' communication during exploratory activities mediated by the use of touchscreen dynamic geometry environments (DGEs). Six pairs of... more

Fig. 1 a Page | of the “area-accumulating function” sketch shows the area under a constant function, b Page
2 shows the area under a linear function, e Page 3 shows the area under a quadratic, d Page 4 shows the area
under the sine function, e A sketch related to the definition of a derivative and f a sketch related to derivative

functions
mathematical notations in ,J* f(t, the sketch showed the value of ‘A’ instead. For
instance, Fig. la shows that the area under the function “/(t) = 1” is “A= 4.70” when
the bounds are set to “a= 0” and “x= 4.70”, and the traces in green represent the
corresponding area-accumulating function when ‘x’ was dragged from x=0 to its
current position. Each page differs by the particular function, f(t), shown. Page 1 shows
the function f()= 1 (Fig. la), Page 2 shows f(t) = t (Fig. 1b), Page 3 shows f(t)=
(Fig. lc), and Page 4 shows f(t) = sin(t) (Fig. 1d). The sketch is designed to be
compatible with students’ prior knowledge relating the “green traces” as the graph of
“accumulated area under f” from ‘a’ to ‘x’. To facilitate students’ exploration of “area
accumulation”, when f(x) > 0 and when f(x) <0, the areas were colour-coded different-
ly. Figure le, f shows the screenshots and design consideration of two other dynamic
sketches mentioned in this paper and are related to the definition of a derivative and
derivative functions, respectively. The former (Fig. le) offered two draggable points on
the touchscreen-DGE, both of which were points of the given function and allowed the

Fig. 2 a, b Selected snapshots and screenshots of Ivy’s dragging (Turn 1.3)
devices.

Next, she dragged ‘x’ continuously from x = | to x =3 and uttered, “One times two is
two, one times three is three” (Fig. 2b). It was clear that Ivy was stating how the area of
the rectangles could be calculated numerically. The timing of her utterance
corresponded with the very state of the rectangle as ‘x’ was dragged continuously,
suggesting that she was coordinating her speech and hand movements on the
touchscreen. Thus, Ivy was talking about the change in area under f(t) = 1 numerically,
from x=1 to x=2, then x=3. Furthermore, the change in area was communicated
discretely, one moment at a time, as opposed to continuously, indicating that Ivy’s
gestures functioned as a static visual mediator to speak about mathematical objects (x =
1, x=2 and x=3). However, interestingly, her gestures in between her utterance “one
times two is two” and “one times three is three” were performed while she was
dragging continuously on the touchscreen, rather than the traditional “point and hold”.
Compared with the deictic gesture that Ivy performed at the beginning of Turn 1.3,
some differences were observed: First, her hand, with her pointing finger, was moving
steadily while she spoke, and second, her pointing finger was touching the iPad during
this span. These characteristics were mobilised through the use of touchscreen-DGEs,
namely touchscreen capability and continuous-dragging affordances.

The act of touchscreen-dragging on the DGE was so inextricably linked to Ivy’s

Fig. 3 a, b Selected snapshots and screenshots of Ana and Tammy’s dragging (Turns 2.1—2.5)

Fig. 4 Jay dragged ‘x’ rapidly, thereby quickly forming a set of green traces in the form of F(x)=o!* sin()dt =
cos(x) — (— cos(0)) = — cos(x) + 1

Jarier in their exploration, Jay and Katie’s discourse centred around the movement of the
sreen traces. As evident in Tum 3.1, Jay shifted his discourse from attending to the
novement to the general shape of the green traces. While on the page showing f= f°,
hen f(t) = sin(t), he referred to these functions in their algebraic forms, “degree 3” and
‘sine”, and predicted that the green traces ought to be in “degree 2” and “negative cosine”,
espectively. These words used suggest that he was attending to the green traces as one
ybject—the area-accumulating function, A(x). He then verified this proposed relationship
yetween f(x) and A(x) by tracing the shape of A(x) through his “rapid-dragging” routine.
[his routine allowed him to reveal the shape of the green traces sooner. In addition, by
lragging ‘x’ back and forth, more green traces could be left on the screen, which made the
shape of the green traces more visible. Both of these characteristics of the “rapid-dragging”
outine suggest that, in this episode, Jay was engaging in a mathematical discourse about
sncapsulating the set of ordered pairs (x, A(x)) into a singular object, the area-accumulating
unction, A(x). Sfard (2008) defines encapsulation as the act of assigning a noun or
yronoun (signifier) to a specific set of discursive objects, so that some of the stories about
he members of this set that have, so far, been told in plural may now be told in singular.

Jay communicated the green traces as one function both algebraically (using
words like “degree 2” and “cosine”) and graphically (through his rapid-dragging
outine). He also was actively predicting and verifying the relationship between
he two fimnctionse The reneated nee of ranid-dracoinga wae ahcerved hy other

Fig. 5 Yee dragged a point along f(x) = 3x— 2 while observing the slope of f(x) and “derivative of f”
The words “doesn’t change” and “always” suggest that Yee had been observing the
change in the slope of tangent to f(x) = 3x— 2 at different points on f(x). To say that the
“derivative of fis always three” , one needs to compare the derivative of f(x) across
different values of ‘x’. This was achieved through Yee’s dragging of ‘x’ before his
utterance in Tur 4.1. Besides comparing, Yee also was reasoning about why “the
slope doesn’t change”. Evident in the word “because”, he was providing a mathemat-
ical explanation in that the derivative always was three, a constant; thus, the slope does
not change in the presence of dragging ‘x’.

In response. Melissa communicated the invariance of the “‘tangent slope” of f/x) =

Fig. 6 Melissa responded while she dragged the a on f(x)
Examining Technology-Mediated Communication Using a Commognitive...

Fig. 7 Ivy said the area “becomes negative” while Larry was dragging ‘x’ toward the left of the y axis
In Turn 5.1, Ivy used the conjunction “because”, followed by the question marker
“right”, with a high intonation at the end to question whether a causal relationship
existed between the sign of ‘x’ and the sign of “area”. She seemed to notice that when
the sign of ‘x’ was negative, the area accumulated under /(t) = ¢ on the interval [0, x]
was negative. Thus, her questioning can be taken as a conjecture as to why “the area
becomes negative”. At this point, it was clear that Ivy already assigned the green trace
as the graphical representation of area.

Tt bh eee. Be Tes OT Ae Rm its Aa eee me UV ae bh aise Oke Meee t5

Examining Technology-Mediated Communication Using a Commognitive...
Fig. 8 Larry dragged ‘x’ toward x= 0 and said “when it’s at zero”

Examining Technology-Mediated Communication Using a Commognitive...
Fig. 9 Two students simultaneously dragging on the touchscreen-DGE

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A Systematic Review on Task Design in Dynamic and Interactive Mathematics Learning Environments (DIMLEs

by Mustafa Çevikbaş

2021, A Systematic Review on Task Design in Dynamic and Interactive Mathematics Learning Environments (DIMLEs

Task design constitutes a growing core of research in mathematics education. In particular , task design in Dynamic and Interactive Mathematics Learning Environments (DIMLEs) has become very popular, although it remains under-researched.... more

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The Future of Secondary School Geometry

by Michael de Villiers

1996

This paper discusses some renewed interest in geometry research as well as the Van Hiele theory of learning geometry, the USEME teaching experiment in 1977/78, and some implications for teaching of new developments such as dynamic geometry.

(meet in the same point) (Figure 1b). connecting opposite vertices are concurrent circle, then the three lines (the diagonals) three points of intersection of the opposite If a hexagon is inscribed in a circle, then the Brianchon's theorem

One of the important concepts in the classification of border patterns and types. An obvious property of any tiling is that of a repetition of the pattern. If a

Slightly adapted version of Plenary presented at the SOSI Geometry Imperfect Conference, 2-4 October 1996, UNISA, Pretoria. Figure 3: Examples of fractals

Figure 5: Gergonne point & Gergonne-Soddy-Incenter line Slightly adapted version of Plenary presented at the SOSI Geometry Imperfect Conference 2-4 October 1996, UNISA, Pretoria. The Gergonne point G of a triangle is the point of concurrency of the three line

Figure 6: Van Aubel rectangle-generalisation In Figure 7, EG is always congruent to FH. Also KM is perpendicular to LN where

Slightly adapted version of Plenary presented at the SOSI Geometry Imperfect Conference 2-4 October 1996, UNISA, Pretoria. Just a brief perusal of some recent issues of mathematical journals like the

By using task-based interviews, Burger & Shaughnessy (1986) characterize

For example, in a triangular tessellation pattern such as shown in Figure 8.

and explanation that the opposite angles of a parallelogram are equal.

ire conceptually important for analysing the geometric properties embedded in Process versus product teaching in geometry The distinction between "processes" and "products" in mathematics education is a Proposals of MASA regarding the South African Mathematics Project, stated:

process of proving also has its own product, namely a proof, which should b Jue to limitations of space, we shall here mainly focus on the handling of The distinction between some of the main processes and products of formal made. In addition, they're not even allowed to try their own hand at baking! Benchara Blandford wrote (quoted in Griffiths & Howson, 1974: 216-217): Traditional geometry education of this kind can be compared to a cooking

Figure 12: Construction & measurement sychologically, constructions like these are extremely important for t ansition from Level 2 to Level 3. It helps to develop an understanding of t ifference between a premisse and conclusion and their causal relationship; ther words, of the logical structure of an "if-then" statement. Logically each 1e above statements can be rewritten in this form. For example, the Il. atement could be rewritten as: "If a quadrilateral has both pairs of oppos des parallel and one pair of adjacent sides equal, then it is a rhombus Cie. h 1 sides equal, perpendicular bisecting diagonals, etc)". Smith (1940) report arked improvement in pupils' understanding of "if-then" statements by letti p pup g y ee a a) a ee ee ee ee ee ee Psychologically, constructions like these are extremely important for the

also extremely valuable in constructing counter-examples for false conjectures. Figure 14: Pupil research in geometry However, the development of dynamic geometry has also necessitated a radical

Slightly adapted version of Plenary presented at the SOSI Geometry Imperfect Conference. 2-4 October 1996, UNISA, Pretoria. Formulation The line segments consecutively connecting the midpoints of the adjacent sides Deductive explanation A deductive analysis shows that the inscribed quadrilateral is always a rectangle,

Figure 17: Dynamic transformation of parallelogram into this particular area is needed. understand this even at Van Hiele Level 1 (Visualization), but further research The ability to quickly and efficiently transform geometric configurations with 17. In fact, it seems quite possible that pupils would be able to accept anc

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Genese instrumentale du deplacement en geometrie dynamique chez des eleves de 6eme

by Angela Maria Restrepo

2008

Je tiens tout d'abord à remercier à Mme. Colette Laborde et Mme. Sophie Soury-Lavergne pour m'avoir guidée dans cette merveilleuse étape de ma vie, pour m'avoir donné le goût de la recherche et m'avoir encouragée à chaque pas. Cette thèse... more

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Dual Generalizations of Van Aubel's theorem

by Michael de Villiers

1998, The Mathematical Gazette, pp. 405-412

This paper generalizes Van Aubel's theorem to similar rectangles and similar rhombi on the sides of a quadrilateral. Dynamic Java sketch at: http://dynamicmathematicslearning.com/aubelparm.html

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A Hexagon Result and its Generalization via Proof

by abc def

2007

This paper presents the discovery of a hexagon result on Geometer's Sketchpad and its generalization via proof for any 2n-gon. The result is : If ABCDEF is a hexagon with opposite sides parallel (not necessarily equal), then the... more

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The Effectiveness of an Enrichment Program Using Dynamic Geometry Software in Developing Mathematically Gifted Students' Geometric Creativity in High Schools

by Mohamed El-Demerdash

It goes without saying that students can learn and develop their creative potential if appropriate programs are used that successfully teach them the creative skills and operations necessary. However, mathematics in general and geometry... more

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Effects of Using Dynamic Geometry Activities on Eighth Grade Students’ Achievement Levels and Estimation Performances in Triangles

by Cahit Aytekin and

In this study, we administrated dynamic geometry activities which provide students opportunities to explore and estimate geometric figures to connect measurement estimation with geometry. The aim of the study is to investigate effects of... more

As an example for the activities in the study first two activities were described in detail. The first two activities address to first objective which is about to link sum or difference of lengths of two sides of triangle with length of third side. In these activities, some hints are given to make GeoGebra usage easy for students, to eliminate the difficulties which may result from the usage of GeoGebra, and to focus on mathematical objectives. In the first activity, a triangle ABC with |AB| =6 cm and |AC| =8 cm is given (Figure 1). Figure 1. First activity: linking sum or difference of lengths of two sides of triangle with length of the other side

In this triangle, length of |BC| changes dependently a slider. Students are asked to change length of |BC| freely by using slider and note the |BC| and perimeter of the ABC triangle for ten different positions of slider. Afterward, they try to answer questions in worksheet according to their observations. In this activity, students are supposed to investigate the relationships between lengths of sides and triangular inequality. The purpose of the second activity is to assess students’ learning about triangular inequality concept (Figure 2).

In this study, Triangles Achievement Test (TAT) and Estimation Test in Triangles (ETT) were used as data gathering tool. Both the TAT and ETT were developed by the researchers who are specialist in elementary mathematics education. In figure 3 and figure 4 some sample questions of the TAT and ETT were given respectively. The real versions of the tests were developed in Turkish. The examples of items given here were translated from Turkish to Enalish.

When the results for students’ estimation and academic performances in triangles were considered separately, there were significant differences, using a Bonferroni adjusted alpha level of .025, in both students’ estimation performance scores (F (1, 58) = 7.26, p = 0.009, partial eta squared = 0.111 or Cohen’s f* = 0.125) and academic performance scores (F (1, 58) = 7.124, p = 0.010, partial eta squared = 0.109 or Cohen’s f? = 0.122) (Table 2).

Table 2, MANOVA Results; Test of Between-Subjects Effects

Table 3. Mean Scores of TAT and ETT In addition, it was revealed that students in dynamic geometry supported instruction group achieved slightly higher levels of academic performance (M = 72) than students in tradition instruction group (M =59.67).

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Place Kicking Locus in Rugby

by Michael de Villiers

1999, Pythagoras, No. 49, pp. 64-67.

This paper examines the best position with the largest viewing angle from which to take a place kick in rugby after a try has been scored, as well as exploring the locus of this point.

The place kicking problem in rugby is a lovely problem that I first came across in a presentation by a well loved doyen of mathematics education in South Africa, namely, Tickey de Jager (then teacher at Rondebosch Boys High School, now retired). The problem also appears in his book (De Jager, 1996: 158-161), as well as in Picciotto (1999: 19-24) where it is contextualised (a little differently) within soccer. With the upcoming World Cup Rugby it seems appropriate to revisit this problem as an example of the application of mathematics to the real world.

Many kickers seem to think that the further they go back (within their kicking limit) the better, but that is not necessarily the case. If we ignore the actual distance to be kicked, the best position is clearly where the angle ACB subtended by the posts along TF is a maximum. But how can this maximum angle be found?

With Sketchpad we can, of course, make a dynamic sketch, measure the angle ACB, and find its maximum value simply by dragging C along line TF. However, the solution can also be found elegantly by imagining a series of expanding circles with AB as chord (see Figure 2). Note that in each of the expanding circular arcs all the inscribed angles are equal (inscribed angles on the same chord). Furthermore, note that these sets of equal angles on the circular arcs decrease as the circles get larger (from the exterior angle theorem for a triangle, angle ADB > angle AEB > angle AFB > angle AGB). It therefore follows that the optimal solution would be found where one of the expanding circles touches the line TF (i.e. is tangent to it).

From the preceding discussion, it should now also be clear that our model is based or an over-simplification of reality, as we have not accounted at all for the height of the crossbar above the ground. (Basically we had assumed it was at ground, which is perhaps not too bad if one is far away, but the closer one is, the more important 1 becomes). In fact, as shown in Figure 6, the required tangent circles lie on a series o1 planes running through the crossbar with the optimal positions located where these planes cut the ground. The relationship between the optimal distance y on these inclined planes to the corresponding distance z on the ground is given by the theorem of Pythagoras. If we let the height of the crossbar be h, then 2? aa? ah =x -@2_h? which gives us an improved hyperbolic curve on the ground. To simplify matters, let's acauime fb — 7 and draw; hnth the ariaginal hyunerhknla (M\) and the imnraved ane (CM) ar It is certainly tempting to consider painting these hyperbolic curves onto rugby fields so that all a kicker would have to do is to move from the tryline perpendicularly until he meets the curve. However, how well does this mathematical model fit with reality? As can clearly be seen from either Figure 4 or 5, as a try approaches the posts, the hyperbola rapidly approaches an upright post. This means that if a try is scored very close to the posts, the largest (optimal) angle is also very close to the posts. Clearly, his is impractical as the ball must also be kicked over the crossbar, and if one is too close to the posts, the angle of elevation becomes high, increasing the difficulty of the kick. (In the extreme case, consider a try being scored directly behind one of the upright posts, and where the optimal position is given at the base of the upright!). It would therefore appear that our model is only likely to give reasonably good results for tries scored further from the posts.

It also seems natural to ask: where is the optimal position to kick from if a try is scored inside the goal posts, rather than outside? If we ignore the height of the crossbar (the angle of elevation), the largest kicking angle is obviously on the tryline (with an angle of 180°). Again this is completely impractical as it would be impossible to kick the ball over from a position directly below the crossbar. Therefore, the kick should not be taken too close to the tryline (as the angle of elevation is then too high), but also not too far away (as the kicking angle then becomes too small). The kicker therefore needs to find some balance between the angle of elevation and the kicking angle, depending on his own preferences or strengths.

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Conjecturing, refuting and proving within the context of dynamic geometry

by Michael de Villiers

2014, Learning and Teaching Mathematics

Two convincing geometry conjectures of points appearing to be collinear are given that easily pass a normal 'drag-test' with dynamic geometry software. Another related conjecture is proved using a spiral similarity, but counterexamples to... more

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Geoproof, a user interface for formal proofs in geometry

by Julien Narboux

2007, Mathematical User-Interfaces Workshop

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Interactive geometry labs – combining the US and Russian approaches to teaching geometry

by Valery Ryzhik and

2008, International Journal of Continuing Engineering Education and Lifelong Learning

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... more

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A generalisation of the Spieker circle and Nagel line

by Michael de Villiers

2006, Pythagoras 63, pp. 30-37

This paper generalizes the incircle of the median triangle (Spieker circle) to an inscribed conic, together with an associated generalization of the Nagel line. Dynamic sketch at:... more

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Exploring mathematical patterns using dynamic geometry software

by Atara Shriki

Our experience indicates that many students view mathematics as a collection of theorems and facts that were discovered by intelligent mathematicians and all they have left to do is to study them carefully. We believe it is essential to... more

In our experience, many students view mathe- matics as a collection of theorems and facts that were discovered by intelligent mathemati- cians and all they have left to do is to study them carefully. We believe it is important to help students change this view (Lavy & Shriki, 2003). One possible way is to enable students to modify the attributes of a given situation, and then pose problems that concern the newly generated situation (Brown & Walter, 1993). In this paper we wish to demonstrate the simplicity of such a process, and exemplify the idea that even a fundamental situation can serve as a trigger for discovering unknown patterns. Looking at a triangle, one can think of In our experience, many students view mathe- present an example of surprising and unex-

the simplicity of such a process, and exemplify

the sides m times its length? In Figure 4, BF = ka, CE = lb, AD = mc.

Adding, this explains the result A; = 5Ao. area(ABEF) = 2area(AABC); area(AHGD) = 2area(AADC); area(AHAE) = 2area(AABD); and area(AFCG) = 2area(ABCD).

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Geometría dinámica y razonamiento científico: Dúo para resolver problemas

by Camilo Sua

2019, Educación Matemática

Resumen: En este artículo presentamos la resolución de un problema de geometría por parte de una pareja de estudiantes de grado noveno de educa-ción básica secundaria, que está relacionado con el lugar geométrico de los centros de las... more

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The evaluation of a pre-service mathematics teacher’s TPACK: A case of 3D shapes with GeoGebra

by Dr Ipek Saralar-Aras

2018, International Journal for Technology in Mathematics Education

This paper describes a pre-service mathematics teachers’ technological pedagogical content knowledge (TPACK) during her school experience. This study focuses on how the participant taught different views of three-dimensional objects in a... more

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Problem posing in a dynamic geometry environment and the development of mathematical insights

by Atara Shriki

Engagement in problem posing activities is among the significant activities of teaching and learning mathematics. This engagement might become richer when technology is involved. In this paper, we describe the experience of teachers who... more

The third example and Figure 3 are taken from Irina's portfolio: could clearly see on the graph the point from which I have started". The second example and Figure 2 are taken from Gail's portfolio: would change. But no! And this was the amazing part, because I was certain that the pattern is related to

the measures of the angles of the original triangle. But the graph showed that only the relations between "Several weeks ago I assigned my students the following problem: ABCD is a square. Point E lies on side

equation)...Indeed, as you can see, if |EC|=a, |EB|=ka, than f(k) =2k+1".

The problem posing approach embedded in the WIN strategy enabled each teacher to engage in problems The problem posing a pproach embedded in the WIN strategy enabled each teacher to engage in problems that relate to his/her own domain of interest (Mason, 2000). As can be seen from the teachers' works, the dynamism feature of the DGS and the WIN strategy are interwoven in a natural way, and in fact stimulate and nurture each other. Specifically, the WIN question and the need to determine which components of a figure are fixed and w the teachers had to ac hich are unfixed appear to go hand in hand, since in order to progress in their work, knowledge that a figure consists of relationships among its components, and that a functional dependency exists among them (Holzl, Healy, Hoyles & Noss, 1994). nd so on) would also yield some regularity, and perhaps there is a 'meta rule' that governs everything" and nurture each other. Specifically, the WIN question and the need to determine which components of a figure are fixed and which are unfixed appear to go hand in hand, since in order to progress in their work,

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Saber suficiente no es suficiente: un estudio de los comportamientos metacognitivos al resolver problemas de demostración con el apoyo de la geometría dinámica

by Camilo Sua

2017

Los problemas de demostración, vistos como una instancia de la resolución de problemas, demandan en el individuo que los afronta que ponga en juego distintos conocimientos y habilidades instrumentales cuando cuenta con el apoyo de la... more

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Interactive Whiteboard in Mathematics Education

by Piroska Biró

Picture 2. shows the online solving of the exercises and the plotting functions. A lesson of practice followed, where students solved a series of problems in Pnine nf triton than epnmnarad and rhackad thair reaciite with the haln nf WR A pupil dictated the solutions of the tasks to another one, who typed it. After ach solution, the problems were explained, supplemented and corrected.

Picture 3. Picture of the board — Gay Lussac’s second law. The new lesson introduced Gay Lussac’s second law. The students were taught in an inductive way. The teacher used animation to let students find out and formulate the law. The students were active during the lesson, they were deeply involved in the topic, since the animation showed the changes accompanying physical phenomena verv well. Their interest was also verified bv the number of questions they asked.

Learning organisation consisted in frontal work, individual work, and then common discussion, group work in the framework of a game. By using the interactive whiteboard, we could save a lot of time, and it was possible to practise more. The lessons started by revising general concepts; students could get acquainted with the characteristics of the quadrangle in an inductive way. Their knowledge was broadened starting from the special quadrangles towards the more general ones. The problems were taken from the Realika online database, and they were solved by students in groups.

The sole device used was IWB. Learning organisation included frontal work, individual work, and then common discussion, group work. It is important to highlight the constructing and modifying options of the diagrams made by the program Geogebra — diagrams which would be very difficult to do using traditional methods.

Most of the students know and use some dynamic geometric program, and they have had the opportunity to try and use other programs as well at school. The personality of the teachers is thought to have a great influence on what programs their students know. The teacher controls the learning process at school, and the students start reading about and using certain mathematical programs on the teacher’s advice. Graph 1. shows the frequency of the programs used in mathematics lessons. It must be remarked that it is a major advantage of the school that teachers have internet access in every classroom, and, as the following diagram indicates, they are fond of using it. Graph 1. shows the frequency of the programs used in mathematics lessons

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Geometria Dinâmica em tablets: Estudo de caso com o aplicativo GeoGebra

by Larissa Moreira

Recursos para Geometria Dinâmica favorecem práticas pedagógicas investigativas. Assim, é importante que a formação de professores contemple o uso dos mesmos. Nesse contexto, este artigo visa analisar um estudo de caso promovido na... more

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Computer Assisted Education's Effects of Learning the Eighth Grade Math Subjects with Geometer's Sketchpad on Students' Performance Grades and Academical Achievements and Students' Opinions: A Mixed Method Study

by Firat Hayyam Sabuncu

2021, Article

In this study, it was aimed to show the effect of computer assisted learning on the academic achievement, performance levels and computer attitudes of 8th grade students in "Transformation Geometry" subject and determine students'... more

One of the students used the following expressions on the theme of "individual study”:

Sabuncu, Firat Hayyam, and Ipek, Jale. (2021), Computer Assisted Education’s Effects of Learning the Eighth Grade Math Subjects with Geometer’s Sketchpad on Students’ Performance Grades and Academical Achievements and Students’ Opinions: A Mixed Method Study. In: Education Quarterly Reviews, Vol.4 Special Issue 1: Primary and Secondary Education, 307-322.

Table 1: Mean Success Points of Classes for Previous Years According to the Data Taken from the e-School System As seen in Table 1, it has been determined that 8-A class has the mean of 74,624 and 8-B class has the mean of 74,359. Based on these data, it could be thought that the academic achievement levels of 8-A and 8-B classes were close to each other. Since the same mathematics teacher has been teaching in these two classes, it was thought that they could be suitable classes for the study. In order to determine whether the 8-A and 8-B classes were statistically equivalent or not, the Transformation Geometry Achievement Test was applied to both classes as a pretest and the data sets were analyzed. The findings obtained are given in Table 2.

Table 2: Independent Sample T-Test for Equivalence of Control and Experimental Groups According to the results of the Independent Sample T-Test, no statistically significant difference was found between academic achievement test pretest score averages of the control group and the experimental group (t (40) = - 0.55, p = 0.585, r = 0.04). The effect size value shows that there is a small difference between the mean scores of the groups (r = 0.04). It was determined that the groups chosen before the study were statistically equivalent to each other. Based on these data, 8-A class was assigned to the experimental group and 8-B class was assigned to the control group randomly.

The subject of “transformation geometry” was chosen because of the fact that reflection, symmetry, rotation and translation are among the basic concepts of geometry and it was determined that students have misconceptions about it in the literature. The dynamic geometry software, GSP, was chosen because it is an interactive teaching tool and enables constructivist teaching environments. Transformation geometry sub-learning areas were investigated before the implementation of the study, which was designed using a pretest-posttest control group quasi-experimental design. All the outcomes related to the subject have been determined by examining the curriculum and making use of the studies on sub-learning areas. Considering the opinions of the experts who have worked in the field, it was decided that a 3-week curriculum of 12 hours in total was sufficient for the students to achieve the relevant outcomes. The table of transformation geometry outcomes is given in Table 3. Before the implementation, information about the use of GSP was given to the math teacher of the school by t researcher and several practices were carried out together. Transformation geometry achievement test a computer attitude scale were applied to the both groups as a pretest. Then, 3 hours of training was given by t! math teacher in drawing points, segments and polygons, calculating side lengths and areas, creating a coordine system and determining the coordinates of geometric shapes in order to introduce the GSP software to t experimental group students and to gain them skills in using it in the IT Laboratory. There were 20+1 compute in the laboratory. After the implementation started, the subject of Transformation Geometry was taught to bo groups simultaneously by the same math teacher at the school. The control group was taught in the classroo using traditional teaching methods. The experimental group was taught the same subject in the IT Laborato1 The lessons were taught by math teacher with daily lesson plans, worksheets and GSP drafts prepared by tl researcher in accordance with the curriculum. Worksheets containing paper-pencil activities, thinking activiti and GSP activities were distributed to each student before the lesson. GSP activities were also copied to tl desktop of student computers by the researcher before the lesson started. Students created what was asked them in the worksheets and wrote their findings and generalizations on the worksheets. When it was the turn the relevant GSP activity in the worksheets, the students opened these drafts on their computers and worked « them. After each activity on the worksheets was completed, the students discussed the findings they obtain and the students who volunteered solved it on the smart board. After the existing worksheets were finished, ne worksheets were distributed to the students. The researcher was not present in the IT laboratory during t lesson, but he was ready outside in order to immediately intervene in the malfunctions that may occur in tl computers or the smart board and to ensure the effective use of the course time.

Table 4: The Quantitative Method of the Research The experimental implementation was carried out in 12 lesson hours for 3 weeks. In the both groups, the subjects were started and completed at the same time. After the implementation, transformation geometry achievement test was applied to both groups as posttest. In addition, computer attitude scale was applied to the experimental group students as a posttest.

Table 5: ANCOVA Analysis of the Pretest Corrected Posttest Scores According to Groups According to ANCOVA results given in Table 5, no statistically significant difference was found between the pretest-corrected posttest scores of the students studying in the control and experimental groups (F (1,42) = 0.049, p> .05, n? = .001). The finding of Eta-square (7) .001, which is the effect size, can be interpreted as 1% of the variance of achievement test scores originates from GSP supported education. This finding means that GSP supported instruction has a very low effect on Transformation Geometry Achievement Test scores. It was determined that all the assumptions required for the analysis (the need of a linear relationship between the dependent variable and covariate, the scores of the dependent variable for each of the groups are normally distributed and their variances are homogeneous, the regression slopes -coefficients- are equal within the groups) were met before ANCOVA Covariance Analysis was performed.

Table 6: Mann-Whitney U Test for the Computer Attitude Scale Pretest Results of the Groups

Table 7: Wilcoxon Signed Ranks Test of the Experimental Group's Computer Attitude Scale Before and After Implementation * Based on negative ranks. According to the results of the Wilcoxon Signed Ranks Test given in Table 7, the use of CAI in mathematics eaching has no effect on computer attitude (T = 48.50, p = 0.058, z = -1.893, r = -0.41). The effect size value shows that there is a moderate difference between pretest and posttest (r = -0.41).

Table 8: Wilcoxon Signed Ranks Test Regarding Pretest and Posttest Mean Scores of the Experimental Group's Computer Attitude Scale "Self-Confidence in Using Computer" Sub-Factor * Based on negative ranks. As seen in Table 8, a statistically significant difference was found in the attitude scores of the experimental group students before and after the implementation, belonging to the "self-confidence in using computer" sub- factor determined in the Computer Attitude Scale (p = 0.009, p <,05, r = -0.56). Considering the rank totals of the difference scores, it is seen that the observed difference is in favor of the posttest mean scores. This finding means that there is a positive increase in students’ self-confidence attitudes while using computers. The effect size value indicates a moderate difference (r = -0.56).

Table 9: Experimental Group Students’ Opinions Regarding the Learning of Geometry Topics Before Implementation

Table 10: The Views of the Experimental Group Students on Geometer's Sketchpad “What are your positive thoughts about Geometer's Sketchpad? Can you explain the reason?” The finding obtained from answers to this question are given in Table 10. It was concluded that the lessons were taught using the traditional teaching method in the classroom, various implementations were opened from the smart board and the subjects were learnt by making use of some educational websites and DGSs were not used.

Table 12: Thoughts of Students on Computer Assisted Mathematics Teaching Ine of the students used the following expressions regarding the theme "satisfaction":

Table 13: Opinions of Students Related to Situations in Daily Life with Transformation Geometry Concepts “Do you think this software will contribute to teaching mathematics? Can you explain why?" The findings obtained from answers to this question are given in Table 14. Based on the answers given by the students, it could be said that they could make connections between the concepts they have learnt about transformation geometry with the situations they encountered in daily life, they could better understand the geometry of the various movements of objects, in short, they realized that there was mathematics in situations that they can encounter at any moment in daily life and that mathematics was a part of life.

“Finally, do you want to add anything or do you have any suggestions about education? If so, what are they?" The findings obtained from answers to this question are given in Table 15. Based on the answers given by the students, it could be said that GSP software made the learning environment student-centered, provided better learning opportunities and helped the information they learnt to be more permanent due to the visual environment and implementation opportunities provided to students. CAI would contribute to students’ development of a more positive perspective on mathematics lesson, help students overcome their fear of mathematics and make the lesson more enjoyable. Based on the answers given by the students, it could be said that the implementation of CAI was more efficient, it should be applied to all students in other lessons, and it provided easier learning of the subjects due to its

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How Argumentation and GeoGebra Support Students’ Understanding of Nets of Prisms? Pages 270-276

by Şule Şahin Doğruer

2018, ILTER CONGRESS

It is important to support an instruction that promote spatial ability and geometrical understanding with visual imagery in teaching geometry. Research has revealed that many times students have difficulties to develop strong conceptual... more

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Proving Basic Theorems about Chords and Segments via High-School-Level Geometric Algebra

by James Smith

We prove the Intersecting-Chords Theorem as a corollary to a relationship , derived via Geometric Algebra, about the product of the lengths of two segments of a single chord. We derive a similar theorem about the product of the lengths of... more

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Tareas Digitales: Recurso didáctico para favorecer la argumentación

by Camilo Sua and

2017

El trabajo tiene origen en el análisis de las tareas digitales que, como resultado del auge actual de la tecnología, han venido desarrollando las editoriales colombianas y el Ministerio de Educación Nacional para el área de matemáticas y,... more

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Procesos de conjeturación y justificación: el rol de los programas de geometría dinámica

by Camilo Sua

2017

El presente trabajo, evidencia una investigación de estudio de caso desarrollada en los años 2015, 2016 y 2017. Desarrollada con un grupo de tres estudiantes, de grado noveno de una institución privada sin ánimo de lucro de la ciudad de... more

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Saber más no implica resolver mejor

by Camilo Sua

2018, 22 Simposio SEIEM

Los problemas de demostración, vistos como un ejemplo de la resolución de problemas, demandan poner en juego distintos conocimientos y habilidades en el uso de la geometría dinámica cuando se cuenta con el apoyo de esta. Apoyados en dos... more

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Spherical Geometry: Integrated a teacher education course with touchscreen-based technology

by Mina Sedaghatjou

2018

In this exploratory case study, we describe how teacher candidates with minimal mathematics knowledge of spherical geometry construct, discuss, manipulate and share the three-right angled triangle on a sphere using GeoGebra software. The... more

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HOW THE ACTIVITY OF PROVING IS CONSTITUTED IN A CYPRIOT CLASSROOM FOR 12 YEAR OLD STUDENTS

by Maria Pericleous

2017

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The Effectiveness of Kinesthetic Approach in Developing Mathematical Function Graphs Recognition and Understanding at University Level

by Mohamed El-Demerdash and

This research paper investigates our belief as educators that kinesthetic approach can be used to develop students' recognition and understanding of mathematical function graphs with the help of a competitive dancing game. Given that some... more

This research paper investigates our belief as educators that kinesthetic approach can be used to develop students' recognition and understanding of mathematical function graphs with the help of a competitive dancing game. Given that some students need to use other senses as a preparation to abstract and symbolic thinking, we developed Augmented Reality kinesthetic digital resources around standard mathematical function graphs using Microsoft Kinect sensor and some programming of Unity 3D Development Kit, Microsoft Kinect (SDK) Software Development Kit and CindyScript the programming language associated with Cinderella Dynamic Geometry Software, linked through a UDP (User Datagram Protocol) connection. The software recognizes students' body gestures as input representations for mathematical function graphs. Aiming at verifying the resources effectiveness in developing recognition and understanding of mathematical function graphs, a quasiexperimental design, a preexperimental one group (pretest intervention posttest), is considered. The test consists of 40 items that are equally distributed among the two levels of achievement: recognition and understanding. 20 items are designed at each level in isomorphic pairs that are administered randomly as whether pretest or posttest. The items type for the test are multichoice questions of 4 choices. The test items were written in verbal and nonverbal, symbolic and graphical ways. The setting of the experimental study was at CFAI Savoie Chambéry a training college for careers in industry for young people with 51 students of the first year of BTS Maintenance of Systems and BTS Technical Assistance engineers. The quantitative data collected through the pre and post test does not allow us to draw a clear conclusion on the effectiveness of using the kinesthetics resources in developing students' recognition and understanding as the means difference was negative. However, the reason of that is not related to the data itself but on how it was collected. On the other hand, the qualitative data gathered from the students and their teacher shows that the resources were very engaging and pleased students as they played. The results are not conclusive, the experimentation needs to be repeated in a context where the didactical contract ensures the motivation for the post test in order to investigate the resources effectiveness in terms of both recognition and understanding.

corresponds to body gestures, and the green one corresponds to the goal graph which the student should perform to obtain the positive score during the game. On the left side of the figure above we show Cinderella plotting the functions in accordance with what the player (right side) is performing in front of Kinect sensor. Figure (1): Cinderella plots the function graph in accordance to body gesture detected by the Kinect sensor. fractional function, trigonometric functions, exponential function, logarithm function.

Figure (2): Matching body gestures to some mathematical function graphs.

Table (1): Specification of the Achievement Test Procedure of Experimental Study

Table (4): Pre-Post Tests Results performance.

Table (2): Results of Pretest Results of the Posttest

Table (3): Results of Posttest Results of the Pre-Post Results

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Digital Resources to Enhance Creative Mathematical Thinking in a Biomathematics Context

by Mohamed El-Demerdash and

In this research, we present the design of digital resources, carried out within the framework of the MC Squared project (http://mc2-project.eu). These resources are the so-called " c-book units " (" c " for creative), which are digital... more

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Flashback to the Past: a 1949 Matric Geometry Question

by Michael de Villiers

2015

This paper explores a 1949 geometry exam question, generalizing it by 'looking back' at the solution in Polya style, as well as considering some interesting variations of it, including an optimization problem that is solved purely... more

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Some Reflections on the Van Hiele theory

by Michael de Villiers

This paper gives a review of research on the Van Hiele Theory over the past 30 years, and highlights some important issues regarding theoretical implications for designing learning activities in dynamic geometry contexts, as well as... more

halfturn of the grey triangle around the midpoint of the side AB maps angle A onto angle

Invited plenary presented at the 4 Congress of teachers of mathematics of the Croatian Mathematical Society, Zagreb, 30 June —2 July 2010, as well as at the National Mathematics Congress in Namibia, Swakopmund, 21-23 May 2012. Also presented as public lecture in the Program Post-Graduate Studies in Mathematics Education, Pontifical Catholic TInivercity af San Panlo Rrazil 19 Anril 9010 In the first case, pupils can again see that angle A = angle D due to a saw being formed

error) to ‘discover’ that they always add up to 180°.

Formal Van Hiele Level 3 definitions in textbooks are often preceded by an activity

was an important factor (e.g. calling a square a special rectangle).

dragging, students also explore special cases (rhombus, square). Investigate

Construct and connect the midpoints of the sides of a kite. (The result generalizes to any quadrilateral with perpendicular diagonals).

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Characteristics of Different Learning Environments in Geometry Classrooms

by B. Webre

This study used in-depth interviews of selected teachers to collect qualitative data to address the following research question: What characterize different learning environments in geometry classrooms? The main findings include: Since... more

Figure 1. A problem situation Figure 2. Investigating the problem situation Investigation (Investigating the problem situation to generate conjecture): In Figure 1,

Figure 3. The circle going through A, B, and Y Figure 4. The situation constructed for proving

Figure 6. The investigation that Chris did Figure 5. The problem situation by Chris finding the maximum angle, she constructed a situation similar to Figure 6.

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MESH SIMPLIFICATION VIA A VOLUME COST MEASURE

by International Journal of Computer Graphics & Animation (IJCGA)

2015, International Journal of Computer Graphics & Animation (IJCGA)

We develop a polygonal mesh simplification algorithm based on a novel analysis of the mesh geometry. Particularly, we propose first a characterization of vertices as hyperbolic or non-hyperbolic depend-ing upon their discrete local... more

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From the Fermat point to the De Villiers points of a triangle

by Michael de Villiers

2009, Proceedings of the 15th Annual AMESA Congress, pp. 1-8.

On the basis of an often-observed (but not generally true) duality between circumcentres and incentres, it was conjectured in 1996 that the following might be true from a similar result for circumcentres (Kosnita’s theorem), namely: The... more

of triangle ABC (see the diagram at right) and constructing the straight lines A'B, B'C and Slightly modified from Meyer, JH & Van Biljon, A, (2009), Proceedings of the 15" Annual AME: Congress, University of Free State, Bloemfontein, pp. 1-8.

that will give us further insight leading to an interesting, and important generalization. concurrent at the centroid.” Let’s consider a non-traditional proof based on areas, but

Figure 3: Proof of Fermat-Torricelli generalization

In order to prove this result we will use the following lemma, which is stated here without proof.

Using the aforementioned generalization of the Fermat-Torricelli point, it was now very

are concurrent. The proof follows similarly from the Fermat-Torricelli generalization. Concluding comment with the incentres of the triangles BCI,, CAI,, and ABI, (J; are the excentres of AABC)

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Πατσιομίτου, Σ. (2015). Η ανάπτυξη ικανότητας εργαλειακής αποκωδικοποίησης νοητικών αναπαραστάσεων των μαθητών ως μη γλωσσική εγγύηση στην ανάπτυξη της γεωμετρικής τους σκέψης. Περιοδικό «ΝΕΟΣ ΠΑΙΔΑΓΩΓΟΣ», 5ο τεύχος, σσ. 29-60.

by STAVROULA PATSIOMITOU (Δρ. Σταυρούλα Πατσιομίτου)

2015

εργασία με κατοχύρωση πνευματικών δικαιωμάτων

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Aprendizagem de Funções Reais utilizando Geometria Dinâmica

by Filipe Hasche

2008

Neste artigo é feita uma síntese de uma experiência de ensino de Funções Reais utilizando Geometria Dinâmica. Relatam-se aqui alguns episódios dessas atividades ligados a reflexões sobre pesquisas acerca do ensino e aprendizagem de... more

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Oner, D. (2009). The role of dynamic geometry software in high school geometry curricula: An analysis of proof tasks. International Journal for Technology in Mathematics Education, 16(3), 109-121.

by Diler Oner

2009, International Journal for Technology in Mathematics …

In this study, I examine the role of dynamic geometry software (DGS) in curricular proof tasks. I investigated seven US high school geometry textbooks that were categorised into three groups: technology-intensive, standards-based, and... more

This view of learning has been enabled by the immediate availability of a very powerful set of computer tools, namely DGS, in classrooms. DGS realises the vision described above by enabling heuristic explorations through its most defining ‘dragging’ feature (Goldenberg and Cuoco, 1998). Due to this feature, the construction dynamically responds to the changes made by maintaining the invariant. Kaput (1992) states that conjecturing and more inductive approaches to geometric knowledge are facilitated by this aspect, as students can recognise patterns and easily check the generality of their hypotheses. Figure | The Geometer’s Sketchpad construction of a right triangle ABC and three squares on its sides

One might think this is merely one occasion. However, what Sketchpad allows is that one can easily drag vertices 4, B or C to different places and still see that this relationship holds. Although it is not possible to represent the dynamic experience on paper, the reader can imagine that Figure 2 Same construction, vertex A is dragged

Figure 3 Same construction, vertices C and B are dragged

Nevertheless, it has often been argued that using the drag mode to test myriad cases may reduce or even eliminate any need for deductive proof (Allen, 1996; Chazan, 1993a; Hadas et al., 2000; Hoyles and Jones, 1998; Laborde, 2000). When students can generate their own empirical evidence Allen (1996) is one of the strongest critics of DGS. For him DGS tends to blur the distinction between illustration and proof. A dissection proof of the Pythagorean Theorem exemplifies this. In this DGS-based construction, students can cover the largest square by using the parts of the smaller squares, which are partitioned (Figures 4, 5 and 6). Allen argues that such an illustration cannot be considered as proof, since no previous theorems are invoked. Figure 4 The dissected Pythagoras illustration first screen

Figure 5 The dissected Pythagoras illustration second screen

Figure 6 The dissected Pythagoras illustration third screen

Figure 7 Experimental/explorative activity percentages across the three types of curricula Below I examine the role of DGS using three research questions within the three types of curricula defined earlier:

paper and pencil-based explorations. It is even implied that paper-pencil is preferred to DGS-based explorations: “Using the Dissection of Squares worksheets or the Sketchpad demonstration will speed up the investigation, but the use of many different triangles drawn by the students [by hand] strengthens the inductive conclusion” (p. 462). Secondly, DGS use mostly appears in the teacher’s edition (in 15 of the activities, or 48%). DGS is included in the student text in only six activities, mostly as a link to an online Geometer’s Sketchpad sketch and its drawing. For instance, in one of these activities students investigate the theorem “the sum of the lengths of any two sides of a triangle is greater than the length of the third side” (p. 214). Using a compass and a straightedge, students are to construct triangles with given sets of segments. Prior to conjecturing, students are asked if they were able to construct these triangles and discuss their results. After students are told to state their conjecture, the text reads: “You can also think of it [the triangle inequality conjecture] in another way: The shortest path between two points is along the segment connecting them. In other words, the path from A to C to B can’t be shorter than the path from A to B (p. 214). This explanation is followed by four screen shots of a DGS construction showing a triangle ABC, the lengths of the sides, AB, AC and CB, and the sum AC + CB, as one of the vertices (point C) is being dragged to different places (see Figure 8 below representing one of these screens). By looking at these, students are expected to notice that the distance from A to C to B is always greater than or equal to the distance from A to B.

The proof activities were sampled by first counting the number of lessons meant to be covered in one class period. On average, the textbooks in the study have about 102 lessons that cover geometry content. (The term

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Dynamic Geometry Sofware

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