ࡱ> z|y`x6bjbjss BPH,I ,,,@!!!8!$"<@i|L"L""""b=b=b=!i#i#i#i#i#i#i$khmGi,^@=@^=^@^@Gi "">\i.D.D.D^@ ","!i.D^@!i.D.DRYd. ,g"@" H5;!@Meh,ri0ikefn Bfn<gfn,gb=>r.Dt>\>b=b=b=GiGiC^b=b=b=i^@^@^@^@@@@D d@@@D @@@   AN EXAMPLE OF CREATIVITY DEREK HOLTON One of my favourite problems for working with students aged from 10 upwards is the Six Circles Problem. I use one response, that has been made by many children over the years, to part of the solution of this problem, to illustrate my answers to some of the questions posed on the web site. 1. The Six Circle Problem: Is it possible to put the numbers 1 to 6, one per circle, so that the numbers in the three circles on each side of the equilateral triangle, have the same sum? If so, how many ways of doing this are there?  Its worth trying this problem for yourself before reading on as it will help you to gain insight into what is happening in the discussion below. 2. A Creative Moment: After experimenting students generally find four solutions the ones shown below. The question is now, how can you show there are no more solutions?  The younger children who havent been corrupted by algebra have to think about this for a while. They usually see that the sum of the numbers on any one side cant be too big because there arent enough big numbers to go round the three sides of the triangle. For example, it is clear that you cant have a side sum of 100 or 20 or even 15. But why is 12 the highest side sum there is? And why is 9 the smallest side sum? Producing an argument that is more than hand waving takes some effort. They usually flounder for a while on this before someone says 1 has to go somewhere. The biggest sum on the side where 1 is is 1 + 6 + 5 = 12. So you cant get a sum bigger than 12. It seems to me that this is a genuine creative idea. It is nothing like anything they have met before. Nothing they have done before has been this subtle. To get to this point they have had to think about all the addition that they have done in the past and somehow focus that on all possible sums of three of the numbers from 1 to 6. They have to isolate the 1 along with the largest two numbers in the set of 6 and see that it isnt possible to get a bigger sum than 12. Nowhere in the regular school curriculum have they seen anything like this before. I would like to submit that this is an example of mathematical creativity. 3. A Discussion of Creativity: I want to now look at the questions that are posed on the DD 9 web site and answer them in relation to the moment of creativity above. I should note that what I have written here are my views as a mathematician who is interested in education. I have no research, but a reasonable amount of experience, to back what I am saying. 3.1 What is mathematical creativity? 3.1.1 Is mathematical creativity a property of a person, a problem, a solution, a process or a teaching technique? The answer to this question for the specific moment in section 2 is probably yes but it is a property of more than these things. It has to be a property of a person because not every student is able to formulate the idea. It has to be a property of the problem because this idea is meaningless in another situation. It has to be a property of a particular solution because this problem can be solved in a number of other ways none of which require this idea. It has to be a property of the teaching technique that I use because I cant guarantee that it would necessarily occur if another teacher were in charge of the group - they wouldnt necessarily know that by forcing the students to think about the limitations of the side sum a solution could be obtained. Indeed until I have had quite a lot of experience with a problem I dont always know the best way to approach it. Actually I have often tried to stop students going down what I considered to be blind alleys and they have managed to find a solution through their wrong route. But this creative idea is also a property of the age of the students and what mathematical experience they have had. As I said before, students corrupted by algebra wont take this approach to the problem. They will want to formulate the problem algebraically and will go down a quite different route to the solution. I remember a bright 15 year old who lost his creativity after he had done calculus and algebra courses at university. For a while he would only blindly use those techniques in every problem he tackled rather than thinking about the problem and seeing what he could do for himself. 3.1.2 How does mathematical creativity relate to general concepts of mathematics, mathematical problem solving, problem posing, research, and creativity? So what is not creativity? If a teacher has just explained why 3x + 7x = 10x along with similar examples, then a student who can see that 5x + 8x = 13x is not being creative. They are just following a rule, an algorithm. There is nothing novel in that calculation. The same is true for any process that is taught in the traditional way. This suggests that mathematical creativity is less likely to occur in a closed situation. New ideas are more likely when there is an open situation. 3.1.3 Should it be something new to the world or can it be just new to the creator? I have now seen tens of students achieve the creative idea above. (As well as hundreds of students who didnt.) I believe that it was a creative idea for each one of them. They each invented the idea from the problem and the teaching situation. They didnt know of the previous creative events. Teachers will see the same sort of thing in class. Each year students will have a moment of inspiration that previous students have also had. 3.1.4 Is it enough to have ideas that are novel and innovative or must creative mathematics be applied to mathematical problem solving? Surely you cant have novel and innovative ideas in a vacuum. They must be in some context. If they are mathematically novel ideas then they have to be in the context of mathematics. Since all mathematics involves a problem, you can only be creative in a problem situation. But the question seems wrong to me. Is there something that is called creative mathematics that you walk around with ready to apply? Or does creativity occur because you are trying to solve something that you dont have an algorithm for and you desperately want to find a solution to. Does creativity come by accident? Are you a creative person only because you have been shown to have creative ideas in a number of previous situations? 3.2. Which mathematics students can and should be creative? 3.2.1 Can all mathematics students be creative? Is mathematical creativity dependent on mathematical talent or is it a distinct trait? Often teachers who sit in on the problem solving sessions I give in their classes, are surprised by the students who have been creative. Just because you can pick up algorithms quickly and solve routine mathematical problems does not mean that you can be creative. Some creative people appear to be disinterested in routine problems. However, the more mathematics that you know and the more links that you can see between what you know, the more likely you are to be able to put it all together in a novel way. If you have no mathematical knowledge it is hard to see how you could be mathematically creative. But its not easy to walk into a class, even one that you know well, and say this student will be creative today. Potentially they all can be. 3.2.2 Is an in-depth knowledge of mathematics a prerequisite for becoming mathematically creative? It depends what you mean by in-depth. In the Six Circles example students know little more than how to manipulate small numbers. But they know all that they need to know. 3.3. How do the teacher, resources and the environment affect the mathematical creativity of the student? 3.3.1 How should we prepare teachers to foster mathematical creativity in all students? As with most other things, teachers should be taught in the way that we would like them to teach. So we need to put them in challenging situations where they can be creative and enjoy the experience. This means giving them the opportunity to solve problems that may or may not be directly linked to the curriculum that they will teach. 3.3.2 What might teachers, students parents, or others do to foster (or inhibit) creativity? Students need to be given space to think. There is probably much that they have to be told because they cant create everything for themselves but they need to be given open situations to explore at regular intervals and this needs to be shown to be valued. Opportunities can be given to the child both in and out of school. 3.3.4 How might mathematical creativity be made an explicit goal/critical area in mathematic education? It has to become part of the curriculum and assessed. 3.3.5 Will a focus on mathematical creativity distract from other critical areas of mathematics education? Thats hard to say. What are the other critical areas? There is a good chance that creativity will support those other areas. 3.3.6 Should mathematical concepts and skills be learned creatively or should they be memorized before students are encouraged to be creative? I dont think that you can tell a child that today they will be creative. I think that creativity tends to just happen when the situation is right. Im all for creative learning (though Im not quite sure what that is) but I suspect that some things just have to be memorized. 3.3.7 What are examples of good practice, literature, investigations, and problems that can be useful for promoting mathematical creativity? There are a lot of things in the literature that can be cited. 3.3.8 What methods of instruction might stimulate students to create new problems, solve problems uniquely, conduct research work in mathematics, etc.? Im not sure what solving problems uniquely means. Again though this is all about open environments. 3.3.9 Does the use of technology (Computer-Algebra Systems, Dynamic Geometry Systems) promote or inhibit students mathematical creativity? Like anything else it depends on who is using the technology. I have recently seen many examples of calculators that have enhanced creativity. This was possible because the calculator was use to stimulate the class to generate conjectures and produce a need to follow through and solve the problems. 3.4. How might we recognize and assess mathematical creativity? 3.4.1 How might assessment be used to promote rather than inhibit mathematical creativity so that all students might be creators and not simply consumers of knowledge? Assessment has to have open questions that allow students to be creative or students wont value creativity. This is especially true where high stakes assessment is involved. It is likely that it is easier to have such assessment as the school part of such exams as such questions generally need more time than is available on an external exam. 3.4.2 What are the effects of standardized or standards-based assessment on mathematical creativity? In New Zealands recent experience the effects of standards-based assessment at the senior level of secondary school have not been positive. This is because of the focus on the exam and the way that the standards approach has compartmentalised the curriculum so that a student is unable to solve a geometry problem using algebra.  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