Organization and methods of teaching mathematics in a technology rich world Thomas Lingefjärd University of Gothenburg

First, we might want to ask our selves: What is the reason of teaching everybody mathematics? Technical jobs (critical to the development of our economies) Everyday living (e.g. figuring out mortgage, being skeptical of government statistics) Logical mind training / logical thinking (mathematics is a great way to learn logic)

Posing the right questions Identifying patterns Convert from real world to mathematical formulation Computation Convert from mathematical formulation BACK to real world What IS mathematics?

The problem? In mathematics education, we’re spending about 80% of the time teaching students to do caclulation by hand. Mathematics is not equal to calculating, mathematics is a much broader subject than calculating. In fact, mathematics has been liberated from calculating.

Should we have to “Get the basics first”? Are the “basics” of driving a car learning how to service or design the car? Are the “basics” of writing learning how to sharpen a quill?

People confuse the order of the invention of the tools with the order in which they should use them in teaching. Just because paper was invented before computers, it doesn’t necessarily mean you get more to the basics of the subject by using paper instead of a computer to teach mathematics.

What about this idea that “Computer dumb mathematics down” … that somehow, if you use a computer, it’s all mindless button-pushing. But if you do it by hand it’s all intellectual. Do we really believe that the mathematics that most people are actually doing in school practically today is more than applying procedures to problems they don’t really understand for reasons they don’t get? … What’s worse … what they’re learning there isn’t even practically useful anymore. It might have been 50 years ago, but it isn’t anymore. When they’re out of education, they do it on a computer.

In the book: Improving Mathematics at Work: The Need for Techno- Mathematical Literacies, by Celia Hoyles, Richard Noss, Phillip Kent, and Arthur Bakker the authors argues:

What are the mathematical knowledge and skills that actually matter for the world of work today? Has technology reduced the necessary knowledge to the most basic arithmetic? This book argues that there has been a radical shift in the nature of mathematical skills required for work.

Examining how information technology has changed mathematical requirements, the idea of Techno-mathematical Literacies (TmL) is introduced to describe the emerging need to be fluent in the language of mathematical inputs and outputs to technologies and to interpret and communicate with these, rather than merely to be procedurally competent with calculations. The authors argue for careful analyses of workplace activities, looking beyond the conventional thinking about numeracy, which still dominates policy arguments about workplace mathematics. Throughout their study, the authors answer the following fundamental questions:

What mathematical knowledge and skills matter for the world of work today? How does information technology change the necessary knowledge and the ways in which it is encountered? How can we develop these essential new skills in the workforce?

Shifting Assessment in a World with WolframAlpha

Estimate when the rate of change of unemployment in India (measured in percent) was the highest

Shifting Assessment in a World with WolframAlpha Estimate when the rate of change of unemployment in Sweden (measured in percent) was the highest

Shifting Assessment in a World with WolframAlpha Describe the view of the curves in a mathematical language and then make an analyze of why the curves shows such a different behavior for the two countries.

One key issue in teaching are about how to organize the teaching. Students may work with collective tasks one by one in class, two by two, or in smaller groups. The teacher might address the whole class, group by group or individual students. Another key issue is the method with which one tries to achieve a goal, eg. the teaching and/or learning of certain concepts or procedures. In certain cases the method is given. You cannot learn to cooperate if you not are allowed to practice it, you cannot learn to present a solution if you never are required to do so.

Organization and methods Method Passive reception Organization Lecture

Organization and methods Methods Passive reception Repetitive Reproductive Elaborative Reflection Analyzing Investigating Constructing … Organization Lecture Group work Active discussion Two and two Seminary Role play Interview Self education … Views of learning and knowledge

What kind of organization or method is suitable for the following task? Find the equation of a circle with center on the line y = 3x and which is tangent to the y- axis on (0,2).

What kind of organization or method should be suitable for the following task? Find the equation of a circle with center on the line y = 3x and which is tangent to the y-axis on (0,2). Explore what will happen if you change the slope and/or the point… Hypotheses? Conjectures? Findings?

As good as all students know how to film something with her or his mobile phone or digital camera. This film can easily be transferred to a computer and split by, for instance, Virtual Dub.

Teaching idea: Let the students film themselves while doing some activity and then split the film and use some carefully selected frames it to pose a question to their peers.

A video film image by image through Virtual Dub

Will the ball go into the basket? An example of Mathematical modeling with student’s own data

To import the split images into GeoGebra, add points where the position of the ball is, put the coordinates into the spreadsheet and do a curve fit is just a 15 minutes activity!

The students could/ should validate GeoGebra´s ways for finding the best fit by using Excel and/or Maxima for finding their second degree polynomial through the same set of data points...

The students could/ should transform the data in a coordinate system which fits the original situation in the basket ball hall better...

Students could use technology as a way to test conjectures at many different levels or for problem solvingconjectureslevels problem solving