Reverse

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.

Iceberg A task is the tip of a learning iceberg. There is always more to a task than is recorded on the card.

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 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: 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 common ones. 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. (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. 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 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 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)].

Whole Class Investigation Tasks 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: only one person can move at a time. a person can move one position in either direction. a person can jump over one other person in either direction. Of course we will interpret jumping as standing up and politely moving around. Any ideas? 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... (We would welcome examples of your students' publishing). 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: MWA Space & Logic Years 9 & 10

From The Classroom

Fax 1

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.

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 this depth which we refer to as the 'iceberg' of the task, the challenge is:

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

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.

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.

Keep smiling,
Doug Williams

PS: Disclaimer

Fax 2

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.

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?

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.

Keep smiling,
Doug.

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