Learning mathematics practically in the classroom is not enrichment, it is at the core of empowering an individual’s understanding of the subject.
Mathematics educators have suggested that students should receive opportunities to use and apply mathematics and engage in mathematical modelling (1). Constructivists state that students do not merely learn by being told a set of instructions to follow and that they would benefit from opportunities to model mathematics in real world situations. Jo Boaler (2) states that
Students do not only learn knowledge in the mathematics classrooms, they learn a set of practices and these come to define their knowledge… Students [who] engage in mathematical modelling… develop a deeper, conceptual knowledge of mathematics…. that can have value beyond the classroom.
I am great believer in classroom experiments to help students confront any fears and anxieties they may have about mathematics. The classroom can at times create only boundaries for many students to real life mathematics. The primary focus of experimenting in lessons is to develop engaging and non-threatening mathematics that will capture students’ imaginations because they are innately entertaining. When students complete the practical part of a lesson, they will: have a better appreciation of what mathematics is; feel like they have been involved in experimenting with and understanding mathematics; and have sense of mathematical ownership and curiosity.
Delivering mathematical concepts in an experimental way engages and reinforces learning. It puts the ideas learnt into a setting and allows time for those ideas to be developed without any of the mathematical anxieties which can sometimes occur.
A length of rope is a great prop to start a mathematics lesson. Mixing this up with a magical constant called the ‘Rendezvous Constant’ [3], a name that suggests feelings of adventure, is a great place to start the lesson.
Mark a number line with a piece of chalk from 0 to 10, in 30cm divisions, on the ground. Ask three people to stand at points on the line. Set the class the challenge of where to place the fourth person on the line. For the fourth person to be standing at the Rendezvous point, the sum of the three distances to this fourth volunteer from the other three points needs to be a total of 15.
To perform this practical experiment, use a 900cm rope and mark a number line 0 to 10 with 30cm divisions on the ground. Once the four people have taken their positions on the line you can test to see if the distances add to the length of the rope. Start at the Rendezvous point (y) person and go out to one of the three students on the line, and then back to the y point, repeating this for all three students. If there is any surplus/shortage of rope when this is completed it will give a clue to the students as to what is needed to be done to find the correct Rendezvous point. Repeating this practical experiment a few times reinforces the concepts. The class will then be ready to investigate their findings in groups – maybe by using counters on a piece of paper.
This simple computer code will help the students check and test their conjecture:
10 a = rnd(10)+1 : b = rnd(10)+1 : c = rnd(10)+1
20 PRINT “Position of a, b, c “;a,b,c
30 FOR i = 0 TO 10 STEP 0.1
40 d = sgn(a-i)*(a-i)+sgn(b-i)*(b-i)+sgn(c-i)*(c-i)
50 IF d < 15.2 AND d > 14.8 THEN GOTO 80
60 NEXT i
80 PRINT “Rendezvous position on the line “;i
The computer code gives an approximate location of the Rendezvous point with three points on the line. Students can edit this code to investigate at what happens for more points on the line.
Eventually the students can move to algebra to verify their approximate solutions exactly by working with modulus linear equations.
For example, with three counters at 0, 1 and 10 gives the Rendezvous equation is
The solution can be found from
giving the Rendezvous point as 6 on the line (y = 6).
A general definition for the Rendezvous Constant is given by Gross’s Theorem:
For any collection of points x1 , x2 ,….., xn ∈ E , there is a point y ∈ E for which the average distance from y to x1 , x2 ,….., xn is a(E) , i.e.
The amazing fact is that although y varies with the collection of points, the Rendezvous Constant (a(E)), works for all collections points and no other constant works.
For a straight line 0 to 10 the Rendezvous Constant (a(E)) is always five – the middle of the line. The reason our target number was 15 in the experiment above is that with three people on the line, 3×5 gave 15. If you had four people on the line the target number would be twenty, 4×5.
There are Rendezvous Constants for all shapes. Regular polygons and circles offer some wonderful opportunities to work on experiments and mathematical proof. The Rendezvous Constant is an intriguing place for young minds to meet and explore magical mathematics. I have always found that students of all ages and abilities are fascinated by this result.
1] A. Schoenfeld, Learning to think mathematically: problem solving, metacognition, and sense making in mathematics, Handbook of Research on Mathematics Teaching and Learning, pp 334-371,New York: MacMillan 1992
[2] J. Boaler, Mathematical modelling and New theories of learning, Teaching Mathematics and its Applications, 20, no. 3, 2001
[3] S. Humble, Rendezvous Constants, Mathematical Gazette July 2008 [accessed from the website http://drmaths.org]
