The effectiveness of `what if not' strategy coupled with

International Journal of Mathematical Education in Science and Technology

ISSN: 0020-739X (Print) 1464-5211 (Online) Journal homepage: http://www.tandfonline.com/loi/tmes20

The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquirybased geometry classroom Ruti Segal, Moshe Stupel, Avi Sigler & Jay Jahangiril To cite this article: Ruti Segal, Moshe Stupel, Avi Sigler & Jay Jahangiril (2018): The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquiry-based geometry classroom, International Journal of Mathematical Education in Science and Technology, DOI: 10.1080/0020739X.2018.1452302 To link to this article: https://doi.org/10.1080/0020739X.2018.1452302

Published online: 18 Apr 2018.

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The effectiveness of ‘what if not’ strategy coupled with dynamic geometry software in an inquiry-based geometry classroom Ruti Segala,b , Moshe Stupelc , Avi Siglera and Jay Jahangirild a Shaanan Religious College of Education, Haifa, Israel; b Oranim College of Education, Tivon, Israel; c Shaanan Religious College of Education & Gordon College of Education, Haifa, Israel; d Kent State University, Kent, OH, USA

ABSTRACT

ARTICLE HISTORY

In this paper we present the results of a study which was carried out in an inquiry-based teaching and learning environment with the use of ‘what if not’ methodology coupled with the integration of dynamic geometry software. The vast majority of the students reported that they perceived themselves as participants rather than spectators. Most of the prospective teachers came to the conclusion that the implementation of the findings of this study in their future teachings was a good idea and that it will raise the students’ motivation and enhance and deepen the knowledge pool of the learners.

Received  June  KEYWORDS

Geometry task; dynamic geometry software; what if not; prospective teachers

1. Introduction For the past two decades, enquiry- and inquiry-based teaching have been extensively researched and implemented in the teaching of mathematics. It was widely expected that enquiry- and/or inquiry-based teaching strategies impart scientific knowledge and foster students’ research expertise and confidence. The idea, as appealing as it is in theory, appeared to be not as fruitful in implementation. It caused the students to react to the class activities with a level of discomfort and some students put up a noticeable resistance [1]. There are reports of students’ resentments, frustrations, as well as challenges that instructors face in their choice of appropriate and stimulating geometry problems [2]. There are more critical reports pointing out that the concepts have disadvantages and may fail if they are applied without the appropriate preparations [3–5]. We would be in remiss if we do not mention those studies that reported success by implementing active learning in classrooms. In an experimental study, Freeman et al. [6] meta-analyzed 225 studies that reported data on examination scores or failure rates when comparing student performance in undergraduate STEM (science, technology, engineering, and mathematics) courses under traditional lecturing versus active learning. They concluded that active learning to some degree increases students’ performances while lecturing increases failure rates by a large margin. According to Santos-Trigo [7], ‘A salient feature of a problem solving approach to learn mathematics is that teachers and students CONTACT Ruti Segal

[email protected], [email protected]

©  Informa UK Limited, trading as Taylor & Francis Group

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develop and apply an enquiry or inquisitive method to delve into mathematical concepts and tasks’ (p. 20).

2. Power of technology in an inquiry-based environment Silva [8] states that: We live in a society where technology is in a rapid evolution, with new tools arriving at the consumer market every year. It is more than natural that these tools are also offered at the school level. Two main reasons can be stated in favor of this: First of all, the school has never been as efficient as the society desires and so new approaches are normally welcome (at least by most people); secondly, if the school is to prepare students for real life and for some professional activity, then teaching should somehow incorporate the technological tools that students will find someday in their adult life. (p. 3)

Integration of dynamic geometry software (DGS) in the teaching and learning of mathematics helps learners to check or test the many possible aspects of problems by exploring examples. The ability of DGS to rapidly generate a variety of examples, to store and restore numerous cases, and to provide qualitative feedback, offers the learners valuable information about the mathematical concepts under the study; information that constitutes bases for hypotheses testing, prepare for the proof as well as venues for generalizations [9,10]. The drag mode in DGS is a cognitive tool that provides students with integrative means of expressing their geometrical thoughts in visual-dynamic ways which might contribute to the formation of intellectual thinking and abstract knowledge.

2.1. Description of the activity In a deliberate attempt, we investigated the effectiveness of ‘what if not’ (WIN) strategy coupled with DGS in an inquiry-based geometry classroom. In the framework of our activity study, we used ‘WIN’ strategy as a prelude to what if more (WIM), what if less (WIL) and what if instead (WII) inquiries and then implemented the seven dragging modalities identified by Arzarello et al. [11], (also [12]). Our DGS of choice was GeoGebra. In choosing the configuration of the research, we made sure that: (i) It conveys surprise and stimulates curiosity. (ii) It constitutes a special case of a more general case. (iii) It admits inquiry in three subsections of WIN strategy: • WIM: The case that includes additional features than the original configuration, • WIL: The case that considers the absence of certain features in relation to the original configuration, • WII: The case that explores other possible features in place of the original configuration. (iv) It allows the exposition of a hypothesis with the help of the technological tool. (v) It includes elements of ‘prove’ and ‘find’ (according to [13]). (vi) It tolerates non-frustrating success for all prospective teacher students. (vii) It is well investigated in order to prevent dead ends or divergence.

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We thoroughly researched the implementation of the suggested configurations, discussed the assessment of prospective teachers’ achievements which included the numbers and percentages of prospective teachers (PT) who arrived at the correct hypothesis and also provided justifications or proofs, the types of questions they managed to pose (indicating their research skills) and the numbers and percentages of PT who arrived at the correct hypothesis but could not provide justifications or proofs. 2.2. The students Sixty-one sophomore undergraduate middle-school and high-school mathematics PT took part in the study. 2.3. Duration The course, in the framework of which the research took place, was taught over two semesters and dealt with the didactics of math teaching. 2.4. The activity process By integrating investigational activities in enquiry-based teaching, the PT experienced activities that combined WIN strategy with DGS. The students worked in pairs and were asked to work on a given problem, make hypotheses or conjectures, test and prove or disprove their hypotheses or conjectures, make conclusions, suggest new problems and possible solutions and then reflect on their insights and feelings concerning the inquiry process. They then were asked to record their investigation processes that included the use of technology and present them in class. These assignments including the class presentations and the follow-up discussions constituted essential bases for the investigation. 2.5. The configuration In the given figure, the internal angle bisectors of the parallelogram ABCD intersect at points M, N, P and Q. (a) Prove that MNPQ is a rectangle. (b) Prove that AB || CD || MP and BC || AD|| QN. (c) Calculate the lengths of MP and QN in terms of the sides of the parallelogram.

(1) Using the WIN method and GeoGebra software, formulate your hypotheses or conjectures and provide justifications. (2) Try to formulate new questions deriving from your inquiry.

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Following the completion of the above task, a class discussion took place addressing all the ideas for developing the task using the WIN method and the use of DGS. The PT worked in pairs, proposed several lines of inquiry, and opted to focus on the inquiry regarding the transformation of the given figure (here, the quadrangle) into other related geometrical variations (different quadrangles). Here we note that it was for students to discover the changing of the configuration of the internal quadrangles (i.e. MNPQ) as the outer quadrangle (i.e. ABCD) changed. Then they would discuss, debate and try to justify such identifying relations between the angles and the sides of the internal quadrangle vs. the exterior quadrangle.

2.6. Collecting data and analysis In this study, we focused on the data, which presented an evolution of mathematical knowledge and research abilities of the PT as well as on the feelings of the PT while executing their mathematical assignments.

2.7. The data came from three sources The first source was students’ log. This contained the comprehensive notes that the PT prepared during their inquiry-based coursework, including the hypotheses made (with the use of DGS) and all the explanations and justifications of their proofs or disproof. The second source contained the prospective teachers’ reflections of the activities, including students’ feelings, attitudes, thoughts and reflections. The third source was the recordings of the class discussions during the inquiry process. After the data collection, we proceeded to a process of analytical induction, according to Goetz and LeCompte [14], which cross-analyzed the data as a whole in order to identify issues and patterns. This generated an initial statement concerning the influence of DGS on problem solving and problem posing.

2.8. Possible hypotheses based on our preliminary analysis The following three responses were expected from the students. The corresponding quantitative analysis of the students’ responses is demonstrated in Table 1. Response 1: (a)

If ABCD is a rectangle, then MNPQ is a square.

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(b)

If ABCD is an isosceles trapezoid, then MNPQ is a kite with two opposite right angles.

(c)

If ABCD is a trapeze, then MNPQ is a quadrangle with two opposite right angles (see link to applet below).

(d)

If ABCD is a quadrilateral, then MNPQ is a cyclic quadrangle.

(e)

If ABCD is a quadrilateral then the angles MNPQ are the arithmetic means of adjacent angles of ABCD.

Response 2: (a)

If ABCD is an isosceles trapezoid then, AB||CD || MP and BC, AD, and QN are concurrent.

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(b)

If ABCD is a trapezoid AB||CD || MP and BC, AD, and QN are concurrent.

(c)

If ABCD is a quadrilateral then AD, BC, and MP are concurrent.

(d)

If ABCD is a quadrilateral then the angle between QN and MP equals to ∠A+∠C . 2

(e)

If ABCD is a cyclic quadrangle, then QN⊥MP.

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Response 3: (a)

If ABCD is a parallelogram, then MP = QN = |AB − BC|.

(b)

If ABCD is an isosceles trapezoid, then MP = |(AB+DC)−2AD| 2

(c)

If ABCD is a trapezoid, then and MP = |(AB+DC)−(AD+BC)| 2 QN = (AD+BC)−(AB+DC) ∠C+∠D

(d)

If ABCD is a quadrangle then MP = |(AB+DC)−(AD+BC)| and ∠C+∠B

2 sin

QN = (e)

2

2 sin 2 |(AD+BC)−(AB+DC)| . 2 sin ∠B+∠A 2

If ABCD is a quadrangle then ∠M · ∠N · ∠P · ∠Q ≥ ∠A · ∠B · ∠C · ∠D

Link: applet of trapezoid bisectors, enquiring by PT (Before running the application, download the GEOGEBRA free software)

3. Results The following table summarizes the frequency of the responses within the framework of the PT research work integrating technology (GeoGebra Software). 3.1. Remarks Most PT were able to generate the hypotheses in sections 1a–1c. The DGS was instrumental in helping PT to generate their hypotheses and then proceed with the deductive proofs. In sections 1d and 1e, the PT calculated the angles using the DGS and tried to find a connection between the sizes of the respective components of the given figures. A large

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Table . Results within the framework of the PT research work.

Item

Number and percentage of PTs with correct hypotheses and proofs.

a

61 61 (100%)

WIM

ABCD rectangle → MNPQ square

b

61 61 (100%)

WII

ABCD isosceles trapezoid → MNPQ kite with two opposite right angles.

c

58 61 (95%)

WIL

3 61 (5%)

ABCD trapezoid → MNPQ quadrangle with two opposite right angles

d

53 61 (87%)

WIL

8 61 (13%)

ABCD quadrilateral → MNPQ a cyclic quadrangle

e

45 61 (74%)

Creating tools for continuation of enquiry.

6 61 (10%)

ABCD quadrilateral → MNPQ angles are the arithmetic means of the adjacent angles of ABCD

a

35 61 (%)

WII

16 61 (26%)

ABCD isosceles trapezoid → AB||CD || MP and BC, AD and QN are concurrent.

b

35 61 (%)

WIL

16 61 (26%)

ABCD trapezoid → AB||CD || MP, and BC, AD and QN are concurrent.

c

6 61 (%)

WIL + creating tools.

40 61 (66%)

ABCD quadrilateral → DC, AB and MP are concurrent.

d

5 61 (8%)

Creating tools.

20 61 (33%)

ABCD quadrilateral →the angle . between QN and MP equals ∠A+∠C 2

e

10 61 (%)

WIM + loading and constructing a question.

35 61 (57%)

ABCD cyclic quadrangle → QN⊥MP

a

55 61 (90%)

WIM

4 61 (7%)

ABCD parallelogram → MP = QN = |AB − BC|

b

50 61 (82%)

WII + creating tools

4 61 (7%)

ABCD isosceles trapezoid →MP = |(AB+DC)−2AD| 2

c

48 61 (79%)

WIL + creating tools

3 61 (5%)

ABCD trapezoid → MP = |(AB+DC)−(AD+BC)| 2 QN = (AD+BC)−(AB+DC) ∠C+∠D

d

4 61 (7%)

WIL + creating tools

5 61 (8%)

The type of questions (research skills).

Number and percentage of PTs with correct hypotheses without proofs.

Expected results

2 sin

ABCD quadrangle → MP = |(AB+DC)−(AD+BC)| ∠C+∠B QN =

e

3 61 (5%)

Dynamic process

50 61 (82%)

2

2 sin 2 |(AD+BC)−(AB+DC)| . 2 sin ∠B+∠A 2

ABCD quadrangle → ∠M · ∠N · ∠P · ∠Q ≥ ∠A · ∠B · ∠C · ∠D

majority of the PT using the software managed to formulate correct hypotheses and most of them were able to prove their hypotheses deductively. In sections 2a and 2b, only about half of the PT presented the correct hypotheses and about a quarter of the PT were able to prove their hypotheses. In sections 2c–2e and 3d, most of the PT had difficulty to reach a correct hypothesis, even with the help of DGS.

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In sections 3a–3c, a large majority of PT reached correct hypotheses while working with the DGS, as well as presenting deductive proof but needed the instructors’ help.

3.2. WIN coupled with DGS generated interest and curiosity Most of the PT evinced great interest and curiosity while undertaking the inquiry tasks. The questions they raised throughout the investigative process were signs of their high motivation for continuing to explore, to discover and even to prove, although they did not succeed every time in proving their hypotheses (based on working with GeoGebra). At every stage, they discovered something new, which encouraged them to ask follow-up questions. They were intrigued during the research process and the activities encouraged inquisitive thinking.

3.3. Examples of prospective teacher’s feedback (with no alterations) • ‘The use of the GeoGebra software infused me with confidence, and it has been, for me, a valuable tool in determining the research question’. • ‘This activity made me think of all the theorems that I have learned and I had to recall them’. • ‘I have learned how to present, during lessons, a research activity, integrating a technology’. • ‘It took us a long time to prove some of our hypotheses. But the process was significant, and when we finished we felt proud and satisfied’. • ‘I feel a responsibility to pass my experience to my pupils in order to raise their motivation for learning’. • ‘We could easily observe the relations between the exterior and internal angles, between the lengths …’.

3.4. Examples of future research proposed by PT The students applied the WIN method with problem positing (including the integration of a DGS) and problem solving (according to the proofs they provided during their enquiry processes), and then proposed further possible directions. Here are some of their suggestions: In a given parallelogram ABCD.

• Let the points M, N, P and Q be the mid-points of the sides AB, BC, CD and DA, respectively. What can be said about the quadrangle MNPQ? (WII)

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• What can be said about the quadrangle EFGH generated by the perpendicular bisectors of the sides of the parallelogram ABCD? (WII)

• What can be said about the quadrangle EFGH generated by lines that divide the interior angles of the quadrangle ABCD in the ratio of 1:2 (instead of 1:1)? (WIM) • What can be said about the quadrangle EFGH generated by lines that divide the interior angles of the quadrangle ABCD in the ratio of m:n (instead of 1:2)? (WIM)

4. Discussion This research investigated the power of integration of a DGS (GeoGebra, in our case) coupled with the implementation of WIN strategy in the inquiry-based teaching. The target audience of this study were the prospective mathematics teachers in general and prospective geometry teachers in particular. Using the DGS as a tool in the learning process helped the PT to create a variety of examples that, in many cases, were surprising to them. The implementation of WIN strategy motivated them to pursue the justification of the geometrical hypotheses leading to formal proofs. As a testimony to our case, here we briefly cite two related recent studies. Lavy [15] presents a study of problem posing that integrates WIN strategy and DGS with PT. She reports that for teachers’ who work on a problem, posing and solving them through technological tools reinforces their self-confidence, their sense of belonging to the learning process in the classroom, and their possibilities of actively integrating into the classroom discussion. Leikin [16] presents three types of geometry problem posing integrating WIN strategy and technology. It is claimed that this allows teachers and students to experience meaningful mathematical activities and discovery of new mathematical facts, starting from one simple problem. Our research and investigation presents an affirming evidence of how students started with a simple problem and then used the WIN strategy to create and solve a wide range of problems (problem posing and problem solving) using the integration of technology. We observed that in an inquiry-based teaching environment, the students were all active and all reported an experience of success and significant influence on the motivation to explore, to investigate and to discover. The experience widened the students’ technological, pedagogical and mathematical knowledge and reduced their fear of exploration. Most PT were inspired to implement the inquiry-based teaching and learning in their future classroom teachings.

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Disclosure statement No potential conflict of interest was reported by the authors.

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