Axioms to Theorems
This is a great activity to try with all IB Mathematics students to help understand some of the ideas and issues associated with axioms and theorems which is, of course, at the heart of Mathematical knowledge. This activity could be used in a mathematical studies classroom or with the whole year group as part of some subject specific ToK sessions. It is practical, engaging and can prompt some very useful discussion.
- Clear a space that is big enough for 16 chairs arranged in a 4 by 4 grid with enough room for students to sit on them. (if you enough students, 32+, then make 2 grids.
- Give each of 16 students a piece of A4 paper, four are marked with a '1', four with a '2', four with a '3' and four with a '4'
- Ask the students to sit down on the grid and hold there numbers in front such that each row, each column and each 2 x2 corner of the grid contains a 1, 2, 3 and 4 as per the rules of the well known 9 x 9 puzzle.
Questions and Discussion
This can be a really interesting exercise for group dynamics and it can take a while for students to complete the puzzle. Once they have, and you have checked that it is correct, the key question to ask is 'Have you found a unique solution?' If you have enough students for a second group then obviously this is a good moment to compare. The group will likely be split and making both the conjectures that it is a unique solution or that it isn't. It is good to introduce this term here and explore what needs to be done to prove it one way or another. Students will likely then move on to demonstrate that there is another solution.
Are there a finite number of solutions to the problem? - This question is nice because of the opportunity it gives students to use reasoning to give an answer.
'As there are a finite number of people and a finite number of places to sit, there must be a finite number of permutations, only some of which will be correct solutions and so there must be a finite number of solutions.
This is a nice moment for making clear the ideas of ToK. Students will usually arrive at this logical deductive reasoning by themselves. It is worth pausing to ask whether or not they feel there is any room for doubt left so as to examine the certainty of the claim
Now repeat the exercise only this time direct some of the students to their seats based on the pattern below.
Make sure that the students in these seats understand that they are fixed and cant move. Again, if you have two groups then give them both the same task.
Questions and Discussion
Having found and checked the solution, ask - 'Have you found a unique solution?' Again, time should be taken to examine and prove conjectures that are made.
Students will eventually conclude that they have a unique solution and probably begin an attempt at a proof by deduction. If they are able to do this then that is a bonus.
It is now important to establish the difference between the two exercises and focus on the difference being the fixed starting points.
There are a number of ways that the discussions above may have gone but all paths should lead to this point. The exercise is meant to model the ideas of axioms to theorems. In the second case, the starting points are models for the 'Axioms', the unquestioned truths on which we would base our solution piece by piece. The solution then becomes the 'Theorem' which has ben proved by logical deduction starting with the axioms.
This is a model of how mathematical knowledge can develop and of how it is only as certain as the axioms on which it is based. In the first case there were no axioms and so a number of possible theorems.
Reflection and Records
Having done the activity, you can use the following Human Sudoku Reflection to help students keep records off and reflect about what they have done and could take away from this activity.