Integrating ICT in mathematics education: Curricula challenges inthe 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 donot 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 educationalpractices with the rapid innovations in technology. Emergence of tools that increase interactivity inlearning and facilitate distributed learning and collaboration sharply contrasts conventional curriculaprovisions and practices which have been relatively stablefor several decades. Integrating ICT in ed-ucation would also probably remedy the apparent lack of relevance of school mathematics for mostlearners. Nevertheless, from a semiotic epistemological perspective, ICT would only mediate learn-ing despite the potentials to provide prompt feedback, personalize instruction and express inherentgenerality of mathematical concepts. Besides, given the embedded cognitive hierarchies, computers,instructional software, calculators and multimedia are likely to imply greater instructional challengesthan the constructivist reforms. In Kenya for example, lackof curriculum coherence; poor articulationwithin the system of education; inadequate teacher preparation and professional development; and thetendency for individualistic rather than collaborative learning are critical challenges in integrating ICTin mathematics education. A dynamic geometry software,Dr. Geo, is used to illustrate the challengesbased onSimilarity of Figures. The Government policies one-society are noted for the potential toaddress the challenges of infrastructure development. However, there is need for collaboration in theintegration 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 definethenext practicesin education, and enable young Kenyans to be competitive in aglobalized society.

1 Introduction

Rapid developments in information and communications technologies continue to influence economic andsocial development (Richards, 2008) and afford hitherto unforeseen comfort to end-users. Advances intechnology have not been without controversy due to perception of automation as an affront to labourand anxiety over the requisite skills for its integration inmost professions. Evidently, technology hasrevolutionized the banking industry, for exampleM-pesaand has enhanced efficiency in architecturaldesigns, e.g.Archicad1 andArtlantis.2

∗A paper for the 1st Regional Conference on e-Learning:Increased access to education, diversity in applications andmanagement 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 technologicalinnovations, notwithstanding decades of inefficiency in education. For instance, educators in Kenya, asin other countries, have been concerned with students’ performance, low motivation and negative attitudetowards mathematics, attributed partly to curriculum thatappears 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 studentsand 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 enhancedfocus on collaborative and cooperative learning. The proposed reform visions may be achieved within anICT integrated curriculum, which would also provoke deepermathematical reasoning. However, a numberof challenges have to be overcome before digital technologies can be effectively integrated in mathematicseducation. These include curriculum coherence, inappropriate pedagogical practices, inadequate teacherpreparation and professional development, and lack of appropriate infrastructure. In this paper, we outlinethese challenges after an illustration based onSimilarity of Figures, and argue for enhanced collaborationin the design and implementation of ICT integrated mathematics education.

2 Integrating ICT in school curricula

Technology has been used in mathematics inAnalysis(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 andLeron, 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 mathematicalrepresentations that enhance generality of mathematical concepts, and provide opportunities for counter-examples, unlike in paper and pencil environments. Technology also enhances curiosity that may driveinventions as illustrated in computational mathematics, (see, for example Borwein and Bailey, 2003).

Abramovich (1999)’s use of spreadsheets in generalizing Pythagorean Theorem demonstrates howcomputers 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 effectsand 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 ofaccurate diagrams, simulation, drag effects, and when coupled with after-shadows or trace facilities revealmathematical properties which may be difficult to achieve on paper. The multiple representations incomputer applications and prompt feedback (Ainley, Bills and Wilson, 2005; Laborde, 2001) illuminatethe critical challenge for mathematics educators, as Gausscited in Borwein (2005) observed, ishow toarrive at the solutions. That is, mathematics education hasto transcend the novelty and curiosity in the useof ICT so that these are used as learning tools.

In Kenya, besidesComputer Studiesin the secondary curriculum, ICT has largely focused on com-puter literacy and efficiency in computations. Recent introduction of calculators in mathematics educationis a major step although its efficacy on students’ learning is yet to be investigated. The focus on accurateanswers is inadequate in mathematics given that results maybe obtained without understanding how toarrive at them. Moreover, mathematics instruction must transcend novelty, fun and the awe experiencedin ICT applications if the aroused interest is to be sustained. Consequently, the challenge for mathematicseducators is how and when the various computer applicationsand other ICT are integrated in the schoolcurriculum (cf. Laborde, 2001). In the next sections, we illustrate challenges that ought to be addressedas ICT is integrated in mathematics education. Screenshotsfrom a dynamic geometry software,Dr. Geo3

are given and the reader is encouraged to attempt especiallyTasks 3, 4 and 5 before considering our partialexplanations.

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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 andtheir 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 scalefactors and (g) real life applications (p. 20). Three conjectures and three tasks are used to explore conceptsthat 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 topolygons (Figure 1b) such that, polygons with relatively parallel sides are similar. The hypothesis may betested using triangles to verify its truth. A counterexample (Figure 1c) generated by the drag facility inDr. Geoshows the distortion when parallelism is maintained. The drag facility allows one to investigateembedded 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), andthe involved realia furtherdisproves 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 theunderstanding of similarity of figures beyond definitions and formulae. One has to consider the inherentmathematical 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 eitheranswer? Similarity in figures involves ratio of corresponding sides, corresponding ratios of sides andequality of angles, but how would these be linked to the preceding conjecture? The embedded propertiesyet to be revealed are evident in the next task.

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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 geometryenvironments, drag facility allows for faster simulation of different orientations of quadrilaterals, therebyconfirming the generality of the conjecture. While this may arouse interest among learners, inability tounderstand the embedded mathematical properties may allude to a mythical perception of mathematicsand 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 processof producing the parallelogram in Figure 4b. While the forward process is fairly easy, the reverse processmay have some cognitive challenges. In fact, the task may be challenging both in paper & pencil anddynamic 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 obtaina midpoint of a line segment? What are the properties of the midpoint? In school mathematics, midpointsare obtained from perpendicular bisectors of lines. Instruction emphasizes on use of arcs, which in ouropinion 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 isthe line segment. Completing the task requires use of appropriate constraints, and an understanding thatparallelogram 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 thecircles. In the preceding task, it may be useful to justify the result in Figure 4a. Auxiliary lines joining theopposite vertices of the quadrilateral are necessary in thejustification. Similar arguments are embeddedin 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 aparallelogram.

Task 5 Construct a parallelogram whose acute angle has a constant valueθ

The task introduces possibly the need for global thinking with diagrams. The use of such a task may alsoaid in diagrammatic reasoning (Hoffmann, 2005), and perhaps point to the need for thinking outside thediagram (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 mathematicsteachers reveals that the problem solving skills involved in the latter task may have not been acquired inteacher preparation courses.

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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 suchthat BE= 14cm and BF= 21cm. CE and DF intersect at G. Find the area of quadrilateralAEGD

A D

CB

G

F

E

Figure 6: Area and Similarity

In the following sections, we reexamine the affordances ofDr. Geo in solving the problems andoutline 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 thecounter-example. Parallelism definesangles, thus equality of angles is a necessary but insufficient con-dition for similar figures. Without measuring the sides of polygons and comparing ratios, the connectionbetween 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 joiningthe 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, trianglesABC and ADE are similar. The following pair of converse propositions may be considered.

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Proposition 7 In △ABC, D and E are points on AB and AC respectively. Prove that ifAB:AD= AC:AE,then DE//BC

Proposition 8 In △ABC, D and E are points on AB and AC respectively. Prove that ifDE//BC, thenAD: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,

ADAB

=AEAC

(4)

⇒AD

AD+ DB=

AEAE+ EC

(5)

⇒ AD � AE+ AD � EC = AD � AE+ AE � DB (6)

⇒���AD�EC

��AD � AE

=�

�AE�DB

AD���AE

(7)

∴ 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

2BC. Similar argument holds for the invariant parallelogramin Figure 7c.

In the preceding examples and explanations, reasoning withthe 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 propertiesof the diagrams. Moreover, there is a reciprocalrelationship 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 efficienttesting of conjectures and affords multiple orientations and visualizations that may trigger mathematicalgeneralizations. Such reasoning requires supportive curricula; otherwise the embedded generality maynot be achieved even in dynamic geometry applications. In the rest of the paper, we outline curriculachallenges to be overcome before ICT can be effectively integrated in mathematics instruction.

4 Discussion

4.1 Curriculum Coherence

Secondary mathematics curriculum is apparently congestedat 68 topics (KIE, 2002), which together withperceived 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 primarymathematics. Although the syllabus aims at developing students’ logical and critical thinking, there is noprovision 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 mutuallyisolated content. The analysis of intended emphasis in Kenya Certificate of Secondary Education (KCSE)and the syllabus objectives based onBloom’s Taxonomy of Learning Objectivesreveal that while the syl-labus emphasizes lower order objectives, KCSE examinationrequires higher cognitive levels of analysis,

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synthesis and evaluation. Moreover, difficulty in the syllabus has more to do with lack of connectionsamong 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 precedinggrades. 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 mathematicalstructures; as Otte (2005) also argues that "there is no reasoning from particulars to particulars.. . . toknow 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 fromSimilarity and Enlargementin Form Two,they integrate various content areas, includingGeometrical constructions(Form 1); Angles and planefigures(Form 1);Area of plane figures(Form 2)Circles, chords and tangents(Form 3);Loci (Form 4), andTransformations(Form 2 and Form 4). That the current syllabus provision on Similarity and Enlargementis inadequate for solving the problems indicates lack of rigour in the curriculum. It is also doubtful ifCircles, chords and tangentsor Loci as presented in the KIE (2002) syllabus would help in solvingtheproblems, an indication of lack of progression and focus in the curriculum. In fact, the last two tasks mayeven be challenging to students in tertiary education. Thus, further simplification of the curriculum maynot address the illusive ease in secondary mathematics.

ICT environments may give generalized results, for instance shaded area in Figure 6, but the natureof 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’ cognitivehierarchies, 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 andWilensky (2007) also argue

The composite nature of mathematical representations is often covert – one can use these conceptswithout 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, maynot develop a sense of understanding, because they do not have opportunities to build on the embeddedideas, 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 anda shift from the result-oriented pedagogy. As Gauss observed, it is the’how’ in mathematical reasoningthat 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 ineducation.

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 conceptof articulation as used by Ng’ethe, Subotzky and Afeti (2008) in formal tertiary education settings needsto be extended to include how skills acquired in the so-called commercial colleges can be utilized. Animplication is for a shift from the low regard for education offered outside conventional classrooms. Be-sides, the exam-driven education at all levels of educationremains an obstacle to meaningful education.Nevertheless, there is an emerging trend where university students, for example architects, are pursuingcourses in graphic design– evidence that employers are beginning to value proficiency in relevant ICTapplications. The trend is a pointer for the need to considercompetency-based curriculum in education.

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More significantly, integrating ICT in mathematics education would be challenged by assessmentframeworks which have traditionally been used to select andplace students into the limited employmentand higher education opportunities. In fact, examinationshave been a major obstacle to the realization of,especially constructivist, reforms in education. Moreover, examinations would be inadequate in assessingstudents’ complex problem-solving skills in technology-enabled instruction without bold, perhaps radical,changes in assessment. Furthermore, technology in mathematics education has considerable implicationsfor teacher preparation and continuing professional development, some of which are outlined in the nextsection.

4.3 Teacher preparation and professional development

Innovations in education are dependent of teachers’ attitudes, beliefs and conceptions. Presently, thereis a gulf between initial teacher preparation and reform-driven roles that teachers are expected to play ininstruction. 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 developindependent 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-solvingabilities 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 ofconceptions of mathematics, learningand 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 ondeeperunderstanding of mathematical concepts. Moreover, the functions in dynamic geometry software are basedon structural properties, embedded in terms of programming"primitives" (Laborde, 2005) or kernels, thusunderstanding of mathematical structure would be necessary for understanding themagical behaviourof mathematical software. The development in computational mathematics, together with software likeMathematicaandMapleare already challenging the conception of proof in number theory (Borwein andBailey, 2003; Borwein, 2005).5

Computer software and technological tools require deeper understanding of operations, syntax andfamiliarity with the embedded functions. Most mathematicsteachers in Kenya lack proficiency in digitaltechnologies, and the urgent focus should be on improving their literacy in ICT. Furthermore, integratingtechnology 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 perceptionof technology from being amaster, servantor partner to extension of self.

When skills and knowledge are limited to a range of operations, say computer literacy, the technologyis taken to be amaster. On the other hand, technology is seen as aservantwhen it enhances efficiency,for example use of calculators in computations. Instruction in ICT environments that involve provision ofnew tasks or alternative approach to existing tasks is possible when technology is taken as apartner. Theillustrations in this paper possibly reflect this view. Viewing technology as anextension of selfimplies agreater 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 buildingin 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 lackof significant evidence on how technology supports learningin everyday classrooms (Samwelsson, 2006;Taylor and Corrigan, 2007). Thus, if technology is to be usedto engage students, enhance higher orderthinking skills and facilitate deeper understanding of mathematics, then continuous collaboration amongvarious stakeholders in education is necessary.

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4.4 Individualistic versus collaborative learning

Formal education is characterised by stiff competition – evident examination rankings and rush for limitedhigher educational opportunities. Moreover, the society has effectively nurtured the culture of valuing theproductoverprocessof education. Thus, advocacy for cooperative and collaborative learning has largelybeen 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", ICTskills have rarely been shared to maintain competitive edgeover contemporaries. Nevertheless, thereis an emerging global trend of sharing skills, enhanced collaboration and availability ofsharewareandopen sourcecomputer software. In addition, collaborative learning intechnology 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 collaborativelydeveloped, but also provide opportunities for learners to collaborate and interact across diverse locations.Design ofe-content and general integration of ICT in mathematics education must significantly involvemathematicians, technicians, curriculum developers, teachers and education evaluators (see also Laborde,2001; Ogwel, 2007).

Moreover, borrowing fromdesign research(cf. Abrahamson and Wilensky, 2007; Cobb, 2000) alsoScherer and Steinbring (2006), development ofe-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. Andgiven the resource constraints in Kenya, products developed in other regions may be subjected to the localcontexts, in any case, reinventing the technological-wheel would be uneconomical.

We contend that while integrating ICT in mathematics is longoverdue, evidence on how they promotestudents’ learning in regular school settings is necessary. Thus, collaborative dissemination of research re-sults to inform practice and policy need to also incorporatevoices of the major stakeholders, and must alsobe accessible and comprehensible, especially to educationpractitioners. Moreover, multimedia includingvideos, 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 bothbest practicesandnext practicesin education (Hannon, 2008), theconservative nature of formal education does not support such practices. Consequently, there is danger ofrevolution 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 theVision 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 provisionof computer hardware and software, e.g. Com-puter Aid International (Richards, 2008), Computer for Schools ande-governance are policy measuresthat confirm the government’s regard for a knowledge-based economy. In addition, laying of the fibreoptic network would increase connectivity and minimize cost of accessing theInternet. Consequently,educators have to utilize this window of opportunity to improve quality of education and align educationalprovision 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, promptfeedback during problem-solving and inherent generality of concepts in mathematics. These have beenillustrated usingDr. GeoandInkscapeto attempt problems on similarity of figures. Although full integra-

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tion of ICT in mathematics education is desirable, there areinherent curricula challenges in the educationsystem 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 oftechnology in education; proficiency in use of ICT; minimum levels of mathematical understanding; andinappropriate 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, educatorsin the design and research on use of technology in mathematics education. In particular, educators haveto urgently pick up the challenge if they are to remain relevant within the rapid revolution in educationalcourseware.

Notes

1graphiso f t.com/products/archicad2http : //www.artlantis.com3http : //www.o f set.org/drgeo4http : //www.inkscape.org5See Centre for Experimental and Constructive Mathematics at http : //www.cecm.s f u.ca

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