Are we really Numerate? How numbers lead us up the garden path!Our Politicians and Business Leaders talk about the need for us all to be both literate and numerate when we leave school and as such literacy and numeracy are key subject components of the National Curriculum and beyond.But are we infact teaching the literacy and numeracy required for success in the real world? I think not and I will be posting my thoughts on this over the next few weeks on this blogHere I make a start, looking at Numeracy. I call the numeracy that we learn at school and in our universities “Math 1.0″ (And in general this is the only form of numeracy we are taught, so most if not all of our leaders are only numerate to the level of Math 1.0)This is a Math that is useful, but only in the very simple domain of counting and manipulating pure numbers. This domain is what Donald Wheeler (one of the few statisticians in the world who seems to understand this stuff) calls “Math World” a strange world that has very little bearing on everyday reality. It is very misleading in fact when, in the Real World, we use Math 1.0 to manipulate, interpret and compare measurements
It was Walter Shewhart (the man who has been called the father of quality) who said “Data is meaningless outside of its context”. Using my language here, he could have rephrased this as “Data is meaningless unless processed using Math 2.0″ (Math 2.0 is a way of working with numbers that keeps the important context ‘in view’)
Math 1.0 is the Math of the Counters. Math 1.0 works for the abstract Math-World. Math 2.0 however is needed for Real-World Problems. Many of our “number experts” (mathematicians and statisticians for example) base their life-long working knowledge on this Math 1.0, so they are then part of the problem. Math 1.0 is entrenched in academia and science. It is now one of those implicit unquestioned assumptions (like water is to fish and air to birds) that Math 1.0 is numeracy and that Math 1.0 describes the sole reality of numbers. There will certainly be a few people in very high powerful places who know about Math 2.0 but are happy for the rest of us to just learn Math 1.0. When it comes to comparing things, Math 1.0 does not clarify issues, instead it clouds them. All this means so few people know or understand the limitations of numbers, and therefore that numbers can be used to keep us all in the dark (ages?) about most things. We will never really know whether our Health Service or Schools are getting better or worse using Math 1.0. What is certain is that using Math 1.0 we get into endless debate about the trivia from the data (we can call this “noise”) and we will nearly always be missing the important understandings (we can call this the “signal”). Without Math 2.0 the useless information (noise) is drowning out the important information (signal).So although it seems ludicrous that some, if not most, of our main ‘experts’ in Maths and Statistics use a Math that was devised for the special case of pure numbers and counting and that is strictly NOT applicable to numbers as MEASUREMENT. But it is such experts that write a numeracy curriculum for our schools, universities and accountants that is based on a special case with numbers (the Math of simple counting – Math 1.0). In the real world most of the important numbers we deal with on a day to day basis are to do with measurement, or involve counts that are being used as measures, and so we need to apply “Math 2.0″ in order to interpret these situations. When we use simple Math 1.0 for interpreting data measurements we create problems and misunderstanding. Because we have come to rely on numbers in every facet of life and business (we found we could no longer trust the word of leaders, doctors, scientists etc so we needed their numbers) numbers now heavily impinge on our emotions. We can get very angry when we see numbers we don’t like. The problem is often there is no valid reason to get angry with the numbers, it is Math 1.0 we should be getting angry with. We should be getting angry that we are not taught the ‘numeracy of measures’ at all. Everyday we are all making decisions with sometimes life-threatening or very severe unintended consequences because of a lack of real-world numeracy because we don’t have the skills of Math 2.0No-one is excluded. Politicians, scientists and business leaders all continually make poor decisions when they apply Math 1.0 thinking to real-world Math 2.0 situations, making us depend on numbers in a way that is totally irrelevant, abstract, misleading, artificial, and distorting. Our lack of real-world numeracy Math 2.0 skills is I believe a big part of the problem why so much today seems to be going wrong. We follow the numbers but we don’t understand the numbers and as a consequence we jump to the wrong conclusions and we take actions misguidedly on the numbers and actually then make matters worse rather than better. (Deming called action based on misguided interpretation of data – tampering and he devised the funnel experiment to help us understand how tampering makes matters worse)So at this point you may be asking what is this Math 2.0, why isn’t it taught in school and what difference would it make? I will some outline the key differences between Math 1.0 and Math 2.0 next time but here is a taster.Math 1.0 is an artificial world where lines have no thickness, parallel lines can’t meet and numbers are absolute. When we use Math 1.0 there is only one correct answer and it is not possible to have variation in the answer. (in the real world however variation is always present)Math 2.0 on the other hand is a real-world Math where lines have thickness, parallel lines can meet and most importantly measured numbers are never absolute. As variation exists in all things Math 2.0 does not ignore its effects (whereas Math 1.0 assumes random variation does not exist)So here is a little teaser to see if you are working from Math 1.0 or Math 2.0MATH 1.0Math 1.0 2 + 2 = 4 YES this is absolute, there is only one answer and that is 4Math 1.0 implies that this answer is the same whether we are using simple counts or measures. So 2 inches plus 2 inches will always equal 4 inchesMATH 2.0Math 2.0 – when simply counting, the results are the same as for Math 1.0So 2 + 2 = 4 this is absolute, there is only one answer and that is 4However when adding together measures or comparing measures: 2 + 2 = 4 but only on the average (so each time we take measurements and add them together the answer can vary either side of the number 4 by an amount which Math 2.0 can reliably approximate )this scenario would be more precisely written as:2 (v1) + 2 (v2) = 4 (v3) where v1,v2,v3 is the variation (plus minus 3-standard deviations) that is inherent in each measurementThis brings me on to a further significant difference between Math 1.0 and Math 2.0. In Math 1.0 there is ALWAYS significance in any change of number and therefore there is value in comparing just two data points. So if something measured 20 last month and 23 this month Math 1.0 says there is a change (an improvement if good stuff, a worsening if the measure is bad stuff. So as Math 1.0 is the math of pure counting if we have 20 apples in one basket and 23 apples in another it is clear that the second basket has (three) more apples in it. Math 2.0 would come to the same conclusion. However if the tree in your garden produced 23 apples this year and 20 last year we actually need Math 2.0 when seeking to make a decision about whether this difference means the tree yield is improving? For we are now not looking at the pure count of the apples we are seeking to use the numbers to give us knowledge about the tree. Now instead of apples and tree performance think of pupil exam success and school performance. And then by way of extrapolation think school success and league tables.In Math 2.0 we CANNOT KNOW IF THERE IS A DIFFERENCE between 20 and 23 unless we have more data (and then a lot of the time Math 2.0 will show there will be no likely significant change). Math 2.0 tells us that comparing just two data points is ALWAYS meaningless (and of course it can provide the evidence for this). Each time we just compare two data points we are viewing the data outside of its context.If only journalists were schooled in Math 2.0 we would not have so many meaningless, stupid headlines in our papers. But there again, they probably wouldn’t sell so many newspapers, so you could see that their bosses would be quite happy that their journalists are only numerate to Math 1.0 level. Using Math 2.0 many headlines in the newspapers would read “Probably no change in the trade figures this month” rather than something that appears very dramatic like “4% fall in trade figures throws UK back into recession”. Which of these two headlines would make you buy the newspaper – the first one (‘probably no change” so nothing much is happening - a quite likely scenario using Math 2.0) or the second headline derived using the inappropriate use of Math 1.0 ?Have I grabbed your attention? If you already know what I mean by Math 2.0 great, please post your own examples here about how Math 1.0 misleads, If you think I am a raving lunatic and it simply can’t be possible that we are being taught the wrong numeracy at school for making sense of the real-world, then please follow, watch and learn. And if you still think I’m being stupid tell me so.In the next two articles in this series I will be comparing in much more detail some of the differences between Math 1.0 and Math 2.0 and seeking to impress upon the sceptics out there that this is really important stuff.Next Time: Maths and Science leading us up the (wrong) garden path
I think this confuses Maths and Statistics. I think your Maths 1.0 = maths, and Maths 2.0 = Statistics.Numeracy is something else again. There is no accepted definition, and different people mean different things by it. Usually it is used to mean the user’s psychological grasp of ‘basic’ arithmetic. The quotes are around basic because I think these ideas, such as ratio and proportion, are not simple. And many people who have strong opinions about numeracy could not describe what natural numbers, integers, rationals, irrationals and real numbers are.If you mean ‘the use of quantitative data’ for Maths 2.0, I agree it is a very difficult subject.This is not entirely relevant, but it is fun. Imagine a rope tied around the Earth at the Equator. Now imagine the rope is stretched so it is 10 feet above the ground, everywhere. How much do you think you’d need to stretch it by?
Barry,As you know I am fully in favour of what you call Math 2.0.What I would say about your Math 1.0 is that, like Newtons Laws, it still has its uses. You can get a rocket to the moon using Newtons Laws but you would have difficulty designing a computer chip that operates at the atomic level. Knowing that there is a limit of usefulness and knowing that limit is important.Likewise Math 1.0 is useful for buying your groceries and even running the store. If you want to lead the store, in troubled times, on a growth path or into new markets can be done with Math 1.0 after a fashion. If you use Math 2.0 you can get further, quicker with less risk.Alan
I agree totally Alan that Math 1.0 is useful in the right context and indeed we need to learn Math 1.0 and the manipulation of pure numbers before we can appreciate and use Math 2.0. Another way of looking at it is that Math 1.0 is a necessary component of Math 2.0. The problem is that Math 1.0 is not sufficient to properly understand the limitations of numbers when used in the area I have called ‘measurement’
Walter good to hear from you. I know I can always rely on you for some stimulating discussion.I appreciate that there is not really a word for what I am trying to describe and numeracy comes closest (if you know a better word in this context please let me know.Wikipedia describes Numeracy as the ability to reason with numbers. My argument would be that we cannot possibly reason with numbers when we ignore the fact that numbers used for measurement contains variation and variation of a much higher degree than we can intuit. Therefore to be able to reason with numbers we need tools that show us the limitations of the numbers we are reasoning with these tools may sometimes be found within the Math 1.0 domain but are not used appropriately.Wikipedia goes on to say that to be numerically literate, a person has to be comfortable with logic and reasoning. Some of the areas that are involved in numeracy include: basic numbers, orders of magnitude, geometry, algebra, probability and statistics.The word statistics is a stumbling block here as it can be based on different thinking.You say that I am confusing Maths and Statistics and that you think Maths 1.0 = maths, and Maths 2.0 = Statistics.As I will be outlining in my next few blogs on this topic, the statistics of Math 1.0 is not the same as the statistics of Math 2.0 so watch this space.I am not a statistician but from my layperson perspective I would say that as a simplistic generalisation, the statistics of Math 1.0 seeks to remove uncertainty, make things black and white and takes data out of its context in order to do this. As Donald Wheeler points out (referenced in the blog) this is not how the real world is. It is a very limited abstraction of the real world which as Alan Clark points out in his comment is an abstraction that is useful in some contexts.The statistics of Math 2.0 keeps the uncertainness and the variation ‘in the frame’ and seeks to therefore be more ‘real-world’.Your final question is a real Math 1.0 puzzle isn’t it? (I am sure readers of the blog would like your answer). And this is an interesting thing. I am not especially well versed in the deeper concepts within Math 1.0, I never learnt calculus for example, which can be exceeding useful I believe in certain contexts (it was necessary to land a man on the moon?) The problem as set is a Math 1.0 problem (correct me if I am wrong). Actually it won’t be about the real earth but an abstraction of the real earth (ie a perfect globe with a perfect smooth service and a rope with no width) The Math 1.0 answer I suspect would be an exact number. And here is the first difference between Math 1.0 and Math 2.0A Math 2.0 solution would be a range of possible lengths not a right answer (plus/minus the minimum and maximum possible) and would require knowledge of the variation inherent in the measuring device by which the ten foot from the earth’s surface is measured. Also the length of the rope required to fix around the real earth would of course vary depending on the thickness of the rope. The thicker the rope the shorter the length.
I think a historical perspective might be useful here. The start of maths was concerned with real-world solutions – like working out how much food would be needed to feed the Pharaoh’s army – a maths 2.0 issue. Calculus with Newton and Leibnitz was also concerned with real-world problems, but focussed on an area (the motion of the moon) where maths 1.0 worked extremely well. Further development led to a formal axiomatic system (Hilbert end of the C19) which is the apotheosis of maths 1.0 – a purely logical system which has no direct connection with the real world. And Maths 2.0 (I think) got going in the middle of teh C19, with Florence Nightingale trying to make use of data.My problem – the circumference of a circle is 2 PI R. Take R as the radius of the Earth, and h as the height you want the rope above teh ground. Then the initial rope length is 2 PI R, and the stretched length is 2 PI ( R + h). The difference is 2PI(R+h) – 2PIR = 2PIh.So to have the rope 10 feet above the ground, you’d stretch it by a bit over 60 feet. Most people guess much more than this.What is weird is that R makes no difference. So if we altered this to say putting the rope round the orbit of Pluto, then moving it 10 feet out, we’d still only need to stretch it by 60 feet. Counter-intuitive.
Walter, your comment on intuition supports the case for using numbers, measurement and prediction in the first palce. They are required to balance intution because of the comlexity of existence.Intuition may be based on valid experience, however, it is filtered by individual perceptions, emotions and assumptions at the point someone makes a prediction. To take the example of the rope around the globe, perception is altered depending where you stand.By way of illustration consider a tennis ball on the ground with your rope, of the same thickness, wrapped around it. If you were to try to predict the increase in the length of the rope required for it tobe 10 ft from the ball you may be able to make a close approximation to the required length.Doing the same thing with a rope around the globe would depend whether you were considering it indoors, standing outside looking at the rope on the ground or orbiting earth in a spacecraft.Barry and I both value George Box’s insight that ‘all models are wrong, but some are useful’. If someone’s perception filters sensory information leading to incorrect assumptions, models, then their prediction or forecast in a situation is may be wrong.Math 2.0 for me means moving away from black and white to shades of grey. Our social conditioning as part of a highly complex social system can often lead us to be very uncomfortable with the uncertainty of a random universe. We can predict that a massive asteriod, of the size of the one that wiped out the dinosaurs, is highly likely to strike the earth again. We don’t exactly know when. Our models predict that it is highly likely the earth will be destroyed as the Sun becomes a supernova. We do not know what other events may occur in the mean time that may change our current prediction of 5 billion years.
What’s the difference between Maths 2.0 and Probability and Statistics?
WalterYou ask for my explanation of the difference between Math 2.0 and Statistics. It will be next week before my next full blog on the topic where I will begin to clarify this question. But to give you something to contemplate on before then is the following. Both Math 1.0 and Math 2.0 have their own form of statistics. In general the statistics of Math 1.0 (that in common usage) either ignores variation and uncertainty or marginalises it. Math 1.0 purpose appears to be to convert the uncertain (shades of grey) into the certain, whereas the purpose of Math 2.0 is to make the uncertainty certain or to put another way to make uncertainty visible (I was going to use the word ‘transparent’ here but interestingly in this context could mean the exact opposite to my intended meaning)Can I give you an example from my days as a Clinical Biochemist. When two methods are compared using Math 1.0 linear regression analysis variation was always ignored in the baseline method (x axis). This simple regression approach, characteristic of Math 1.0 thinking, led to erroneous conclusions about the performance of new methods. This issue was not addressed until the use of what became known as the Deming RegressionDeming regression is an errors-in-variables model which tries to find the line of best fit for a two-dimensional dataset. It differs from the simple linear regression in that it accounts for errors in observations on both the x- and the y- axis. This led to clinical chemical method performance assessment that was closer to Real-World experience. The Deming Regression method fits into my Math 2.0 because it seeks to approximate the random variation in all measures whereas Math 1.0 often ignores or marginalises (as unimportant) random variation in measurement. My Math 2.0 embraces Donald Wheelers concept of Real World Maths (but there are some additions in Math 2.0) so if you want to get a feel of some of the differences between the statistics of Math 1.0 and Math 2.0 Wheeler’s work will provide useful background (his website http://www.spcpress.com)