When I was writing about stupidity I was close to including a refusal to use mathematics. Where most of the time I mean arithmetic. I really do not understand why it is that we so readily accept that, for example, it is acceptable for someone in the public eye to say “I could never do maths” followed so often by a statement which illustrates a failure to make logical statements.
It could be that the habit of testing mathematical statements for truth, “This really is equal to that” never percolated. It could be that a failure at an early age to be satisfied with ‘getting things right’ leads to be happy not getting things right, as an expression of the status quo.
For those people who don’t know whether they fall into either camp, I’ll give some examples of cases where I think using maths would be appropriate:
1. Shopping at the supermarket and keen to underspend, you see Scotch Eggs at 30p each, 75p for two and £1 for 3. You want some, maybe three or four. What do you do? ¹
2. A group walking on Dartmoor have a planned route for say 30km and, after two hours it is nine o’clock and they have covered 5km. When do they expect to finish?
3. You’re driving in a 20mph zone. You are expected to be able to stop in a short distance; standard tables suggest 40ft or about 13 metres. Assuming those standard conditions apply and that there may be a circumstance in which you’d be expected to stop, how fast would you be going at the notional impact point if instead you were doing 26mph? Put it another way, how far beyond the expected stopping point would you actually stop?
4. A similar group on Dartmoor are not enjoying walking on wet heather in springtime. One of the group observes that they all walk faster when on any sort of track, including a sheep track. If they changed from going directly to their intended target and instead took the next available acceptable sheep trail, how far off the desired direction could they go and still be going faster in the desired direction? Try for a guess in multiples of five degrees before you go work it out.
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you’re supposed to go find a piece of paper and do some figuring at this point. I’ll bet most don’t.
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1. The single eggs are significantly cheaper. The packets of two are the most expensive. I am not the only one to notice this. Why then do people choose the other offerings? Mostly because they don’t want to think about the numbers, I suggest.
2. Continuing at the same rate, they would expect to finish this route in twelve hours, ten more hours from noticing the time, so a 19:00 finish. Which, for many groups, means that they will not actually finish this route. Which in turn means that they have a problem that should have been addressed before they started. It is likely that they have someone inexperienced added to the group unexpectedly who has a problem, so that this has not been factored into the plan. If completing the route is important then (i) this needs to be communicated and there are consequences such as a late finish or someone dropping out or (iii) the route needs to be modified to accommodate what today’s group can achieve. In a sense, it needs to be clear what it is that constitutes a success for the day. From experience, such a group would make far better time if they ceased stopping to regroup and found a speed that their slowest member can walk at all day. Boring but true. I still do about 5kph on rough ground, but my stopping is under a minute an hour. But then I can (and do) do arithmetic. Incidentally, this was a fairly common problem at PMC and generally a 19:00 finish was okay, where a 21:00 finish was dubious and was often dealt with by abandoning that route, though our action rather depended upon the weather and which training weekend it was.
3. If you're a driver, you are supposed to already know your stopping distances. It was easy in the bad old imperial days of miles and feet, because the numbers worked nicely; for speed S mph, the total stopping distance was made up of thinking distance S feet plus braking distance of S²/20. At 20mph that is 20+400/20=40 feet. At 26mph you need 26+676/20=60 feet (a tad less) and, at 40mph you need twice that, 40+1600/20=120 feet. At 26mph and at the 40 foot mark, where you’d be expected to be stationary, you’d have been actually braking for 14 feet, so the speed at that instant would be whatever needs 20 feet (what’s left of the 60 feet from 26mph) to stop in, S²/20=20 or √(20x20)=20mph. At impact you’d be only a shade under the declared speed limit. So the effect is as if you’d been behaving properly in terms of speed and not braked at all. At an initial 27mph you’d be at almost 22mph at the expected stationary point and starting at 30mph, over 26mph. Even at an initial 23mph, you reach the critical point at almost 14mph, faster than most can run. Very scary. So why do you NOT obey the limit in a 20 zone?
4. Exciting, an excuse to use geometry. Say speed on the Dartmoor-standard rough stuff is R and about 3kph. Say speed on a sheep track is S and about 4kph. Differences of 0.5 to 1.0 kph are normal, based on a lot of observation by me. Bigger differences are possible, but once it is learned that this approach works, the calculations are moot (because it is now a strategy). Draw a right-angle triangle with the longest side S and the middle length side R. The cosine of the angle between these is R/S. For 3 (direct, R) and 4 (indirect, S) we get an amazing 40º at which the choices are equivalent in time. For S=3.5 and R=3 we get 31º. For 5 and 4.5 we get 26º. In other words, you can look quite a long way ‘off-direction’ for any surface that will be quicker to walk on and that such a choice will succeed in comparison to plugging away at a compass direction. You don’t do this if working/walking in bad visibility. Most of us do open-ground walking in relatively nice weather. This simple calculation—once seen, it is not as if you’re going to refine it—shows that there will be gain a long way off target, up to about a quarter turn (which would be 22.5º, if we were being fussy about exact numbers). Obviously you will need to compensate with another such track in a while, but this applies also to very short-term choices (i.e. within several paces) as it does to long stretches. I have noticed recently that on wet ground the ‘path’ (wet, soft and muddy) is actually slower than using the ‘rough’ (drier, perhaps the same knee lift, but more solid); I am aware that on Lakeland hills, some stony paths are slower than the adjacent grass (and now the suggested off-line policy may be seen to be bad for one’s personal erosion coefficient). I also notice these states are significantly changed by the current degree of slope. The situation is similar but different if running.
Those examples barely use GCSE maths. What they require is a willingness to look at the numbers and see that there is a useful question to ask. You agree, I hope, that the maths above is pretty easy. My point is that too often no such a question is posed. What used to drive me to distraction (rage, wanting to be elsewhere) is not just the failure to recognise the message in the numbers, but (far more often) that there is a refusal to act in line with the result. A sort of ‘Whatever’, a “Yes, but this doesn’t apply to me”, which could be translated as really not caring at all. That is not dissimilar from the comment I used to get that numbers themselves were simply too much; symbols with no meaning. How do people get themselves into such a mess?
On Dartmoor, where such evidence often results in simply recognising that “Today is going to be a long day”, these are non-dangerous results. The group will eventually learn that they go at <this speed> and that this means in turn a ‘day’ is <that long>. Eventually they will address underlying problems, all which could be classed as preparation — from being fitter, to having better kit, to having a lot less kit, to sharing the weight around to balance abilities, to picking better routes, to waking up to the challenge as a whole. Oh, I missed out simply losing the people who refuse to let the numbers inform decision-making.
On the road, I find the ignorance of risk astonishing. I agree that frequently one is left wondering why a piece of road has a particular speed limit. I picked the 20mph zone deliberately, since there it is common for other users of pavement and road to act on the assumption that 20mph means a 40foot stopping distance. Typically, young pedestrians learn that this is a ‘safe’ place to act in ways that elsewhere would be grossly irresponsible. We call this ‘safe’, yet every day you drive through such a district you are aware of the significant numbers of road users that exceed the designated limit by so much that they would still be over the speed limit at the point of impact. This might be declared criminal ignorance.
For the walking questions (I wrote posers, but that will be misunderstood in a bad way), It may be relevant to quote the African saying in translation “If you want to go quickly, travel alone; if you want to go far, travel together”. For long distance walks, the trick is to reduce the stopping to a minimum.
More advanced forms of these same questions:
31. Assuming you are aware that many people exceed the posted speed limit, then for speed limit L, at what expected stopping distance do you not only fail to stop in time but are still exceeding the posted speed limit at the notional impact point ( the place where the expected stopping distance is used up)? Does that inform your driving style? Might you use that to inform others? [I give you enough excuse here for the use of algebra, but the problem is not can you do that, but can you set up the algebra to be done, the difficult bit that goes first.]
32. Speed vs Safety essay 156, uses the term marginal speed. If you have 10% excess over the posted limit, what is your residual speed at the point when you ‘should’ have stopped?
33.(and 34) If out walking on open ground, are you aware of your changes of speed? Of direction? Are you aware of the many micro-decisions you make that constitute such walking? How do you modify those choices with regard to what you can see? This might be asking you to assess your own capacity for use of feedback. Does it matter how much you lift your feet in taking a step? Does that answer vary with the ground? If it does, do you use that information in making short-term route choices?
Oh, you complain, that’s not maths.
Well of course it isn’t: Maths is the tool for doing other stuff, rather like language is for communication.
I am sure that the hard parts are:
• recognising what would lend itself to such a problem. Indeed, often, recognising that there is a problem at all.
• acquiring the data (but I note that very often the data is in front of us already, begging the question to be asked)
• setting up the maths to be done. I often said to classes that the world doesn’t actually need very many (super) mathematicians. What is does need is a lot of people who recognise that there is a problem to pose and that these people be able to communicate what that problem is so that a solution can be found. And then to communicate the solution (interpret is a word often used, with connotations of spin) back to those who are expected to make non-mathematical decisions based upon something better than an unexplained reaction.
Why do we do maths? Mostly because we need it to explain so many things. I am noticing increasingly a connection between an inability to explain (many things) and a low ability to do maths. Where perhaps what I mean is an ability to think in mathematical terms, for there are many people with an acceptable qualification in maths who persist in the refusal to apply any such thinking. It is one thing to fail to spot some possible mathematical application. It is entirely something less to deny the evidence and to refuse the result.
DJS 20160419
Happy birthday, JP
I resisted the inclination to use medical risks, though I am not entirely sure why. A rich field: perhaps I am open to suggestions. What would be appropriate?
What is the point in Maths? “A point has no size, but simply marks position." See Essay 31
See several other pages on this site, not all in Lower School Extension work:
pensions walking, fuel tiling going to Cambridge Seeds
1 Within a week of writing this, Asda has been reported on the radio as having been taken to task for this pricing technique (on the scotch eggs, specifically), along with several other examples. It turned out that several purveyors were accused of similar practices and told to make it stop. Damn, it has been a fruitful source of persuading people to do some arithmetic reasoning. What was wrong with having a loss leader? Why is it a bad thing to offer customers good deals if they can only be bothered to read the price-per labels?
31: Let L be the Limit, your fast speed is F, your stopping distance is supposed to be L² / 20 + L and in practice it is F² / 20 + F, so the difference in distance, ΔD, is L² / 20, becasue you're doing the speed limit at the critical point where you should have stopped.. Multiplying all through by 20, we have
F² - L² + 20 (F - L) = L² Rearranging, F² + 20F = 2L² + 20L = 100∂. I've picked 100∂ for convenience. Solving for F, F=10 [√(∂ + 1) - 1]. Giving:
L 20 30 40 50 60 70 mph
∂ 12 24 40 60 84 112
F 26 40 54 68 82 96 mph
T n/a 0 48 57 66 75 mph
F is the speed that leaves you STILL over the speed limit at the expected stopping point (which you might think of as expected impact point or first non-impact point). Example: doing 26 in a 20 limit leaves you doing 20 at the notional impact point, braking.
T is that speed that leaves you doing 30 mph at the expected stopping point, the notional impact point. At 30mph there is a 50% chance that the impact with a pedestrian willl result in death.
Let 100ß be defined as 900 + L² - 20L; then T=10 [√ß + 1) - 1] and we can add a row reading, T≥30.
Look what that means at motorway speeds! Just 5mph over the limit and you’ve a 50% chance of killing someone at the expected non-impact point. If you think of having a two-second gap to the car in front, then, if you're stupid enough to be exceeding the speed limit you need more than a two-wecond gap, so that, on seeing the need to reduce speed, you’re substantially under the posted speed limit at what might be called the expected reaction point, when you are exactly two seconds behind the car in front and (by then) doing the same speed. Now, if you assume that the car in front is at the posted speed limit, you soon realise how oimportant that two-second gap is. At 70mph the amount of time at which you have to make a decision or to correct a decision is really very small. You could set up some examples to explore this. My conclusion is that behaving much as the cars around you is important and that that two-second gap represents your safety margin, such that, if someone moves into the gap, you want to do something about making that gap. bigger again. It is very easy to become confused about speed and time and distance. Where you can, remove distance from the problem.
Example: Your speed is 80mph or so on the motorway and others around you are near the speed limit. Let us suppose that you're doing exactly 80 and traffic in front is doing exactly 70. How much extra time do you need the time gap to be so that, when you've braked to the slower speed, you have a two second gap? I made the answer about ¾ of a second, and, assuming I'm right that means that a mere 10mph difference at motorway speeds requires your safety gap to go up by around 50%. At a 20mph excess you need more like double the previous margin, between 3.5 and 4 seconds. The issue here is that 4 seconds is probably a lot further than the slower car driver is looking in the mirror, so at this speed difference you are at risk of being not ignored but unnoticed. It is the difference in speeds that causes little mistakes to turn into large accidents.
A 5mph difference in speeds is 2.2 m/s which means it takes about 2 seconds to pass the length of the car. So you could estimate speed difference by estimating the time it takes for a car to go past. Do experiment: I think we are very bad at estimating relative speeds and we should be using estimates of walking and running to measure these differences. Because the events occur at speed, we alklow ourselves to be easily confused as to what it is we think we're seeing. a 20mph difference is scary.
I've driven on German autobahns, where there is no effective speed limit. If your car is small, so that 70-75 is quite fast enough, then someone going 50-100% faster that is needing to look a very long way ahead and having to guess whether or not you have seen (or even are able to see) them in the mirror, should you, the slower motorist, want to pass another vehicle. Obviously this becomes critical if you move into the fastest lane and, under certain traffic densities is a dangerous manouevre. One partial solution is to find another vehicle that wishes to travel at your sort of speed, set your interval at 3-5 seconds at watch all the more carefully at the reaction to any change of lane. The third time I drove in Germany I had a far more powerful car and saw this from the other side; I found a succession of other drivers who I allowed to 'break trail', so that they did the acceleration and braking (often quite severe), while I sailed along relatively peacefully in their wake. But a two-second gap is too small; 4-6 seconds is very comfortable.
32: Let L be the Limit, your speed is S=1.1*L, your stopping distance is supposed to be L²/20+L and in practice it is S²/20+S, so the difference in distance, ΔD, is (S²-L²/20 +(S-L) = (S-L)(S+L+20)/20. So your residual speed, R, is when R²/20 = that difference,
so R²= (S-L)(S+L+20) = (0.1 L)(2.1L+20) = 0.21 L² + 2 L
L 20 30 40 50 60 70 mph
S 22 33 44 55 66 77 mph
ΔD 6 12 21 31 44 58 ft
R 11 16 20 25 30 34
33&4 People who walk a lot, just as people that run a lot, can spend a surprising amount of time thinking about numbers. Marathon runners with a target will be doing speed / time calculations a lot of the time; distance walkers use the information to calculate not only when they expect to be somewhere, but to know where they are right now. Using the evidence (data) you have to inform what you do is, to my mind, necessary and obvious feedback. Ignoring the information requires a sort of ‘don’t care’ attitude that might be explained by “I’m not out here to think, so I’m going to not-think; it takes as long as it takes”. Which is fine until you change your mind about how much you want to do. I’ve seen far too many people still committed to a route long after they should have had a rethink, in a position where turning round is too much loss of face (somehow), even if eminently sensible. This requires a loss of objective, in a sort of ‘why are we here?’ way. These days, I’m out in the hills to enjoy myself without getting so tired that I’m more unsafe than usual when I try to drive home. On Dartmoor days, the objective was to keep the kids safe as they became more and more tired. Which was fun up to the point where the staff were as tired as the kids.
The question posed encourages you to recognise just how much time you spend doing thinking while walking (or don’t at all). You could concentrate far more on where you put your feet. The many micro-decisions that constitute walking on rough ground are positive mental stimulus (and more of us should do it). I note that for many, what happens is that they slow down until their usual mental capacity spent on walking (a tiny amount) fits with the ground. This makes for very slow days and very short days in terms of distance. Those who run in the hills exhibit a very different set of skills, especially in terms of what they will accept as sites for foot placement.
