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    Integrating ict in_education Integrating ict in_education Document Transcript

    • Integrating ICT in mathematics education: Curricula challenges in the Kenyan system of education∗ Ateng’ Ogwel† Centre for Mathematics, Science and Technology Education in Africa I have had my results for a long time: but I do not yet know how to arrive at them Carl Friedrich Gauss (1777-1855) Abstract Out-of-school application of information and communications technologies (ICT) in modelling, design and in enhancing professional efficiency reveals an urgent need to align formal educational practices with the rapid innovations in technology. Emergence of tools that increase interactivity in learning and facilitate distributed learning and collaboration sharply contrasts conventional curricula provisions and practices which have been relatively stable for several decades. Integrating ICT in ed- ucation would also probably remedy the apparent lack of relevance of school mathematics for most learners. Nevertheless, from a semiotic epistemological perspective, ICT would only mediate learn- ing despite the potentials to provide prompt feedback, personalize instruction and express inherent generality of mathematical concepts. Besides, given the embedded cognitive hierarchies, computers, instructional software, calculators and multimedia are likely to imply greater instructional challenges than the constructivist reforms. In Kenya for example, lack of curriculum coherence; poor articulation within the system of education; inadequate teacher preparation and professional development; and the tendency for individualistic rather than collaborative learning are critical challenges in integrating ICT in mathematics education. A dynamic geometry software, Dr. Geo, is used to illustrate the challenges based on Similarity of Figures. The Government policies on e-society are noted for the potential to address the challenges of infrastructure development. However, there is need for collaboration in the integration ICT in mathematics education; enhanced teacher professional development, and continu- ous research on students’ learning based on ICT environments. It is only then that ICT would define the next practices in education, and enable young Kenyans to be competitive in a globalized society. 1 Introduction Rapid developments in information and communications technologies continue to influence economic and social development (Richards, 2008) and afford hitherto unforeseen comfort to end-users. Advances in technology have not been without controversy due to perception of automation as an affront to labour and anxiety over the requisite skills for its integration in most professions. Evidently, technology has revolutionized the banking industry, for example M-pesa and has enhanced efficiency in architectural designs, e.g. Archicad1 and Artlantis.2 ∗ A paper for the 1st Regional Conference on e-Learning: Increased access to education, diversity in applications and management strategies, November 18–20, 2008: Kenyatta University † Email 1: ogwel26@yahoo.com; Email 2: ogwel26@gmail.com
    • Despite the developments, formal education in most countries has been slow to adopt technological innovations, notwithstanding decades of inefficiency in education. For instance, educators in Kenya, as in other countries, have been concerned with students’ performance, low motivation and negative attitude towards mathematics, attributed partly to curriculum that appears irrelevant to most learners. Ironically, students’ poor performance contrasts sharply with skills they acquire out of school in ICT environments. Envisaged reforms in mathematics education advocate for use of authentic tasks that engage students and promote development of problem-solving skills; and linking instruction to everyday life. In addition, there is an envisaged shift in instruction from teacher-centred to student-centred practices with enhanced focus on collaborative and cooperative learning. The proposed reform visions may be achieved within an ICT integrated curriculum, which would also provoke deeper mathematical reasoning. However, a number of challenges have to be overcome before digital technologies can be effectively integrated in mathematics education. These include curriculum coherence, inappropriate pedagogical practices, inadequate teacher preparation and professional development, and lack of appropriate infrastructure. In this paper, we outline these challenges after an illustration based on Similarity of Figures, and argue for enhanced collaboration in the design and implementation of ICT integrated mathematics education. 2 Integrating ICT in school curricula Technology has been used in mathematics in Analysis (Moormann and Grob, 2006), Algebra (Abramovich, 1999; Ainley, Bills and Wilson, 2005; Dreyfus and Hillel, 1998), Statistics (Abrahamson and Wilensky, 2007), Geometry (Cobo, Fortuny, Puertas & Richard, 2007; Healy and Hoyles, 2001; Laborde, 2001). Internet is increasingly being used to enhance collaborative and interactive learning (Cazes, Gueudet, Hersant and Vandebrouck, 2006; Cress and Kimmerle, 2008; Resta and Rafferriere, 2007) also (Lavy and Leron, 2004). ICT enhances efficiency of mathematical thought, enables learners to make conjectures and imme- diately test them in non-threatening environment (Laborde, 2001). ICT also offer multiple mathematical representations that enhance generality of mathematical concepts, and provide opportunities for counter- examples, unlike in paper and pencil environments. Technology also enhances curiosity that may drive inventions as illustrated in computational mathematics, (see, for example Borwein and Bailey, 2003). Abramovich (1999)’s use of spreadsheets in generalizing Pythagorean Theorem demonstrates how computers may be used to learn concepts in geometry and algebra, just as Ainley, Bills and Wilson (2005) give insights in the use of spreadsheets. Use of expressive media with computational and visual effects and convenient user interfaces has also advanced use of technology in instruction (Ioannidou, Repenning, Lewis, Cherry and Rader, 2003). For example, Dynamic Geometry software enables construction of accurate diagrams, simulation, drag effects, and when coupled with after-shadows or trace facilities reveal mathematical properties which may be difficult to achieve on paper. The multiple representations in computer applications and prompt feedback (Ainley, Bills and Wilson, 2005; Laborde, 2001) illuminate the critical challenge for mathematics educators, as Gauss cited in Borwein (2005) observed, is how to arrive at the solutions. That is, mathematics education has to transcend the novelty and curiosity in the use of ICT so that these are used as learning tools. In Kenya, besides Computer Studies in the secondary curriculum, ICT has largely focused on com- puter literacy and efficiency in computations. Recent introduction of calculators in mathematics education is a major step although its efficacy on students’ learning is yet to be investigated. The focus on accurate answers is inadequate in mathematics given that results may be obtained without understanding how to arrive at them. Moreover, mathematics instruction must transcend novelty, fun and the awe experienced in ICT applications if the aroused interest is to be sustained. Consequently, the challenge for mathematics educators is how and when the various computer applications and other ICT are integrated in the school curriculum (cf. Laborde, 2001). In the next sections, we illustrate challenges that ought to be addressed as ICT is integrated in mathematics education. Screenshots from a dynamic geometry software, Dr. Geo3 are given and the reader is encouraged to attempt especially Tasks 3, 4 and 5 before considering our partial explanations. 2
    • 3 Dynamic geometry software in mathematics education The following illustration on Similarity and Enlargement is drawn from Form two of the secondary math- ematics syllabus (KIE, 2002). The syllabus provides for seven content areas, viz (a) Similar figures and their properties, (b) construction of similar figures, (c) properties of enlargement, (d) construction of ob- jects and images under enlargement (e) enlargement in Cartesian plane (f) linear, area and volume scale factors and (g) real life applications (p. 20). Three conjectures and three tasks are used to explore concepts that we believe are achievable within the syllabus. 3.1 Parallelism and Similarity Conjecture 1 Parallelism defines similar figures in mathematics The conjecture is based on construction of concentric circles (Figure 1a) and squares and extended to polygons (Figure 1b) such that, polygons with relatively parallel sides are similar. The hypothesis may be tested using triangles to verify its truth. A counterexample (Figure 1c) generated by the drag facility in Dr. Geo shows the distortion when parallelism is maintained. The drag facility allows one to investigate embedded mathematical properties and challenge the apparently plausible conjecture. Graphic software, a b c Figure 1: Parallelism and Similarity e.g., Inkscape4 that allows one to stretch photographs (Figures 2 and 3), and the involved realia further disproves the conjecture. The approach in the figures has been used in some of the textbooks in Kenya (e.g., Owondo, Kang’ethe and Mbiruru, 2004). However, we contend that the example may not aid in the understanding of similarity of figures beyond definitions and formulae. One has to consider the inherent mathematical properties beyond the visualization. Questions that may elicit deeper reasoning include: Is Figure 2: Elephant 1 Figure 3: Elephant 2 the conjecture true for all triangles? Regular polygons? And what would be the explanation for either answer? Similarity in figures involves ratio of corresponding sides, corresponding ratios of sides and equality of angles, but how would these be linked to the preceding conjecture? The embedded properties yet to be revealed are evident in the next task. 3
    • 3.2 Invariant Parallelogram Conjecture 2 Midpoints of quadrilaterals form the vertices of a parallelogram (Figure 4a) The conjecture can be investigated in conventional mathematics classrooms, but requires accurate con- structions and several drawings figures to warrant a generalization. On the contrary, in dynamic geometry environments, drag facility allows for faster simulation of different orientations of quadrilaterals, thereby confirming the generality of the conjecture. While this may arouse interest among learners, inability to understand the embedded mathematical properties may allude to a mythical perception of mathematics and technology. The variation of the task is presented below. Task 3 Given a parallelogram, construct an enscribed quadrilateral Mathematical understanding requires thinking back and forth, and Task 3 is meant to reverse the process of producing the parallelogram in Figure 4b. While the forward process is fairly easy, the reverse process may have some cognitive challenges. In fact, the task may be challenging both in paper & pencil and dynamic geometry environments. In order to appreciate the complexity of the task, one has to reflect on a b c Figure 4: Parallelogram → Quadrilateral the process of constructing the parallelogram. Questions to aid the reflection include: how does one obtain a midpoint of a line segment? What are the properties of the midpoint? In school mathematics, midpoints are obtained from perpendicular bisectors of lines. Instruction emphasizes on use of arcs, which in our opinion may be insufficient to solve Task 3. An understanding of arcs as parts of circles is necessary; therefore the midpoint obtained by perpendicular bisection is the centre of a circle whose diameter is the line segment. Completing the task requires use of appropriate constraints, and an understanding that parallelogram is a generalization of all quadrilaterals. Thus, there will be no unique quadrilateral. Our examples so and explanations are yet to significantly link these tasks to similarity, except for the circles. In the preceding task, it may be useful to justify the result in Figure 4a. Auxiliary lines joining the opposite vertices of the quadrilateral are necessary in the justification. Similar arguments are embedded in the next task, where a unique parallelogram is desired. Conjecture 4 The midpoints of two opposite sides and diagonals of a quadrilateral from vertices of a parallelogram. Task 5 Construct a parallelogram whose acute angle has a constant value θ The task introduces possibly the need for global thinking with diagrams. The use of such a task may also aid in diagrammatic reasoning (Hoffmann, 2005), and perhaps point to the need for thinking outside the diagram (see also Laborde, 2005). Similar reasoning may be called for when solving the other problem (Task 6) which is given without explanation. Previous interaction with 40 secondary school mathematics teachers reveals that the problem solving skills involved in the latter task may have not been acquired in teacher preparation courses. 4
    • a b Figure 5: Conditional construction of parallelogram Task 6 ABCD is a square of side 42 cm. E and F are points on AB and BC such that BE = 14cm and BF = 21cm. CE and DF intersect at G. Find the area of quadrilateral AEGD A D E G B C F Figure 6: Area and Similarity In the following sections, we reexamine the affordances of Dr. Geo in solving the problems and outline challenges that may not be addressed with such a dynamic software. On parallelism and simi- lar figures, lack of equality of ratios in corresponding sides or corresponding ratio of sides explains the counter-example. Parallelism defines angles, thus equality of angles is a necessary but insufficient con- dition for similar figures. Without measuring the sides of polygons and comparing ratios, the connection between similarity and enlargement enables one to investigate the condition through a centre of enlarge- ment. That is similarity in regular polygons and all triangles is because of the concurrency of lines joining the vertices, precisely, concurrency of cevians (see Figure 7a). a b c Figure 7: A reflection on the tasks The invariant parallelogram involves some geometrical proof. As illustrated in Figure 7b, triangles ABC and ADE are similar. The following pair of converse propositions may be considered. 5
    • Proposition 7 In △ABC, D and E are points on AB and AC respectively. Prove that if AB:AD = AC:AE, then DE//BC Proposition 8 In △ABC, D and E are points on AB and AC respectively. Prove that if DE//BC, then AD:DB = AE:EC The corresponding segments are AB ∼ AD; AC ∼ AE; and DE ∼ BC. An auxiliary segment DF is intro- duced in the diagram such that DF//EC, with F on BC. The two propositions may be proved thus: Case I ∠BFD = ∠BCE (1) ⇒ ∠BDF = ∠BAC (2) ⇒ ∠ABC = ∠ADE (3) ∴ DE//BC Case II: By the introduction parallelogram DFCE, △ADE ∼ △ABC, (AAA). Thus, AD AE = (4) AB AC AD AE ⇒ = (5) AD + DB AE + EC ⇒ AD AE + AD EC = AD AE + AE DB (6) $$EC AD ¨¨DB AE ⇒ ¨ = (7) ¨ AE AD AD¨¨ AE ∴ EC : AE = DB : AD Both propositions are on properties of parallelograms, equality and parallelism of opposite sides. In partic- ular, if D and E are the midpoints, then DE = 1 BC. Similar argument holds for the invariant parallelogram 2 in Figure 7c. In the preceding examples and explanations, reasoning with the geometrical properties in the dia- grams is essential. Use of auxiliary lines, like the diagonals transforms the task from triangle to parallelo- gram, and the auxiliary triangle in Figure 5b are necessary in demonstrating equivalence in mathematics. The transformations do not alter the structural properties of the diagrams. Moreover, there is a reciprocal relationship between the context of the task and mathematical concepts (Abrahamson and Wilensky, 2007; Steinbring, 2005). Technology, like other media, does not directly communicate the inherent mathemati- cal relationships. The symbolic relationships are interpreted, but the graphic software allows for efficient testing of conjectures and affords multiple orientations and visualizations that may trigger mathematical generalizations. Such reasoning requires supportive curricula; otherwise the embedded generality may not be achieved even in dynamic geometry applications. In the rest of the paper, we outline curricula challenges to be overcome before ICT can be effectively integrated in mathematics instruction. 4 Discussion 4.1 Curriculum Coherence Secondary mathematics curriculum is apparently congested at 68 topics (KIE, 2002), which together with perceived difficulty of content explains students’ poor performance. Efforts to improve students’ achieve- ment have been characterized by removal of "difficult" content and inclusion of content from primary mathematics. Although the syllabus aims at developing students’ logical and critical thinking, there is no provision for mathematical proofs. Furthermore, our analysis (Ogwel et al, in preparation) and the pre- ceding illustration indicate that poor performance in mathematics could be due to repeated and mutually isolated content. The analysis of intended emphasis in Kenya Certificate of Secondary Education (KCSE) and the syllabus objectives based on Bloom’s Taxonomy of Learning Objectives reveal that while the syl- labus emphasizes lower order objectives, KCSE examination requires higher cognitive levels of analysis, 6
    • synthesis and evaluation. Moreover, difficulty in the syllabus has more to do with lack of connections among the content– evidence of weak coherence, than any particular content area. Curriculum is coherent if it is rigorous, progresses from particular to general and has inherent struc- tures that link various content areas (Schmidt, Wang and McKnight, 2005). Schmidt, Wang and McKnight (2005) attribute difference in students’ achievement between the US and top countries (Singapore, Korea, Japan, Hong Kong, Belgium and Czech Republic) in Third International Mathematics and Science Study (TIMSS) to curriculum coherence. They observe that mathematics curriculum in the high achieving coun- tries is sequenced hierarchically based on inherent mathematical structure and logic. Besides, mathemat- ics curriculum in these countries progresses such that senior grades have rigorous content than preceding grades. On the other hand, mathematics curriculum in the US is characterized by endlessly repeated con- tent arbitrarily assigned to grades. Isolated content would not promote understanding of mathematical structures; as Otte (2005) also argues that "there is no reasoning from particulars to particulars. . . . to know implies, in any case, to relate a particular to a general; it implies to generalize" (p. 10). Although the previous tasks are assumed to come from Similarity and Enlargement in Form Two, they integrate various content areas, including Geometrical constructions (Form 1); Angles and plane figures (Form 1); Area of plane figures (Form 2) Circles, chords and tangents (Form 3); Loci (Form 4), and Transformations (Form 2 and Form 4). That the current syllabus provision on Similarity and Enlargement is inadequate for solving the problems indicates lack of rigour in the curriculum. It is also doubtful if Circles, chords and tangents or Loci as presented in the KIE (2002) syllabus would help in solving the problems, an indication of lack of progression and focus in the curriculum. In fact, the last two tasks may even be challenging to students in tertiary education. Thus, further simplification of the curriculum may not address the illusive ease in secondary mathematics. ICT environments may give generalized results, for instance shaded area in Figure 6, but the nature of curriculum would determine whether the result could be understood and justified. As Yerushalmy (2004) also argues, "technologically-supported curricular change can lead to change in students’ cognitive hierarchies, though such change may have as much to do with curriculum as it have to do with technology." Moreover, one may be proficient is using technology without understanding the mathematical structures. For instance, one may obtain the invariant parallelogram using dynamic geometry software (Figure 4a) without understanding the inherent structure in the proof of propositions 7 and 8. As Abrahamson and Wilensky (2007) also argue The composite nature of mathematical representations is often covert – one can use these concepts without appreciating which ideas they enfold or how these ideas are coordinated. The standard mathe- matical tools may be opaque – learners who, at best develop procedural fluency with these tools, may not develop a sense of understanding, because they do not have opportunities to build on the embedded ideas, even if each of the embedded ideas is familiar or robust. (p. 28) Consequently, integrating ICT in mathematics education calls for a re-examination of the curriculum and a shift from the result-oriented pedagogy. As Gauss observed, it is the ’how’ in mathematical reasoning that would be of educative value than answers readily computed by the machines. Furthermore, a co- herent curriculum would ensure smooth transition beyond secondary school, an aspect of articulation in education. 4.2 Articulation in the education system Lack of curriculum connection affects both secondary and tertiary education. Whereas secondary edu- cation does not include proofs, tertiary mathematics education requires formal reasoning. The concept of articulation as used by Ng’ethe, Subotzky and Afeti (2008) in formal tertiary education settings needs to be extended to include how skills acquired in the so-called commercial colleges can be utilized. An implication is for a shift from the low regard for education offered outside conventional classrooms. Be- sides, the exam-driven education at all levels of education remains an obstacle to meaningful education. Nevertheless, there is an emerging trend where university students, for example architects, are pursuing courses in graphic design– evidence that employers are beginning to value proficiency in relevant ICT applications. The trend is a pointer for the need to consider competency-based curriculum in education. 7
    • More significantly, integrating ICT in mathematics education would be challenged by assessment frameworks which have traditionally been used to select and place students into the limited employment and higher education opportunities. In fact, examinations have been a major obstacle to the realization of, especially constructivist, reforms in education. Moreover, examinations would be inadequate in assessing students’ complex problem-solving skills in technology-enabled instruction without bold, perhaps radical, changes in assessment. Furthermore, technology in mathematics education has considerable implications for teacher preparation and continuing professional development, some of which are outlined in the next section. 4.3 Teacher preparation and professional development Innovations in education are dependent of teachers’ attitudes, beliefs and conceptions. Presently, there is a gulf between initial teacher preparation and reform-driven roles that teachers are expected to play in instruction. Moreover, teachers are rarely supported in the implementation of reform visions and profes- sional development courses appear to offer generic solutions that do not easily transfer to regular practice. Generally, teachers are expected to design purposeful tasks, provide opportunities for students to develop independent thinking, elicit and incorporate students’ diverse conceptions in instruction, and validly eval- uate learning. Requisite skills for designing tasks that potentially engage students and promote problem-solving abilities are rarely developed in teacher education courses. Moreover, most instructional activities, includ- ing practice exercises are derived from textbooks, and inadequate time has been seen to hinder teachers’ adaptations of such tasks. In addition, designing purposeful learning environments imply deeper under- standing of mathematics. Integrating ICT in mathematics would also require change of conceptions of mathematics, learning and theoretical perspectives. We agree with Ioannidou, Repenning, Lewis, Cherry and Rader (2003) that ICT should not only enhance efficiency in learning the existing curricula, but must focus on deeper understanding of mathematical concepts. Moreover, the functions in dynamic geometry software are based on structural properties, embedded in terms of programming "primitives" (Laborde, 2005) or kernels, thus understanding of mathematical structure would be necessary for understanding the magical behaviour of mathematical software. The development in computational mathematics, together with software like Mathematica and Maple are already challenging the conception of proof in number theory (Borwein and Bailey, 2003; Borwein, 2005).5 Computer software and technological tools require deeper understanding of operations, syntax and familiarity with the embedded functions. Most mathematics teachers in Kenya lack proficiency in digital technologies, and the urgent focus should be on improving their literacy in ICT. Furthermore, integrating technology in education requires teachers to be confident users of technology (Taylor and Corrigan, 2007). Following Goos (2005), integrating ICT in mathematics education would require transition in perception of technology from being a master, servant or partner to extension of self. When skills and knowledge are limited to a range of operations, say computer literacy, the technology is taken to be a master. On the other hand, technology is seen as a servant when it enhances efficiency, for example use of calculators in computations. Instruction in ICT environments that involve provision of new tasks or alternative approach to existing tasks is possible when technology is taken as a partner. The illustrations in this paper possibly reflect this view. Viewing technology as an extension of self implies a greater efficiency in designing tasks, deeper understanding of mathematics, using variety of technologies, and seamlessly integrating them in instruction. To reach this level, teachers would need capacity building in technology. Although teachers may use technology in their learning, they cautiously integrate it in regular in- struction (see Barak, 2006, for example). Reluctance to integrate ICT in education is probably due to lack of significant evidence on how technology supports learning in everyday classrooms (Samwelsson, 2006; Taylor and Corrigan, 2007). Thus, if technology is to be used to engage students, enhance higher order thinking skills and facilitate deeper understanding of mathematics, then continuous collaboration among various stakeholders in education is necessary. 8
    • 4.4 Individualistic versus collaborative learning Formal education is characterised by stiff competition – evident examination rankings and rush for limited higher educational opportunities. Moreover, the society has effectively nurtured the culture of valuing the product over process of education. Thus, advocacy for cooperative and collaborative learning has largely been confined to reform literature as the classrooms remain invariably individualistic. Moreover, the ICT industry has not been spared from the competitive culture, where computer pro- ficiency has determined placement in prestigious careers. Given the limited "greener pastures", ICT skills have rarely been shared to maintain competitive edge over contemporaries. Nevertheless, there is an emerging global trend of sharing skills, enhanced collaboration and availability of shareware and open source computer software. In addition, collaborative learning in technology enhanced environments, like the Internet, and the conception of learning as a social activity calls for a redefinition of school in- structional practices. Consequently, e-content in mathematics education must not only be collaboratively developed, but also provide opportunities for learners to collaborate and interact across diverse locations. Design of e-content and general integration of ICT in mathematics education must significantly involve mathematicians, technicians, curriculum developers, teachers and education evaluators (see also Laborde, 2001; Ogwel, 2007). Moreover, borrowing from design research (cf. Abrahamson and Wilensky, 2007; Cobb, 2000) also Scherer and Steinbring (2006), development of e-content and design of tasks would only be the begin- ning of a cyclic and iterative process of designing content, testing it in classrooms, analysing students’ interactions and modifying the content. Data generated from such studies may be analysed from mul- tiple theoretical perspectives to provide a richer and holistic interpretation of educational process. And given the resource constraints in Kenya, products developed in other regions may be subjected to the local contexts, in any case, reinventing the technological-wheel would be uneconomical. We contend that while integrating ICT in mathematics is long overdue, evidence on how they promote students’ learning in regular school settings is necessary. Thus, collaborative dissemination of research re- sults to inform practice and policy need to also incorporate voices of the major stakeholders, and must also be accessible and comprehensible, especially to education practitioners. Moreover, multimedia including videos, photos and audio data would supplement the print media in disseminating such research studies. Web blogs also have potentials for effective dissemination of research results and sharing of experiences. Although ICT holds the key to both best practices and next practices in education (Hannon, 2008), the conservative nature of formal education does not support such practices. Consequently, there is danger of revolution in education influenced from outside without critical reflection of educators. 4.5 Government policies Infrastructure development is necessary as the country aspires for industrial development in line with the Vision 2030 and Millennium Development Goals. Electricity, telephone services and security are nec- essary for technology to be integrated in rural development. So far, zero-rating of tax on computers, encouragement of public private partnerships in provision of computer hardware and software, e.g. Com- puter Aid International (Richards, 2008), Computer for Schools and e-governance are policy measures that confirm the government’s regard for a knowledge-based economy. In addition, laying of the fibre optic network would increase connectivity and minimize cost of accessing the Internet. Consequently, educators have to utilize this window of opportunity to improve quality of education and align educational provision to the global trends, if the Kenyan youth are to be competitive in a globalized society. 5 Concluding Remarks In sum, we have considered some affordances of technology in mathematics education, including effi- ciency in computation and geometrical constructions, opportunities for multiple representations, prompt feedback during problem-solving and inherent generality of concepts in mathematics. These have been illustrated using Dr. Geo and Inkscape to attempt problems on similarity of figures. Although full integra- 9
    • tion of ICT in mathematics education is desirable, there are inherent curricula challenges in the education system that have to be tackled for optimal technology-enabled education. These include inadequate cur- riculum coherence; poor articulation due to outdated assessment frameworks; teacher beliefs on use of technology in education; proficiency in use of ICT; minimum levels of mathematical understanding; and inappropriate pedagogical practices. There are indications of beginning collaboration in this field and supportive government policies. Nevertheless, there is paucity of evidence on how students learn mathematics within digital technologies. Consequently, there is need for continuous collaboration among mathematicians, technicians, educators in the design and research on use of technology in mathematics education. In particular, educators have to urgently pick up the challenge if they are to remain relevant within the rapid revolution in educational courseware. Notes 1 graphiso f t.com/products/archicad 2 http : //www.artlantis.com 3 http : //www.o f set.org/drgeo 4 http : //www.inkscape.org 5 See Centre for Experimental and Constructive Mathematics at http : //www.cecm.s f u.ca References Abrahamson, D. & Wilensky, U. (2007). Learning axes and bridging tools in technology-based design for statistics. International Journal of Computers for Mathematical Learning, 12(1), 23–55. Abramovich, S. (1999). Revisiting an ancient problem through contemporary discourse. School Science and Mathematics, 99(3), 148–155. Ainley, J., Bills, L. & Wilson, K. (2005). Designing spreadsheet-based tasks for purposeful algebra. International Journal of Computers for Mathematical Learning, 10(3), 191–215. Barak, M. (2006). Instructional principles for fostering learning with ICT: Teachers’ perspectives as learners and instructors. Education and Information Technologies, 11(2), 121–135. Borwein, J. & Bailey, D. (2003). Mathematics by experiment: Plausible reasoning in the 21st century. Natick, Massachusetts: A K Peters. Borwein, J. M. (2005). The experimental mathematician: The pleasure of discovery and the role of proof. International Journal of Computers for Mathematical Learning, 10(2), 75–108. Cazes, C., Gueudet, G., Hersant, M. & Vandebrouck, F. (2006). Using e-exercise bases in mathematics: Case studies at university. International Journal of Computers for Mathematical Learning, 11(3), 327–350. Cobb, P. (2000). The importance of situated view of learning to the design of research and instruction. In J. Boaler (Ed.), Multiple perspectives on mathematics teaching and learning (pp. 45–82). Westport, CT: Ablex. Cobo, P., Fortuny, J. M., Puertas, E. & Richard, P. R. (2007). AgentGeom: A multiagent system for pedagogical support in geometrical proof problems. International Journal of Computers for Math- ematical Learning, 12(1), 57–79. Cress, U. & Kimmerle, J. (2008). A systemic and cognitive view on collaborative knowledge building with wikis. International Journal of Computer-Supported Collaborative Learning, 3(2), 105–122. Dreyfus, T. & Hillel, J. (1998). Reconstruction of meanings for function approximation. International Journal of Computers for Mathematical Learning, 3(2), 93–112. Goos, M. (2005). A socio–cultural analysis of development of pre–service and beginning teachers’ peda- gogical identities as users of technology. Journal of Mathematics Teacher Education, 8(1), 35–59. Hannon, V. (2008). Should educational leadership focus on ’best practice’ or ’next practice’? Journal of Educational Change, 9(1), 77–81. 10
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