Reverse

Reverse: A Conversation

This 'conversation' was initiated by a question from Michael Palme, Brigidine College, New South Wales, to Sue Davis, Task Centre Project Executive Officer. The conversation was carried out by fax. The text is quite long and is best understood by engaging in the problem as you read, so you may find it more fruitful to print the material and investigate it at leisure. The information also encourages development of a whole class investigation.

Fax 1

Dear Michael,

Sue has passed your question about Reverse to me. I manage the Task Centre Project on behalf of Curriculum Corporation.

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. You will find support for this and other principles of the Task Centre Project at the Task Centre Home Base: http://www.blackdouglas.com.au/taskcentre

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, as evidenced by scores of publications that reside in libraries around the world, with the new mathematics they generated as a result of trying. 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:

It illustrates the need to select from the Strategy Toolbox (see Working Mathematically document), and almost 'cries out' for application of the Try a simpler case strategy.
It is not skill dependent, ie: you don't have to know how to multiply or measure angles and the like, in order to work on it. Therefore the door is open to a student less confident with the skills of mathematics to show that they are capable of higher order mathematical thinking.
The solution of the initial problem is open to at least two 'movement systems'. Mathematicians like it when there is more than one answer to a problem. Two answers could mean that twice the mathematics can be generated. From a teaching point of view it could mean a chance to applaud the different ways your kids think.
The movement systems become a dynamic visual pattern that can be described in natural language.
The natural language description can be used to predict the number of moves for other numbers of counters. This is a step towards generalisation; the real algebraic achievement. Symbolic algebra is not a pre-requisite for algebraic thinking; rather it is the servant of algebraic thinking.
The pattern in the natural language description of each movement system can lead to a deeper generalisation that allows us to predict the number of moves for any number of counters. However, the description of these structures has multiple layers. These layers are easily lost if we focus on the 'combination number' which is the total of the parts.
There is a symbolic algebra generalisation that describes the number of moves for each movement system, but because the logic has multiple layers, so must its symbolic representation. A professional mathematician is likely to be challenged both by the task of finding the algebraic representation for each movement system, and the task of explaining the result to others.
The algebra of the problem doesn't explain Why does it work?. It only allows us to describe and predict. Re-examining the dynamic visual pattern encourages a search for the logical structure underlying it. If there is an algebraic rule to describe a situation, there will always be a logic reason - rather than symbol manipulation reason - why the symbols in the rule are as they are.

We think that Reverse is a great problem for students because, even within all these complexities which the project refers to as the 'iceberg' of the task, the challenge is:

easy to state,
easy to start, and
provides multiple exit points

at which student achievement to date can be celebrated.

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 Time 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.

Now to the meat of the Reverse.

Strategy

Most 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 Mathematically document) 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:

The growing recognition that you reach a stage where you are facing the same problem but with one fewer counters. That is, for example, the 5 gets to the right hand end where you want it and you are left with five cells (instead of six) and the counters 1, 2, 3, 4 ready to reverse their order.
Dipping into the Strategy Toolbox (which some classrooms display as a Strategy Board) to select the Try a simpler case strategy.

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 Systems

Perhaps 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 that are common among those who have accepted the challenge of this task.

All left first:
- So _, 1, 2, 3, 4, 5 becomes 1, 2, 3, 4, 5, _ after five moves.
- Then jumping the 4 to the right first, followed by alternating slides and jumps produces 5, _, 1, 2, 3, 4 after a further eight moves.
- The 5 can be left out for now while the simpler problem of _, 1, 2, 3, 4 is tackled by repeating the system.
Thirteen moves to get to this stage.
Jump first then alternate slides right and jumps left:
- The only first jump move in _, 1, 2, 3, 4, 5 is into the empty space to give 2, 1, _, 3, 4, 5.
- Now the only slide right move is 1 into the empty space to give 2, _, 1, 3, 4, 5.
- This now determines the next jump left and so on to reach 2, 3, 4, 5, _, 1 after eight moves.
- The 1 is now in the correct position and four more moves slides the counters back to _, 2, 3, 4, 5, 1 where the 1 can be ignored and the sub-problem _, 2, 3, 4, 5 can be tackled in the same way.
Twelve moves to get to this stage.

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 Patterns

Wherever 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.

Each system can now be tracked to keep a count of how that total is made up. I will follow the System 2 path in the rest of this explanation because it would seem appropriate to mathematicians to use the most efficient process.

System 2
Eight moves to the left and four slides back to complete Stage 1. Then six and three for Stage 2; four and two for Stage 3; and two and one for Stage 4.

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.

Five Counters
Stage Left Moves Slides Back

1 8

4

2 6

3

3 4

2

4 2

1

Totals 20 10

30

**Five Counters**
Stage	Left Moves	Slides Back
1	8
		4
2	6
		3
3	4
		2
4	2
		1
Totals	20	10
	30

Clearly there is a pattern here that embeds the simpler cases. If it was used to predict the next case the table would be:

Six Counters
Stage Left Moves Slides Back

1 10

5

2 8

4

3 6

3

4 4

2

5 2

1

Totals 30 15

45

**Six Counters**
Stage	Left Moves	Slides Back
1	10
		5
2	8
		4
3	6
		3
4	4
		2
5	2
		1
Totals	30	15
	45

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: playing with the problem to collect and organise data discussing and recording as notes and diagrams seeking and finding a pattern making and testing an hypothesis applying their skill tool box applying their strategy tool box asking the mathematician's questions What if... and Can I check this another way?.

Extending the Algebra

It 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.

It seems the Slides Back column always begins one less than the number of counters and its total is the sum of the natural numbers to this point.
A closer look at the Left Moves column shows that this is the same natural number sequence twice more, ie: 10 = 2 x 5, 8 = 2 x 4 etc.
So the Total looks like it is 3 x [sum from 1 to (n - 1)], where n is the number of counters
... and the resolution of the problem is starting to look annoyingly simple given the initial struggle to find a successful way into it.

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 Structure

The 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:

For 1 to finish in the right hand cell, it must slide one less cell than the number of counters (n-1) because it is already occupying one of the cells.
In the process of sliding to the right, each other counter must jump over 1 just once. There are (n-1) counters that have to jump because 1 doesn't jump itself.
When 1 has made it to the right hand end, the (n-1) counters each have to slide back one space to set the problem up for the next phase.

That is three lots of (n-1), and it explains where the 3 comes from and where the (n-1) starting point comes from.

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)].

Are We Done?

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 on the Task Centre Home Base, 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?

Keep smiling,
Doug Williams

PS: Disclaimer

If Andrew Wiles can solve the world's hardest maths problem and still finish up on the cover of Time magazine even though his solution was flawed, I reserve the right to be wrong.

Fax 2

Dear Michael,

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 the Home Base 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 logical 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 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:

3 Counters 4 Counters 5 Counters

JL JL

SL SL

JR JR

SL SL

JR JL

SR

JR

SR

JL

SL

JR

SR

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.

All my information about Fermat's Last Theorem comes from a book called 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 today with 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 (which may stretch on and off over several sessions) so it supported the 'discovery' of a generalisable system. 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.

Keep smiling,
Doug.

Follow this link to Task Centre Home page.

3 Counters	4 Counters	5 Counters
JL	JL
SL	SL
JR	JR
SL	SL
JR	JL
	SR
	JR
	SR
	JL
	SL
	JR
	SR