Lecture Notes
Contents
1. how long would a user take and how can we redesign things to be easier Later lectures will look at how matrices can be used how codes e g the maths of codes like Morse code can be used and how symmetry can be used The lectures will give plenty of examples that out perform proprietary products Because of their creative and practical design element all the lectures will appeal strongly to designers and manufacturers particularly those designing or making highly interactive products such as mobile phones car radios and even aircraft cockpits But the lectures will also appeal to the rest of us everyday device users we who have to put up with using ticket machines photocopiers and mobile phones In all lectures the maths is not difficult and you will go away able to do some device design or analysis of your own You ll see why these things are hard to use and you ll wonder why industry does not use maths to make the world easier for us Summary of the lecture demonstrations The first lecture showed how devices such as mobile phones video recorderrs and even games can be simulated on a computer and showed how various sorts of manual and description can be generated automatically for instance to provide a web site documenting how to work the devices In turn these sorts of description can be converted into complete and correct user manuals We also showed how code as device manufacturers might need to actually build a real fully w2. Philips video recorder Now compare the JVC with a Philips VCR The Philips VR502 looks better in this sort of analysis though it has fewer states it doesn t have as many features Model Philips VR502 Number of states 18 Number of edges 143 which is 46 732 complete Probability a button does nothing 0 272727 27 2727 of the time Average recovery overhead for a random press 1 46465 excluding the press Average cost to get anywhere after a random press 1 98457 excluding the press Average number of button presses to get from anywhere to anywhere 2 04902 Number of single button presses with direct length 1 undo 79 55 2448 Average undo cost for single button press 2 01389 Average recovery length from single over run 2 25 Maximum over run length length 6 Simplifying the JVC video recorder If the Philips is easier in the sense we ve explored then can we redesign the JVC to make it easier We ve already pointed out how the tail of states on the JVC make many things harder Let s delete them and see what happens Model Reduced JVC HR D540EK VCR Number of states 14 Number of edges 50 which is 27 4725 complete Probability a button does nothing 0 571429 57 1429 of the time Average recovery overhead for a random press 0 723214 excluding the press Average cost to get anywhere after a random press 1 94133 excluding the press Average number of button presses to get from anywhere to anywhere 2 07692 Number of single bu
3. shut your eyes and wished I want to take the goat and the wolf across the river then about 62 of the time you couldn t do it for instance because the goat was on the other side or was eaten or something else had gone wrong you hadn t noticed with your eyes shut What about other benefits If Mathematica can use the definitions to run full interactive simulations of the devices surely it or rather mathematics more generally can do more Yes it can Later lectures will develop the ideas further and will show how mathematics can be used constructively to help improve designs With tools like Mathematica the mathematics becomes quite easy to use in fact the mathematics is quite straight forward and could easily be embedded inside familiar design tools The ideas would then move from the research field into industry Conclusions Mathematics can underpin the design process Mathematics is very creative and helps improve design Mathematics is easy to use with the right tools doesn t have to be Mathematica Mathematics can be used for simulation manuals design and manufacture About the author Harold Thimbleby is Gresham Professor of Geometry and Director of the UCL Interaction Centre UCLIC He is one of the first Royal Society Wolfson Research Merit Award Holders More details on his lecture series plus many background papers and references are available at http www uclic ucl ac uk usr harold gresham Email him at h th
4. the light bulb we can ask Mathematica to make a simulation that can be used Play Operate Forward Rewind Pause Record Stop Eject Tape in At least the photograph is more realistic than the light bulb s The big black bar would show the video recorder s state but as it has started off nothing is lit up to see We next show this device as a circular graph The advantage of a circular embedding is that no two lines are ever drawn on top of each other so we can be certain that we are looking at everything It helps if we write the names of the states too but there are so many that the drawing gets quite messy Even though the diagram is messy and we haven t put the states in a useful order around the perimeter you can see little facts like the two states on with tape in and off with tape in are very easy to get to from almost anywhere you can clearly see the cluster of arrow heads hitting each of these states from almost every other state K eine Be Er i ty gt An alternative way of drawing the graph like one way we drew the wolf goat cabbage problem is to rank the states so that the machine off with its tape out is shown at the far left and each column of states is the same distance from off Thus the further right you go in the drawing the harder things are to do at least if you start from off The long tail of states each increasingly hard to get to makes this JVC device hard to use We ve drawn the gr
5. 31 169 85 170 108 buttons gt Off On Dim labels gt dim off on fsm gt 2 3 0 0 3 1 2 0 1 indicators gt Dim Off On startState 32 manualRange 3 3 manual 1 gt in dim light in the dark in bright light manual 2 gt are in dim light are in the dark are in bright light manual 3 gt dim dark bright notes gt Null Simple light bulb with dim mode Null action gt press pressed pressing beeping gt never positions gt 0 0 0 0 bulbpicture eps Graph gt Graph lt 6 3 Directed gt Running an animation of a device Mathematica can take the sort of description shown above and create an animation The animation works just like the device is intended to We could use it to test the device out on users or to see whether it works as we intended Mathematica is fully programmable and all sorts of realistic features could be provided Here we just give a simple animation that shows a row of buttons underneath an indicator panel For fun the indicator panel is in green rather like a LED display ri Dim off on Din At least if you were reading this text in a Mathematica document Mathematica could run the simulation You can click the buttons and they work which Mathematica shows by changing the text in the display above The light bulb picture happens to be a static picture but if we wanted to spend more time programming Mathematica we could get it an
6. 40 minutes play a tape fast backward gt pause recording but stop in 240 minutes play a tape gt pause recording but stop in 240 minutes rewind a tape gt pause recording but stop in 240 minutes So the hardest operations are all ones that end up with the video recorder doing something in 240 minutes in fact the state at the extreme right of the last diagram we drew The JVC has 8 buttons and 28 states so if the buttons were used optimally to make doing anything as brief as possible the average would be less namely about logs of 28 which is about 1 6 This is a lot less than what the 3 8 the JVC achieves Another way of calculating this is to see that 1 state namely the one you start from can be reached with zero button presses 8 states can be reached with 1 press which leaves 19 states that can be reached in 2 presses The exact average cost is therefore Ox1 1x8 2x19 28 1 64 Of course we might get some data that says users tend to do some things more often than others and we ought to weight these actions more than ones that aren t used much we ll look at these sorts of real world issues in the second lecture in the series 1 60245 We can conclude the JVC was not designed to minimise button presses to do things regardless of other concerns One would therefore have expected some other advantage for the JVC design decisions such as the buttons more often meaning the same things like PLAY always meaning play Let s chec
7. 66667 excluding the press Your random press can give you a bit more information but has it made your task harder Compare this figure with the mean cost between states Average number of button presses to get from anywhere to anywhere 1 the average cost of doing anything from anywhere the mean cost taking everything as equally likely Number of single button presses with direct length 1 undo 6 100 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec 6 of 7 how often if you make a mistake can it be undone directly with one button press Average undo cost for single button press 1 If you make a single button press mistake on average what does it cost to recover Over run errors do not happen on this device An over run error occurs when a button is pressed once too often and goes to another state On this device pressing a button twice always leaves you in Over run errors don t really happen on light bulbs or at least not this one Imagine a typical gadget with nasty rubber keys that you aren t sure you ve pressed hard enough Suppose you press OFF and the device has not gone off Maybe the device is slow maybe the lights tend to stay on for a bit or maybe you didn t press OFF hard enough and until you press it properly it isn t going to switch off So you press it again If in fact this is the second press of OFF we will call it an over run error Maybe an ove
8. Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec Design creativity with maths Gresham Geometry Lecture 26 September 2002 Harold _ Thimbleby Introduction to the 2002 3 Geometry Lecture Series Gadgets like mobile phones and car navigation systems are often difficult to use Most of the time we cope trying to ignore their more confusing and specialised features Yet occasionally in some situations trying to use a complex system can be dangerous and expensive not just tedious A traveller trying to use an automatic ticket machine has to find their destination ticket class and type of ticket enter cash confirm collect the tickets and change all under the pressure of having to catch the train on time as well People under these circumstances make predictable mistakes such as leaving their change behind Such errors can be fixed by changing the design Trying to use a mobile phone or even the radio or navigation system while driving a car can be so distracting to be dangerous Ideas like voice control are not going to change the underlying complexity as anyone who has been frustrated by telephone voice menus will know Deeper ideas are needed A nurse using a syringe pump to provide automatic drug injections is under extreme pressure to do the right thing in a distracting environment yet errors can have unfortunate consequences for patients Errors caused through ignorance about how to use the equipment ma
9. aph going left to right rather than top to bottom as with the wolf problem because it fits on paper better this way Doozo Ae aw P z r F a eS Ar Tet im A C7 i Standby l press ra 2 presses ASS 3 presses SaN x US 4 presses si 5 presses 6 presses et 7 presses D lk N 8 presses D EHS See ee 9 presses Pe a lt IO praises SA 1 presses 12 presses 13 presses 14 presses 4of7 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec 5 of 7 ie 15 presses 16 presses 17 presses 18 presses It s interesting to look at the average length of best paths between any two states For the lecture notes we ve hidden the simple Mathematica code that works out this number we ve just shown the result 3 86905 This means that if you know how to use this device perfectly and few people do on average to do anything will take you almost 4 button presses With such bad averages it s interesting to know what the hardest operations are fast forward gt pause recording but stop in 240 minutes off with tape in gt pause recording but stop in 240 minutes off with tape out gt pause recording but stop in 240 minutes on with no tape gt pause recording but stop in 240 minutes play a tape fast forward gt pause recording but stop in 240 minutes pause playing a tape gt pause recording but stop in 2
10. component is everything that can be done transporting the wolf or the man alone across the river but from any of these states it isn t possible to go back to a state where the eaten cabbage or goat exists again Another component is when you have everything Another strongly connected component includes the start state and every state with all objects cabbage goat and wolf present one side or the other of the river Since by symmetry getting the objects to one side of the river is the same as getting them to the other side the start and end states must be in the same strongly connected component Below we ve used Mathematica to summarise the four states of one of the strongly connected components this one with only the man and wolf present 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec It s fun to get Mathematica to draw a transition diagram with the strongly connected components pulled apart to make them clearer While interesting this drawing probably doesn t help solve the problem I ve redrawn the same graph in a different way below Each row in this new diagram of states is the same distance from the start state which is at the top so for example the second row shows all 7 possible states that can be reached in one canoe trip across the river Whereas the previous diagram didn t really help us in this one an optimal solution to the problem is represented as a path down the dia
11. get anywhere This is because sometimes it is much harder to get back to where you were But the difference isn t much and if you can find out something useful about where you are what the device is doing by pressing a button on the JVC this could is a good strategy Sometimes pressing a button does nothing For example on the JVC if you press PLAY when it is playing nothing happens Suppose we modify the JVC so that a user can tell if a button will do something For example each button might have a little light that comes on only if the button works These useful buttons would be easy to find in the dark Now if we press a button at random it will always do something How does this change the numbers Average cost to get anywhere after a working random press 4 04582 excluding the press Your random press will give you a bit more information but has it made your task any easier It s worse So on the JVC you re better off not playing with the buttons to see what they do But you re only better off if you know what it is doing and that would require the indicator lights to tell you what it was doing We ve already seen they are inadequate So on the JVC the user is in a quandary you can t always tell what state it is in and experimenting to find out makes any task harder Of course to be fair once you ve experimented and found out where you are you can now use the JVC properly which you can t do when you don t know what it is doing A
12. gram that only goes only from top to bottom if it ever turned back upwards or even went left or right without going down in the diagram it would be going to a state that would have been easier to reach on a shorter route more directly from the top We ll use exactly the same form of graph drawing to look at a video recorder later in these notes again the diagram allows one to read off good ways of using a device and conversely the shape of the diagram gives a designer a good idea of how efficiently users can work the device Because we ve drawn a rather small dense graph there isn t enough space to show the names of the state of the wolf goat and cabbage in each state The problem is tricky because sometimes cabbages or goats get eaten and there is then no going back In the diagram some arrows are one way if a canoe trip is taken that corresponds to one of these arrows it is a one way trip in terms of the states that can be reached If the cabbage gets eaten by the goat no states with cabbages are accessible any longer If something like a DVD player was like this it would be very tedious to use Of course video recorders are sometimes like this when you accidentally record over a favourite programme Well we can use Mathematica to automatically find the states that cannot be got out of delete them and hence make a simpler version of the problem where nothing can go wrong We make a new set of easy states and then draw the new graph T
13. his one has a pleasing elegant structure and we have derived it completely automatically simply by deleting the nasty states in the original problem All we have to do is define what we mean by nasty in this case delete all states that are not in the strongly connected component containing the start state and then run Mathematica to fix the design One might want to do similar things with actual devices make it impossible for the user to get stuck with them If we decide what criteria we want we can redesign automatically The big wolf goat cabbage graph was a bit obscure without any state names so we ve defined some names for the states The states are named symbolically so that they are easier to understand thus cwmll with nothing after the Il river symbol means that the cabbage wolf and man are all together on the left side of the river and the goat must have been eaten since it is on neither side of the river As Mathematica arranged this diagram the further right one goes the harder it is to get there from the starting point with everything on the left of the river Interestingly the goal of the problem with everything on the right of the river is the hardest state to get to it s shown on the far right A JVC video recorder Now we look at a JVC video recorder the JVC HR DS540EK 3 of 7 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec What would this device look like As with
14. imated too On paper of course nothing will work The wolf goat and cabbage problem In the lecture one of the fun examples was the wolf goat and cabbage problem The problem requires that the goat is never left alone with the cabbage and for the wolf never to be left alone with the goat in either case something will get eated Like a device it has states which correspond to various combinations of wolf goat and so on being on each side of the river and it has actions which correspond to the canoe carrying one or more things across the river We first show this problem as a simple graph Ters F Ate ot j aed FAS 4 Se P lt T i SEN 3 So TA N IS In this diagram the problem has effectively been changed to finding a route following arrows from one circle to another starting at the circle labelled Start and going on to the finish of the game at the state End There are some sets of circles which you can get to and once there you can move around in the set freely but if you take a way out of the set you can t get back you can get stuck if you make wrong decisions and get into one of these sets The sets of circles are called strongly connected components and Mathematica can easily find them The problem has 12 strongly connected components some with 4 states some with 8 and quite a few 8 in fact with only state One component is when the cabbage and goat have both been eaten The
15. imbleby ucl ac uk Converted by Mathematica October 3 2002 7 of 7 1 26 03 5 09 PM
16. instance if the JVC model is on with a tape in and you make it go fast forward it won t tell you anything has happened on with tape in gt fast forward fast forward gt on with tape in play a tape fast forward gt on with tape in play a tape fast backward gt on with tape in rewind a tape gt on with tape in on with tape in gt rewind a tape It looks like the video should have had indicators for fast forward and rewind states It doesn t Rather than carry on writing special Mathematica code for each idea we have we ll write a single function that prints out some interesting facts about any device Here is the information for the simple light bulb The first time we use the Mathematica function we ll ask it to explain what everything means but to save space below we won t print this explanation again Model Light bulb the model type Number of states 3 how many things can be done with this device Number of edges 6 which is 100 complete In a complete graph you can do anything in one step so if this figure is 100 the device cannot be made faster to use Probability a button does nothing 0 333333 33 3333 of the time chance a random button press does nothing Average recovery overhead for a random press 0 666667 excluding the press if you make a random press how hard on average is it to get back Compare this figure with the mean cost Average cost to get anywhere after a random press 0 6
17. k this idea out next Some buttons have names like OPERATE and PLAY that seem to have helpful names We can look at the design of the system and find out how likely buttons are to do things We ve taken a very simple approach here but we can see for example that the OPERATE button makes the JVC device on 44 of the time other times most of the time OPERATE makes the device inoperative off For the JVC HR D540EK VCR Play when itdoes something always achieves on tape in Operate whenitdoes something mostly achieves on 44 444444444444443 of the time orward whenitdoes something alwaysachieves fast forward on tape in Rewind when itdoes something always achieves on rewind tape in Pause when itdoes something always achieves on pause tape in Record when itdoes something always achieves on tape in Stop Eject when itdoes something mostly achieves on 44 444444444444445 of the time Tape in when itdoes something always achieves on tape in What this means is that when the PLAY button does something it will leave the video with the on and tape in lights on Of course if the video was off PLAY would do nothing That s not very surprising but some of the other buttons meanings are When buttons are pressed on a device it should give feedback that something has happened Do some buttons not give decent feedback The ambiguous actions are shown below for
18. orking device can be generated automatically For technical people who missed the lecture the demonstrations showed several finite state automata how they could be modified and simulated how their structure allows usability questions to be answered as well as intelligent help provided and how they can be used to generate HTML for web based manuals or Javascript for executable implementations The FSAs were defined in Mathematica code but the simulations were done in a program I wrote which parsed the Mathematica The lecture will be recorded on the Gresham College web site and you will be able to get the video demonstrations from http www gresham ac uk under the Geometry lectures Overview of these notes These notes show how Mathematica can analyse descriptions of interactive devices by writing text drawing pictures and by doing numerical analyses Mathematica is a sort of mathematician s word processor all the text and pictures here were created in it and all the information shown about devices whether pictorial numerical or textual was worked out by Mathematica from definitions of the devices The results have not been touched up everything is automatic and the same sorts of results could be worked out for other devices as desired for instance if a new device was being designed The descriptions of devices used here are ones that we demonstrated working in the lecture Exactly the same definitions can also be used for generating u
19. r run of OFF will switch this device back on again On our simple light bulb since OFF only switches the bulb off switching it off when it is off keeps it off and the same for all the other buttons Pressing DIM when the bulb is dim keeps it dim pressing it twice still keeps it dim Pressing ON when the bulb is on keeps it on pressing it twice still keeps it on Hence the summary information above says over run errors do not happen As we can see below the JVC device has some curious properties If we have an over run error e g we wanted to get to the video to play a tape but we pressed PLAY once too often perhaps because we didn t notice when the device got where we wanted it perhaps it is too slow or doesn t provide decent feedback then on the JVC it takes 2 3 presses on average to get back to where we wanted to be or 3 3 including the error On the other hand to get from anywhere to anywhere takes on average 3 9 presses so an over run error is practically the same as getting completely lost an over run error puts you about as far away on average from where you want to be as you can be On the other hand we have 3 hands if you make a completely random button press it only takes 1 8 presses to recover on average or 2 8 including the error But this is easier than an over run error There are three main reasons for this i some random presses do nothing and therefore cost nothing to recover from ii most random pre
20. ser manuals intelligent help or for building complete systems These brief notes don t exhaust all the possibilities of course Since some people reading these notes will not be interested in any details of how Mathematica works the Mathematica instructions themselves have not been printed Anyone who wants the Mathematica code can email Harold_Thimbleby for it A very simple device The first and simplest device we ll consider is a simple light bulb with three states off dim and fully on This can be drawn as a transition diagram with three circles one for each of the states and with arrows between them showing how one could change the state You can think of the diagram as a game board when you press a button you move along the right arrow to a new circle for clarity I haven t written down arrow names As it happens this bulb allows any state to go to any other state directly so every line is a double headed arrow but this is rarely the case with more complex devices Most device descriptions are quite big but the light bulb is simple enough so that we can show it in its entirety You can see below how Mathematica has got names for the states descriptions of how it works and how to draw it on screen for working simulations 1 of 7 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec 2 of 7 modelType gt Light bulb indicatorLoc 3 120 45 218 70 buttonLocs gt 170 1
21. sses don t get you as far away as an over run iii if a button worked to get you to this state it is likely to work to get you away from it in other words over run errors are likely Model JVC HR D540EK VCR Number of states 28 Number of edges 106 which is 14 0212 complete Probability a button does nothing 0 526786 52 6786 of the time Average recovery overhead for a random press 1 78125 excluding the press Average cost to get anywhere after a random press 3 91964 excluding the press Average number of button presses to get from anywhere to anywhere 3 86905 Number of single button presses with direct length 1 undo 38 35 8491 Average undo cost for single button press 3 76415 Average recovery length from single over run 2 30986 Maximum over run length length 9 The joke about three hands feeble as it was reminds me that the remote control for this video recorder is completely different from the unit itself We haven t space to show it here but it s very obvious from any drawing of the transition diagram Making it different doubles the learning the user has to do to make good use of the device and almost doubles the size of the user manual If you make a random press you may find out more about the device It s a tempting thing to do you walk up to something What does it do The only way to find out is to press a button and see what happens On the JVC if you press a button at random you may have made it harder by a bit to
22. tton presses with direct length 1 undo 25 50 Average undo cost for single button press 1 6875 Average recovery length from single over run 1 Maximum over run length length 1 Wolf and goats again 1 26 03 5 09 PM Lecture Notes file G4laptop Organised Web New 20web 20site gresham Lec Finally we look at the wolf goat and cabbage problem again for comparison Many of the figures are infinity because once something has gone wrong e g the goat has been eaten there is nothing you can do the average of impossible and anything else is still impossible on average Unlike the interactive devices like video recorders we ve analysed pressing buttons at random to try and better understand what s going on does not help Model Wolf goat amp cabbage problem Number of states 36 Number of edges 94 which is 7 46032 complete Probability a button does nothing 0 626984 62 6984 of the time Average recovery overhead for a random press Infinity excluding the press Average cost to get anywhere after a random press Infinity excluding the press Average number of button presses to get from anywhere to anywhere Infinity Number of single button presses with direct length 1 undo 70 74 4681 Average undo cost for single button press Infinity Average recovery length from single over run Infinity Maximum over run length length Infinity The rather high probability 62 that a button does nothing really means in this case that if you
23. y be reduced by better training but skill based errors can only be reduced by better design In fact most user ignorance can be better dispelled by simplifying designs than by more training The 2002 3 Geometry lectures will explore the underlying theories of system design so that we can see why things are difficult to use and how they can be made better and easier to use The slogan for the series of lectures is design creativity with maths The maths we cover is simple stuff and surprisingly effective in leading to improved designs thus helping make systems much easier and more reliable to use I am not aware that these important mathematical techniques are used by industry but I hope these lectures will show how easy and useful they are The first lecture introduces graphs mathematical objects that are basically just dots and arrows and therefore easy to draw and understand We can imagine web sites to be graphs the web pages are the dots and the links between the pages are the arrows We can also imagine devices like mobile phones to be graphs Immediately any phone is like a web site Which means more constructively that we can simulate and test one on the web very easily or we can write its user manual as a web site Such a manual would be complete and correct unlike most real mobile phone manuals We can do lots of other things with graphs like measuring how long it takes to get across them and this gives insights into design
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