Task 184 ... Years 4 - 10
SummaryEasy to state and easy to start this problem is simply about five numbered counters arranged in a line from 1 to 5 with a blank space to the left of number 1. Counters can slide one space in either direction, if there is an empty space, or jump over one counter in either direction, if there is an empty space to land in. The challenge is to reverse the order and still have the blank at the left end.
A Special History
The 'conversation' below in the From The Classroom section was initiated by a question from Michael Palme, Brigidine College, New South Wales, to Sue Davis in the early days of the Mathematics Task Centre Project. Sue was Executive Officer of the project. The conversation was carried out by fax - email was not widespread at the time - between Michael and Doug. Williams, Project Manager. The aspects of the problem that pertain directly to its solution have been extracted and placed in the Iceberg section. The remaining discussion explores the problem in the context of a class learning to work like a mathematician.
IcebergA task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.
StrategyMost people first tackle this problem using a trial and improve strategy. This will sometimes lead to a solution, ie: the pieces will change position as required under the rules. However, it can also lead to frustration, and, if a solution is found, it is often not easy to reproduce. This is a very important phase. It is the play stage of a mathematician's work (see working like a mathematician above) and it is providing data that has yet to be organised to support solution of the problem.
Two realisations work together to extend beyond this initial play:
Students who bail out of the problem (for now) at this point can be complimented on gathering data, finding a key condition in the problem and making a wise selection of strategy.
Movement SystemsPerhaps using the case of five counters, or perhaps through exploring the simpler cases, students begin to talk about a system for moving the counters. There are at least two common ones.
Students who bail out of the problem (for now) at this point can demonstrate the system they use on several starting numbers of counters. They can be complimented on finding and describing a key solution process, and recognising that we don't all think the same way, so there is often more than one way to tackle a problem.
Visual Patterns and Number PatternsWherever there is a visual pattern there will be a number pattern and vice versa. In this case the visual patterns are dynamic and have several sub-steps that each follow their own rules. Exploring the various movement systems a little more deeply can be the next layer of the iceberg. Comparing the two above shows that they don't both take the same number of moves. System 1 takes 36 moves and System 2 takes 30. Comparing further with other possible student-designed systems shows that System 2 always gives the minimum number of moves. (Editor's note: Further discussion below about this statement.)
Each system can now be tracked to keep a count of how that total is made up. Using the System 2 path in the rest of this explanation seems appropriate because mathematicians usually try to use the most efficient process.
That is 8 + 4 + 6 + 3 + 4 + 2 + 2 + 1 = 30
Looking at this vertically and dividing it into the two parts may be more instructive. It is also an application of the Make a list or a table strategy.
Clearly there is a pattern here that embeds the simpler cases. If it was used to predict the next case the table would be:
That hypothesis can be checked with the materials. If it works, which it does, the pattern can be used to predict the minimum number of moves for any starting number of counters. Any student who describes how to do that has just 'done' algebra.
Students who bail out of the problem for now at this point can be complimented on engaging with several aspects of the Working Mathematically process, for example:
Extending the AlgebraIt is about now that a mathematician might ask, What else can I learn from this problem? A possible challenge arising from this question is: Given any number of counters can we work out the total of moves without actually doing the problem. It would be somewhat tedious for example to have to do all the moves for 100 counters on a 101 board. The table above provides a guide.
All that is necessary now is to find a rule for summing the natural numbers to (n - 1). For some mathematicians this is answered by asking Have I seen a similar problem before?; for some it is mere application of a skill that has become part of their toolbox; and for some this is a new problem that now has to be investigated before the current problem can be completed. By whatever means, a symbolic algebra expression for the minimum number of moves can eventually be written.
However, with or without the symbolic algebra any student who reaches this level should be complimented on being a great mathematician, and, in fact, a great algebraist.
Underlying StructureThe numbers above are very nice and it is very satisfying to be able to control them with a symbolic expression. But, why are they as they are? Algebra was invented to make sense of the world. Any algebraic equation should directly relate to, and be explainable in, the context in which it was developed. Why do the numbers have these patterns? Why is the final expression 3 times something?
One look at these questions (and we have already seen that other mathematicians will see things from a different view) suggests:
The next phase is a repeat, but it begins with (n-2) counters; the next with (n-3); the next with (n-4) and so on to the last two counters. Each phase is a three times situation. The total is the sum of the phases. There is a common factor of 3, so the total must be the sum of the 3 x [sum from 1 to (n-1)].
Whole Class InvestigationTasks are an invitation for two students to work like a mathematician. Tasks can also be modified to become whole class investigations which model how a mathematician works.
Place six chairs at the front of the room and select five students. Give each student a large size card from a set numbered 1 - 5 and seat the students as shown on the task card above.
Our challenge today is to reverse the order of these people so it goes 5, 4, 3, 2, 1 using the rules that:Easy to state, easy to start and heaps of maths to follow as indicated in the iceberg above.
When the students are sufficiently interested in trying to achieve this, ask them to go to their seats and in pairs make a set of numbered 'cards' and a set of 'chairs' by quickly folding and tearing paper. Now they continue the investigation until a solution is found. Those who find it first can demonstrate using the real people and chairs, but as they explain, they must keep their hands behing their backs.
Again students return to their seats and, with their partner, convince themselves that they can do solve the problem as shown.
The way into the iceberg above depends on how the lesson develops to here. It may be students have noticed a movement pattern already, and this can be explored, tabulated and generalised. The question can then be asked, Can we check it another way?.
Or it may be that two different solution methods (or more) have been discovered and each can be followed through as above. In both cases highlight the steps in the Working Mathematically process as they develop.
When the problem has been explored as far as appropriate, draw it together using one of the two lessons titled Learning to Write a Maths Report, in Recording & Publishing. Then encourage the students to prepare their own report as a 'paper like a mathematician', a slide show, video, poster, comic strip or...
At this stage, Reverse does not have a matching lesson on Maths300.
Is it in Maths With Attitude?Maths With Attitude is a set of hands-on learning kits available from Years 3-10 which structure the use of tasks and whole class investigations into a week by week planner.
The Reverse task is an integral part of:
From The Classroom
|Michael initiated a major discussion of this task when he asked a question of Sue Davis, about its usefulness in his task collection. Sue passed the question to Doug. Williams, who consulted with Charles Lovitt, and the resulting exchanges with Michael were by fax as below.|
Reverse is a wonderful problem and it would be a pity if it was to be hidden away, so I am happy to try to guide you with this one. I will do so from within the context of learning to work like a mathematician (Working Mathematically) which is one of the principles behind why a task is included in our set.
Andrew Wiles, the Princeton professor who solved Fermat's Last Theorem in 1994, after it had remained unsolved for over 300 years (and in fact for more like 2500 years), is quoted as saying:
The definition of a good mathematical problem is the mathematics it generates, rather than the problem itself.For hundreds of years mathematicians were well aware that they hadn't solved Fermat's Last Theorem, but they were delighted with the new mathematics they generated as a result of trying, as evidenced by scores of publications which reside in libraries around the world. For your children, Reverse may well be equivalent to the Fermat problem, so perhaps the focus in using it needs to shift from 'the pattern' to 'the mathematics it generates'. All of our tasks can be looked at in this way, which is why we say that one guiding principle for a good task is that it can illustrate the process of Working Mathematically.
Reverse has value in this context because:
We think that Reverse is a great problem for students because, even within all this depth which we refer to as the 'iceberg' of the task, the challenge is:
It took Andrew Wiles 20 years of background thought and preparation (he decided at the age of ten that he would do it) and seven years of isolated, focussed study to solve Fermat's Last Theorem. Even then Andrew's first solution, although it got him on the cover of the New York Times and People magazine, contained a logical error, so it wasn't really a solution at all. It took another year of devoted study in the company of a critical friend to be able to finally solve the problem - and in fact, shorten his proof by scores of pages.
So is the problem finished? I wouldn't dare say yes. No mathematical problem is ever finished. It is only finished for now. However this is where I am bailing out - for now.
Michael, I want to thank you for asking this question. It has engaged me in every aspect of the Working Mathematically process - including publishing for others. Evidence shows curriculum based on this process not only increases student numeracy but hand-in-hand develops student literacy. I hope this will help you with Reverse and with the way you use all your tasks.Now I need a favour from you. It has taken considerable time to prepare this paper, especially since I had no idea of the 'answer' when I first saw your question. Now I know there are several answers and answers on several levels. I would like to place this paper onsite, so others can also make use of it. However, I would like to be true to the context in which the question arose and quote your name and school with your original question to Sue. May I have your permission to do that?
Let me assure you that when I began thinking about a response to your original query, I also did not expect the response you received!
To receive a Wow! from you is pleasing. The Task Centre Project is hoping that teachers can stimulate Wow! from kids in the context of mathematics as often as possible. I hope that putting something of our discussion on site will be one more small step in achieving that outcome. Thank you for permission to do that.
Thank you too for the extra information about the data you have gathered. I did reserve the right to be wrong and I see that I must now delete or modify my sentence:
Comparing further with other possible student-designed systems shows that System 2 always gives the minimum number of moves.System 2 in my previous fax is indeed the minimum of the two I presented, but clearly not necessarily the minimum overall. You have achieved two results in significantly fewer steps. On the other hand, I have presented you with a complete movement system/ visual pattern/ natural language situation that is generalisable to any number of counters. The logic underpinning why the movement system works guarantees the generalisability of the algebra.
The challenge for you is to find a similar justification for your approach, and at the moment you say you have drawn a blank. What you have done is certainly not wrong - it may be unfruitful for generalisation, but it is not wrong. Mathematicians frequently travel blind alleys, experience extreme frustration and put problems on the back burner. Fermat was put on that particular burner for just over 100 years after Sophie Germain made her contribution to its partial solution. No other mathematician, male or female, could take it any further.
The challenge in Reverse is Can you generalise? and if you can't, then the question is simply, Can you generalise using another way?
|Don't give up yet. I simply don't know whether your data will generalise. However, I do know that if it does, then there will be a movement system that is at the basis of it. I have tried to analyse the movement system for your data. I don't have much more time I can spend on this, but to date I have classified the moves to see if I can see a pattern. If, for example, JL = jump left and SR = slide right, then I see this...
Firstly please check that I have these correct. Secondly, if they are correct, I think I could make a prediction for the first four rows of Column 5, but I have no idea what to predict for the other rows. Two pieces of data is shaky ground for predicting a pattern. I think you would have to solve the 5 case and then see what steps it took.
But that is just the problem. How do you make the 5 case work with a movement system that is generalisable - first in language and eventually in symbolic algebra?
We are back to square one. I have shown this can be done with one system. You have yet to show that your data leads to a system, or you have yet to decide that it doesn't and follow another path.
Perhaps when it gets to the site, some readers might also like to join us in this challenge.
All my information about Fermat's Last Theorem comes from a book titled Fermat's Enigma, Simon Singh, Walker and Company, New York. It is a remarkable story and in it you will find that Andrew Wiles faced moments like this for seven years. It is part of a mathematician's work to not know the answer.
In discussion with Charles today, a colleague who has spent time on this problem in the past, we concluded that, although it is possible to define a process that will always solve the problem, still for particular numbers of counters there may be aberrations that will achieve the outcome in fewer moves.
In the classroom we find such dysfunctions exciting. We would structure a lesson so it supported the 'discovery' of a generalisable system - it may stretch on and off over several sessions. When we had used the lesson to (once again) illustrate how a mathematician works, we would offer a further voluntary challenge of finding a solution for particular numbers of counters in fewer moves. For in-house use we would also name some of these solutions just the way mathematicians name theorems. For example, the two shorter ways you have illustrated could be the Palme or Brigidine solutions.