conjecture

In our last Low Floor High Ceiling article, we had looked at Squaring the Dots... a series of questions on counting the dots inside squares of different sizes and orientations drawn on dotted paper with the dots as lattice points. The focus of the activity was to tilt squares and try to find a general formula for the number of dots inside the square of a particular tilt, as the side of the square changed.

I first came across the idea of a conjecturing classroom during my teacher training (PGCE) in the UK. We were watching a video clip of 6 and 7 year old students in a mathematics class. The classroom looked ordinary, children sitting in rows, a bit of chatting and minor disruptions, books and pencils and erasers strewn across desks, the teacher at the front without any spectacular resources, holding an unfussy whiteboard pen.

Most problems in our textbooks are procedure oriented and repetitive; they can be solved in a mechanical fashion. There is very little scope for reasoning, investigating, discovering, predicting. Nor is there any scope for challenge and creativity. Children need exposure to problems requiring higher order thinking skills. All children deserve such experiences - the challenge and enjoyment of interesting problems in mathematics.
 
In this note, we offer an explanation to the observations made in the ‘Origamics’ article (November 2013 issue of At Right
Angles).

Haga’s Origamic activities require students to explore simple, geometric properties found when we fold paper in prescribed ways. The aim of these activities is to give students easy-to-explore paperfolding puzzles so that they can experience a micro-version of the three stages of mathematical research: exploration, conjecture and proof. In this article, we take up another ‘origamics’ exploration by Dr. Kazuo Haga from the chapter 'Intrasquares and Extrasquares' of his book.

Haga’s Origamic activities require students to explore simple, geometric properties found when we fold paper in prescribed ways. The aim of these activities is to give students easy-to-explore paperfolding puzzles so that they can experience a micro-version of the three stages of mathematical research: exploration, conjecture and proof.
Here we look at one such activity from the chapter “X-Lines with lots of Surprises”.
The perils of teaching by example...
Constructive teaching encourages students to recognize patterns and build appropriate theory with its roots in conjecture. What are the dangers in this method? What precautions should the teacher take to avoid conveying the impression that a result is true simply because it has been observed in all the examples considered?

A letter from the editors about  Viviani’s theorem and  its corollary - the ‘cousin’ to Viviani’s theorem - that was shared in Issue I-2 of At Right Angles. 

Written by practicing mathematicians, the common thread in this section is the joy of sharing discoveries and the investigative approaches leading to them.This time it has Lagrange, Fair Game, Viviani, Paper foldings among others.

17292 registered users
6659 resources