ICT (information and communications technology) is an umbrella term that includes older technologies like radio, television, as well as newer technologies like mobile phones, computers and other digital devices. I prefer to use the term educational technology. This includes any technology that is used in an educational context. It can be older educational technologies, for example the OHP, blackboard, books, as well as digital technologies, e.g. calculator, mobile phones, computers, and internet. It therefore includes software that are used in an educational context, for example GeoGebra, PhET, Microsoft Word, Cabri,... “The use of technology has a long history in mathematics education. Many societies, for example, introduce arithmetic with an abacus, for two reasons. First, the abacus supports computation. Second, the abacus presents a tangible image of mathematics, which helps students understand difficult concepts”. - Center for Technology in Learning, SRI International (2007)
Educational technology use can be categorized as the use of technology by the teacher (personal use), learners’ use, and the use of technology to enhance teaching and learning in the classroom. This is not a clear cut categorisation because of the overlap of some categories. However, I find it a useful categorisation to structure this presentation. The use of technology can make a teacher (personal use) more productive and their work more professional. It can also improve communication and sharing of work with other colleagues. (e.g. Typing a maths test in Word and use the same structure again)The use of technology can make learners more productive and improve communication between them. (e.g. Mobile phones: Face book, MiXit, … , use of calculator enable them to focus on more important content)The use of technology can enhance visualisation and conceptual development and make immediate feedback possible. (dynamic geometry & the use of clickers)The most important aspect in terms of learners’ understanding is that technology creates new possibilities in terms of visualisation in mathematics. According to the Van Hiele theory visualisation is an important component / stage in conceptual geometric conceptual development.
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CAS: Computer algebra system
Technology can save us time, make us more productive. The dull, routine work should be done by computers (Schrenko 1994). Instead of spending a significant amount of classroom time on computation, students can focus on higher level skills such as "decision making, reflection, reasoning, and problem solving" (National Council of Teachers of Mathematics). They can use technology to manipulate variables, solve equations and to construct graphs. Using technology learners could spend:Less attention to procedural and algorithmic skills and tedious calculations AND more emphasis on deep conceptual understanding.For example when the focus of the lesson is the transformations of graphs: the mechanics of sketching graphs could constitute a distraction for the learners. The last question is important: What is the real heart of mathematics?Your view about the use of technology will depends on your answer!
Visualisation is the ability to draw mental images: seeing it in your mind. Some prefer the term spatial reasoning which consist of the “cognitive processes by which mental representations for … objects, relationships, and transformations are constructed and manipulated” (Clements & Battista 1992)According to the Van Hiele theory visualisation is an important stage in conceptual geometric conceptual development. Using technology can help represent mathematics in ways that help students understand concepts. ICME Technology group view “technology as inspiring and driving visualisation in mathematics education. Thus, research and development regarding innovations with technology that allow us to visualize things that in the past have not been easy to visualize…”While many researchers suggested that visualisation aids conceptual development, there is still some way to go to understanding this relationship. Love (1995 P 125) suggests that in geometry the relationship between "mental objects and physical images is an especially difficult one".
Show Cabri program + examplesShow GeoGebra: 1) Graphs 2) Sliders & graphs 3) transformation geometry 4) Euclidean Geometry GeoGebra 4: Stats & Linear programmingLazy teachers: see my website: I created applets based on the high school curriculum.Mobile devices provides “flexible and timely access to learning resources, instantaneous communication, portability, active learning experiences and the empowerment and engagement of learners, particularly those in dispersed communities” (JISC, 2011).GeoGebraMobile: Convert Java to Java scripts: Implication is that learners can use it in the classroomsTAM: Technology adaption is in perceived usefulness and perceived ease of use
There are many other powerful and useful mathematical software that you can use in your classroom.
“Even though technology could be an effective tool to assist lecturers, it is essential that the mathematical content and pedagogy is not compromised” (Laurillard 1993). In some cases if we DO NOT use technology, the mathematical structure becomes clear and the solution more understandable. In other cases if we DO use technology, the mathematical structure becomes clear and the solution more understandable.
The use of GeoGebra in this course enhanced students’ development in van Hiele levels 1 to 3 of geometric development but not on the highest level. GeoGebra enhance visualisation and conceptual understanding but does not improve formal deductive reasoning skills or rigor. The heart of the problem lies within the nature ICT it selves, in this case dynamic geometry software. Although GeoGebra and Cabri 3D can enhance learners understanding of basic concepts, it cannot and are not designed for proof construction and the learning of axiomatic systems. In fact, these kind of skills cannot be teach with the aid of any software. This does not mean these software are useless for the teaching of formal proofs, it will instil the curiosity of students, give them the opportunity to discover relationships, and encourages them and make them more susceptible to try to do proofs.
“Allow calculator use when computational labor can get in the way of the purpose of the lesson. When learning how to perform the computation is the purpose of the lesson, calculators may be a bad idea” Goldenberg (2000).
“Basic skills are still important but we must determine how much valuable instructional time we should devote to helping students become proficient with lengthy or tedious calculations”. (Goldenberg 2000)"Education technology is neither inherently effective nor inherently ineffective; instead, its degree of effectiveness depends upon the congruence among the goals of instruction, characteristics of the learners, design of the software, and educator training and decision-making, among other factors" ( Schneider M., SIIA, 2000)“Allow calculator use when computational labor can get in the way of the purpose of the lesson. When learning how to perform the computation is the purpose of the lesson, calculators may be a bad idea”. (Goldenberg 2000)Efforts to replace teachers or to free up some teaching time have led to the use of technology for drill and practice purposes and that is negatively related to academic achievement (Wenglinsky). “These skills [computational skills] allowed students to secure jobs and to become informed citizens in an industrial society. However, with advances in technology, such computational skills are no longer as important. Instead, students need to develop critical-thinking skills to interpret data appropriately and to use technology to solve more complex problems. Thus, changes in our society have led to a change in what we value in mathematical skills”. (Confrey & Lachance 2000, p. 232)
Wenglinsky (2005) suggests that a lesson should not be planned around a computer, but a computer be used to enhance a lesson. “Decisions about what is or is not obsolete content must be made thoughtfully, attending not just to what technology can do, but to a careful analysis of what students need to be able to do— especially how they need to be able to reason” (Goldenberg 2000). Using calculators can obscure the learners’ ability to see patterns, and insight into why this pattern holds.
Technology integration should not be the aim, but be seen as a tool to make a teachers more effective or to improve the teaching and learning of mathematics. “Think foremost about what you want for your children—the goals of the particular classroom, and the needs of each particular student—and after deciding on your goals, then assess whether the tools are bringing you closer or distracting you away (Goldenberg 2000).
Teachers use blackboards and OHPs because they are easy to use. Teaching is already complex task; teachers do not have time and energy to waste on complex technological systems. “The same is true of teaching. While some teachers enjoy the high adventure of experimenting with novel tools, many others feel more creative when their attention is not divided between their craft—students, thinking, and subject matter—and what may seem like low-level technological details. For these teachers, the advantages of a new tool (even if they agree it has advantages) can be
Technology in maths education
1<br />Gerrit Stols<br />Teaching and learning of Mathematics using technology: <br />opportunities and issues<br />UNIVERSITY OF PRETORIA<br />
What is mathematical educational technology?<br />ICT (information and communications technology) is an umbrella term that includes<br />I prefer to use the term educational technology. This includes any technology that is used in an educational context. <br />2<br />
3<br />Why should we use technology in maths education?<br />
4<br />Technology for productivity and professionalism<br />
How can technology make a teacher more productive and professional?<br />5<br />
6<br />Technology: opportunities for learners<br />
7<br />Opportunities for learners (outside classroom) <br />
Should learners use technology for computation and graphing?<br />The use of technology should results in the learning of more mathematicsANDalso more important mathematics. <br />Goldenberg (2000): “A well-designed lesson has a central idea and focuses students’ attention on it, without distraction by extraneous ideas or procedural details”. <br />“Allow calculator use when computational labor can get in the way of the purpose of the lesson”.<br />“Teachers can focus less on memorizing facts and performing routine calculations and more on developing ideas, exploring consequences, justifying solutions, and understanding connections – the real heart of mathematics” (Heid, 1988).<br />8<br />
Visualisation (the main focus of maths software) <br />Visualisation is the ability to draw mental images: “seeing it in your mind”. <br />Some prefer the term spatial reasoning.<br />Visualisation conceptual development (Van Hiele theory) <br />Technology is inspiring and driving visualisation in mathematics education - ICME Technology group<br />10<br />
Goldenberg (2000)<br />Physical manipulatives in lower grades:<br />provide visual and experimental supports<br />serve as temporary physical stand-ins for mathematical ideas,<br />In the higher grades:<br />dynamic software can provide interactive “virtual manipulatives”<br />“Students who watch carefully as they drag and distort geometric objects onscreen begin to learn how to perform the same kinds of experiments in their minds”.<br />11<br />
12<br />Examples of opportunities for teaching & learning<br />
Other software for visualisation purposes <br />GeoGebra (free)<br />Graph (free)<br />Yenka (limited free use)<br />Cabri 3D (licence) <br />Autograph (licence)<br />Geometer’s Sketchpad (licence)<br />13<br />
Two main concerns about the use of technology: <br />Using technology might focus T&L on the visual and procedural level<br />Worksheet generator<br />GeoGebra / Cabri 3D / applets <br />Using technology might impede learners’ ability to justification and proof results<br />Learners are satisfied that the conjecture is true (after dragging): They do not see the need for proofs<br />The use of technology (going direct to the answer) may hide the mathematical structure, and therefore impede the ability of the learner to justify and proof more general cases. <br />14<br />
A study of the 3rd year in-service geometry teachers at UP (van Hiele)<br />Level 1 (visualisation): At this level, students make decisions based on perception, not reasoning. <br />Level 2 (Analysis): Students see figures as collections of properties. <br />Level 3 (Abstraction): Students perceive relationships between properties and between figures. <br />Level 4 (Deduction): At this level, students should be able to construct proofs such as those typically found in a high school geometry class. <br />Level 5 (Rigor): Students at this level understand the formal aspects of deduction. <br />15<br />
Possible pitfalls<br /><ul><li>Microsoft Worksheet generator: focus on procedural skills
Dynamic maths software: focus on the visual levels / no need for proofs
Calculator / CAS: focus on the answer obscures the details and structure (that might enhance understanding and ability to prove)
Calculator / CAS: lack of basic procedural skills</li></ul>rather <br /><ul><li>Focus on technology: not on mathematics teaching & learning
A ruler, pencil and compass can sometimes create better results than advance technology.
Learning how to use software: you might waste some valuable teaching and learning time</li></ul>16<br />
Issues<br />Is the ability to calculate, manipulate, and graph still important? YES<br />Should they be the focus of teaching? NO<br />Technology, in terms of conceptual development, can help us create mental images, do explorations, finding patterns, but it cannot develop deductive reasoning and rigor (construct proofs and do more formal deduction). It have the potential, if not wisely used to do the opposite. <br />Technology changes curriculum, assessment and what we regard as important mathematical skills<br />17<br />
Using technology to graph, using CASS, & stats might raise the question: “Why do we do maths?” <br />If even your Word processor can draw graphs, factorise, solve equations, integrate and differentiate, why is this the focus of T&L in many classes?<br />Teachers and learners will ask: “What is the purpose of maths?<br />My view: Mathematics is about exploring and proving relationships, finding patterns, and searching for structure AND using these discoveries to solve real life problems.<br />Like technology, procedural knowledge is just a tool to serve this purpose.<br />If technology obstructs this aim, don’t use it.<br />If using technology helps in this endeavour, use it!<br />18<br />
Conclusion<br />“Technology can improve teaching and learning, but just having technology doesn’t automatically translate to better instructional outcomes" (SIIA 2000).<br />Using technology requires a sound content knowledge and a deep understanding of the spirit of maths and the curriculum.<br />Not everything that can be done with technology should be done. Using technology “imposes the burden of judgment” (Goldenberg 2000).<br />Technology use has the potential to make a good teacher even better BUT a bad teacher even worse. Technology cannot enhance teaching by itself.<br />19<br />
Why don’t all teachers use technology?<br />Reasons why teachers are reluctant to use technology:<br />it is expensive<br />they believe using it will take more time<br />they believe it is difficult to use (no support)<br />possible negative impact on the classroom management<br />they believe that technology is not reliable<br />they just do no see the need: satisfied with their learners’ results <br />To implement technology, teachers should have a mastery of:<br />the mathematics content, the curriculum, <br />the pedagogical skills, <br />the technology ...<br />20<br />
21<br /> “The question, then, is not whether to use technology, but how to use it in ways that support the mathematics learning of every student. If students do not learn appropriate ways to use technology in school, they will surely find inappropriate ways to use it outside of school” - Cathy Seeley, NCTM President<br />http://school-maths.com<br />email@example.com<br />