1UNIVERSITY OF PRETORIA

Gerrit Stols

Teaching and learning of Mathematics using technology: opportunities and issues

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What is mathematical educational technology?

• ICT (information and communications technology) is an umbrella term that includes

• I prefer to use the term educational technology. This includes any technology that is used in an educational context.

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Opportunities

Outside class:

Teacher

Productivity

Professionalism

Communication

Personal use:

Learners

Productivity

(computation & graphing)

Communication

In class: Teaching & Learning

Visualisation

Assessment

Why should we use technology in maths education?

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Opportunities

Teacher

Productivity

Professionalism

Communication

Learners

Productivity

(computation & graphing)

Communication

Teaching & Learning

Visualisation

Assessment

Technology for productivity and professionalism

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How can technology make a teacher more productive and professional?

Productivity & Professionalism

Microsoft Excel

Calculation of marks

Presentation of marks

Analyses marks

MS Word

Prepare papers/worksheets

Construction & drawings

MS Worksheet generator

TabletPC Interactive Whiteboard

Inking and handwriting in Word & PowerPoint

InkSeine

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Opportunities

Teacher

Productivity

Professionalism

Communication

Learners

Productivity

(computation & graphing)

Communication

Teaching & Learning

Visualisation

Assessment

Technology: opportunities for learners

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Computation (CAS, graphing, calculating)

Focus & productivity

GeoGebra (CAS & graph)

MSWord

MS Math

Communication Social Media

MiXit

Face book

Twitter

Opportunities for learners (outside classroom)

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Should learners use technology for computation and graphing?

• The use of technology should results in the learning of more mathematics AND also more important mathematics.

• Goldenberg (2000): “A well-designed lesson has a central idea and focuses students’ attention on it, without distraction by extraneous ideas or procedural details”. – “Allow calculator use when computational labor can get in the

way of the purpose of the lesson”.

• “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).

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Opportunities

Teacher

Productivity

Professionalism

Communication

Learners

Productivity

(computation & graphing)

Communication

Teaching & Learning

Visualisation

Assessment

Technology drives visualisation

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Visualisation (the main focus of maths software)

• Visualisation is the ability to draw mental images: “seeing it in your mind”.

• Some prefer the term spatial reasoning.

• Visualisation conceptual development (Van Hiele theory)

• Technology is inspiring and driving visualisation in mathematics education - ICME Technology group

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Goldenberg (2000)

• Physical manipulatives in lower grades:– provide visual and experimental supports – serve as temporary physical stand-ins for mathematical

ideas,• In the higher grades:

– dynamic software can provide interactive “virtual manipulatives”

• “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”.

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Visualisation (representation)

Conceptual understanding

Cabri 3D

GeoGebra / Autograph / GSP /

Graph

Applets: http://school-maths.com

Assessment Effective learning environment

Formative: Clickers

Summative: CBT

Examples of opportunities for teaching & learning

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Other software for visualisation purposes

• GeoGebra (free)• Graph (free)• Yenka (limited free use)• Cabri 3D (licence) • Autograph (licence)• Geometer’s Sketchpad (licence)

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Two main concerns about the use of technology:

• Using technology might focus T&L on the visual and procedural level– Worksheet generator– GeoGebra / Cabri 3D / applets

• Using technology might impede learners’ ability to justification and proof results

• Learners are satisfied that the conjecture is true (after dragging): They do not see the need for proofs

• 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.

A study of the 3rd year in-service geometry teachers at UP (van Hiele)

• Level 1 (visualisation): At this level, students make decisions based on perception, not reasoning.

• Level 2 (Analysis): Students see figures as collections of properties.

• Level 3 (Abstraction): Students perceive relationships between properties and between figures.

• Level 4 (Deduction): At this level, students should be able to construct proofs such as those typically found in a high school geometry class.

• Level 5 (Rigor): Students at this level understand the formal aspects of deduction.

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Possible pitfalls

• Microsoft Worksheet generator: focus on procedural skills• PowerPoint: cannot adapt to learners need • 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

rather • 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

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Issues

• Is the ability to calculate, manipulate, and graph still important? YES

• Should they be the focus of teaching? NO

• 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.

• Technology changes curriculum, assessment and what we regard as important mathematical skills

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Using technology to graph, using CASS, & stats might raise the question: “Why do we do maths?”

• 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?

• Teachers and learners will ask: “What is the purpose of maths?

• My view: Mathematics is about exploring and proving relationships, finding patterns, and searching for structure AND using these discoveries to solve real life problems.– Like technology, procedural knowledge is just a tool to

serve this purpose.– If technology obstructs this aim, don’t use it.– If using technology helps in this endeavour, use it!

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Conclusion

• “Technology can improve teaching and learning, but just having technology doesn’t automatically translate to better instructional outcomes" (SIIA 2000).

• Using technology requires a sound content knowledge and a deep understanding of the spirit of maths and the curriculum.– Not everything that can be done with technology should be done.

Using technology “imposes the burden of judgment” (Goldenberg 2000).

• Technology use has the potential to make a good teacher even better BUT a bad teacher even worse. Technology cannot enhance teaching by itself.

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Why don’t all teachers use technology?

• Reasons why teachers are reluctant to use technology:– it is expensive– they believe using it will take more time– they believe it is difficult to use (no support)– possible negative impact on the classroom management– they believe that technology is not reliable– they just do no see the need: satisfied with their learners’

results

• To implement technology, teachers should have a mastery of:– the mathematics content, the curriculum, – the pedagogical skills, – the technology ...

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“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

gerrit.stols@up.ac.za

http://school-maths.com