Category: Algebra

The mathematics of fountain design: a multiple-centres activity

The mathematics of fountain design: a multiple-centres activity by Marshall Gordon describes a study about a variation of differentiated instruction in mathematics which caters to students of all ability needs. The lesson is about transforming quadratic equation. The article was published online in January 4, 2013 in the journal Teaching Mathematics and its Application. Here’s the abstract of the article. Use the email below to get access to the journal article.

Teachers of mathematics recognize the difficulty of reaching every student when the range of student abilities puts a considerable strain on the classroom discussion and time. In a response to the problem, students are grouped so that those with greater mathematical aptitude help those who have difficulties. While this approach is to be appreciated, it tends to mean that the more able students have less opportunity to explore further their own initiatives in mathematics, while those who have more difficulties find themselves on the receiving end with little opportunity to be in the role of enriching the mathematics experience for everyone, including themselves. A ‘multiple-centres’ approach is designed to overcome these problems. In this variation of differentiated instruction, all students get the chance to engage the material from a vantage point and at a level they find interesting and challenging as a consequence of their selecting extensions of the teacher’s initial focus problem. This article will present some findings of 11th year (roughly Fifth Form) average mathematics students at a US Independent School in transforming the standard quadratic equation to represent fountain parabolic trajectories, which was the teacher’s focus problem, along with some multiple-centre investigations they chose. A further set of opportunities with commentaries providing additional centres for student inquiry are included.

 

 

 

 

Abstraction and Algebra

Just how essential abstraction and algebra are? Here’s what Caleb Gattegno says on the importance of abstraction and algebra: 

Nobody has ever been able to reach the concrete. The concrete is so ‘abstract’ that nobody can reach it. We can only function because of abstraction. Abstraction makes life easy, makes it possible. Words, language have been created by man, so that it does not matter what any reader  evokes in his mind when he reads the word red, so long as when we are confronted with a situation, we are using the same word even for different impressions. Language is conveniently vague so that the word car, for example, could cover all cars, not just one. So anyone who has learn to speak,  demonstrate that he can use classes. There are no words without concepts. …

Therefore how can we deny that children are already masters of abstraction, especially the algebra of classes, as soon as they use concepts, as soon as they use language, and that they of course bring this mastery and the algebra of classes with them when they come to school. […]

… The essential point is this: that algebra is an attribute, an essential power, of the mind. Not just of mathematics only. Without algebra we would be dead, or if we have survived so far, it is partly thanks to algebra – to our understanding of classes, transformations, and the rest …

(Gattegno, 1970, pp 23-5)

I think the ‘algebra of classes’ is a bit of a stretch but I completely agree that our young learners are very much capable of making abstraction, and yes, to engage in algebraic thinking.

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